Keywords: 2D quantum gravity, Liouville theory, Mabuchi action, string susceptibility, spectral cutoff regularization, renormalization

erms to both the partition function and the n-point functions of the same order in the perturbative expansion. Finally, it seems that among the initial 15 counterterms possible to insert, 4 stay undetermined and contribute to the string susceptibility. One can stress that the allowed counterterms are either local (as tted) or measure-like. The measure action coming from the regularization gives indeed rise to non-local terms. The local-VII ity of the global Liouville, measure and counterterm action is however assured through the counterterms. This hints at an understanding of the measure-like counterterms as a renormalization of the measure, which has been overlooked before. In the last chapter these results are generalised to the coupling to non-conformal matter by considering the Mabuchi action in addition to the Liouville action, in the case of the torus.

Our discussions have sharpened my scientic mind and improved my methods. Moreover, Adel has carefully followed the present work. We have performed together a signicant part of the calculations and he read the present manuscript, allowing me to improve it through his suggestions.

I am also grateful to Semyon Klevtsov. I thank him for his invitation at the Simons Center for Geometry and Physics, for his help in nding a post-doc position and for the position he oered me in Cologne. I am also indebted to Frank Ferrari, who has pushed my post-doc applications and allowed me to have a post-doc position in Bruxelles in fall. This work would not have been possible without the support of the heads of the LPTENS. Therefore, I would like to thank Constantin Bachas and Costas Kounnas for having welcomed me at the LPTENS, and allowed me to work in good conditions, especially during the renovation of the building. I would also like to thank all the members of the LPTENS and especially Marc-Thierry Jaekel who helped me with the computers as well as Gérard Toulouse who gave me access to his library in our common oce. I am also grateful to the LPTENS secretary sta, Viviane Sebille and Sandrine Patacchini, who have always been very helpful in the huge administrative troubles I had during my PhD.

I am also grateful to the Capital Fund Management foundation and to the Dephy fondation for their nancial support of my work.

My gratitude extends to my rapporteurs Vincent Rivasseau and Semyon Klevtsov for accepting to read my manuscript and providing me with instructive comments. I also thank my examinateurs François David, Frank Ferrari and André Neveu.

On the personal side, I also have a special thank for all of my friends: my physicist friends Thomas Bernet, Pierre Clavier, Harold Erbin, Hanna Grönqvist, Lucien Heurtier, Pierre Ronceray and Corentin Schreiber, as well as Astrid Cadet and Morgane Schreiber who are part of the group and Corinne Delacroix, my PhD sister. I would also like to thank Elise Massot and Lucien Xu, Anthony Arnault and Nicolas Desplan and Marine Richard and Alexandre Pasquier. I also have a special thought to Hanna Chêne and Sylvain Rouch-Seize. Finally, I would especially like to thank Cédric Mezrag for his inestimable help in the last year, and without whose support this thesis would not have been the same.

Finally, I would like to express my gratitude to my family which supports me both in every day life and in hard times, and especially to my parents Mireille and Alain Leduc.

I would like to dedicate this thesis to my grandmother Pierrette Piveteau and to my late grandparents Emilienne and Georges Leduc and Guy Piveteau, whose absence I deeply regret. Thanks to all of you.

III

Résumé / Abstract

Le but de cette thèse est d'étudier la gravité quantique bidimensionnelle. Nous nous intéressons plus particulièrement aux approches dans le continu. Ces dernières reposent principalement sur l'action de Liouville qui décrit le couplage entre théorie conforme et gravité. Si cette action, bien connue, est très bien comprise, la mesure de l'intégrale fonctionnelle sur l'espace des métriques pose plus de problèmes. Toutefois, sous l'hypothèse simplicatrice d'une mesure de champ libre, la dépendance en l'aire de la fonction de partition de la gravité quantique en présence de matière conforme a pu être établie, permettant ainsi d'accéder à un exposant critique : la susceptibilité de la corde. Malgré l'hypothèse assez forte sur la mesure d'intégration, cette formule (dite KPZ), a été conrmée par des calculs issus de méthodes discrètes, et ce dans plusieurs cas particuliers. Grâce à une nouvelle méthode de régularisation spectrale en espace courbe, cette mesure d'intégration a récemment pu être proprement dénie. En considérant les surfaces de Kähler de genre quelconque (qui coïncident avec l'ensemble des surfaces à deux dimensions), un calcul perturbatif de la fonction de partition à aire xée a ainsi pu être mené à une boucle, en utilisant à la fois les actions de Liouville et de Mabuchi. L'action de Mabuchi intervient en correction au premier ordre si l'on couple de la matière non conforme à la gravité. La susceptibilité de la corde ainsi calculée correspond, dans le pur cas conforme, au développement de la formule KPZ à une boucle. Ce résultat n'est que peu surprenant, puisque les eets induits par le caractère non-trivial de la mesure d'intégration sur les géométries n'apparaissent qu'à partir de deux boucles.

An de déterminer les conséquences de cette mesure d'intégration, un calcul perturbatif de la fonction de partition a aire xée jusqu'à trois boucles a été conduit dans cette thèse. Nous avons premièrement considéré l'action de Liouville et des surfaces de genre quelconque, puis ces résultats ont été généralisés dans la cas du tore à une action couplant les actions de Liouville et de Mabuchi. Nous avons accédé à l'expression complète de la fonction de partition à aire xée à deux boucles, puis nous avons calculé les divergences dominantes quadratiques et logarithmiques à trois boucles. La régularisation de la mesure d'intégration génère une action de mesure, ajoutant ainsi des vertex à ceux issus du développement de l'action de Liouville (et de Mabuchi). Si, à deux boucles, les diagrammes à considérer ne sont qu'au nombre de quatre (dont dix sous-diagrammes), à trois boucles, il faut calculer 29 types de diagrammes totalisant plus de 200 contributions de sous-diagrammes. Des divergences étant présentes, il faut renormaliser les actions (en y ajoutant des contre-termes). An de les xer, nous avons calculé la fonction à deux points complète à une boucle et requis que celle-ci soit à la fois nie et indépendante de notre choix de régularisation. Ceci nous à permis de xer la plupart des contre-termes et d'obtenir une fonction de partition à aire xée nie et indépendante de la régularisation à deux boucles. La susceptibilité de corde ainsi obtenue est compatible avec le résultat KPZ, mais plus générale car dépendante d'une constante libre. Nous avons donc poursuivi le calcul à trois boucles an d'obtenir la contribution complète des contre-termes à la fonction de partition à aire xée ainsi qu'aux fonctions à n-points au même ordre dans IV V le développement perturbatif. Au bout du compte, il apparaît que parmi les 15 contretermes possibles, 4 restent indéterminés et contribuent à la susceptibilité de la corde. On peut attirer l'attention sur le fait que les contre-termes sont soit locaux (comme voulu) soit similaires à la mesure, cette dernière contenant en eet des termes non-locaux en 1/A. En revanche, il est possible de garantir la localité de l'action globale (somme des actions de Liouville, Mabuchi, de la mesure et des contre-termes) grâce aux contre-termes. Ceci nous suggère d'interpréter les contre-termes comme une renormalisation de la mesure d'intégration.

VI

This thesis is devoted to the study of two-dimensional quantum gravity. After giving a general introduction displaying dierent approaches of quantum gravity, we highlight the continuous ones, mainly based on the so-called Liouville action which universally describes the coupling of any conformal eld theory to gravity. While the Liouville action is relatively well understood, the appropriate functional integral measure is complicated, however. Nevertheless, a formula for the area dependence of the quantum gravity partition function in the presence of conformal matter has been obtained, under the simplifying assumption of a free-eld measure. It has then been possible to derive relevant critical exponents for these theories, one of them called the string susceptibility. Notwithstanding its non-rigorous derivation, this formula, often referred to as the KPZ formula, has since been veried in many instances and has scored many successes.

However, the recent development of ecient multi-loop regularization methods on curved space-times opened the way for a precise and well-dened perturbative computation of the xed-area partition function in the Kähler formalism. The string susceptibility was therefore previously computed in this framework up to one loop for surfaces of arbitrary genus using a somewhat more general quantum gravity action including the Liouville and Mabuchi actions; the latter corresponds to possible couplings to non-conformal matter.

For conformal matter only, the one-loop KPZ result was reproduced. This was to be expected since the non-trivial nature of the quantum gravity integration measure only shows up at two and higher loops.

In this thesis, a rst-principles computation of the xed-area partition function up to three loops, considering the Liouville action, is performed. Among other things this allowed us to appreciate the role of the non-trivial quantum gravity integration measure. Our computation allows us to get the full expression of the partition function at two loops and to access both the leading quadratic and logarithmic divergences at three loops, as well as precisely discuss the diverging structure of the string susceptibility. This was done for Riemann surfaces of any genus. The regularization of the complicated integration measure leads to a measure action adding vertices to the ones coming from Liouville. While the two-loop computation only requires to take into account 4 types of diagrams subdivided in 10 subdiagrams, the three-loop calculation is equivalent to consider 29 types of diagrams consisting of more than 200 subdiagrams. As expected, the presence of divergences requires renormalization to obtain a nite result in the end. Therefore a counterterm action has been added and the full one-loop two-point Green's function computed. This allows to x the relevant counterterms, resulting in a nite and regularization independent xed-area partition function at two loops. The resulting string susceptibility is more general than the KPZ result, although compatible. It indeed depends on one of the unxed counterterms. Therefore, the three-loop computation has been performed for the purpose of accessing to the complete contribution of the countert-Introduction La fantaisie et la liberté d'imagination ne s'acquièrent pas comme ça, qu'il y faut du temps, de l'obstination, de la sévérité, de la rigueur, des mathématiques, de la raison.

Philippe Sollers.

Quantum gravity remains one of the most thrilling scientic challenges of the 21th century. General relativity has deeply changed our conception of time and space a century ago and has been successfully experimentally veried using dierent techniques, including the recent discovery of gravitational waves [START_REF] Abbott | Observation of gravitational waves from a binary black hole merger[END_REF], and from them the rst observation of merging black holes. However, physics occurring near the center of black holes or in the early universe involves regions where the curvature becomes large and having at our disposal a theory of quantum gravity undoubtedly is unavoidable [START_REF] Penrose | Gravitational collapse and space-time singularities[END_REF][START_REF] Hawking | The Singularities of gravitational collapse and cosmology[END_REF]. Quantum gravity appears to be the missing piece in the jigsaw puzzle of our understanding of basic interactions.

Due to the complexity of the topic (see e.g. [410]), two-dimensional theories can be considered as insightful starting points to unravel the quantum aspect of gravity. First, one may think to look at pure quantum gravity. Yet, at 2D, pure quantum gravity does not contain any propagating degrees of freedom. For this reason, and of course keeping in mind that our universe does contain matter elds, a theory of quantum gravity coupled to matter is highly desirable. Since Polyakov's seminal paper [START_REF] Polyakov | Quantum Geometry of Bosonic Strings[END_REF], conformal matter coupled to quantum gravity on two-dimensional manifolds has been studied intensely [START_REF] Di Francesco | 2-D Gravity and random matrices[END_REF][START_REF] Kawai | Quantum gravity and random surfaces[END_REF], both in the discretized approach [1421] and in the continuum approach, known as Liouville quantum gravity [2227]. While the matrix models give a non-perturbative denition, the Liouville theories oer a more transparent physical interpretation. In the continuum approach, most of the computations have been done within the conformal gauge. When the conformally coupled matter is integrated out, one ends up with the Liouville action S L [g 0 , g] as an eective gravity action.

One of the simplest, yet interesting objects to study in this quantum gravity is the partition function at xed area Z[A]. One way to dene this partition function Z[A] on a Riemann surface of genus h, with metric g of area A, is to choose the conformal gauge with a background metric g 0 and a conformal factor σ such that g = e 2σ g 0 . Then,

Z grav = Dg e -µ 2 c A Z mat [g] = dA Z[A] ,
and Z[A] can be formally written as

Z[A] = Dσ e -κ 2 8π S L [g 0 ,g]-µ 2 c A δ A -d 2 x √ g 0 e 2σ , κ 2 = 26 -c 3
,
where c is the matter central charge 1 and the -26 accounts for the gauge xing (ghosts).

The delta-function restricts the integration to metrics of area A. The measure Dσ for 1 For reference on 2D Conformal Field Theory, see e.g. [START_REF] Di Francesco | Conformal eld theory[END_REF].

VIII IX the conformal factor can be derived from the standard metric on the space of metrics and is a complicated non-at measure. Many of the diculties in dealing with this quantum gravity theory originate from this measure being non-trivial. One of the main diculties when computing the partition function lies in the complicated non-at measure Dσ for the conformal factor.

Khnizik, Polyakov and Zamolodchikov studied two-dimensional gravity for genus zero in the light-cone gauge instead [START_REF] Knizhnik | Fractal Structure of 2D Quantum Gravity[END_REF]. Using the relation with an SL(2) current algebra they derived a remarkable formula relating the scaling dimensions ∆ of conformal primary operators coupled to gravity and their undressed conformal dimensions ∆ (0) :

∆ -∆ (0) = √ 25 -c - √ 1 -c 2 24 ∆(1 -∆) ,
known as algebraic KPZ relation. The scaling of the partition function is obtained from the dressing of the identity operator (∆ (0) id = 0) and leads to a scaling

Z[A] ∼ e -µ 2 c A A γstr-3 ,
where γ str = ∆ id is the string susceptibility. The previous equation then yields γ str = 2-2

√ 25-c √ 25-c- √ 1-c
. This formula gives the correct scaling for certain random lattice models corresponding to specic values of c (see [START_REF]Dynamically triangulated random surfaces[END_REF] and references therein).

On the other hand, working in the conformal gauge, and using several simplifying assumptions, specically on the measure Dσ, as well as consistency conditions, ref. [START_REF] Distler | Conformal Field Theory and 2D Quantum Gravity Or Who's Afraid of Joseph Liouville?[END_REF] and [START_REF] David | Conformal Field Theories Coupled to 2D Gravity in the Conformal Gauge[END_REF] have extended these remarkable formulae to surfaces of arbitrary genus. In particular they found the following formula for the string susceptibility

γ str = 2 + 2(h -1) √ 25 -c √ 25 -c - √ 1 -c ,
referred to as the KPZ formula. More recently, ref. [3036] have given more rigorous, though more abstract alternative derivations of these formulae for c ≤ 1 (and h = 0).

An alternative, more physical derivation can be found in [START_REF] David | Another derivation of the geometrical KPZ relations[END_REF]. For recent probabilistic constructions of the free eld Liouville measure on peculiar geometries, see [3840].

Probably the most puzzling property of these formulae is that they only seem to work for c ≤ 1, since for c > 1 (actually 1 < c < 25) they turn complex. This is the so-called c = 1 barrier (see e.g. [START_REF] David | Simplicial quantum gravity and random lattices[END_REF][START_REF] Correia | A Note on the c = 1 barrier in Liouville theory[END_REF] and references therein), where tachyons appear in the Liouville theory [START_REF] Gervais | Novel Triangle Relation and Absence of Tachyons in Liouville String Field Theory[END_REF]. It has been argued that, for c > 1, the two-dimensional geometry is dominated by congurations that no longer are smooth and that the surface develops spikes and a fractal character (see e.g. [START_REF] David | Conformal Field Theories Coupled to 2D Gravity in the Conformal Gauge[END_REF]4448]), sometimes considered as a branched polymer phase in which the surfaces collapse to tree-like objects (see e.g. [4952] and references therein).

However, the recent development of ecient multi-loop regularization methods on curved space-times opened the way for a precise denition of the measure Dσ as it follows from the usual metric on the space of metrics, and thus for a rst-principles quantum eld theory computation of Z[A], on a Riemann surface of arbitrary genus h.

X

This regularization [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF] is a generalization of the ζ-function regularization that works at one loop, to a regularization scheme that works for multi-loop computations on curved manifolds. Its basic objects are the heat kernel and generalized heat kernels dened on the manifold for which exist well-known formulae for the asymptotic small t" behaviour.

More precisely, to compute the partition function at xed area one can conveniently reparametrize the metric in the Kähler formalism, in terms of a xed background metric, the area A and the Laplacian (in the background metric) of the Kähler potential, so that the area appears explicitly as a coordinate" on the space of metrics. Since all the metrics are Kähler's in two dimension, this parametrization is complete. Then, the measure on the space of metrics is given in terms of dA, the standard at measure on the space of Kähler potentials, and various non-trivial determinants. Expanding these determinants and the interactions in the Liouville action in powers of 1 κ 2 generates the loop-expansion. Then, one can regularize the determinants and the propagators with the spectral cuto regularization [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF]. The rst-principles computation has been initiated within this framework in [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF] with the computation of the string susceptibility up to one loop, considering a somewhat more general quantum gravity action including the usual Liouville action for the coupling to conformal matter and the Mabuchi action for nonconformal matter. Indeed, it has been shown [START_REF] Ferrari | Gravitational Actions in Two Dimensions and the Mabuchi Functional[END_REF] that the Mabuchi action is involved in the rst-order mass correction to the Liouville action. For conformal matter only, the one-loop computation was in agreement with the one-loop semi-classical expansion of the KPZ formula. However, it is only beyond one loop, starting at two loops, that the regularized measure plays a role. Thus, it is from this order in the perturbation theory that one can expect to see a dierence with the KPZ formula, derived under the simplifying assumption of a free eld measure [START_REF] Distler | Conformal Field Theory and 2D Quantum Gravity Or Who's Afraid of Joseph Liouville?[END_REF][START_REF] David | Conformal Field Theories Coupled to 2D Gravity in the Conformal Gauge[END_REF]. In order to highlight the consequence of regularizing the integration measure in a proper way, the aim of this thesis is to continue the rst-principles computation of the xed-area partition function to higher order.

In Chapter 1, we present dierent approaches to two-dimensional quantum gravity. We discuss some suitable eective gravitational actions, such as the Liouville and Mabuchi actions. Then, we present the one loop computation [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF]. We also introduce in this chapter the Kähler formalism and the general smooth spectral cuto regularization developed in [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF]. This regularization scheme amounts to rst replacing each propagator G(x, y) = n 1 λn ψ n (x)ψ * n (y) by K(t i , x, y) = n e -t i λn λn ψ n (x)ψ * n (y) where the ψ n and λ n are the eigenfunctions and eigenvalues of the relevant dierential operator, then substituting t i = α i Λ 2 and integrating ∞ 0 dα i ϕ(α i ). Thus, Λ is the UV cuto and ϕ(α) is a fairly general regulator function. This yields a regularized propagator G reg (x, y) = n f (λn/Λ 2 ) λn ψ n (x)ψ * n (y) with an almost arbitrary f that is a Laplace transform of the almost arbitrary ϕ. For large Λ, the t i are small and to evaluate the diverging, as well as the nite parts of any loop diagram, it is enough to know the small t asymptotic of the K. Of course, K(t, x, y) is related to the heat kernel K(t , x, y) on the manifold, which has a well-known small t asymptotic expansion. We end this chapter by discussing the background independence of our formalism. Indeed, in any theory of quantum gravity, physical results should not depend on the arbitrarily introduced back-XI ground metric. However, the regularization scheme is based on inserting a cuto e -t i λn in the sums of the eigenvalues, and both the eigenfunctions and the eigenvalues depend on the choice of the background metric. Thus, the regularization is expected to induce an explicit background dependence in our computations.

Chapter 2 is devoted to the study of the Liouville theory on a Riemann surface of arbitrary genus, up to three loops in the perturbation theory. It is based on [START_REF] Bilal | 2D quantum gravity on compact Riemann surfaces and twoloop partition function: A rst principles approach[END_REF] and [START_REF] Leduc | 2D quantum gravity at three loops: a counterterm investigation[END_REF]. First, we perform the full two-loop computations of the pure Liouville gravity in the general case of a surface of arbitrary genus. The loop expansion is an expansion around the classical solution of the Liouville equation of motion, thus around a background metric of constant curvature. We explicitly write the non-trivial measure in the Kähler formalism and we expand the measure and the Liouville action up to the order 1 κ 2 which corresponds to the two-loop contributions to ln Z[A]. Of course, the measure vertex itself is already a one-loop" eect. The expansion of the Liouville action in terms of the Kähler potential yields the propagator and various n-point vertices. For the twoloop computation we nd a cubic and two quartic vertices, with derivatives acting in various ways. Thus, we have to consider 3 types of two-loop diagrams and one oneloop measure diagram. They get contributions from 10 subdiagrams. We explicitly write their regularized expressions and discuss that they can depend on the dimensionful quantities A and Λ only through the dimensionless combination AΛ 2 . Incidentally, this argument also explains why it is possible to obtain an explicit form for γ str . Indeed, γ str is the coecient of ln AΛ 2 and, hence, it is related to short-distance singularities which, in turn, are determined by local quantities like heat kernel coecients. The regularized xed-area partition function depends on the cuto Λ and the area A through divergent terms of the form AΛ 2 , ln AΛ 2 , ln AΛ 2 2 and AΛ 2 ln AΛ 2 . While the rst term only contributes to the divergent cosmological constant (which can be adjusted by a corresponding local counterterm), and the coecient of the second term determines γ str , the third and fourth terms are unwanted, non-local divergences. Quite non-trivially, all contributions to the third term added up to zero! However, the AΛ 2 ln AΛ 2 divergences remain. Then, one must introduce local counterterms in addition to the cosmological constant. Such local counterterms also contribute to the two-loop partition function, via one-loop diagrams. In particular, they can cancel the AΛ 2 ln AΛ 2 divergences, while they could not have cancelled the ln AΛ 2 2 divergences. The precise coecients of the counterterms are determined up to regulatorindependent nite constants by requiring the two-point function of the Kähler elds, or equivalently of e 2σ e 2σ , to be both nite and regulator-independent. Then, the partition function automatically becomes also both nite and regulator-independent. Yet, two nite renormalization constants on which the two-loop contribution to γ str depends remain undetermined. By a locality argument, one of these renormalization constants can be xed, precisely to the value consistent with the KPZ formula for γ str .

However, the other renormalization constant has no particular reason to be xed to the KPZ value. Remarkably, all values of this parameter are consistent with a background independent partition function at xed area. This means that we actually have (at least) a one-parameter family of quantum gravity theories that we can consistently dene in XII this Kähler approach (at least up to the two-loop order of perturbation theory).

The presence of this free parameter is intriguing and a natural question is whether the structure of the counterterm action introduced at two-loops is enough to also cancel the divergences at three (and higher) loops or whether new counterterms, with additional undetermined nite renormalization constants are required. Thus, in the last section of Chapter 2, we present the three-loop study of the Liouville quantum gravity on surfaces of any genus [START_REF] Leduc | 2D quantum gravity at three loops: a counterterm investigation[END_REF]. We expand the measure and the Liouville action up to the threeloop order 1 κ 4 and add the previous counterterm action. This leads to new vertices: sextic, quintic, quartic and cubic (from the measure). Thus, in order to compute both the leading quadratic and logarithmic three-loop divergences of ln Z[A], we have now to consider 29 types of diagrams consisting of more than 200 subdiagrams. Although the summed leading logarithmic divergence ln AΛ 2 3 cancels quite remarkably, the leading divergence AΛ 2 ln AΛ 2 2 has a non-vanishing coecient. Therefore, genuine three-loop counterterms are required. We compute their full contribution to the three-loop part of the partition function and then to equivalent order 1 κ 4 to the n-point function. Indeed, even if, in principle, one would need to compute the full one-loop three (and four)-point functions and the two-loop two-point function, and then to require them to be nite and regulator independent in order to really determine the coecients of the three-loop counterterms, we can still discuss the degrees of freedom of the counterterms by comparing the independent combinations in which they appear. Then, while the full three-loop computation is beyond the scope of this thesis, the full counterterm parts already encodes some interesting information. For consistency, we also include the two-loop counterterms that could not be completely xed by the previous study. It seems that among the initial 15 counterterms possible to insert, 5 stay undetermined and contribute to the string susceptibility, 4 more than at the two-loop order. We end this chapter by highlightening the relation between the regularized measure and the counterterms.

In Chapter 3, we present similar results for the mixed Liouville and Mabuchi theory in the more specic case of the torus. This chapter is based on [START_REF] Leduc | 2D Quantum Gravity at Three Loops with Liouville and Mabuchi action[END_REF]. We compute once again the full two-loop partition function. Counterterms are also required, as it should have been expected since the leading singularity comes from the Liouville action. However, quite remarkably, the leading logarithmic divergence ln AΛ 2 2 once more cancels out between the diagrams, thus allowing a renormalization through only local counterterms! Then, we check that the three-loop leading divergence ln AΛ 2 3 also cancel and we compute the contribution of the new counterterms to both the partition function and the n-point function. Then, we briey discuss the nite part of the two-loop two-point function, so that we can reduce the number of the remaining free parameters in our theory.

Ma cohabitation passionnée avec les mathématiques m'a laissé un amour fou pour les bonnes dénitions, sans lesquelles il n'y a que des à-peu-près.

Stendhal in Vie de Henry Brulard.

Two-dimensional quantum gravity

In this chapter, we rst briey recall the derivation of the KPZ result for the string susceptibility:

Z[A] ∼ e -µ 2 c A A γstr-3 , (1.1.1)
γ str = 2 + 2(h -1) √ 25 -c √ 25 -c - √ 1 -c , (1.1.2)
and the discrete results conrming this formula. Then, we focus on the specicities of our approach. We rst discuss the eective gravity action one may consider, based on [START_REF] Bilal | 2D quantum gravity on compact Riemann surfaces with non-conformal matter[END_REF] and [START_REF] Ferrari | Gravitational Actions in Two Dimensions and the Mabuchi Functional[END_REF], and the Kähler formalism. Then, to complete the presentation of our framework, we present the smooth spectral regularization techniques developed in [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF].

This regularization is the one used throughout this thesis to obtain the results presented in Chapter 2 and 3. In order to illustrate this formalism, we then recall the one-loop computation of the xed-area partition function of two-dimensional quantum gravity done in [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF]. Finally, we discuss the background independence of our regularization, and give some key formula to check the background (in)dependence of our results.

The DDK argument for the scaling of the xed-area partition function

Since we will compare our results with the KPZ relation (1.1.2), we nd it useful to briey present the DDK argument [START_REF] Distler | Conformal Field Theory and 2D Quantum Gravity Or Who's Afraid of Joseph Liouville?[END_REF][START_REF] David | Conformal Field Theories Coupled to 2D Gravity in the Conformal Gauge[END_REF] that gives this scaling of the partition function in terms of the area. This argument is amazingly simple and yields a result that has been cross-checked by other methods, at least for some specic models at certain values of the central charge (see section 1.1.2). On the other hand, as already pointed out, it relies on various simplifying assumptions that cannot be the full truth. (1. 1.3) In this conformal gauge, the Liouville action takes the simple form

S L [g 0 , g] ≡ S L [g 0 , σ] = d 2 x √ g 0 σ∆ 0 σ + R 0 σ , (1.1.4)
and the area A can be written as A = d 2 x √ g 0 e 2σ . Instead of using the correct nontrivial measure Dσ in the xed-area partition function, DDK use a at free-eld measure D 0 σ. At the same time they argue that, in the quantum theory, the coecient κ 2 in front of the Liouville action should be renormalized to κ 2 and that the denition of the area can no longer simply be d 2 x √ g 0 e 2σ : the coecient in the exponential must also be renormalized so that it becomes d 2 x √ g 0 e 2ασ which should be a conformal primary of weight [START_REF] Abbott | Observation of gravitational waves from a binary black hole merger[END_REF][START_REF] Abbott | Observation of gravitational waves from a binary black hole merger[END_REF] with respect to the Liouville action. Thus

Z DDK [A] = D 0 σ e -κ 2 8π S L [g 0 ,g]-µ 2 c A δ A -d 2 x √ g 0 e 2ασ . (1.1.5)
To determine the coecient α, one can switch to standard normalizations by setting σ = κ σ. Then,

κ 2 8π S L [g 0 , g] = 1 4π d 2 x √ g 0 1 2 σ ∆ 0 σ + κ 2 R 0 σ , (1.1.6)
which represents a standard free-eld action with a background charge with solution

α = κ 2 4 1 -1 - 8 κ 2 , (1.1.8)
where the sign has been chosen to match with the semi-classical limit κ 2 → ∞. The central charge of this free Liouville theory with background charge is

c L = 1 + 3 κ 2 .
(1.1.9)

Background independence requires the total central charge of the matter (c), ghosts (-26) and the Liouville theory to vanish:

c -26 + c L = 0 ⇒ κ 2 = 25 -c 3 = κ 2 - 1 3 .
(1.1.10)

TWO-DIMENSIONAL QUANTUM GRAVITY 3

To obtain the area dependence of Z DDK [A], one simply changes the integration variable in (1.1.5) from σ to σ = σ -b with some constant b. The at measure D 0 σ is invariant by translation, while the Liouville action changes as

S L [g 0 , σ] = S L [g 0 , σ ] + 8π b (1 -h) ,
(1. 1.11) and the delta scales as

δ A -d 2 x √ g 0 e 2ασ = e -2αb δ e -2αb A -d 2 x √ g 0 e 2ασ .
(1.1.12)

Putting things together, and letting e 2αb = A A 0

, one gets

Z DDK [A] = A A 0 -1-κ 2 2α (1-h) e -µ 2 c (A-A 0 ) Z DDK [A 0 ] , (1.1.13)
from which we read

γ str = 2 - κ 2 2α (1 -h) = 2 + 2(h -1) √ 25 -c √ 25 -c - √ 1 -c . (1.1.14)

Results from the discrete approaches

General references for the discrete models lattice models known as dynamical triangulations as well as matrix models can be found for instance in [START_REF] Di Francesco | 2-D Gravity and random matrices[END_REF]. Dynamical triangulations are a discretization of quantum geometries. Indeed, it was proposed in e.g. [START_REF] Ambjorn | Diseases of Triangulated Random Surface Models, and Possible Cures[END_REF][START_REF] Boulatov | Analytical and Numerical Study of the Model of Dynamically Triangulated Random Surfaces[END_REF]6063] that the integral over the internal geometry of a two-dimensional surface can be discretized as a sum over randomly triangulated surfaces. This way, the lattice spacing plays the role of a UV regulator, allowing the theory to be nite.

Another approach consists in expressing the two-dimensional quantum gravity partition function coupled to certain matter systems as the free energy of an associated hermitian matrix model. From it, many quantities can be computed exactly, as this matrix realization can be usually solved using large N techniques. It is interesting to note that matrix models originate from QCD, when in the 70s, t'Hooft [START_REF] Hooft | A Planar Diagram Theory for Strong Interactions[END_REF] realized that planar graphs with a large number of colors could be considered as Feynman diagrams for matrix models. He also noticed that the size of the matrices can be used as an expansion parameter.

Both these discrete models reveal themselves to be in agreement with the Liouville theory. Indeed, considering the KPZ formula on the sphere:

γ KPZ str = 2 -2 √ 25 -c √ 25 -c - √ 1 -c , (1.1.15)
specic values of it have been exactly derived from dynamical triangulations (see e.g. [START_REF] Knizhnik | Fractal Structure of 2D Quantum Gravity[END_REF][START_REF]Dynamically triangulated random surfaces[END_REF] and references therein) and matrix models (see e.g. [START_REF] Di Francesco | 2-D Gravity and random matrices[END_REF][START_REF] Kostov | O(n) Vector Model on a Planar Random Lattice: Spectrum of Anomalous Dimensions[END_REF] and references therein). To exemplify, let us stress the value c = 0 corresponding to pure gravity and 1.2. THE GRAVITATIONAL ACTION 4 yielding γ str = -1 2 . The case describing a coupling with the Ising model, i.e. c = 1 2 has also been derived [START_REF] Boulatov | The Ising Model on Random Planar Lattice: The Structure of Phase Transition and the Exact Critical Exponents[END_REF] and provides a string susceptibility γ str = -1 3 . The cases c = -2 , 0, give [START_REF]Dynamically triangulated random surfaces[END_REF][START_REF] Kazakov | Critical Properties of Randomly Triangulated Planar Random Surfaces[END_REF] γ str = -1 , 0, respectively. All these values are in agreement with (1.1.15).

A continuum limit that includes the sum over topologies of two-dimensional surfaces was then dened [START_REF] Douglas | Strings in Less Than One-Dimension[END_REF][START_REF] Gross | Nonperturbative Two-Dimensional Quantum Gravity[END_REF][START_REF] Brezin | Exactly Solvable Field Theories of Closed Strings[END_REF] for certain matter systems coupled to 2D quantum gravity.

Before closing this section, let us add that other approaches to quantum gravity have been developed more recently, like for instance Tensor Field Theory [6769] or Group

Field Theory [7072]. Details of these approaches are beyond the scope of this thesis.

The gravitational action

Ever since the seminal paper by Polyakov [START_REF] Polyakov | Quantum Geometry of Bosonic Strings[END_REF] it has been known that the coupling of conformal matter to gravity in two dimensions gives rise to the Liouville action [START_REF] Seiberg | Notes on quantum Liouville theory and quantum gravity[END_REF]7377] as the eective gravitational action. More precisely,

-ln Z (c) mat [g] Z (c) mat [g 0 ] = - c 24π S L [g 0 , g] , (1.2.1)
where Z

(c) mat is the partition function of conformal matter of central charge c and g and g 0 two metrics of area A and A 0 related in the conformal gauge by the conformal factor σ: g = e 2σ g 0 . Being dened as the logarithm of a ratio of partition functions computed with two dierent metrics, it is clear that the Liouville action satises a co-cycle identity

S[g 1 , g 2 ] + S[g 2 , g 3 ] = S[g 1 , g 3 ] , (1.2.2)
which is a fundamental consistency condition any gravitational action must satisfy [START_REF] Ferrari | Gravitational Actions in Two Dimensions and the Mabuchi Functional[END_REF].

For more general matter (plus ghost) partition functions, one denes a general gravitational action as

-ln Z mat [g] Z mat [g 0 ] = S grav [g 0 , g] . (1.2.3)
Any gravitational action dened this way automatically satises the co-cycle identity (1.2.2). The simplest example of such a gravitational action is the cosmological constant action

S µ [g 0 , g] = µ 2 c d 2 x( √ g - √ g 0 ) = µ 2 c (A -A 0 ) . (1.2.4)
This action must actually be added to the Liouville action in (1.2.1) as a counterterm to renormalize the divergences.

Other gravitational actions than the Liouville or cosmological constant actions can be constructed and have been studied mainly in the mathematical literature, like the Mabuchi and Aubin-Yau actions (see e.g. [7883]). These latter functionals crucially involve not only the conformal factor σ but also directly the Kähler potential φ and do admit generalizations to higher-dimensional Kähler manifolds. In the mathematical literature they appear in relation with the characterization of constant scalar curvature 1.2. THE GRAVITATIONAL ACTION 5 metrics (see e.g. [START_REF] Mabuchi | K-energy maps integrating Futaki invariants[END_REF][START_REF] Phong | Lectures on Stability and Constant Scalar Curvature[END_REF]8489]). Their roles as two-dimensional gravitational actions in the sense of (1.2.3) have been discussed in some detail in [START_REF] Ferrari | Gravitational Actions in Two Dimensions and the Mabuchi Functional[END_REF][START_REF] Ferrari | Random geometry, quantum gravity and the Kähler potential[END_REF]. In particular, ref. [START_REF] Ferrari | Gravitational Actions in Two Dimensions and the Mabuchi Functional[END_REF] has studied the metric dependence of the partition function of non-conformal matter like a massive scalar eld and shown that a gravitational action dened by (1.2.3) contains these Mabuchi and Aubin-Yau actions as rst-order corrections (rst order in m 2 A where m is the mass and A the area of the Riemann surface of metric g) to the Liouville action.

The purpose of this (quite technical) section is rst to derive a few results valid exactly at nite m. Then, doing a small mass expansion allows us to retrieve some of the results of [START_REF] Ferrari | Gravitational Actions in Two Dimensions and the Mabuchi Functional[END_REF] and in particular the involvement of the Mabuchi and Aubin-Yau actions in the rst order in m 2 A. This section is quite extensively based on [START_REF] Bilal | 2D quantum gravity on compact Riemann surfaces with non-conformal matter[END_REF].

Ideally one would like to study some general matter action where non-conformal terms ∼ a i O i have been added to some conformal theory and obtain exact results in these couplings a i . We are much less ambitious and simply study a single massive scalar eld with action

S mat [g, X] = 1 2 d 2 x √ g g ab ∂ a X∂ b X + m 2 X 2 = 1 2 d 2 x √ g X(∆ g + m 2 )X . (1.2.5)
Here ∆ g is the Laplace operator for the metric g, dened with a minus sign, so that its eigenvalues are non-negative:

∆ g = - 1 √ g ∂ a (g ab √ g∂ b ) . (1.2.6)
Maybe not too surprisingly, some of our massive formula look somewhat similar to those that can be found in [START_REF] Ferrari | Gravitational Actions in Two Dimensions and the Mabuchi Functional[END_REF] for the massless case. However, let us insist that our results are exact in m and valid for any nite mass. Nevertheless, we will write them in a way that immediately allows for a small mass expansion, thus recovering the Liouville action in the zero mass limit and the Mabuchi and Aubin-Yau actions as the rst-order corrections.

Massive versus massless matter

One should keep in mind that adding the mass term is not just a perturbation by some operator that has a non-zero conformal weight. This is due to the zero mode of the scalar eld that is absent from the action for zero mass but obviously plays an important role for non-zero mass. In particular, this means that the relevant quantities of the massive theory are not simply given by those of the massless theory plus order m 2 corrections. This is most clearly exemplied by the Green's function G(x, y) of the operator ∆ g + m 2 .

We let

(∆ g + m 2 )ψ n (x) = λ n ψ n (x) , n = 0, 1, 2, . . . . (1.2.7)
In general, if B is any quantity dened for m = 0, we will denote by B (0) the corresponding quantity for m = 0. Clearly, the eigenfunctions ψ n do not depend on m (i.e.

ψ n = ψ (0) n ), while λ n = λ (0)
n + m 2 . The eigenfunctions which may be chosen to be real, are orthonormalized as

d 2 x g(x) ψ n (x)ψ k (x) = δ nk .
(1.2.8)

THE GRAVITATIONAL ACTION 6

As it is clear from (1.2.6), ∆ g always has a zero mode and, hence,

λ 0 = m 2 , ψ 0 = 1 √ A .
(1.2.9)

The Green's function for m = 0 is given by

G(x, y) = n≥0 ψ n (x)ψ n (y) λ n , (∆ g + m 2 ) G(x, y) = 1 √ g δ(x -y) .
(1.2.10)

For m = 0 these denitions must be modied. Since λ (0) 0 = 0, obviously, the zero mode must be excluded from the sum and then

G (0) (x, y) = n>0 ψ n (x)ψ n (y) λ (0) n , ∆ g G (0) (x, y) = 1 √ g δ(x -y) - 1 A . (1.2.11)
The subtraction of 1 A on the r.h.s. ensures that, when integrated d 2 x √ g . . ., one correctly gets zero. We will consistently put a tilde over the various quantities we will encounter if the zero mode is excluded from the sum. 1 In particular, using (1.2.9), we can write for the massive Green's function

G(x, y) = 1 m 2 A + G(x, y) , (∆ g + m 2 ) G(x, y) = 1 √ g δ(x -y) - 1 A .
(1.2.12)

The smallest eigenvalue contributing in

G is λ 1 = λ (0) 1 + m 2 with λ (0) 1 > 0 being of order 1 A [53]. Thus if m 2 A 1 one can expand 1 λn = ∞ r=0 (-) r m 2r (λ (0) n ) r+1
, resulting in

G(x, y) = 1 m 2 A + G (0) (x, y) + ∞ r=1 (-m 2 ) r G (0) r+1 (x, y) , (1.2.13)
where

G r (x, y) = n≥0 ψ n (x)ψ n (y) λ r n , G (0) r (x, y) = n>0 ψ n (x)ψ n (y) (λ (0) n ) r . (1.2.14)
Clearly, the massive Green's function does not equal the massless one plus order-m 2 corrections since there is a crucial

1 m 2 A term in (1.2.13).
Writing X = n≥0 a n ψ n , the matter partition function is dened as

Z mat [g] = D g Xe -Smat[g,X] = ∞ n=0 da n √ 2π e -1 2 n≥0 λna 2 n = det(∆ g + m 2 ) -1/2 .
(1.2.15)

1 Except for determinants missing the zero mode, where we will write det , following the usual notation.

THE GRAVITATIONAL ACTION 7

In the massless case, since λ (0) 0 = 0, the integration over a 0 would be divergent and instead one replaces it by a factor √ A (see e.g. [START_REF] Ferrari | Gravitational Actions in Two Dimensions and the Mabuchi Functional[END_REF]). Thus

Z (0) mat [g] = D (0) g Xe -S (0) mat [g,X] = √ A ∞ n=1 da n √ 2π e -1 2 n>0 λ (0) n a 2 n = det ∆ g A -1/2 . (1.2.16)
Of course, the determinants det and det are ill-dened and need to be regularized. We will use the very convenient zeta function scheme to regularize-renormalize them (see e.g. [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF][START_REF] Ray | R Torsion and the Laplacian on Riemannian manifolds[END_REF][START_REF] Hawking | Zeta Function Regularization of Path Integrals in Curved Space-Time[END_REF]). The spectral ζ-functions are dened as

ζ(s) = n≥0 λ -s n , ζ(s) = n>0 λ -s n , (1.2.17)
and similarly for ζ (0) (s). By Weil's law (see e.g. [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF]), the asymptotic behaviour of the eigenvalues for large n is λ n ∼ n A and, hence the spectral ζ-functions are dened by converging sums for Re s > 1, and by analytic continuations for all other values. In particular, they are well-dened meromorphic functions for all s with a single pole at s = 1 with residue 1 4π (see e.g. [53]). A straightforward formal manipulation shows that ζ (0) ≡ d ds ζ(s)| s=0 provides a formal denition of -n≥0 ln λ n , i.e. of -ln det(∆ g +m 2 ):

Z mat [g] = exp 1 2 ζ (0) , Z (0) mat
[g] = A 1/2 exp 1 2 ( ζ (0) ) (0) . ( 1

.2.18)

There is a slight subtlety one should take into account, see e.g. [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF]. While the eld X is dimensionless, the ψ n scale as A -1/2 ∼ µ where µ is some arbitrary mass scale (even if m = 0), and the a n as µ -1 . It follows that one should write D g X = n µ dan √ 2π

. This

results in Z mat = n λn µ 2 -1/2
, so that ζ (0) should be changed into

ζ (0) → ζ (0) + ζ(0) ln µ 2 (1.2.19)
in the previous expressions of Z mat and Z (0) mat [53]. This writing may seem inhomogeneous since µ encountered in the logarithm is a mass scale. However, this seemingly inhomogeneity is just a trick since this combination stands for -n≥0 ln λn µ 2 . In section 1.2.5, we will get back with quantities of seemingly traceable homogeneity.

The regularization-renormalization of determinants in terms of the ζ-functions may appear as rather ad hoc, but it can be rigorously justied by introducing the spectral regularization [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF]. The regularized logarithm of the determinant then equals ζ (0) + ζ(0) ln µ 2 plus a diverging piece ∼ AΛ 2 (ln Λ 2 µ 2 + const), where Λ is some cuto. This diverging piece just contributes to the cosmological constant action (1.2.4), and this is why the latter must be present as a counterterm, to cancel this divergence.

Thus,

S grav [g 0 , g] = - 1 2 ζ g (0) -ζ g 0 (0) + [ζ g (0) -ζ g 0 (0)] ln µ 2 ,
(1.2.20)

S (0) grav [g 0 , g] = - 1 2 ln A A 0 - 1 2 ( ζ (0) g ) (0) -( ζ (0) g 0 ) (0) + ζ (0) g (0) -ζ (0) g 0 (0) ln µ 2 ,
where the rst line refers to the massive case and the second line to the massless one.

1.2. THE GRAVITATIONAL ACTION 8

The Kähler formalism

Consider a compact Riemann surface with xed complex structure moduli. As already mentioned, up to dieomorphisms, any two-dimensional metric g on the surface may be written in conformal gauge as g = e 2σ g 0 where g 0 is a reference metric that can be chosen to be the constant curvature metric associated with some area A 0 . Moreover, in two dimensions all the metrics are Kähler's, so that rather than writing the metric g in terms of this g 0 and the conformal factor σ, one can also parametrize it in terms of g 0 , the area A = d 2 x √ g and the Kähler potential φ as follows:

g = e 2σ g 0 , e 2σ = A A 0 1 - 1 2 A 0 ∆ 0 φ , (1.2
ω = A A 0 ω 0 + iA∂ ∂φ (1.2.22)
between the volume (Kähler) forms ω and ω 0 of the metrics g and g 0 . Often in the following we will use a rescaled (constant curvature) metric g * of area A, with corresponding Laplace operator ∆ * and Ricci scalar R * given by

g * = A A 0 g 0 , ∆ * = A 0 A ∆ 0 , R * = A 0 A R 0 = 8π(1 -h) A .
(1.2.23)

In particular, since A 0 ∆ 0 = A∆ * , eq. (1.2.21) can also be written as

e 2σ = A A 0 1 - 1 2 A∆ * φ ⇔ g = g * 1 - 1 2 A∆ * φ . (1.2.24)
Consequently, one has

d 2 x √ g = d 2 x √ g * = A , d 2 x √ g * R * = 8π(1 -h) .
(1.2.25)

As compared to the DDK approach (cf section 1.1.1), where in the quantum theory the area is computed as

d 2 x √ g 0 e 2ασ (
S M [g 0 , g] = d 2 x √ g 0 2π(h -1)φ∆ 0 φ + 8π(1 -h) A 0 -R 0 φ + 4 A σe 2σ , (1.2.26)
while the Aubin-Yau action takes the form

S AY [g 0 , g] = -d 2 x √ g 0 1 4 φ∆ 0 φ - φ A 0 . (1.2.27)
As already mentioned, they both satisfy the co-cycle identity (1.2.2) and were shown [START_REF] Ferrari | Gravitational Actions in Two Dimensions and the Mabuchi Functional[END_REF] to appear as accompanying the term of rst order in an expansion in m

δS L [g 0 , g] = 4π(1 -h) δA A - A 4 d 2 x √ g ∆R δφ , δS M [g 0 , g] = 2 δA A -d 2 x √ g R - 8π(1 -h) A δφ , δS AY [g 0 , g] = 1 A d 2 x √ g δφ .
(1.2.29)

Thus the Liouville and Mabuchi actions obviously admit the constant scalar curvature metrics as saddle-points at xed area. Although not obvious from the previous equation, the variation of the Aubin-Yau action when restricted to the space of Bergmann metrics is similarly related to metrics of constant scalar curvature [START_REF] Phong | Lectures on Stability and Constant Scalar Curvature[END_REF].

Heat kernel and zeta functions

Before going forward and computing δS grav as dened in (1.2.20), one must introduce useful quantities such as the heat kernel and the local zeta functions (see e.g. [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF]9395] and the review [START_REF] Vassilevich | Heat kernel expansion: User's manual[END_REF]).

The heat kernel and the integrated heat kernel for the operator ∆ g + m 2 are dened in terms of the eigenvalues and eigenfunctions (1.2.7) as

K(t, x, y) = n≥0 e -λnt ψ n (x)ψ n (y) , K(t) = d 2 x √ g K(t, x, x) = n≥0 e -λnt .
(1.2.30)

THE GRAVITATIONAL ACTION 10

The corresponding K, K (0) and K (0) are dened similarly. The heat kernel K is the solution of

d dt + ∆ g + m 2 K(t, x, y) = 0 , K(t, x, y) ∼ 1 √ g δ(x -y) as t → 0 . (1.2.31)
Note that it immediately follows from either (1.2.30) or (1.2.31) that the massless and massive heat kernels are simply related by K(t, x, y) = e -m 2 t K (0) (t, x, y) .

(1.2.32)

Since the eigenvalues λ n of ∆ g + m 2 are positive, it is clear from (1.2.30) that K(t, x, y) is given by a converging sum and remains nite even as x → y as long as t > 0. For t → 0 however, one recovers various divergences, and in particular

∞ 0 dt K(t, x, y) = G(x, y) (1.2.33)
exhibits the short distance singularity of the Green's function which is well-known to be logarithmic in two dimensions.

The behaviour of K for small t is related to the asymptotics of the eigenvalues λ n for large n, which in turn is related to the short-distance properties of the Riemann surface. It is thus not surprising that the small t asymptotic is given in terms of local expressions of the curvature and its derivatives. Indeed, one has the well-known small t expansion :

K(t, x, y) = e -( 2 /4t)-m 2 t 4πt a (0) 0 (x, y) + a (0) 1 (x, y) t + a (0) 2 (x, y) t 2 + O(t 3 ) (1.2.34)
where 2 ≡ 2 g (x, y) is the geodesic distance squared between x and y. For small t, the exponential forces 2 to be small (of order √ t) and one can use normal coordinates around y. This allows one to obtain quite easily explicit expressions for the a r (x, y) in terms of the curvature tensor and its derivatives. They can be found e.g. in [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF]. Here, we will only need them at coinciding points y = x, where

K(t, y, y) = 1 4πt 1 + R 6 -m 2 t + . . . . (1.2.35)
Let us note that in the massless case and if the zero mode is excluded one has instead

K (0) (t, y, y) = 1 4πt 1 + R 6 - 4π A t + . . . . (1.2.36)
The spectral regularization scheme developed in [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF] and used is this thesis also involves the heat kernel so that we will come back to these quantities later in section 1. For s = 0, -1, -2, . . ., Γ(s) has poles and the value of ζ(s, x, y) is entirely determined by the singularities of the integral over t that arise from the small t asymptotic of K. As shown above, the latter is given by local quantities on the Riemann surface. Explicitly (see e.g. [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF])

ζ(-k, x, x) = (-1) k k! a 1+k (x, x) 4π , k = 0, 1, 2, . . . (1.2.39)
so that in particular,

ζ(0, x, x) = R(x) 24π - m 2 4π and ζ(0, x, x) = R(x) 24π - m 2 4π - 1 A .
(1.2.40)

On the other hand, the values for s = 1, 2, 3, . . . or the derivatives at s = 0 cannot be determined just from the small t asymptotic. To access them requires the knowledge of the full spectrum of ∆ g + m 2 . Clearly, ζ(1, x, y) = G(x, y) is singular as x → y, while for s = 1, ζ(s, x, y) provides a regularization of the propagator. More precisely, it follows from (1.2.38) that ζ(s, x, x) is a meromorphic function with a pole at s = 1 and that the residue of this pole is

a 0 (x,x) 4π = 1 4π . Thus [53] G ζ (x) = lim s→1 µ 2(s-1) ζ(s, x, x) - 1 4π(s -1) (1.2.41)
is well-dened. (Here µ is an arbitrary scale.) This is an important quantity, called the Green's function at coinciding points. One can give an alternative denition of G ζ by subtracting the short distance singularity from G(x, y) and taking x → y.

Variation of the gravitational action

From (1.2.20), one directly reads the variation of the gravitational action:

δS grav [g 0 , g] = - 1 2 δζ (0) + δζ(0) ln µ 2 , δS (0) grav [g 0 , g] = - δA 2A - 1 2 δ( ζ (0) ) (0) + δ ζ (0) (0) ln µ 2 .
(1.2.43)

THE GRAVITATIONAL ACTION 12

Our goal is to compute δζ (0) ≡ δζ g (0) and δζ(0) ≡ δζ g (0) and express them as exact dierentials so that one can integrate them and obtain the nite dierences ζ g 2 (0)-ζ g 1 (0) and ζ g 2 (0) -ζ g 1 (0), thus accessing S grav [g 1 , g 2 ].

To do so, we need to study how the eigenvalues λ n and eigenfunctions ψ n change under an innitesimal change of the metric. Since g = e 2σ g 0 , the Laplace operator ∆ g and hence also ∆ g + m 2 only depend on the conformal factor σ and on g 0 : ∆ g = e -2σ ∆ 0 and thus under a variation δσ of σ one has

δ∆ g = -2δσ∆ g ⇒ ψ k |δ∆ g |ψ n = -2λ (0) n ψ k |δσ|ψ n = -2(λ n -m 2 ) ψ k |δσ|ψ n , (1.2
δψ n = -ψ n |δσ|ψ n ψ n -2 k =n λ n -m 2 λ n -λ k ψ k |δσ|ψ n ψ k . (1.2.46)
Let us insists that this is rst-order perturbation theory in δσ, but it is exact in m 2 .

From (1.2.45) one immediately gets,

ζ g+δg (s) = n≥0 1 (λ n + δλ n ) s = ζ g (s) + 2s n≥0 λ n -m 2 λ s+1 n ψ n |δσ|ψ n . (1.2.47)
Note that λ 0 = m 2 and, hence, there is no zero mode contribution to the second term.

Thus,

δζ(s) = δ ζ(s) = 2s d 2 x √ g δσ(x) ζ(s, x, x) -m 2 ζ(s + 1, x, x) .
(1.2.48)

For m = 0, the term in brackets could have been equally well written as ζ(s, x, x)m 2 ζ(s + 1, x, x), but the writing in terms of the ζ is valid for all non-zero and zero values of m. It follows that

δζ (0) = 2 d 2 x √ g δσ(x) ζ(0, x, x) -2m 2 d 2 x √ g δσ(x) lim s→0 ζ(s + 1, x, x) + s ζ (s + 1, x, x) , δζ(0) = -2m 2 d 2 x √ g δσ(x) lim s→0 s ζ(s + 1, x, x) . ( 1
δζ (0) = 1 12π d 2 x √ g δσ(x)R(x) - δA A -2m 2 d 2 x √ g δσ(x) G ζ (x) + 1 4π - 1 4π ln µ 2 δζ(0) = - m 2 2π d 2 x √ g δσ(x) .
(1.2.52)

The result for δζ(0) can also be directly obtained from (1.2.40), using the fact that

d 2 x √ g R(x) = 8π(1 -h) (1.2.53) is a topological invariant. Since G ζ = G ζ + 1 m 2 A
, we arrive at two equivalent expressions for δζ (0) + δζ(0) ln µ 2 :

δζ (0) + δζ(0) ln µ 2 = δ ζ (0) + δ ζ(0) ln µ 2 = 1 12π d 2 x √ g δσ(x) R(x) - δA A -2m 2 d 2 x √ g δσ(x) G ζ (x) + 1 4π = 1 12π d 2 x √ g δσ(x) R(x) -2m 2 d 2 x √ g δσ(x) G ζ (x) + 1 4π . (1.2.54)
As it stands, this result is exact in m and holds whether m 2 A is small or not. Let us insists that the G ζ and G ζ appearing on the right-hand side are the massive ones. The rst writing is the appropriate one to study the small m 2 asymptotic, as G ζ has a smooth limit for m → 0. However, writing the metric g = e 2σ g 0 amounts to xing the dieomorphism invariance.

δS (0) grav [g 0 , g] = - 1 24π d 2 x √ g δσ(x) R(x) = - 1 24π δS L [g 0 , σ] . ( 1
The corresponding Faddeev-Popov determinant that arises gives also a contribution to S grav in terms of the Liouville action but with a coecient + 26 24π (see e.g. [11, 97, 98]), so that one gets the overall contribution:

S (0) grav [g 0 , g] ghost + conf matter = 26 -c 24π S L [g 0 , σ] . ( 1
δS grav [g 0 , g] = - 1 24π δS L [g 0 , g] + m 2 d 2 x √ g δσ(x) G ζ (x) + 1 4π . (1.2.57)
The task ahead is then to rewrite the second term on the r.h.s as the variation of some local functional.

Let us compute δG ζ (x). In order to do so, we rst establish a formula for δG(x, y) under a variation δg = 2δσ g of the metric and thus under a corresponding variation δ∆ g = -2δσ∆ g of the Laplace operator. One can then either use the denition (

d 2 x √ g G(x, z) 2 = d 2 x √ g n,k≥0 ψ n (x)ψ n (z)ψ k (x)ψ k (z) λ n λ k = n≥0 ψ n (z)ψ n (z) λ 2 n = ζ(2, z, z) . (1.2.62) (Note that ζ(2, z, z) = G 2 (z, z) is nite.) It follows that δ d 2 x √ g G ζ (x) = 2 d 2 x √ g G ζ (x) δσ(x) -2 m 2 d 2 z √ g ζ(2, z, z) δσ(z) + 1 2π d 2 x √ g δσ(x) . ( 1
δS grav [g 0 , g] = δ - 1 24π S L [g 0 , g] + m 2 2 d 2 x √ g G ζ (x) + m 4 d 2 x √ g ζ(2, x, x)δσ(x) .
(1.2.64)

Note that we can replace G ζ by G ζ in the second term. Indeed, their dierence is

m 2 2 d 2 x √ g 1 m 2 A = 1
2 , whose variation vanishes.

Next, we use (1.2.38) to rewrite the last term as m 4 ∞ 0 dt t d 2 x √ g K(t, x, x)δσ(x),

and establish a formula for the variation of the integrated heat kernel K(t). Since λ 0 = m 2 its variation vanishes and we have

δK(t) = δ K(t) = - n>0 t e -λnt δλ n = 2 n>0 t e -λnt (λ n -m 2 ) d 2 x √ g ψ 2 n (x) δσ(x) = -2 t d dt + m 2 d 2 x √ g K(t, x, x) δσ(x) = -2 t e -m 2 t d dt d 2 x √ g K (0) (t, x, x) δσ(x) , (1.2.65)
where we used (1.2.45) and (1.2.32). It then follows that

1 2 ∞ 0 dt t e m 2 t -m 2 t -1 δ K(t) = - ∞ 0 dt e m 2 t -m 2 t -1 d dt + m 2 d 2 x √ g K(t, x, x) δσ(x) = m 4 ∞ 0 dt t d 2 x √ g K(t, x, x) δσ(x) = m 4 d 2 x √ g ζ(2, x, x) δσ(x) = m 4 d 2 x √ g ζ(2, x, x) δσ(x) - 1 A d 2 x √ g δσ(x) , (1.2
δS grav [g 0 , g] = δ - 1 24π S L [g 0 , g] + 1 2 ln A A 0 + m 2 2 d 2 x √ g G ζ (x) + 1 2 ∞ 0 dt t e m 2 t -m 2 t -1 K(t) .
(1.2.67)

Note that in the last term the t-integral is convergent both at t = 0 and at t = ∞.

S grav [g 0 , g] = - 1 24π S L [g 0 , g] + 1 2 ln A A 0 + m 2 2 d 2 x √ g G ζ (x; g) - √ g 0 G ζ (x; g 0 ) + 1 2 ∞ 0 dt t e m 2 t -m 2 t -1 K(t; g) -K(t; g 0 ) .
(1.2.68)

Thus we have expressed the variation of the gravitational action as the variation of a sum of (local) functionals that are all perfectly well-dened without any need of analytical continuation (contrary to the initial ζ (0)).

It will be useful to rewrite (1.2.61) to obtain the variation of G ζ in terms of quantities that all have well-dened limits as m → 0.

Recall that G = 1 m 2 A + G and G ζ = 1 m 2 A + G ζ .
Thus (1.2.61) can be rewritten as

δ G ζ (x) = - 4 A d 2 z √ g G(x, z) δσ(z) -2 m 2 d 2 z √ g G(x, z) 2 δσ(z) + δσ(x) 2π .
(1.2.69)

Using (1.2.28), we can replace δσ in the second term by -A∆δφ/4. Indeed, the δA/2A piece doesn't contribute since G has no zero mode. Using the dierential equation (1.2.12) satised by G, the second term may then be rewritten as

- 4 A d 2 z √ g G(x, z) δσ(z) = d 2 z √ g δφ(z) ∆ z G(x, z) = δφ(x) -δS AY [g 0 , g] -m 2 d 2 z √ g G(x, z) δφ(z) , (1.2.70)
with δS AY given in (1.2.29). Thus

δ G ζ (x) = δσ(x) 2π +δφ(x)-δS AY [g 0 , g]-m 2 d 2 z √ g G(x, z)δφ(z) + 2 G(x, z) 2 δσ(z) .
(1.2.71)

In exactly the same way we also get

δ G(x, y) = 1 2 δφ(x) + δφ(y) -δS AY [g 0 , g] (1.2.72) -m 2 d 2 z √ g 1 2 G(x, z) + G(y, z) δφ(z) + 2 G(x, z) δσ(z) G(z, y) .
While these are exact relations valid for all m, they are written in a way that makes the small mass expansions obvious.

Small mass expansion

Eq. (1.2.67) is non-perturbative in m. However, it is also written in a way that immediately allows for a perturbative expansion in m, since G ζ and K have smooth limits as m → 0. The m 2 → 0 limit of (1.2.67) also exhibits an extra 1 2 ln A A 0 term which only gets removed for m 2 = 0 due to the dierence in the denitions (cf (1.2.20)).

THE GRAVITATIONAL ACTION

Let us determine the order m 2 A-correction S

(1)

grav to S (0) grav . Since K(t) = e -m 2 t K (0) (t) and λ (0) n ∼ 1 A it follows that ∞ 0 dt t e m 2 t -m 2 t -1 K(t) = O((m 2 A) 2 ) . (1.2.73)
Next, it follows from (1.2.12) and (1.2.13), upon subtracting the short-distance singularity and letting x → y, that

G ζ (x) = G (0) ζ (x) + ∞ r=1 (-m 2 ) r G (0) r+1 (x, x) , (1.2.74) so that d 2 x √ g G ζ (x) = d 2 x √ g G (0) ζ (x)+O(m 2 A
). Thus, one reads the terms of order m 2 A in the gravitational action from (1.2.68):

S (1) grav [g 0 , g] = m 2 2 AΨ G [g] -A 0 Ψ G [g 0 ] = m 2 A 2 Ψ G [g] -Ψ G [g 0 ] + (A -A 0 ) m 2 2 Ψ G [g 0 ] , (1.2.75)
where, following [START_REF] Ferrari | Gravitational Actions in Two Dimensions and the Mabuchi Functional[END_REF], we have introduced

Ψ G [g] = 1 A d 2 x √ g G (0) ζ (x; g) . (1.2.76)
The variation of G ζ was given in (1.2.71) and that of G

(0) ζ immediately follow as δ G (0) ζ (x) = δσ(x) 2π + δφ(x) -δS AY [g 0 , g] ,
(1.2.77) so that

G (0) ζ (x; g) = G (0) ζ (x; g 0 ) + σ(x) 2π + φ(x) -S AY [g 0 , g] .
(1.2.78)

This relation has been derived before in [START_REF] Ferrari | Gravitational Actions in Two Dimensions and the Mabuchi Functional[END_REF]. Using (1.2.21), it is then straightforward to obtain (as in [START_REF] Ferrari | Gravitational Actions in Two Dimensions and the Mabuchi Functional[END_REF])

Ψ G [g] -Ψ G [g 0 ] = 1 8π d 2 x √ g 0 4 A σe 2σ -2πφ∆ 0 φ -4πφ ∆ 0 G (0) ζ (x; g 0 ) . (1.2.

79)

For genus h = 0 and choosing g 0 to be the round metric on the sphere, G

ζ (x; g 0 ) is a constant, and one directly gets the Mabuchi action for h = 0. More generally, for arbitrary genus, it has been shown in [START_REF] Ferrari | Gravitational Actions in Two Dimensions and the Mabuchi Functional[END_REF] that

Ψ G [g] -Ψ G [g 0 ] = 1 8π S M [g 0 , g] + h S AY [g 0 , g] -d 2 x √ g c φ , (1.2.80)
where g c is the canonical metric on the Riemann surface. Finally, (1.2.75) becomes

S (1) grav [g 0 , g] = m 2 A 2 1 8π S M [g 0 , g] + h S AY [g 0 , g] -d 2 x √ g c φ + (A -A 0 ) m 2 2 Ψ G [g 0 ] .
(1.2.81)

REGULARIZATION OF THE PARTITION FUNCTION 18

While the last term contributes to the cosmological constant action, the other terms are to be considered as the genuine order m 2 A correction to the gravitational action, and it involves the Mabuchi and Aubin-Yau actions. This motivates the choice made in this thesis to also consider the Mabuchi action in addition to the Liouville action in the quantum gravity partition function.

One can straightforwardly obtain the expansion in powers of m 2 of the terms in (1.2.67) or (1.2.68). Denoting the term of order m 2 A r by S (r) grav , we have

S grav [g 0 , g] = ∞ r=0 S (r) grav [g 0 , g] , (1.2.82)
with

S (r) grav [g 0 , g] = (-) r+1 2r m 2r d 2 x √ g G (0) r (x, x; g) -d 2 x √ g 0 G (0) r (x, x; g 0 ) , r ≥ 2 .

Regularization and computation of the partition function at xed area

The quantum gravity partition function may be written in terms of the so-called partition

function at xed area Z[A] Z grav = Dg e -µ 2 c A Z mat [g] = dA Z[A] , (1.3.1)
with

Z[A] = Dg e -µ 2 c A Z mat [g] δ Σ h d 2 x √ g -A = Dg e -Sgrav[g 0 ,g] δ Σ h d 2 x √ g -A , (1.3.2)
and Σ h is a compact Riemann surface of genus h. Some suitable gravitational actions have been discussed in the previous section. In Chapter 2, we will only consider the

REGULARIZATION OF THE PARTITION FUNCTION

Liouville action while in Chapter 3 and in the remaining part of this chapter both the Liouville and the Mabuchi actions will be considered. As previously stated (see section 1.2.5), xing the conformal gauge amounts to replacing Dg by Dσ Z ghost [g] and results in a term + 26 24π S L [g] in the gravitational action. Thus, following [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF] we write

Z[A] = Dσ e -κ 2 8πε S L [g 0 ,g]-β 2 ε S M [g 0 ,g]-µ 2 c A δ Σ h d 2 x √ g -A , (1.3.3)
where ε is a loop counting parameter (to be set to 1 in the end) and κ 2 = 26-c 3 plays the role of the coupling constant of the Liouville theory of central charge c and includes the ghost contribution.

Note that, as discussed in [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF], a further subtlety arises in the case of the sphere, h = 0, because the gauge-xing (1.2.21) then is incomplete. An additional gauge xing of the residual SL(2, C)/SU(2) group of dieomorphisms acting non-trivially on the conformal factor σ and Kähler potential φ must be performed. The result is to project out the spin-one modes of φ in its decomposition in terms of the spherical harmonics Y m l , which produces an overall factor of A 3/2 in the partition function coming from the Faddeev-Popov determinant. It is implicitly assumed in the rest of this thesis that h > 0, in order not to explicitly deal with this complication.

The measure on the space of metrics

We consider the quantum gravity integration measure Dg. This measure can be derived from a choice of metric on the space of metrics. It is generally assumed that this metric should be ultralocal and, hence, of the form ||δg|| 2 = d 2 x √ g δg ab δg cd g ac g bd +a g ab g cd for some constant a > -1/2 (see e.g. [START_REF] Polyakov | Quantum Geometry of Bosonic Strings[END_REF][START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF][START_REF] Polchinski | Evaluation of the one loop string path integral[END_REF]). The authors of [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF] argue that ultralocality may not be the most important condition for a theory of quantum gravity, and that background independence is a more fundamental requirement. (Then, other metrics satisfying the constraint of background independence could be used in the path integral, such as the Donaldson-Semmes-Mabuchi metric [START_REF] Mabuchi | Some symplectic geometry on compact Kähler manifolds[END_REF][START_REF] Donaldson | Symmetric spaces, Kähler geometry and Hamiltonian dynamics[END_REF][START_REF] Semmes | Complex Monge-Ampère and Symplectic Manifolds[END_REF].) Nevertheless, we use the above with g = e 2σ g 0 to get

||δg|| 2 = 8 (1 + 2a) ||δσ|| 2 , ||δσ|| 2 = d 2 x √ g 0 e 2σ (δσ) 2 . (1.3.4)
The integration measure Dσ over σ is determined from this metric [START_REF] Distler | Conformal Field Theory and 2D Quantum Gravity Or Who's Afraid of Joseph Liouville?[END_REF]. It is not the measure of a free eld, because of the non-trivial factor e 2σ . Instead of σ, we will use equivalently the variables (A, φ), using the relation (1.2.21). Following [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF], we have

||δσ|| 2 = (δA) 2 4A + ||δσ|| 2 A , (1.3.5)
where ||δσ|| 2 A is the metric on the space of metrics for xed area A

||δσ|| 2 A = 1 16 d 2 x √ g (A∆δφ) 2 .
(

||δσ|| 2 A = 1 16 d 2 x √ g 0 e 2σ e -2σ A∆ 0 δφ 2 = 1 16 d 2 x √ g * 1 - 1 2 A∆ * φ -1 A∆ * δφ 2 , (1.3.7)
in terms of the rescaled (constant curvature) metric g * of area A. Formally, (1.3.5) and

(1.3.7) thus induce the non-trivial measure [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF]

Dσ = dA √ A Dφ = dA √ A Det 1 -1 2 A∆ * φ -1 1/2 Det (A∆ * ) D * φ , (1.3.8)
where D * φ is the standard free eld integration measure in the background metric g * deduced from the metric ||δφ|| 2 * = d 2 x √ g * δφ 2 . As previously stated, the notation

Det means that we are not taking into account the zero mode when computing the determinant. Indeed, from (1.2.21), the zero mode of φ is unphysical and must not be included in the integration measure over the Kähler potentials. The measure D * φ can be expressed in the traditional way by expanding φ in eigenmodes of the Laplace operator ∆ * . Following the notation of section 1.2.1, we have

φ = n>0 a n ψ n , ∆ * ψ n = λ (0) n ψ n , d 2 x √ g * ψ n ψ m = δ nm , (1.3.9)
and thus

D * φ = n>0 µ da n √ 2π .
(1.3.10)

One could instead dene an expansion of φ with respect to eigenfunctions normalized with the metric g 0 of area A 0 . Then, the integration measure D 0 φ in the background

metric g 0 is related to D * φ by D * φ = e 1 2
n ln A A 0 D 0 φ [START_REF] Ferrari | Gravitational Actions in Two Dimensions and the Mabuchi Functional[END_REF]. All these relations are formal and must be regularized.

Our starting point for the computation of the quantum gravity partition function at

xed area then is (cf (1.3.3)) Z[A] = 1 √ A Dφ exp - κ 2 8πε S L [g 0 , g] - β 2 ε S M [g 0 , g] -µ 2 c A (1.3.11) ≡ 1 √ A D * φ e -µ 2 c A exp -S measure [g 0 , g] - κ 2 8πε S L [g 0 , g] - β 2 ε S M [g 0 , g] .
The measure action in thus dened as

S measure [g 0 , g] = - 1 2 ln Det 1 -1 2 A∆ * φ -1 -ln Det (A∆ * ). (1.3.12)
This expression is a formal writing and needs to be regularized. . Expanding around this saddle-point up to quadratic order then leads to

S L [g 0 , g] = 4π(1 -h) ln A A 0 + 1 16 d 2 x √ g * φ (A∆ * ) 2 ∆ * + 8π(h -1) A φ + O(φ 3 ) S M [g 0 , g] = 2 ln A A 0 + 1 4 d 2 x √ g * φ A∆ * ∆ * + 8π(h -1) A φ + O(φ 3 ) . (1.3.13)
with respect to the Kähler potential φ. The terms O(φ 3 ) are relevant beyond the oneloop approximation and will be considered in Chapter 2 and 3. Let us insist that, in this expression, the zero mode is excluded. Indeed, as stressed in the previous subsection, the zero mode of φ is unphysical for all h.

As highlighted in [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF], the full dependence of the these actions in the area A comes from the tree-level term in (1.3.13). Indeed, using (1.2.23), all the apparent A-dependence in the second term can be absorbed, and this remains true to all orders in the expansion.

The authors of [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF] conclude that the non-trivial area dependence of the quantum gravity partition function is actually due to the requirement of a proper regularization procedure for the measure in the functional integral.

The loop counting parameter ε introduced in (1.3.3) allows us to write a loopexpansion of the form

ln Z[A] ≡ W [A] = L≥0 W (L) [A] (1.3.14)
where W (L) [A] is the sum of connected vacuum diagrams of order ε L-1 . The tree-level contribution W (0) can be directly read from the tree-level terms in (1.3.13)

W (0) [A] = 1 ε h -1 2 κ 2 -2β 2 ln A A 0 . (1.3.15)
This yields a tree-level string susceptibility

γ str = h -1 2 κ 2 ε -2 β 2 ε + O(1) . (1.3.16)
Note that when considering only the Liouville action, the loop counting parameter can be taken to be directly κ 2 so that

γ L str = h -1 2 κ 2 + O(1) (1.3.17)
which is precisely the KPZ value. Indeed, if one does a loop-expansion of the KPZ result (1.1.2) for the string susceptibility up to one loop, one gets

γ str = h -1 2 κ 2 + 19 -7h 6 + O(κ -2
) . The one-loop computation was made in [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF], starting from (1.3.11). The complicated

determinant factor Det (1 -1 2 A∆ * φ) -1 1/2
in the measure action (1.3.12) is irrelevant at one loop. Its contributions will be carefully studied in the following chapters of this thesis.

From eq. (1.3.13) and (1.3.12) one has [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF],

W (1) [A] = -

1 2 ln A + ln Det (A∆ * ) - 1 2 ln Det A∆ * ∆ * + 8π(h -1) A κ 2 32π A∆ * + β 2 . (1.3.19)
This expression is formal and must be regularized. In section 1.2 the determinants were regularized using the standard ζ-function scheme. However, as discussed in [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF], this method is not adapted when dealing with a product of determinants. This is the socalled multiplicative anomaly [101106]: the ζ-renormalized determinant of a product of operators does not always equal the product of the ζ-renormalized determinants of the operators. As reminded in [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF], the multiplicative anomaly is usually irrelevant since the determinants are dened modulo the addition of arbitrary local counterterms [107111].

However, in the present computation, this anomaly has to be taken into account when performing the change of variables (1.2.21) from the conformal factor σ to the area A and Kähler potential φ if one uses the ζ-function regularization scheme. Otherwise one gets a wrong area dependence of the partition function. This is explained and derived in [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF].

Spectral cuto regularization

The general spectral cuto approach is a powerful regularization scheme developed in [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF]. This is the regularization scheme used throughout this thesis. At one loop it generalizes the zeta function regularization scheme without stumbling into the multiplicative anomaly problem [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF]. It also allows a physical discussion of the divergences. At higher loops it amounts to regulating the propagators in a specic way. In particular, it will allow us in the following to regularize the two-and three-loop Feynman diagrams in our semi-classical expansion of the partition function.

A basic feature of the spectral cuto is to replace any sum over the eigenvalues λ D n of the relevant dierential operator D g by a regularized sum as

n F (λ D n ) → n F (λ D n ) ϕ,Λ = ∞ 0 dα ϕ(α) n e -α Λ 2 λ D n F (λ D n ) . (1.3.20)
The regulator function ϕ is relatively arbitrary, except for a normalization condition which ensures that for Λ → ∞ one recovers the unregulated sum, and regularity proper-

ties as α → 0 or α → ∞ [53]: ∞ 0 dα ϕ(α) = 1 ϕ(α) = α→0 O(α n ) for any n ≥ 0 , (1.3.21) ∞ 0 dα α n ϕ(α) < ∞ for any n ≥ 0 .
Λ is the cuto scale (which is eventually sent to innity). An important physical requirement is that, in the end, all ϕ-dependence should be only in terms that can be changed by the addition of (local) counterterms, while any physical part should be regulator independent 3 . This has been checked on several examples in [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF].

Then, in particular, the regularized propagator 4 is

G(x, y) → G(x, y) ϕ,Λ = ∞ 0 dα ϕ(α) n>0 e -α Λ 2 λ D n ψ n (x)ψ n (y) λ D n . (1.3.22)
The right-hand-side is related to the hatted heat kernel on the manifold. Recalling the expression of the heat kernel (1.2.30), this quantity is dened in [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF] as

K(t, x, y) = n>0 e -λ D n t ψ n (x)ψ n (y) , K(t, x, y) = ∞ t dt K(t , x, y) = n>0 e -λ D n t ψ n (x)ψ n (y) λ D n . (1.3.23)
The integration is convergent at +∞ since5 λ D n > 0 for all n > 0. Of course, K(t, x, y) and K(t, x, y) are symmetric under exchange of x and y and one has the following relations

- d dt K(t, x, y) = D x K(t, x, y) = D y K(t, x, y) = K(t, x, y) , (1.3.24)
as well as

K(0, x, y) = G(x, y) , K(0, x, y) = δ(x -y) [g * (x)g * (y)] 1/4 - 1 A
.

(1.3.25)

3 For the partition function (1.3.2) or (1.3.11), this means that all the residual ϕ-dependence can be absorbed in the cosmological constant µ 2 c , possibly up to an area-independent global normalization constant.

As it is clear from the denitions (1.3.23), for t > 0, K(t, x, y) and K(t, x, y) are given by converging sums and are nite, even as x → y. For t → 0, one recovers various divergences. In particular, for an operator D g = ∆ * -R * + m 2 , K(t, x, y) then yields the logarithmic short distance singularity of G(x, y). Also, since K(t, x, y) and K(t, x, y) do not include the zero-mode, their integrals over x or over y vanish (as is also the case for G(x, y), of course):

d 2 x √ g * K(t, x, y) = d 2 x √ g * K(t, x, y) = 0 . (1.3.26)
Also, from the denition (1.3.23) of K(t, x, y) and K(t, x, y) and the orthonormality of the eigenfunctions ψ n , it is straightforward to get 6

d 2 x √ g * K(t 1 , u, x) K(t 2 , x, v) = K(t 1 + t 2 , u, v) , d 2 x √ g * K(t 1 , u, x) K(t 2 , x, v) = K(t 1 + t 2 , u, v) , d 2 x √ g * d dt 1 K(t 1 , u, x) K(t 2 , x, v) = -K(t 1 + t 2 , u, v) .
(1.3.27)

These formula will be useful in the following chapters. The regularized Green's function (1.3.22) is now seen to be given by

G(x, y) ϕ,Λ = ∞ 0 dα ϕ(α) K( α Λ 2 , x , y) .
I reg n ≡ I n ϕ,Λ = n i=1 ∞ 0 dα i ϕ(α i ) I n (t 1 = α 1 Λ 2 , . . . , t n = α n Λ 2 ) , (1.3.29)
where I n (t 1 , . . . , t n ) is the Feynman diagram (integral) with all propagators G(x i , y i ) replaced by K(t i , x i , y i ). It is obvious from (1.3.29) that the only part of I n (t i ) that contributes is the part that is completely symmetric in all t i . To simplify the notation, we will not write the In our two-and three-loop computations in Chapter 2 and 3, the strategy is then to compute the relevant I n (t 1 , . . . , t n ) and extract the small t i asymptotic. Since we always let

∞ 0 dα 1 ϕ(α 1 ) . . . ∞ 0 dα n ϕ(α n ) in
t i = α i Λ 2 (1.3.30)
the small t i and large Λ asymtotics are, of course, equivalent. To do this, we will once more use the well-known small t asymptotic of the heat kernel K(t, x, y), resp K(t, x, y)

6 The last two lines may also be deduced from the rst relation using (1.3.24).

REGULARIZATION OF THE PARTITION FUNCTION

25

(cf section 1.2.4). Unfortunately, this does not allow us to get the small t asymptotic of K(t, x, y) since by (1.3.23) the latter involves K(t , x, y) for all t ≥ t which are not all small. However, using (1.3.23) and (1.3.25) we can write

K(t, x, y) = G(x, y) - t 0
dt K(t , x, y) .

(1.3.31)

In the second term t ≤ t is small for small t and we get a useful formula if we can also say something about the un-regularized Green's function G(x, y). This is the case if x is close to y, where a short distance asymptotic involving G ζ (x) (dened in (1.2.37) and

(1.2.38) in section 1.2.4) is available.

In view of our later two-and three-loop computations in Chapter 2 and 3 we will now give some more specic formulae in the small t asymptotic for the operators

D * = ∆ * -R * and D M * = ∆ * + 32π A β 2 κ 2 ≡ ∆ * + T.
Indeed, Chapter 2 is dedicated to the study of the Liouville theory coupled to the regularized quantum gravity measure and thus β will be taken to be zero. In Chapter 3, the Mabuchi action will be added but the study will be restricted to the case of the torus, so that in both cases we will be able to work with the eld 7 φ = 1 2 A∆ * φ .

(1.3.32)

Since the zero-mode of φ was excluded anyway, the relation between φ and φ is one-toone. Then,

G(x, y) L = x|(∆ * -R * ) -1 |y , G(x, y) h=1 L+M = x|(∆ * + T) -1 |y . (1.3.33)

Small t and short-distance expansions

We consider the general operator D = ∆ * + ξR * + m 2 . For the Liouville theory, ξ = -1, m = 0 and λ 0 = -R * , while for the Liouville and Mabuchi theory on the torus R * = 0,

m 2 = T = 32π A β 2
κ 2 and λ 0 = T. Recalling the small t expansion of the heat kernel K(t, x, y), we have the following small t expansion of K(t, x, y):

K(t, x, y) = 1 4πt e -2 /4t a 0 (x, y) + a 1 (x, y) t + a 2 (x, y) t 2 + . . . - e -λ 0 t A (1.3.34)
where 2 ≡ 2 A (x, y) is once again the geodesic distance squared between x and y. As previously stated in section 1.2.4, for small t, the exponential forces 2 to be small and we can use normal coordinates around y. Then

2 = (x -y) 2 ≡ z 2 (1.3.35)
7 Using φ instead of φ as the basic integration variable generates a Jacobian determinant that cancels a similar determinant at one loop coming from the Liouville action [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF]. This cancellation could be incomplete if the multiplicative anomaly is present. The correct cancellation thus depends on the regularization scheme. As was discussed in detail in [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF], when using the spectral regularization no multiplicative anomaly occurs and there is no subtlety associated with the change of integration variables from φ to φ.

a 0 (x, y) = a (0) 0 (x, y) , a 1 (x, y) = a (0) 1 (x, y) -m 2 a (0) 0 (x, y) , (1.3.36) a 2 (x, y) = a (0) 2 (x, y) -m 2 a (0) 1 (x, y) + m 4 2 a (0) 0 (x, y) ,
where, following the notation of section 1.2, the a (0) i (x, y) are associated with the massless operator D (0) = ∆ * + ξR * , and (see e.g. the appendix of [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF])

a (0) 0 (x, y) = 1 + R * 24 z 2 + R 2 * 640 (z 2 ) 2 + . . . , a (0) 1 (x, y) = 1 -6 ξ 6 R * + R 2 * 6 1 20 - ξ 4 R 2 * z 2 + . . . , (1.3.37) a (0) 2 (x, y) = R 2 * 60 1 -30 ξ + 90 ξ 2 + . . . . and √ g * = 1 (a 0 (x, y)) 2 = 1 - R * 12 z 2 + R 2 * 480 (z 2 ) 2 + . . . . (1.3.38)
The specic values of the a i (x, y) for the Liouville theory and for the Liouville and Mabuchi theory on the torus can be immediately deduced from these formulae.

In particular, the heat kernel at coinciding points K(t, x, x) is independent of the point x since R * is constant at xed area. Then, for any function f that does not include the zero-mode, this implies that

d 2 x √ g * K(t, x, x)f (x) = 0. In particular, d 2 x √ g * K(t, x, x) φ(x) = 0 . (1.3.39)
To get the small t expansion of K(t, x, y) we use (1.3.31). This yields [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF] K(t, x, y) = G(x, y) -

1 4π k≥0 a k (x, y)t k E k+1 2 4t + t 0 dt e -λ 0 t A , (1.3.40)
where the exponential integral functions E n are dened as

E n (w) = ∞ 1 du u -n e -uw , (1.3.41)
with asymptotic behaviours for small a: [START_REF] Oldham | An Atlas of Functions: with Equator, the Atlas Function Calculator[END_REF][START_REF] Jereys | The Exponential and Related Integrals[END_REF]).

E 1 (a) = -γ -ln a + a + O(a 2 ) and E 2 (a) = 1 + (ln a + γ -1)a + O(a 2 ) (see e.g.
We will be mostly interested in the case when x is close to y where one can use the small distance expansion of the Green's function 8 G(x, y) (see e.g. [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF][START_REF] Bilal | 2D quantum gravity on compact Riemann surfaces and twoloop partition function: A rst principles approach[END_REF]). Thus, in 8 To obtain this expansion, the idea is to rst x the leading singularity so that for x close to y,

∆ x * G(x, y) ∼ δ(x -y) (in normal coordinates around y, g * (y) = 1
). Then, one has to adjust the subleading terms so that (∆ x * -R * ) G(x, y) = -1 A for x = y, and nally, one xes the integration constants in terms of G ζ (y).

normal coordinate and for small t, one gets the following expression for the hatted heat kernel (1.3.40):

K(t, x, y) = 1 4π a 0 (z) 4π G ζ (y) -ln µ 2 z 2 4 -2γ -E 1 z 2 4t -a 1 (z) z 2 4 4π G ζ (y) -ln µ 2 z 2 4 -2γ + 2 + t E 2 z 2 4t + 1 A t + z 2 4 + O(t 2 , tz 2 , z 3 ) . (1.3.42)
As long as t > 0, this has a smooth limit as x → y given by

K(t, y, y) = G ζ (y) - 1 4π ln µ 2 t + γ + t a 1 (y, y) - 4π A + O(t 2 ) . (1.3.43)
Note that the only term in K(t, y, y) that depends on the position y is G ζ (y). Thus,

d 2 x √ g * K(t, x, x)f (x) = d 2 x √ g * G ζ (x)f (x) , d 2 x √ g * K(t, x, x)f (x) = 0 , (1.3.44)
for any function f that does not include the zero-mode.

For later use, let us quote some explicit formulae for the pure Liouville theory on a Rieman surface of genus h ≥ 1, studied in Chapter 2:

K(t, x, y) = 1 4π 1 - z 2 4 R * 4π G ζ (y) -ln µ 2 z 2 4 -2γ - 7 12 R * z 2 -1 + R * 24 z 2 E 1 z 2 4t - 7 6 R * t E 2 z 2 4t + 1 A t + z 2 4 + . . . , K(t, x, y) = e -z 2 /4t 4π t 1 + 1 6 R * t z 2 4t + 7 + . . . - e R * t A , (1.3.45)
as well as

- d dt K(t, x, y) = e -z 2 /4t 4π t 2 1 - z 2 4t 1 + R * 24 z 2 - 7 6 R * z 2 4 + . . . + R * A e R * t . (1.3.46)
and

K(t, x, x) = 1 4πt 1 + 7 6 R * - 4π A t + 41 60 R * - 4π A R * t 2 + O(t 2 ) , K(t, x, x) = G ζ (x) - 1 4π ln µ 2 t + γ + 7 6 R * - 4π A t + O(t 2 ) .
(1.3.47)

One-loop result for the partition function at xed area

Let us recall the derivation of W (1) [A] (1.3.19) made in [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF] in this framework. The

determinant (1.3.19) is of the general form ln Det F 1 (∆) • • • F p (∆) , (1.3.48)
where the operators F i (∆) can be expressed in terms of the Laplacian. In the prescription (1.3.20), the regularized version of (1.3.48) is the sum of the regularized versions of the individual logarithm of determinants:

∞ 0 dα ϕ(α) r e -α Λ 2 λr ln Det F 1 (λ r ) • • • F p (λ r ) = ∞ 0 dα ϕ(α) p i=1 r e -α Λ 2 λr ln DetF i (λ r ) . (1.3.49)
and one has no multiplicative anomaly in this case. This simple result is true because it is the same cuto function ϕ(α)e -α Λ 2 λr that is used to regularize all the determinants.

Here, λ r denotes the eigenvalues of the Laplacian. Now, the only sums required to compute in order to access W (1) [A] are

S 0 = r>0 e -t λr , S 1 a A = r>0 e -t λr ln(λ r + a A ) , (1.3.50) so that ln Det z ∆ + a A ϕ,Λ = ∞ 0 dα ϕ(α) S 0 ln z + S 1 a A (1.3.51)
Following the techniques used in [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF], these sums are evaluated in [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF] such that

S 1 a A = - 1 4π A t + a γ + ln t -ζ 0, a A + O(t) , (1.3.52)
and

S 0 = A 4π t - h + 2 3 + O(t) , (1.3
ζ * s, a A = A A 0 s ζ 0 s, a A 0 , ζ * 0, a A = - h + 2 3 + a 4π ln A A 0 + ζ 0 0, a A 0 , (1.3.55)
so that eq. (1.3.51) can be rewritten as,

ln Det z ∆ + a A ϕ,Λ = ∞ 0 dα ϕ(α) 1 4π AΛ 2 α + a ln zΛ 2 α -γ + h + 2 3 + a 4π ln A A 0 -ln z -ζ 0 0, a A 0 + O(Λ -2
W (1) ϕ,Λ = ∞ 0 dα ϕ(α) AΛ 2 8πα γ -ln Λ 2 -ln κ 2 32πα - 1 2 + 7h -4 6 + 4 β 2 κ 2 ln A A 0 + C[A 0 , κ 2 , β 2 , α] , (1.3.57)
where C is an A-independent irrelevant coecient. The rst term in (1.3.57), which is divergent and cuto dependent, is proportional to A and can thus be absorbed into the cosmological constant. The one-loop string susceptibility is read from the coecient of the ln A term [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF]:

γ one-loop str = 19 -7h 6 -4 β 2 κ 2 • (1.3.58)
As required, it is nite and cuto independent. This expression 9 generalizes the KPZ result (1.3.18). This was to be expected since the non-trivial nature of the quantum gravity integration measure only shows up at two and higher loops.

Background independence

Background independence is the statement that physical quantities should not depend on the choice of the background metric g 0 or g * = A A 0 g 0 . Necessary conditions for this background independence are various co-cycle identities. As an example, consider three dierent metrics, g 1 , g 2 and g 3 related by conformal factors as g 2 = e 2σ 21 g 1 , g 3 = e 2σ 32 g 2 , g 3 = e 2σ 31 g 1 .

(1.4.1)

9 Although the reasoning presented here has been restricted to surfaces of genus h ≥ 1, it is shown in [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF] that this expression is also valid in the case of the sphere.

BACKGROUND INDEPENDENCE 30

One may consider g 3 as the quantum metric and g 2 as the background metric. If one then expresses g 2 in terms of g 1 and uses g 1 as the new background metric, all references to g 2 must disappear. This is only possible if the conformal factors are related by the obvious co-cycle identity σ 32 + σ 21 = σ 31 .

(1.4.2)

In the Kähler formalism, this identity leads to a corresponding identity for the Kähler potentials. Indeed,

g b = e 2σ ba g a ≡ A b A a 1 - 1 2 A a ∆ a φ ba g a (1.4.3) implies the relation 1 - 1 2 A 2 ∆ 2 φ 32 1 - 1 2 A 1 ∆ 1 φ 21 = 1 - 1 2 A 1 ∆ 1 φ 31 .
(1.4.4)

Recalling that ∆ b = e -2σ ba ∆ a then leads to the co-cycle identity for the Kähler potentials

φ 32 + φ 21 = φ 31 , (1.4.5)
up to an irrelevant constant. From (1.4.4), one has the following identity in terms of

φ ba = 1 2 A a ∆ a φ ba : (1 -φ 32 )(1 -φ 21 ) = (1 -φ 31 ) . (1.4.6)
As emphasized in section 1.2, the gravitational action satises the similar co-cycle identity (1.2.2). Let us consider for the present discussion the following gravitational action:

S grav [g 0 , g] = κ 2 8π S L [g 0 , g] + S measure [g 0 , g] + S ct [g 0 , g] .
(1.4.7)

The counterterm action S ct will be shown to be required in Chapter 2 section 2.1.2.

The gravitational action (1.4.7) must satisfy the co-cycle identity (1.2.2). As previously stated in section 1.2, this is true for the Liouville action. Anticipating the rewriting of integration measure D φ exp(-S measure ) in terms of φ in the next chapter (see section 2.1.1), we have

D φ = n>0 µ db n . (1.4.8) with φ = n>0 b n ψ n , ∆ψ n = λ (0) n ψ n , d 2 x √ g ψ n ψ m = δ nm , (1.4.9)
and

S measure [g 1 , φ] = - 1 2 ln Det (1 -φ) -1 = 1 2
Tr ln(1 -φ) .

(1.4.10)

Eq. (1.4.6) immediately implies that S measure also satises the required co-cycle identity, at least naively. Of course, to show that this naive statement is really true, we need to understand how Tr ∼ d 2 x √ g r ψ r ψ r behaves.

To follow the background independence in our computations, we will now study the variations of dierent quantities under an innitesimal change of the background metric.

To be precise, we begin by expressing our quantum metric g 3 in terms of the background metric g * ≡ g 1 . Then we let

g 2 = e 2ω g 1 , g 1 ≡ g * , (1.4.11)
with innitesimal ω and use g 2 as the new background metric. In section 1.2, the innitesimal change was in g 3 = e 2δσ g 1 . Here, the quantum metric g 3 is xed and hence only the background metric g 1 and the conformal factor σ 31 ≡ σ vary. We will simply write

δ B g = 2 ω g * , δ B σ = -ω , (1.4.12)
where g is the background metric varying from its original value g * and δ B gives the innitesimal change of any quantity under this change of background metric. In particular, the following quantities yield

δ B √ g = 2 ω √ g * , δ B R = -2 ω R * + 2 ∆ * ω , δ B ∆ = -2 ω∆ * , (1.4.13)
with R and ∆ respectively the background curvature and Laplacian in the background metric. To simplify, we may assume that g 2 and g 1 correspond to the same area so that by (1.2.24) and (1.4.4)

δ B ln 1 - 1 2 A∆φ = δ B ln 1 -φ = -2ω ⇒ δ B φ = 2 ω(1 -φ) . (1.4.14) Thus δ B d 2 x √ g φ = d 2 x √ g * 2 ω
. Since φ has no zero-mode, the left-hand side vanishes and we see that ω has no zero-mode either:

d 2 x √ g * ω = 0 . (1.4.15)
As a consistency check, it is easy to verify that the area and the genus of the Riemann surface are invariant under δ B :

δ B A = δ B d 2 x √ g = 2 d 2 x √ g * ω = 0 , δ B [8π(1 -h)] = δ B d 2 x √ g R = 2 d 2 x √ g * ∆ * ω = 0 . (1.4.16)
Note that the co-cycle identity for the Liouville action becomes

δ B S L ≡ S L [g 2 , σ 32 ] -S L [g 1 , σ 31 ] = -S L [g 1 , ω] = -d 2 x √ g * R * ω + O(ω 2 ) . (1.4.17)
By (1.4.15) and since R * is constant at xed area, we see that, to rst order in ω, the Liouville action is background independent:

δ B S L = 0 .

(1.4.18)

BACKGROUND INDEPENDENCE 32

Even though the Liouville action is background invariant, the sum of Feynman diagrams it generates at two loops (and at higher order) will not be automatically be background independent. There are two reasons for this. First, the integration measure may or may not be invariant as we will see next. Second, the diagrams have to be regularized. Now, the spectral cuto regularization is based on inserting e -λnt into all sums over the eigenvalues λ n of the operator D. Since the eigenvalues (and eigenfunctions) change under a variation of the background metric, all the regularized quantities have thus an induced additional background dependence. Since δ B e -λrt = -te -λrt δ B λ r , this is an order t ∼ 1 Λ 2 eect but, as usual, when multiplied by some other divergence ∼ Λ 2 it could well lead to some nite additional background dependence.

Let us rst discuss the integration measure n>0 µ dbn √ 2π exp(-S measure ). Once regularized, the measure action (1.4.10) stands as

S measure [g 1 , φ] = 1 2 d 2 x √ g * K(t, x, x) ln(1 -φ(x)) , (1.4.19)
since, from the previous discussion (cf section 1.3.3):

K(t, x, x) = n>0 (ψ (1) n (x)) 2 reg
.

(1.4.20)

Note that the ψ

n are the eigenfunctions of

D 1 = ∆ 1 -R 1 ≡ ∆ * -R * ≡ D * normalized with g 1 = g * . The variation of eq. (1.4.10) is δ B S measure [g 1 , φ] = d 2 x √ g * ω K + 1 2 δ K ln(1 -φ) -K ω = d 2 x √ g * ω K + 1 2 δ K ln(1 -φ) (1.4.21)
since ω has no zero mode and K(t, x, x) is a constant (see (1.3.44)). To determine the variation δ K, we need rst to know how the eigenvalues λ L n ≡ λ (0)

n -R * and the eigenfunctions ψ n behave under an innitesimal change of the background metric. The relevant formulae are once more given by standard quantum mechanical rst-order perturbation theory for δD = D 2 -D 1 = -2ωD * -2(∆ * ω), see e.g. [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF]. There is one slight subtlety though, since the ψ n + δψ n are normalized with respect to g 2 while the ψ n are normalized with respect to g 1 . This implies that ψ n |δψ n is not vanishing but equals -ψ n |ω|ψ n .

One nds

δλ L n = ψ n |δD|ψ n = -2 ψ n | λ L n ω + (∆ * ω) |ψ n , (1.4.22)
and

δψ n (x) = -ω(x)ψ n (x) + m =n ψ m (x) λ L m -λ L n (λ L n + λ L m ) ψ m |ω|ψ n + 2 ψ n |(∆ * ω)|ψ m .
(1.4.23)

Note that the eigenfunctions ψ m are real and, hence, the square bracket in the last line is symmetric under exchange of n and m.

δ B S measure [g 1 , φ] = d 2 x √ g * ω 4πt + 7 24π ∆ * ω - ω A + O(t) ln(1 -φ) .
(1.4.24)

This action does not appear to be background independent, as it is also the case for the counterterm action introduced in the section 2.1.3.

We also need to study how D * φ = n>0 µ dbn √ 2π

transforms. We dene the expansion coecients b

(2)

n and b

(1)

n of φ as φ = n>0 b (1) n ψ (∆ 1 ) n = m>0 b (2) m ψ (∆ 2 ) m . ψ (∆ 1 ) n = ψ n , ψ (∆ 2 ) m = ψ m + δψ ∆ m , (1.4.25)
where the ψ

(∆ i ) n
are the eigenfunctions of the Laplacian ∆ i , rather than of ∆ i -R i . For the constant curvature metric g 1 these are, of course, the same eigenfunctions, but for g 2 this is no longer true. The formula for δψ ∆ n is given by (1.2.46) with the m = 0 and δσ replaced by ω. In any case, we will only need

ψ n |δψ ∆ n = -ψ n |ω|ψ n . Note that ψ (∆ 1 ) 0 = ψ (∆ 2 ) 0 = 1 √ A
is unchanged, and the zero-mode and the non-zero modes transform into each other separately. Thus, for m = 0, b

m = b (1) m + δb m with δb m = n>0 M mn b (1) n , M mn = δψ m |ψ n , (2)
δ B S = n>0 ψ n |ω|ψ n reg = d 2 x √ g * n>0 (ψ n (x)) 2 reg ω(x) = d 2 x √ g * K(t, x, x) ω(x) = 0 , (1.4.28)
and hence

n>0 µ dbn √ 2π
is invariant to rst order in ω.

Les us now remark that even if the regularized measure action (1.4.24) does not seem to be background invariant, this does not mean that the overall computation does not satisfy the background independence requirement. Indeed, as previously stated, the spectral cuto regularization induces a priori a background dependence in all the diagrams, because of the insertion of the e -λnt terms in the propagators. Thus, the background independence of the actions themselves would not have been enough to ensure the background independence of the result of our computation. Of course, it would be extremely cumbersome to follow the background dependence through the individual 1.4. BACKGROUND INDEPENDENCE 34 regularized Feynman diagrams that will appear at two (and higher) loops (see Chapter 2 and 3). Hence, we will give here some relevant formula that would allow, in principle to do this. However, we will content ourselves by studying directly the background (in)dependence of the nal result for the two-loop partition function.

Since we are interested in the dependence in the area of the partition function and anticipating the scaling

G A ζ (x) = G A 0 ζ (x) + 1 4π ln A A 0 , (1.4
δ B G(x, y) = -ω(x) + ω(y) G(x, y) + B + C , (1.4.30) with B = 2 n =0 ψ n (x)ψ n (y) ψ n |(λ n ω + ∆ * ω)|ψ n (λ L n ) 2 (1.4.31) C = n =0 m =n ψ n (x)ψ m (y) + ψ m (x)ψ n (y) λ L m -λ L n 1 λ L n ψ m | (λ L n + λ L m )ω + 2∆ * ω |ψ n .
Upon separating the terms m = 0 from the sum C, we can symmetrize

1 λ L m -λ L n 1 λ L n → 1 λ L m -λ L n 1 2 1 λ L n -1 λ L m = 1 2λ L n λ L m and rewrite C = C 1 + C 2 ≡ 1 2 m =n, n,m =0 ψ n (x)ψ m (y) + ψ m (x)ψ n (y) λ L m λ L n ψ m | (λ L n + λ L m )ω + 2∆ * ω |ψ n + n =0 ψ n (x)ψ 0 (y) + ψ 0 (x)ψ n (y) λ L 0 -λ L n 1 λ L n ψ 0 | (λ L n + λ L 0 )ω + 2∆ * ω |ψ n .
(1.4.32)

The missing terms m = n in the rst sum C 1 are exactly given by the sum B, so that,

recalling n =0 ψ n (x)ψ n (z) = 1 √ g * (x) δ(x -z) -1 A , one has C 1 + B = ω(x) + ω(y) G(x, y) - 1 A d 2 z √ g * G(x, z) + G(y, z) ω(z) +2 d 2 z √ g * G(x, z)∆ * ω(z) G(z, y) . (1.4.33)
To evaluate the sum C 2 we have to remember that λ

L n -λ L 0 = λ L n + R * are the eigenvalues of ∆ * and that ψ 0 = 1 √ A so that ψ 0 | (λ L n + λ L 0 )ω + 2∆ * ω |ψ n = 3λ L n -λ L 0 √ A ω n , where ω n is the expansion coecient in ω = n =0 ω n ψ n . Then, one nds C 2 = - 1 A d 2 z √ g * G(x, z) + G(y, z) + 2 G ∆ (x, z) + 2 G ∆ (y, z) ω(z) , (1.4
δ B G(x, y) = 2 d 2 z g * (z) G(x, z)∆ * ω(z) G(z, y) - 1 A G(x, z) + G(y, z) + G ∆ (x, z) + G ∆ (y, z) ω(z) . (1.4.35)
Note that this has no short-distance singularity as x → y and we can safely take the coincidence limit to get δ B G ζ (x).

The change of the geodesic length (x, y) under the Weyl rescaling of the metric was given in (1.2.59). Then, (as always, to rst order in ω):

δ B ln( 2 A (x, y)µ 2 ) = ω(x) + ω(y) .
(

δ B G ζ (x) = lim x→y δ B G(x, y) + 1 4π ln( 2 A (x, y)µ 2 ) = 2 d 2 z √ g * G(x, z) 2 ∆ * ω(z) - 2 A G(x, z) + G ∆ (x, z) ω(z) + ω(x) 2π . (1.4.37)
This complicated-looking variation simplies when integrated over the Riemann surface, since G, G ∆ and ω have no zero-mode:

δ B d 2 x g(x) G ζ (x) = 2 d 2 x √ g * ω(x) G ζ (x) + 2 d 2 x g * (x) d 2 z g * (z) G(x, z) 2 ∆ * ω(z) , (1.4.38)
and similarly

δ B d 2 x g(x) R(x) G ζ (x) = 2 d 2 x √ g * ∆ * ω(x) G ζ (x) + 2R * d 2 x g * (x) d 2 z g * (z) G(x, z) 2 ∆ * ω(z) . (1.4.39)
We will also encounter expressions involving an integral of R G 2

ζ . We have

δ B d 2 x g(x) R(x) ( G ζ (x)) 2 = 2 d 2 x √ g * ∆ * ω(x)( G ζ (x)) 2 +2 d 2 x g * (x) R * G ζ (x) ω(x) 2π + 2 d 2 z g * (z) G(x, z) 2 ∆ * ω(z) - 2 A G(x, z) + G ∆ (x, z) ω(z) . (1.4.40)
We should remember that we will be interested in the area dependence and that area-

δ B 1 A d 2 x g(x) G ζ (x) = 1 2πA ln A A 0 d 2 x g * (x) ω(x) + A-independent = 0 + A-independent , (1.4.41) δ B d 2 x g(x) R(x) G ζ (x) = 1 2π ln A A 0 d 2 x g * (x) ∆ * ω(x) + A-independent = 0 + A-independent , (1.4.42)
and

δ B d 2 x g(x) R(x) ( G ζ (x)) 2 = 1 π ln A A 0 d 2 x g * (x) ∆ * ω(x) G A 0 ζ (x) + R * π ln A A 0 d 2 x d 2 z g * (x)g * (z) G(x, z) 2 ∆ * ω(z) + A-independent.
(1.4.43)

We conclude that the appearance of the terms

G ζ and R G ζ is compatible with background independence, while the appearance of R G 2
ζ is not 10 . This chapter present a rst-principles quantum eld theory computation of Z[A] on a Riemann surface of arbitrary genus, using the spectral cuto regularization scheme introduced in the previous chapter in section 1.3.3. We consider the Liouville action and the measure action which arises of the regularization of the quantum gravity measure (see section 1.3.1). This work has been initiated with the one-loop computation presented in section 1.3.2 (see [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF]). Now, in section 2.1, we will present the full two-loop computation of Z[A] in a loop-expansion where 1 κ 2 is the loop-counting parameter (see [START_REF] Bilal | 2D quantum gravity on compact Riemann surfaces and twoloop partition function: A rst principles approach[END_REF]). This will give us the two-loop contribution to the string susceptibility γ str . Then, in section 2.2, the three-loop contribution of the partition function is partially computed.

Indeed, we compute the non-vanishing quadratic leading divergence. Thus, counterterms are required and we compute the full contribution of the counterterms to the partition function, so that we can discuss the renormalization of the partition function (see [START_REF] Leduc | 2D quantum gravity at three loops: a counterterm investigation[END_REF]).

Two-loop landscape

In ref. [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF] the partition function Z[A] was computed, with the spectral cuto regularization scheme, up to and including the one-loop contributions, using a more general quantum gravity action that is a sum of the Liouville as well as Mabuchi actions, as presented in section 1.3.2 in the previous chapter. This gave a denite result for γ str which for the pure Liouville gravity reduced to γ 0,1-loop str

= 1 2 (h -1)κ 2 + 19-7h 6
in agreement with the loop-expansion of the KPZ result for the string susceptibility (1.3.18).

Although satisfying, the agreement was, maybe, not too much a surprise. Indeed, the truly non-trivial nature of the determinants coming from the measure over the space of metrics only shows up beyond one loop, starting at two loops. It is thus quite intriguing to try and compute the two-loop contributions to the xed-area partition function. This is what we will do in the present section, based on [START_REF] Bilal | 2D quantum gravity on compact Riemann surfaces and twoloop partition function: A rst principles approach[END_REF].

In the subsection 2.1.1, we expand the Liouville action and the measure up to order are relegated into the appendix. However, the absence of divergences does not x the counterterms uniquely. To x the nite parts of the counterterms requires to impose nite renormalization conditions. While in a massive Minkowski space quantum eld theory it is usually convenient to impose conditions on the mass shell, already in a massless theory it is often more convenient to impose the conditions at an arbitrary scale µ. In the present theory on a curved Riemann surface, there seems just to be no natural nite condition one should impose, rather than any other. Thus there appears to be a whole family of counterterms, depending on two nite parameters. Remarkably, for all choices of these nite parameters, the unwanted AΛ 2 ln AΛ 2 terms cancel in the two-loop contribution to ln Z[A], and the coecient of ln AΛ 2 , which is the two-loop contribution to the string susceptibility, becomes independent of the regulator functions ϕ(α) . This result is given (and discussed) in subsection 2.1.5.

For later comparison, we quote the loop-expansion of the KPZ result up to two loops:

γ str = 1 2 (h -1)κ 2 + 19 -7h 6 + 2(1 -h) 1 κ 2 + O(κ -4
) .

(2.1.1)

Two-loop expansion of the Liouville and measure actions

In the previous chapter, we have expanded the Liouville action up to the relevant order in φ for the one-loop computation. To go further and perform the two-loop computation, let us consider again the Liouville action written in the background metric g * of constant curvature R * and area A.

S L [σ] = d 2 x √ g * σ∆ * σ + R * σ . (2.1.2)
Recalling (1.2.24) in terms of φ (1.3.32), we have

σ -σ cl = 1 2 ln(1 -φ) , σ cl = 1 2 ln A A 0 . (2.1.3)
Since the rst term in S L is not aected by constant shifts of σ and the second term is linear, one obviously has

S L [σ] = S L [σ cl ] + S L [σ -σ cl ] = 4π(1 -h) ln A A 0 + S L 1 2 ln(1 -φ) .
(2.1.4)

Expanding the logarithm in terms of φ one gets

S L [σ -σ cl ] = d 2 x √ g * 1 4 φ(∆ * -R * ) φ + 1 4 φ 2 (∆ * - 2 3 R * ) φ + 1 16 φ 2 ∆ * φ 2 + 1 6 φ 3 ∆ * φ - 1 8 R * φ 4 + O( φ 5
) .

(2.1.5)

When looking at the denition of the quantum gravity partition function (1.3.11) with β = 0, the quantity ε κ 2 can be seen to be a loop-counting parameter. Thus, in the following of this chapter we absorb ε so that the loop-expansion of Z[A] is now done in powers of

1 κ 2 .
To emphasize it, we rescale φ as

φ = κ 4 √ π φ , (2.1.6) so that κ 2 8π S L [σ] = κ 2 2 (1 -h) ln A A 0 + d 2 x √ g * 1 2 φ(∆ * -R * ) φ + d 2 x √ g * √ 4π κ φ 2 (∆ * - 2 3 R * ) φ + 2π κ 2 φ 2 ∆ * φ 2 + 16π 3κ 2 φ 3 ∆ * φ - 4π κ 2 R * φ 4 + O(κ -3
) .

(2.1.7)

The rst term in the rst line provides the classical contribution to the partition function.

The second term of the rst line yields the one-loop determinant studied in [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF]. It also provides a standard propagator for the present two-loop computation. The terms of the second and third line provide the cubic and quartic vertices relevant for the two-loop vacuum diagrams. Clearly, the O(κ -3 ) terms correspond to quintic and higher vertices that can only contribute to three (and higher)-loop vacuum diagrams.

The non-trivial measure does not contribute at one-loop. However, it gives a two-loop (as well as higher loop) contribution. In terms of φ = 1 2 A∆ * φ, eq. (1.3.7) becomes

||δσ|| 2 A = 1 4 d 2 x √ g * (1 -φ ) -1 δ φ 2 , (2.1.8)
so that the measure action (1.3.12) changes in 1

S measure = - 1 2 ln Det (1 -φ) -1 = 1 2 Tr ln(1 -φ) = Tr - √ 4π κ φ - 4π κ 2 φ 2 + O(κ -3
) .

(2.1.9)

1 As previously stated (see [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF], the change of variable in the eld φ = 1 2 A∆ * φ does not induce a multiplicative anomaly when using the spectral cuto regularization scheme.

TWO-LOOP LANDSCAPE 40

More explicitly, in terms of the orthonormal set of eigenfunctions ψ r of the Laplace operator, cf (1.3.9), the trace of any operator O is given by

Tr O = r>0 ψ r |O|ψ r = d 2 x √ g * r>0 ψ r |x x|O|ψ r = d 2 x √ g * r>0 ψ 2 r (x) O(x) , (2.1.10)
(the eigenfunctions ψ r are chosen to be real) and, hence,

S measure = d 2 x √ g * r>0 ψ 2 r (x) - √ 4π κ φ(x) - 4π κ 2 φ 2 (x) + O(κ -3 ) . (2.1.11) Of course, r>0 ψ 2 r (x) = δ(x-x) √ g * (x)
-ψ 2 0 is a formal writing which translates in K(t, x, x) Would it not be simpler to use

Vertices

The Feynman rules are straightforwardly read from the expansions (2.1.7) and (2.1.11).

Note that we normalize our vertices without including any symmetry factors (i.e. a term α φ n in the total action gives a vertex with n legs and a factor -α, not -αn!) so that when evaluating the diagrams one has to count all possible contractions. There is a cubic vertex

= - √ 4π κ (∆ * - 2 3 R * ) , (2.1.12)
where (∆ * -2 3 R * ) is meant to act on the propagator connected to the bold line of the vertex. There are also two quartic vertices, (2.1.16)

= - 2π κ 2 (∆ * -2R * ) , ( 2
We recall that the tilde on G and the prime on the r.h.s. indicate that the zero-mode is not to be included. This propagator can be written explicitly in terms of the eigenvalues λ L n and eigenfunctions ψ n of

D = ∆ * -R * , D ψ n = λ L n ψ n .
(2.1.17

) Since R * = 8π(1-h)
A is constant, D and ∆ * have the same (real) eigenfunctions ψ n , while the eigenvalues simply are related by λ L n = λ

n -R * . Since φ has no zero-mode, the propagator is given by the sum over all non-zero modes as

G(x, y) = n>0 ψ n (x)ψ n (y) λ L n . (2.1.18)
Furthermore, for genus h > 1 one has λ L 0 = -R * > 0, and then one can explicitly subtract the zero-mode contribution

ψ 2 0 λ L 0 = 1 8π(h-1) : for h > 1 : G(x, y) = G(x, y) - 1 8π(h -1) , G(x, y) = n≥0 ψ n (x)ψ n (y) λ L n . (2.1.19)
Note also that for all h ≥ 1, we have λ L n > 0 for n > 0 since the eigenvalues λ (0) n of the Laplacian are positive (for n > 0) and -R * ≥ 0. It is only for h = 0 where λ

L n ≡ λ L l,m = 4π A l(l + 1) -8π
A that we get λ L 0,0 < 0 and λ L 1,m = 0. As already mentioned, for h = 0, these three spin-1 modes are excluded by the SL(2, C)/SL(2, R) gauge xing. In the sequel we will always implicitly assume h ≥ 1.

Two-loop contributions to the partition function

The purpose of the present computation is to pursue the one-loop expansion made in [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF] (see section 1.3). We recall:

ln Z[A] ≡ W [A] = L≥0 W (L) [A] . (2.1.20)
For the present pure Liouville quantum gravity,

W (0) [A] = κ 2 2 (h -1) ln A A 0 (2.1.21)
as can be read o directly from (2.1.7), and [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF] W (1) [A] = 1 -7h

6 ln A A 0 + µ 2 c,div A + c 1 , (2.1.22)
where the divergent piece µ 2 c,div A ∼ Λ 2 A is cancelled by the renormalization of the cosmological constant µ 2 c , and c 1 is an A-independent nite constant that could be eliminated by computing W (1) [A]-W (1)

γ str = h -1 2 κ 2 + 19 -7h 6 + O(κ -2 ) , (2.1.23)
in agreement with (2.1.1). As already mentioned, one of the present goals is to determine the order 1 κ 2 term in this expansion which comes from the two-loop contribution. The

two-loop contribution to W [A] is W (2) [A] = connected vacuum diagrams ∼ 1 κ 2 .
(2.1.24)

This includes the genuine two-loop diagrams made with the vertices of S L , as well as a one-loop diagram made with the vertex from S measure and, as we will see, also one-loop diagrams made with further counterterm vertices.

The vacuum diagrams of order 1/κ 2

= - 8π κ 2 d 2 x √ g * 1 4 G(x, x)(∆ x * -2R * ) G(x, x) + 1 2 (∆ x * -2R * ) G(x, z) 2 z=x + 2 G(x, x) ∆ x * G(x, z) z=x . (2.1.25)
The setting sun diagram: This diagram gets two contributions, one with the two bold lines of the two cubic vertices connected by the same propagator (two possible contractions) and one with the two bold lines not connected by the same propagator (four contractions). Overall, there is also a factor 1 2 from expanding e -S int to second order.

Thus = 4π κ 2 d 2 x g * (x) d 2 y g * (y) G(x, y) G(x, y)(∆ x * - 2 3 R * )(∆ y * - 2 3 R * ) G(x, y) + 2 (∆ x * - 2 3 R * ) G(x, y) (∆ y * - 2 3 R * ) G(x, y) . (2.1.26)
The glasses diagram: This diagram gets four contributions, since for each of the cubic vertices the bold lines can either be contracted with a line from the same vertex (giving a factor of 2) or with a line of the other vertex (factor 1). Again, overall, there is also a factor

1 2 . Thus = 2π κ 2 d 2 x g * (x)d 2 y g * (y) (∆ x * - 2 3 R * ) G(x, x)+2(∆ x * - 2 3 R * ) G(x, z) z=x × G(x, y) (∆ y * - 2 3 R * ) G(y, y) + 2(∆ y * - 2 3 R * ) G(y, z) z=y . (2.1.27)
The measure diagram: The vertex (2.1.15) simply gives

= 4π κ 2 d 2 x √ g * n>0 ψ 2 n (x) G(x, x) .
(2.1.28)

In addition to these diagrams there are also one-loop diagrams involving counterterm vertices that contribute at the same order in 1 κ 2 . They are discussed in the next subsection (section 2.1.3). So far, to summarize:

W (2) [A] = W (2) [A] loops + W (2) [A] ct W (2) [A] loops = i + i i + ii + a `i (2.1.29)
W (2) [A] ct = counterterm contributions at order 1 κ 2 .

Of course, these expressions for the Feynman diagrams are formal writings and have to be regularized. To do so, we use the smooth spectral cuto regularization described in the section 1.3.3 of Chapter 1. In particular, the Green's function G(x, y) are replaced by the K(t, x, y) and the diverging sum r>0 ψ 2 r (x) by the heat kernel at coinciding points K(t, x, x), so that one immediately gets the following expressions for the regularized integrals I 3 : 3 We recall that I(t1, . . . , tn) has to be integrated over ∞ 0 dα1ϕ(α1) . . . ∞ 0 dαnϕ(αn). These integrations are implicitly understood.

I ee= - 8π κ 2 d 2 x √ g * 1 4 K(t 1 , x, x)(∆ x * -2R * ) K(t 2 , x, x) + 1 2 (∆ x * -2R * ) K(t 1 , x, z) K(t 2 , x, z) z=x + 2 K(t 1 , x, x) ∆ x * K(t 2 , x, z) z=x , (2.1.30)
I e= 4π κ 2 d 2 x g * (x) d 2 y g * (y) K(t 1 , x, y) × K(t 2 , x, y)(∆ x * - 2 3 R * )(∆ y * - 2 3 R * ) K(t 3 , x, y) + 2 (∆ x * - 2 3 R * ) K(t 2 , x, y) (∆ y * - 2 3 R * ) K(t 3 , x, y) , (2.1.31) I e e= 2π κ 2 d 2 x g * (x) d 2 y g * (y) × (∆ x * - 2 3 R * ) K(t 1 , x, x) + 2(∆ x * - 2 3 R * ) K(t 1 , x, z) z=x K(t 2 , x, y) × (∆ y * - 2 3 R * ) K(t 3 , y, y) + 2(∆ y * - 2 3 R * ) K(t 3 , y, z) z=y , (2.1.32) I a `d = 4π κ 2 d 2 x √ g * K(t 1 , x, x) K(t 2 , x, x) .
(2.1.33)

One should keep in mind that one can always symmetrize in the t i = α i /Λ 2 because these expressions are multiplied by the symmetric i dα i ϕ(α i ). Thus, we consider two expressions as identical if they only dier by a permutation of the t i .

Scalings

Let us briey discuss some general features about the expected dependence on the area

A of the Feynman diagram (integrals) I r (t 1 , . . . , t n ).
Recall that the eigenfunctions ψ n of D = ∆ * -R * are normalized as

d 2 x √ g * ψ n (x)ψ m (x) = δ nm . ( 2
λ n ≡ λ A n = A 0 A λ A 0 n , ψ n ≡ ψ A n = A 0 A ψ A 0 n , (2.1.35)
where the quantities with a label A 0 are dened as the eigenfunctions and eigenvalues of ∆ 0 -R 0 with respect to the metric g 0 of area A 0 . This immediately implies the scaling relations

K A (t, x, y) = A 0 A K A 0 ( A 0 A t, x, y) , K A (t, x, y) = K A 0 ( A 0 A t, x, y) .
(2.1.36)

The last relation implies in particular that G(x, y) = K(0, x, y) does not depend on A:

G A (x, y) = G A 0 (x, y) . ( 2
I r [t i , A] = I r [ A 0 A t i , A 0 ] . (2.1.38)
Of course, this is true only because these integrals are nite convergent integrals (for t i > 0). In particular, since t i = α i /Λ 2 , we see that the I r cannot depend on A and Λ 2 separately but only on the combination AΛ 2 . On the other hand, the small t shortdistance expansion for K given in (1.3.45) also depends on an arbitrary scale µ introduced when dening G ζ in (1.2.42), so one might wonder whether an additional area-dependence of the form µ 2 A could occur. However, this is not the case. Indeed, in (1.2.42) the scale µ appears in the combination ln[ 2 A (x, y)µ 2 ] where A (x, y) is the geodesic distance between x and y computed with the metric of area A. Thus ln[ 2 A (x, y)µ 2 ] = ln Aµ 2 + . . . where + . . . refers to terms that do not depend on A or µ. It follows that 4π G ζ -ln Aµ 2 does not depend on µ. Since also, as anticipated in the previous chapter ans as we will explain shortly,

G A ζ = G A 0 ζ + 1 4π ln A A 0 , (2.1.39) it follows that 4π G A ζ -ln 2 A (x, y)µ 2 4 = 4π G A 0 ζ -ln A 0 µ 2 + . . . , (2.1.40)
depends neither on A nor on µ. Since G ζ and µ only ever appear in this combination in K it is clear that there is no real µ dependence in the end. Thus

I r [ α i Λ 2 , A] = f r [AΛ 2 , α i α j ] .
(2.1.41)

For example, a term ln Λ 2 A 2 will appear as

ln AΛ 2 2 = ln Λ 2 µ 2 + ln A A 0 + ln µ 2 A 0 2 = ln A A 0 2 + 2 ln Λ 2 µ 2 ln A A 0 + . . . (2.1.42)
where + . . . refers to terms independent of the area A. This structure will be indeed explicitly observed below. Similarly, a term AΛ 2 ln µ 2 A must be accompanied by a term AΛ 2 ln Λ 2 µ 2 . Again, this will be explicitly the case.

G ζ (x) = lim s→1 µ 2(s-1) ζ(s, x, x) - 1 4π(s -1) , (2.1.43)
which is equivalent to saying (2.1.46) is smooth for s → 1 and for y → x. Taking rst the limit y → x (for s > 1) and then s → 1 yields G ζ (x). On the other hand, taking rst s → 1 yields G(x, y) -

µ 2(s-1) ζ(s, x, x) = 1 4π(s -1) + G ζ (x) + O(s -1) . ( 2
1 4π a 0 (x, y)E 1 ( µ 2 2 (x,y) 4
) and then letting y → x yields the relation (1.2.42).

Note for later use that K(t, x, x) in (1.3.47) can now be rewritten as

K( α Λ 2 , x, x) = G A 0 ζ (x) + 1 4π ln AΛ 2 -ln A 0 µ 2 -ln α -γ - 7 6 R * - 4π A α Λ 2 + . . . . (2.1.47)
Evaluating the regularized two-loop integrals

We will now explicitly compute the regularized two-loop integrals. Before going on, we would like to stress that we are free to drop all terms that vanish as Λ → ∞. Also, the area-independent nite terms are without interest since they drop out when computing

Z[A]/Z[A 0 ]
. We do, however, keep the area-independent diverging terms in order to check that, in the end, they only show up in the combinations allowed by the above 2.1. TWO-LOOP LANDSCAPE 47 scaling argument, see (2.1.41). In the following, we will adopt the notation dx for d 2 x g * (x).

The gure-eight and the measure diagrams:

First, ∂ x i K(t, x, x) = 2 ∂ x i K(t, x, z) z=x so that ∆ x * K(t 1 , x, z) K(t 2 , x, z) z=x = 2 ∆ x * K(t 1 , x, z) z=x K(t 2 , x, x) - 1 2 g ij * ∂ x i K(t 1 , x, x) ∂ x j K(t 2 , x, x) = 2 K(t 1 , x, x) + R * K(t 1 , x, x) K(t 2 , x, x) - 1 2 g ij * ∂ x i K(t 1 , x, x) ∂ x j K(t 2 , x, x) . (2.1.48) Also, dx g ij * ∂ x i K(t 1 , x, x) ∂ x j K(t 2 , x, x) = dx K(t 1 , x, x)∆ * K(t 2 , x, x) (2.1.49)
upon integrating by parts. Hence,

I ee = 8π κ 2 dx -3 K(t 1 , x, x) K(t 2 , x, x) - 3 2 R * K(t 1 , x, x) K(t 2 , x, x) . (2.1.50)
From the asymptotic expansions (1.3.47) of K(t, x, x) and K(t, x, x), it is then straight-

forward to get dx K(t 1 , x, x) K(t 2 , x, x) = 1 (4π) 2 Λ 2 α 1 + 7 6 R * - 4π A dx 4π G ζ (x) + ln Λ 2 µ 2 α 2 -γ + c(α i ) + O(ln AΛ 2 /Λ 2 ) (2.1.51) and dx K(t 1 , x, x) K(t 2 , x, x) = 1 (4π) 2 dx 4π G ζ (x) + ln Λ 2 µ 2 α 1 -γ (2.1.52) × 4π G ζ (x) + ln Λ 2 µ 2 α 2 -γ + O(ln AΛ 2 /Λ 2 ) .
Here and in the following, the symbol c(α i ) denotes a generic nite and A-independent constant (but that may depend of the α i ). We always write c(α i ), but this same symbol stands for dierent constants. Using (2.1.39) to explicitly display the area dependence, one can rewrite

dx 4π G ζ (x) + ln Λ 2 µ 2 -γ = A ln AΛ 2 + G 0 , ( 2
G 0 = 4π A 0 d 2 x √ g 0 G A 0 ζ (x) -γ -ln A 0 µ 2 .
(2.1.54) Thus,

I a `d = 1 2πκ 2 AΛ 2 2α 1 ln AΛ 2 + 7 12 AR * -2π ln AΛ 2 + AΛ 2 2α 1 (G 0 -ln α 2 ) + c(α i ) + O(ln AΛ 2 /Λ 2 ) (2.1.55)
and

I ee= 1 2πκ 2 - 3 α 1 AΛ 2 ln AΛ 2 + - 7 2 + 3 ln α 1 AR * + 12π ln AΛ 2 - 3 α 1 AΛ 2 (G 0 -ln α 2 ) - 3 2 R * dx 4π G ζ (x) + ln Λ 2 µ 2 -γ 2 + c(α i ) + O(ln AΛ 2 /Λ 2 ) . (2.1.56) I eeinvolves the structure R G 2
ζ which is not background independent, as was remarked in section 1.4. However, this structure will be seen to cancel against similar contributions from the other diagrams.

The setting sun diagram:

Rewriting (2.1.31) gives

I e= 4π κ 2 dx dy K(t 1 , x, y) × -K(t 2 , x, y) d dt K(t, x, y) t=t 3 + 2 K(t 2 , x, y) K(t 3 , x, y) + 2 R * K(t 2 , x, y) K(t 3 , x, y) + R 2 * 3 K(t 2 , x, y) K(t 3 , x, y) . (2.1.57)
First consider the last term in R 2 * K(t 1 , x, y) K(t 2 , x, y) K(t 3 , x, y). Replacing each of the K(t i , x, y) by the unregulated Green's function G(x, y) one gets a converging nite integral, since the short-distance singularity ∼ (ln (x, y)) 3 is integrable. Now, G(x, y)

does not depend on the area and, thus and use the fact that for small t the heat kernel K(t, x, y) and its derivated d dt K(t, x, y) both exponentially vanish unless (x, y) 2 is of order t or less. Thus we can use normal coordinates z = x -y and use the expressions (1.3.38), (1.3.45) and (1.3.46), so that we can do the various integrations over z explicitly. For convenience, we have listed the relevant integrals in the appendix A. Now, eq. ( 2.1.57) may once more be rewritten (upon symmetrising the t i ) as

R 2 * dx dy K(t 1 , x, y) K(t 2 , x, y) K(t 3 , x, y) = R 2 * dx dy ( G(x, y)) 3 + O (ln AΛ 2 ) 2 /Λ 2 . ( 2
I e= 4π κ 2 dx dy K(t 1 , x, y) × -K(t 2 , x, y) d dt K(t, x, y) t=t 3 + 2 K(t 2 , x, y)K(t 3 , x, y) + 2 R * K(t 2 , x, y)K(t 3 , x, y) - e R *
R * dx dy K(t 1 , x,y) K(t 2 , x, y)K(t 3 , x, y) = R * (4π) 2 dx 4π G ζ (x) + ln Λ 2 µ 2 -γ 2 -2 AR * (4π) 2 ln(α 1 + α 2 ) ln AΛ 2 + c(α i ) + O (ln AΛ 2 ) 2 /Λ 2 ) .
(2.1.64)

We now consider the term in d dt K(t, x, y) t=t 3 . This term involves one more factor of t -1 3 than K(t 3 , x, y) in the previous computation (see (1.3.46)), so that one has to keep the terms O(t i ) or O(z i z j ) in the expansions of K(t i , x, y). The computation is then quite lengthy but straightforward. Note that a term

dx ∂ i G ζ ∂ i G ζ appears.
Integrating once more by parts and using (2.1.39), this equals

dx G ζ ∆ * G ζ = dx G A 0 ζ ∆ * G A 0 ζ . Since dx ≡ d 2 x
√ g * scales as A and ∆ * = A 0 A ∆ 0 this is once more an A-independent constant. Upon using the symmetry under exchange of the α i , the result then is

dx dy K(t 1 , x, y) K(t 2 ,x, y) d dt K(t, x, y) t=t 3 = - R * (4π) 2 dx 4π G ζ (x) + ln Λ 2 µ 2 -γ 2 - 2 (4π) 2 AΛ 2 α 1 + α 2 ln AΛ 2 - 2 (4π) 2 1 6 - α 1 α 2 (α 1 + α 2 ) 2 -ln(α 1 + α 2 ) AR * -4π ln AΛ 2 + AΛ 2 C(α i ) + c(α i ) + O (ln AΛ 2 ) 2 /Λ 2 .
(2.1.65)

The constant C(α i ) is a rather complicated fonction of the α i we choose not to explicitly display since it will be absorbed in the cosmological constant in the end. It comes indeed with the combination AΛ 2 (as allowed by the scaling argument (2.1.41)). Note that one might have expected a leading singularity ∼ AΛ 2 ln AΛ 2 2 , due to the leading

1 t 2 3 singularity of d dt K(t, x, y) t=t 3
multiplying the ln AΛ 2 singularities from each of the two K(t i , x, y). However, this leading term is multiplied by d 2 ze -z 2 (1 -z 2 ) which vanishes. Thus the leading singularity is ∼ AΛ 2 ln AΛ 2 as expected from naive power counting for the present two-loop diagrams.

To compute the last remaining integral, we use (1.3.38)

dx dy K(t 1 , x, y)K(t 2 , x, y)K(t 3 , x, y) = 1 (4π) 2 AΛ 2 α 2 + α 3 ln AΛ 2 + G 0 + ln α 2 + α 3 α 1 α 2 + α 1 α 3 + α 2 α 3 + AR * (4π) 2 7 6 - α 2 α 3 (α 2 + α 3 ) 2 ln AΛ 2 + c(α i ) + O(ln AΛ 2 /Λ 2 ) , (2.1.67)
so that one ends up with the following contribution from the setting sun diagram: Using the by now familiar argument of replacing the K(t 2 , x, y) by G(x, y), one sees that the last term is just an A-independent constant. Indeed,

I e= 1 2πκ 2 3 2 R * dx 4π G ζ (x) + ln Λ 2 µ 2 -γ 2 + 2AΛ 2 α 1 + α 2 ln AΛ 2 + 4 3 - 2α 1 α 2 (α 1 + α 2 ) 2 -3 ln(α 1 + α 2 ) AR * -12π ln AΛ 2 + AΛ 2 C(α i ) + c(α i ) + O (ln AΛ 2 ) 2 /Λ 2 . ( 2
R 2 * dx dy G ζ (x) K(t 2 , x, y) G ζ (y) = R 2 * dx dy G ζ (x) G(x, y) G ζ (y) + O(1/Λ 2 ) = R 2 * dx dy G A 0 ζ (x) G(x, y) G A 0 ζ (y) + O 1/Λ 2 (2.1.71)
where we once again used dx G(x, y) = dy G(x, y) = 0, together with (2.1.39). Since G A 0 ζ and G do not depend on A, this term is just another A-independent constant, up to vanishing terms.

Writing once more K(t 2 , x, y) = K(t 2 , x, y) -e R * t 2

A , we compute

dx dy G ζ (x) K(t 2 , x, y) G ζ (y) = dx G 2 ζ (x)(1 + R * t 2 ) -t 2 G ζ (x)∆ * G ζ (x) - e R * t 2 A dy G ζ (y) 2 + O(t 2
dx dy G ζ (x) d dt 2 K(t 2 , x, y) G ζ (y) = dx R * G 2 ζ (x) -G ζ (x)∆ * G ζ (x) - R * A dy G ζ (y) 2 + O(1/Λ 2 ) . (2.1.73)
As previously seen,

dx G ζ ∆ * G ζ is an A-independent term, so that I e e = 2π κ 2 R * dx G 2 ζ (x) - R * A dy G ζ (y) 2 + c + O(1/Λ 2 ) = 2π κ 2 c + O(1/Λ 2 ) , (2.1.74)
where we used again (2.1.39) to show that the explicitly written nite terms do not depend on A.

It is satisfying that this diagram does not give any non-trivial contribution. Indeed, we expect that the propagator connecting the two loops should carry zero momentum and, since no zero-mode is present, we expect to obtain zero. The absence of a zeromode, of course, was the reason that the K(t i , x, x) could be replaced by G ζ (x), and similarly for y. However, on a non-trivial manifold G ζ (x) is not constant, and the overall contribution does not need to vanish. We also note that each individual integral is nite and area-independent, in agreement with our general argument that A can only appear in the combination AΛ 2 : only divergent integrals can give an area dependence.

TWO-LOOP LANDSCAPE 53

Summing the two-loop and measure diagrams Summing the contributions of the four diagrams, we nd the two-loop contribution for the logarithm of the partition function:

W (2) [A] loops = I ee+ I a `d + I e+ I e e = 1 2πκ 2 2AΛ 2 α 1 + α 2 - 5AΛ 2 2α 1 + AR * 3 2 ln α 1 α 2 (α 1 + α 2 ) 2 -2 α 1 α 2 (α 1 + α 2 ) 2 - 19 12 -2π ln AΛ 2 + AΛ 2 C(α i ) + c(α i ) + O( ln AΛ 2 2 /Λ 2 ) , (2.1.75)
where C(α i ) and c(α i ) are nite area-independent but regulator-dependent constants.

Quite remarkably, the R * dx 4π

G ζ (x) + ln Λ 2 µ 2 -γ 2
terms and thus the (ln AΛ 2 ) 2 divergences appear in exactly the right combination in I ee and I e to cancel in W (2) . The cancellation of these (ln AΛ 2 ) 2 -terms is equivalent to the cancellation of the R( G ζ ) 2 terms. As we have observed in section 1.4 in the previous chapter, the presence of these terms in the logarithm of the partition function would have ruined its background independence. Thus, it is this cancellation which ensures the background independence of the xed-area partition function! Indeed, the remaining structures R G ζ and 1 A G ζ were shown to be background independent, up to irrelevant area-independent terms. Finally, Λ 2 G ζ is background independent modulo adjusting the divergent cosmological constant.

To simplify the notation, we dene

F (α i ) = 8π(1 -h) 3 2 ln α 1 α 2 (α 1 + α 2 ) 2 -2 α 1 α 2 (α 1 + α 2 ) 2 - 19 12 -2π , H(α i ) = 2 α 1 + α 2 - 5 2α 1 .
(2.1.76)

If we normalize with respect to A 0 , i.e we compute ln Z

[A] Z[A 0 ] , we get W (2) [A] loops -W (2) [A 0 ] loops = 1 2πκ 2 AΛ 2 H(α i ) + F (α i ) ln A A 0 + (A -A 0 )Λ 2 2πκ 2 C(α i ) + O (ln AΛ 2 ) 2 /Λ 2 . (2.1.77)
The last line corresponds to a cosmological constant term, and we may always add a corresponding counterterm to the quantum gravity action to cancel this term. The rst line displays an ln A A 0 -term, as expected, although with a rather complicated, regulator dependent coecient, as well as a diverging term ∼ AΛ

W (2) [A] -W (2) [A 0 ] KPZ = 2 κ 2 (1 -h) ln A A 0 -µ 2 c (A -A 0 ) . (2.1.78)
Clearly, as in any quantum eld theory, the contributions at order κ -2 do not only come from two-loop diagrams, but also from tree and one-loop diagrams involving counterterms. Of course, there is no vacuum tree-diagram and the only tree contribution is the cosmological constant counterterm we already mentioned. However, the various n-point functions at one loop (that are ∼ κ -n ) are also divergent quantities and, to make them nite, one has to introduce various one-loop counterterm n-point vertices that will be of order κ -n . In particular, there will be a quadratic counterterm vertex needed to make the renormalized propagator nite, and the one-loop diagram made with this counterterm vertex will contribute at order κ -2 to the partition function. Thus, our next task is to determine this counterterm. This will involve the one-loop computation of the two-point function.

Counterterm contribution to the partition function

Any divergence in W [A] loops that is simply proportional to the area A (and hence also to Λ 2 ) can be cancelled by a counterterm of the form d 2 x √ g * Λ 2 c i ln Λ 2 µ 2 + c i , referred to as cosmological constant. Also, any A-independent constants drop out when computing ln Z

[A] Z[A 0 ] = W [A] -W [A 0 ]
and are thus irrelevant. However, the divergences ∼ Λ 2 dy 4π G ζ = AΛ 2 (ln A A 0 + c) are non-local divergences. Of course, they cannot be cancelled by any local two-loop counterterm. However, they can be cancelled by one-loop diagrams involving local (one-loop) counterterm vertices. The same thing happens on a four-dimensional curved manifold with non-derivative φ 3 and φ 4 couplings [START_REF] Bilal | Multi-Loop Zeta Function Regularization and Spectral Cuto in Curved Spacetime[END_REF]. At present, the only such one-loop contribution to the partition function comes from a diagram having a single propagator connecting the two legs of the counterterm vertex that itself is ∼ 1 κ 2 . This is very similar to the diagram coming from the measure and can thus be seen as renormalizing the measure. In particular, the measure resulted in a vertex ∼ dx K(t, x, x) φ 2 (x) where K(t, x, x) = Λ 2 4πα + 7 24π R * -1 A + . . . is a constant. Thus any counterterm that looks like an (unwanted) mass renormalization can actually be interpreted as a renormalization of the measure. We thus allow a counterterm action of the form

S ct = 8π κ 2 d 2 x √ g * c φ 2 φ(∆ * -R * ) φ + c R 2 R * φ 2 + c m 2 φ 2 .
(2.1.79)

Note that a linear counterterm automatically vanishes since φ has no zero-mode, and higher counterterm vertices only contribute within two-loop vacuum diagrams that are ∼ 1 κ 4 (see section 2.2.3). The counterterm coecients can obviously depend on the cuto Λ and on the regularization functions ϕ(α). They also have an expansion in powers of 1 κ . Due to the explicit factor of 1 κ 2 in front of the counterterm action, we will only be interested here in the lowest order, κ-independent pieces. Note that in terms of the 2.1. TWO-LOOP LANDSCAPE 55 original φ they are of order κ 0 φ 2 , consistent with the fact that these terms originate at one loop. The dependence on Λ is again dictated by dimensional considerations: c φ and c R must be dimensionless and will correspond to at most logarithmic divergences, while c m has dimension of Λ 2 and corresponds to an at most quadratic (times a log) divergence.

Then, with dα i ϕ(α i ) implicitly understood 4 , one has:

c φ (Λ, α i ) = c (1) φ (α i ) ln AΛ 2 + c (2) φ (α i ) , c R (Λ, α i ) = c (1) R (α i ) ln AΛ 2 + c (2) R (α i ) , c m (Λ, α i ) = c (1) m (α i ) Λ 2 ln AΛ 2 + c (2) m (α i ) Λ 2 + c (3) m (α i ) 1
A .

(2.1.80)

Note that counterterms involving explicitly the area A like c

(1)

φ , c (1)
R , c (1)
m and c

(3) m are non-local counterterms that should not occur in any standard QFT. However, we have already observed the similarity between the counterterms and the measure action. The latter actually corresponds to some well-dened values of c

(2)

m , c (2)
R and c

(3) m , so that a non-vanishing counterterm coecient c

(3) m is certainly allowed.

Of course, the counterterm action (2.1.79) cannot be expressed in terms of geometric invariants written using only the metric g and curvature R, contrary to the cosmological constant ∼ d 2 x √ g. This is why it is often considered that such counterterms should not be allowed. However, as repeatedly emphasized, the whole quantization procedure is carried out with respect to some xed background metric g * and already the original Liouville action cannot be written in terms of g and R alone but requires reference to the background metric. As discussed before, the real question is whether the whole quantization procedure is independent of the choice of the background metric. As argued in the section 1.4 of the previous chapter, discussing the background independence of the counterterm action (2.1.79) (or of the measure plus counterterm action) is pointless. Indeed, it misses the crucial contributions from the regularization of the two-loop integrals.

Thus, as already stated, we will instead study the background (in)dependence of the nal contributions to the logarithm of the xed-area partition function W (2) [A].

The one-loop diagram with the counterterm vertex that follows from (2.1.79) gives a contribution W (2) [A] ct to be added to (2.1.75) that is

W (2) [A] ct = 8π κ 2 dx - c φ 2 K(t, x, x) - c R 2 R * + c m 2 K(t, x, x) (2.1.81) = - 1 κ 2 c φ AΛ 2 α + 7 6 R * A -4π + (c R R * A + c m A) G 0 + ln AΛ 2 A 0 µ 2 -ln α ,
where we used the small t expansions (1.3.47) of K(t, x, x) and K(t, x, x), as well as the relation ( 2

c (1) m = c (1)
R = 0 , (desired values) .

(2.1.82)

Below, we will indeed nd that c

(1)

m = c (
W (2) [A] ct = - 1 κ 2 c (2) m AΛ 2 ln AΛ 2 + c (3) m + c (2) R R * A ln AΛ 2 +   c (2) φ α + c (2) m G 0 -ln A 0 µ 2 -ln α   AΛ 2 + c ct (α i ) + O(ln AΛ 2 /Λ 2 )
W (2) [A] loops + W (2) [A] ct = - 1 κ 2 - 1 2π H(α i ) + c (2) m AΛ 2 ln AΛ 2 + - 1 2π F (α i ) + c (3) m + c (2) R R * A ln AΛ 2 + AΛ 2 [. . .] + c(α i ) + c ct (α i ) + O (ln AΛ 2 ) 2 /Λ 2 .
(2.1.85)

We see that c

(2) φ only enters in the cosmological constant part and, hence, does not play any role. Cancellation of the AΛ 2 ln AΛ 2 terms requires

c (2) m = 1 2π H(α i ) .
(2.1.86)

Finally, the physical coecient of ln AΛ 2 should not depend on the choice of the regulator functions ϕ(α i ), which requires

c (3) m + c (2) R R * A ≡ c (3) m + 8π(1 -h)c (2) R = 1 2π F (α i ) + const , (2.1.87)
where const is a true, α i -independent constant. One could be tempted to equate separately the terms proportional to R * A ∼ (1 -h) and those not involving the curvature, resulting in universal, genus-independent counterterm coecients. But since we compute on a surface of xed genus h, this does not really make sense. In any case, it is 2.1. TWO-LOOP LANDSCAPE 57 satisfying to remark that, with the possible exception of c

(3) m , the non-local counterterms c

(1)

φ , c (1)
m and c

(1) R are not required! Let us come back to the issue of background independence. As one sees from (2.1.81), the counterterms can only give rise to contributions in W (2) that involve at most one G ζ (contained in G 0 ). This is obvious since the counterterm action is quadratic in φ and the resulting one-loop diagrams involve at most one K giving rise to one G ζ . For this same reason, only ln AΛ 2 appears in W

(2) ct , and not (ln AΛ 2 ) 2 as did appear in the individual two-loop diagrams. While one could obtain a contribution to W

(2) ct involving (ln AΛ 2 ) 2 by allowing a non-local counterterm coecient c

(1) R , there is no way to produce a term ∼ R( G ζ ) 2 . For the total two-loop contribution to W (2) , the absence of (ln AΛ 2 ) 2 was equivalent to the absence of R( G ζ ) 2 terms. The latter would have violated background independence and, as just shown, it could not have been cancelled by counterterms. We see that the absence of non-local counterterm coecients is a consequence of background independence of the xed-area two-loop partition function W (2) ! After all these considerations, it is now time to actually compute the counterterm coecients. Remarkably, we will nd that they satisfy all desired relations, in particular (2.1.86) and (2.1.87), which are quite non-trivial.

One-loop computation of the two-point function and determination of the counterterm coecients

To actually determine the counterterms, one must do a one-loop computation of the twopoint Green's function and impose some convenient renormalization conditions to not only cancel the diverging parts but to also ensure the result to be regulator independent.

Doing so will constrain the nite contributions.

Note that in ordinary at space quantum eld theory one does not need to compute the full two-point Green's function G (2) (u, v). Instead one rather computes the amputated or 1PI (which is the same at one loop) two-point function in momentum space.

The corresponding contribution of the counterterms can then be read directly from the counterterm action. The analogue of such a momentum space computation is not available, in general, on a curved manifold, 5 and one has to do the computation directly in real space. In this case it seems that the simplest way to correctly take into account all contributions is to compute G (2) (u, v).

We will compute the regularized6 two-point function G (2) (u, v) which we dene so that at tree-level it is just K(t, u, v). At order κ -2 it receives one-loop Liouville contributions and tree-level contributions from the measure and the counterterms:

G (2) (u, v) = K(t, u, v) + G (2) (u, v) 1-loop L + G (2) (u, v) tree measure + G (2) (u, v) tree ct + O(κ -4
) .

(2.1.88)

A rst renormalization condition one clearly wants to impose on this full 2-point function

is niteness if u = v : For u = v : lim Λ→∞ G (2) (u, v) is nite.
(2.1.89)

As will be shown, this condition indeed xes all diverging parts of the counterterm coecients, i.e. c (1) φ , c

R , c (1)
m , as well as c

(2) m .

It turns out to be surprisingly dicult to nd a sensible renormalization condition to

x the nite parts of G (2) and thus the nite parts of the counterterm coecients. The trouble is that there is no analogue of a renormalization condition at some particular value of momentum. The only natural choice seems to be zero momentum, corresponding to the zero-mode of the two-point function. But du G (2) (u, v) = 0 automatically, and this condition is empty. Instead, one might be tempted to try to impose some condition at u = v like e.g.

For u = v : lim Λ→∞ G (2) (u, u) -K(t, u, u) ? = 0 . (2.1.90)
However, this doesn't make sense either. Indeed, the divergence of G (2) (u, v) as u → v will turn out to be dierent from the one of K(t, u, v) : there are additional diverging and additional nite terms. As a matter of fact, absence of the diverging terms would require a non-vanishing counterterm coecient c

(1) φ , which will be excluded by requiring niteness of the two-point function at u = v. Independently of this, the dierence of the nite terms also makes it impossible to impose (2.1.90).

Actually, this problem was to be expected. The usual renormalization of the twopoint function at some nite value of momentum, or equivalently at non-coinciding points, does not, of course yield a nite two-point function at coinciding points, nor does it imply that the loop and counterterm contributions to the two-point function vanish at coinciding points. This is the translation of the fact that to dene composite operators like ( φ(u)) 2 one needs an independent renormalization constant. Of course, in any ordinary at-space quantum eld theory the nite parts of the counterterm coecients can be changed by changing the renormalization conditions this freedom being at the origin of the renormalization group. Hence, one should probably accept that one cannot (completely) x the nite parts of our counterterm coecients. In particular, this implies that, at the two-loop level, the nite coecient of ln A A 0 in W (2) [A] -W (2) [A 0 ] is a parameter that can be adjusted!

The computation of the regularized two-point function G (2) (u, v) for u = v is displayed hereafter. Then, the result for the case of u = v is briey discussed. To keep this subsection readable, many computational details are deferred to the appendix B.1.

TWO-LOOP LANDSCAPE 59

Counterterm contributions

The regularized counterterm contribution to G (2) (u, v) is easy to write down, since it only involves two regularized propagators K(t 1 , u, x) and K(t 2 , x, v) connected by the counterterm vertex as given by (2.1.79):

G (2) (u, v) ct = 8π κ 2 dx - c φ 2 K(t 1 , u, x) K(t 2 , x, v) + K(t 1 , u, x) K(t 2 , x, v) -(c m + c R R * ) K(t 1 , u, x) K(t 2 , x, v) . ( 2
G (2) (u, v) ct = - 8π κ 2 c φ K(t 1 + t 2 , u, v) (2.1.92) - 8π κ 2 (c m + c R R) dx K(t 1 , u, x) K(t 2 , x, v) .
Now, one needs to distinguish two cases. For u = v the function K(t, u, v) is non-singular for t → 0. More precisely, it equals G(u, v) plus terms that are either exponentially small or of order 1/Λ 2 . Since c φ is at most logarithmically divergent, one may replace c φ K(t 1 + t 2 , u, v) by c φ G(u, v) up to terms that vanish as Λ → ∞. Thus,

G (2) (u, v) ct = - 8π κ 2 c φ G(u, v) - 8π κ 2 (c m + c R R) dx K(t 1 , u, x) K(t 2 , x, v) + O(ln AΛ 2 /Λ 2 ) , for u = v .
(2.1. [START_REF] Gilkey | Invariance theory, the heat equation and the Atiyah-Singer index theorem[END_REF] We see that the nite O(Λ 0 ) parts of c m and c R , i.e. c (2) R and c

(3) m only give nite contributions to G (2) ct . Hence, to determine them, one needs to obtain the nite O(Λ 0 )

terms in G (2) 1-loop L and in G (2) tree measure . For u = v however, K(t 1 + t 2 , u, u) is divergent when t i → 0, so that: G (2) (u, u) ct = - 8π κ 2 c φ K(t 1 + t 2 , u, u) - 8π κ 2 (c m + c R R) dx K(t 1 , u, x) K(t 2 , x, u) . (2.1.94)
Note that the integral is a nite smooth function of u even in the limit t i → 0. However, as mentioned above, one needs to keep the t i nite, since the subleading terms, multiplied with the divergent pieces from c m , can lead to nite contributions.

TWO-LOOP LANDSCAPE 60

Total one-loop contribution to G (2) (u, v) for u = v

We will now determine the one-loop, i.e. all order 1/κ 2 contributions to G (2) (u, v).

There are three one-loop diagrams contributing to G (2) (u, v). They are i , i and i (including the regularized external propagators). These diagrams contribute at the same order ∼ 1 κ 2 as the tree contribution from the measure vertex and the one from the counterterms. The counterterm contribution has been determined above in (2.1.93).

The computation of the three two-point one-loop diagrams is quite lengthy, mainly due to the non-symmetric nature of the cubic and quartic vertices. This implies that there First however, we want to highlight the fact that the case u = v and u = v have to be considered separately. Indeed, the i diagram for example involves terms such as K(t 1 , u, x) K(t 2 , x, y) K(t 3 , x, y) K(t 4 , y, v). The terms K(t 2 , x, y) and K(t 3 , x, y) force x and y to be close, due to e -2 (x,y)

4 1 t 2 + 1 t 3 . For u = v, we can assume 2 (u, v) 1 Λ 2 , so
that we can safely consider that x is not close to both u and v and, hence, Taylor expand:

K(t 1 , u, x) = K(t 1 , u, y) + (x -y) i ∂ i y K(t 1 , u, y) + 1 2 (x -y) i (x -y) j ∂ i y ∂ j y K(t 1 , u, y) + . . . . (2.1.95)
This expansion is only valid as long as 2 (u, x) ∼ 2 (u, y)

1 Λ 2 , so that K(t 1 , u, x
) is a smooth function of x in the vicinity of y. This is no longer the case if y → u. In particular, if 2 (u, x) ∼ 2 (u, y) ∼ 1 Λ 2 , all terms in the expansion (2.1.95) are similarly large. If u → v, this occurs, as it will it is discussed in more details in the appendix of [START_REF] Bilal | 2D quantum gravity on compact Riemann surfaces and twoloop partition function: A rst principles approach[END_REF] and in the appendix C for the three-loop computation.

TWO-LOOP LANDSCAPE 61

We now add the contributions of the three one-loop diagrams (B.1.11), (B.1.15), (B.1.21) and the measure contribution (B. 1.22), as well as the counterterm contribution (2.1.93). As before, since these expressions have to be multiplied with dα i ϕ(α i ) for every α i , we may symmetrize all expressions in α i . We get

G (2) u-O-v + G (2) u c v + G (2) u e v + G (2) measure + G (2) ct (u, v) = 1 κ 2 4 G(u, v) 3 2 ln α 2 α 3 (α 2 + α 3 ) 2 -1 - 2α 2 α 3 (α 2 + α 3 ) 2 -2πc (1) φ ln AΛ 2 -2πc (2) φ + 8π G(u, v) G ζ (u) + G ζ (v) - 2 A dx G ζ (x) - 8π A dx G(u, x) + G(x, v) G ζ (x) + 4 dx K(t 1 , u, x) K(t 4 , x, v) 2Λ 2 α 2 + α 3 - 5Λ 2 2α 2 -2πc (1) m Λ 2 ln AΛ 2 -2πc (2) m Λ 2 +R * 3 2 ln α 2 α 3 (α 2 + α 3 ) 2 - 19 12 - 2α 2 α 3 (α 2 + α 3 ) 2 -2πc (1) R ln AΛ 2 -2πc (2) R - 2π A 5 + c (3) m + 2πR * G ζ (x) - 1 A dy G ζ (y) + 8π ( G(u, v)) 2 - 1 A dx ( G(u, x)) 2 - 1 A dx ( G(x, v)) 2 + 1 A 2 dxdy ( G(x, y)) 2 + 8πR * dx ( G(u, x)) 2 G(x, v) + G(u, x)( G(x, v)) 2 - 16πR * A dxdy G(u, x) G(x, y) G(y, v) - 8πR * A dxdy ( G(x, y)) 2 G(u, x) + G(y, v) + 8πR * G(u, v) dx G(u, x) + G(v, x) G ζ (x) - 8πR * A dx G(u, x) + G(v, x) dy G(x, y) G ζ (y) + 8πR 2 * dxdy G(u, x) ( G(x, y)) 2 G(y, v) + G(u, x) G(x, v) G(x, y) G ζ (y) + O(ln AΛ 2 /Λ 2 ) .
(2.1.96)

Finiteness requires c

(1) (2)

φ = c (1) m = c (1) R = 0 , c (2) m = 1 2π 2 α 2 + α 3 - 5 2α 2 . ( 2
m [ϕ] = ∞ 0 dα 2 dα 3 ϕ(α 2 )ϕ(α 3 ) 1 2π 2 α 2 +α 3 -5 2α 2 .

TWO-LOOP LANDSCAPE 62

Next, since there are no more divergent coecients, one may now safely replace the K(t 1 , u, y) K(t 4 , y, v) by the regulator independent G(u, y) G(y, v). We moreover require that the Green's function G (2) (u, v) should not depend at all on the regulator functions ϕ(α), i.e. that (2.1.96) should not depend on the α i . This yields

c (2) φ = 1 2π 3 2 ln α 2 α 3 (α 2 + α 3 ) 2 -1 - 2α 2 α 3 (α 2 + α 3 ) 2 + c φ , (2.1.98) c (2) R = 1 2π 3 2 ln α 2 α 3 (α 2 + α 3 ) 2 - 19 12 - 2α 2 α 3 (α 2 + α 3 ) 2 + c R 2π , ( 2
G (2) u-O-v + G (2) u c v + G (2) u e v + G (2) measure + G (2) ct (u, v) = 1 κ 2 -8π c φ G(u, v) + 8π G(u, v) G ζ (u) + G ζ (v) - 2 A dx G ζ (x) - 8π A dx G(u, x) + G(x, v) G ζ (x) + 8π dx G(u, x) G(x, v) - c m + 5 A + R * - c R 2π + G ζ (x) - 1 A dy G ζ (y) + 8π ( G(u, v)) 2 - 1 A dx ( G(u, x)) 2 - 1 A dx ( G(x, v)) 2 + 1 A 2 dxdy ( G(x, y)) 2 + 8πR * dx ( G(u, x)) 2 G(x, v) + G(u, x)( G(x, v)) 2 - 16πR * A dxdy G(u, x) G(x, y) G(y, v) - 8πR * A dxdy ( G(x, y)) 2 G(u, x) + G(y, v) + 8πR * G(u, v) dx G(u, x) + G(v, x) G ζ (x) - 8πR * A dx G(u, x) + G(v, x) dy G(x, y) G ζ (y) + 8πR 2 * dxdy G(u, x) ( G(x, y)) 2 G(y, v) + G(u, x) G(x, v) G(x, y) G ζ (y) + O(ln AΛ 2 /Λ 2 ) , (2.1.101)
which now is nite and completely regulator independent. As a consistency check, we note that this expression correctly vanishes if integrated over du or over dv.

There seems to be no obvious way to x the remaining constants c φ , c R and c m without imposing some denite value for G (2) (u, v) at some xed u, v. As already discussed, imposing a condition at u = v does not help since in this case new divergences appear 2.1. TWO-LOOP LANDSCAPE 63 that have to be renormalized independently. In any case, our expression (2.1.101) is only valid for u = v.

Total one-loop contribution to G (2) (u, u)

To explicitly see which are the new divergences and nite terms that appear for u = v, we now quote the result for the total one-loop plus counterterm contributions to the twopoint function at coinciding points u = v. This case involves some interesting technical diculties, similar to some one encounters in the three-loop computation and which are discussed in the appendix C. Since we do not use the expression of G (2) (u, u) any further we do not spell out the computation and only give the result:

G (2) u-O-u + G (2) u c u + G (2) u e u + G (2) measure + G (2) ct (u, u) = 1 κ 2 8π K(t i , u, u) K(t j , u, u) + 4π [ĉ 1 (α i ) -2 c φ ] K(t i + t j , u, u) + ĉ2 (α i ) +16π K(t 1 + t 4 , u, u) G ζ (u) - 1 A dx G ζ (x) + R * dx G(u, x) G ζ (x) +4 dx K(t 1 , u, x) K(t 4 , x, u) 2Λ 2 α 2 + α 3 - 5Λ 2 2α 2 -2πc m + R * 3 2 ln α 2 α 3 (α 2 + α 3 ) 2 - 2α 2 α 3 (α 2 + α 3 ) 2 - 19 12 -2πc R - 56π A dx ( G(u, x)) 2 - 16π A dx G(u, x) G ζ (x) + 8π A 2 dxdy ( G(x, y)) 2 +16πR * dx ( G(u, x)) 3 + 8πR * dx ( G(u, x)) 2 G ζ (x) - 16πR * A dxdy G(u, x) G(x, y) G(y, u) + G(u, x)( G(x, y)) 2 - 8πR * A dxdy 2 G(u, x) G(x, y) G ζ (y) + ( G(u, x)) 2 G ζ (y) +8πR 2 * dxdy G(u, x) ( G(x, y)) 2 G(y, u) + ( G(u, x)) 2 G(x, y) G ζ (y) + O(ln AΛ 2 /Λ 2 ) . (2.1.102)
Here, ĉ1 (α i ) and ĉ2 (α i ) are nite coecients that we did not determine. Using the sums given in the appendix C we could compute them explicitly. However, this will not give us any further information about the counterterms. Indeed, as discussed above, one should φ , one clearly cannot require that G (2) (u, u) equals K(t, u, u) in the Λ → ∞ limit. Thus, in the three-loop computation, we will only consider the two-point function at non coinciding point.

not expect niteness of G (2) (u, u) or that it equals K(t, u, u) as Λ → ∞. The rst term in (2.1.102), i.e. 2 K(t i , u, u) K(t j , u, u) is divergent as Λ → ∞.

Two-loop result for the partition function and discussion

The nal result for the logarithm of the partition function is obtained from inserting the values of the counterterm coecients we have determined in (2.1.97)-(2.1.100) into (2.1.85):

W (2) [A] = W (2) [A] loops + W (2) [A] ct = - 1 κ 2 c m + 1 + 4(1 -h) c R ln AΛ 2 + AΛ 2 [. . .] + c(α i ) + O (ln AΛ 2 ) 2 /Λ 2 . (2.1.103)
We see that not only the terms ∼ AΛ 2 ln AΛ 2 have cancelled, moreover the coecient of ln AΛ 2 now is independent of the α i , i.e. independent of the regulator functions ϕ(α) ! It only depends on the two nite renormalization constants c m and c R . As already repeatedly emphasized, we can also adjust the cosmological constant counterterm to cancel any divergence in W that is proportional to the area A. Most importantly, as discussed in section previously, the absence of (ln AΛ 2 ) 2 divergences is equivalent to the absence of the R( G ζ ) 2 term that would not have been background independent. Clearly, all area-dependent terms present in (2.1.103) are background independent.

Subtracting from (2.1.103) the same expression evaluated at area A 0 , we nally get

ln Z[A] Z[A 0 ] κ -2 = W (2) [A] -W (2) [A 0 ] = - 1 κ 2 c m + 1 + 4(1 -h) c R ln A A 0 + (A -A 0 )Λ 2 [. . .] + O (ln AΛ 2 ) 2 /Λ 2 .
(2.1.104)

Equivalently, this shows that the area dependence of the partition function is

Z[A] two-loop+measure+ct ∼ e -µ 2 c A A γ (2) str -3 , (2.1.105) with γ (2)
str = 2 κ 2 -2 c R (1 -h) - c m + 1 2 . (2.1.106)
We see that our careful, rst-principles computation of the 2D quantum gravity partition function has established that, up to two loops, the partition function has indeed the expected form (2.1.105). However, we have also found a maybe unexpected dependence on two nite renormalization constants. By which principle should these counterterm coecients be xed? We observe that our result is compatible with the KPZ scaling, since we get agreement with the (two-loop prediction of the) KPZ formula if we choose

c m KPZ = -1 , c R KPZ = - 1 2 .
(2.1.107)

TWO-LOOP LANDSCAPE 65

In the absence of any principle to x these constants, the area-dependence of the partition function Z[A] appears to involve an arbitrary power of A. One more principle we may invoke is locality. Locality of the counterterms implies that all coecients c

(1)

φ , c (1)
m , c (1)
R which multiply a non-local ln AΛ 2 should vanish. We have indeed found this. But it would also imply that c

(3) m = c m which multiplies a non-local 1 A should equally vanish. On the other hand, such a 1 A -term was already present in the regularized measure action due to the absence of a zero-mode. Indeed, from (2.1.11) and (1.3.47) we read that

S measure = - 4π κ 2 d 2 x √ g * 1 4πt + 7 24π R * - 1 A + . . . φ 2 (x) . (2.1.108) Thus, a 1
A -counterterm should certainly also be allowed. However, we may require that the non-local

κ 2 8π S L + S measure + S ct ⇒ c (3) m ≡ c m = -1 . (2.1.109)
This is precisely the KPZ-value (2.1.107) !

Let us come back on the background independence. As previously emphasized in section 1.4 in the previous chapter, while the Liouville action is background independent, neither is the measure action, nor the counterterm action. This was, of course, to be expected. This must be so, since it is missing the background dependence induced by the regularization. The fact that the regularization explicitly depends on the background metric makes it dicult to trace this dependence through the intermediate steps of the loop computations. Nevertheless, we were able to determine among the structures that could appear in the nal result for the xed-area partition function which ones are background independent and which ones are not. We found that precisely those structures accompanying the divergences (ln AΛ 2 ) 2 are not background independent. However, quite remarkably, the (ln AΛ 2 ) 2 structures and divergences that could not be cancelled by counterterms already cancel among themselves in the two-loop contribution. The remaining structures are all background independent ! To summarize, within the present two-loop, order κ -2 computation, there does not seem to be an obvious criterion why to choose the KPZ-value c R = -1 2 rather than any other value. The DDK reasoning which yields the KPZ value was based on an all-order conformal invariance argument which encodes background invariance. However, our two-loop computation was also background invariant in the end, despite the regularization, and the counterterms did not play any role to ensure that. One may wonder then if a higher loop computation would be able to x the remaining constant c R . Indeed, at the three-loop level, i.e. at order ∼ κ -4 , the counterterm c R φ 2 contributes via two-loop diagrams and thus enters into more complicated divergences which need to be cancelled. However, one could expect genuine three-loop counterterms (and thus 2.2. THREE-LOOP INVESTIGATION 66 even further undetermined parameters) to be required in any case, especially to ensure the strong locality condition. Indeed, the measure action gives a cubic and a quartic vertex at three loops, including non-local 1 A terms. Thus, if the counterterms are to be understood as a renormalization of the measure, a cubic and a quartic structure are to be expected. Regarding the background independence criterion to x these (new) parameters, it will most likely be irrelevant. Indeed, the background dependent structures are related to divergences which will be required to vanish. At three loops for example, the (ln AΛ 2 ) 2 divergence which is related to the background dependent term R( G ζ ) 2 , could be cancelled by contributions from two-loop diagrams involving a counterterm. In any case such a divergence will be absorbed in a counterterm and to ensure both the niteness of the partition function and its background dependence. The only three-loop divergence that could not be cancelled by a counterterm will be the (ln AΛ 2 ) 3 divergence (also related to a background dependent term), whose cancellation among the diagrams will be needed. This is checked in the next section of this chapter, dedicated to the study of the three-loop divergences and of the genuine three-loop counterterms. Also note that if c R indeed remained undetermined to all orders, we would be led to consider that there could be dierent choices of consistent quantizations of this two-dimensional gravity. One could imagine that at least some of these new quantization schemes could be consistent quantum gravities for all matter central charges, thus maybe allowing to go beyond the c = 1 barrier.

Three-loop investigation

The purpose of the present section is to go into the previous study in depth, with the idea of improving our understanding of the counterterms. In the subsection 2.2.1, the Liouville and the measure actions are expanded to the order relevant for the computation of the partition function at three loops. In particular, this leads to new vertices. Then the three-loop vacuum diagrams are enumerated. In the subsection 2.2.2, the allowed divergences are investigated in some detail and the leading divergence ∼ AΛ 2 ln AΛ 2 2 is fully computed. Since it does not cancel, genuine three-loop counterterms are required.

We end this subsection by stating that the (ln AΛ 2 ) 3 divergence cancels, as required for both niteness and background independence. Finally, subsection 2.2.3 is dedicated to the three-loop counterterms that contribute via one-and two-loop diagrams to the threeloop partition function. We discuss the degrees of freedom of these additional parameters. This section is based on [START_REF] Leduc | 2D quantum gravity at three loops: a counterterm investigation[END_REF].

Three-loop expansions of the actions

To pursue the investigation to three loops, 1 κ 4 terms need to be included. Since the Liouville action (2.1.2) comes with a factor κ 2 8π , it has to be expanded up to the 1 κ 6 order.

THREE-LOOP INVESTIGATION 67

One gets:

κ 2 8π S L [σ] = κ 2 2 (1 -h) ln A A 0 + dx 1 2 φ(∆ * -R * ) φ (2.2.1) + dx √ 4π κ φ 2 (∆ * - 2 3 R * ) φ + 2π κ 2 φ 2 ∆ * φ 2 + 16π 3κ 2 φ 3 ∆ * φ - 4π κ 2 R * φ 4 + dx 16π 3/2 κ 3 φ 4 ∆ * - 4 5 R * φ + 2 3 φ 3 ∆ * φ 2 + (8π) 2 κ 4 4 5 φ 5 ∆ * - 5 6 R * φ + 1 2 φ 4 ∆ * φ 2 + 2 9 φ 3 ∆ * φ 3 + O(κ -5 )
where dx stands for d 2 x g * (x). The last two lines yield the quintic and sextic vertices which appear only at three (or higher)-loop computations. They can be grouped as follows. Two quintic vertices

= - 16π 3/2 κ 3 ∆ * - 4 5 R * , = - 16π 3/2 κ 3 2 3 ∆ * (2.2.2)
and three sextic vertices

= - (8π) 2 κ 4 4 5 ∆ * - 5 6 R * , = - (8π) 2 κ 4 1 2 ∆ * , = - (8π) 2 κ 4 2 9 ∆ * . (2.2.3)
Similarly to the two-loop case, the bold parts of the vertices encode the ∆ * acting on one or several propagators. For example, for the two quintic vertices, the ∆ * -4 5 R * in the rst vertex acts on the single propagator connected to the bold line, while in the second one ∆ * may act either on the product of the two propagators connected to the bold part of the vertex on the right or on the three other ones.

The three-loop expansion of the non-trivial measure action (2.1.9) is

S measure = dx K(t, x, x) - 4π κ 2 φ 2 - 32π 3/2 3κ 3 φ 3 - 32π 2 κ 4 φ 4 + O(κ -5 ) , (2.2.4)
where K(t, x, x) is given in (1.3.47). In addition to the quadratic vertex already relevant for the two-loop computation, this action now provides a cubic and a quartic vertex, leading to the following measure vertices up to three loops:

= 4π κ 2 K(t, x, x) , = 16π 3/2 κ 3 2 3 K(t, x, x) , = (8π) 2 κ 4 1 2 K(t, x, x) .
(2.2.5)

The requirement for counterterms has been highlighted by the previous study at two loops, so that both the two-point function and the partition function are nite at

THREE-LOOP INVESTIGATION

68 respectively one loop and two loops. Thus, the two-loop counterterm action (2.1.79) is also to be considered for the three-loop computation, leading to the overall action:

S = κ 2 8π S L + S measure + S ct . (2.2.6)
The vertices relevant up to three loops are summed up in Tab. 2.1.

Order Liouville

Measure CT

1 κ 1 κ 2 1 κ 3 1 κ 4
Table 2.1: Vertices relevant for the three-loop computation Note that all these vertices are normalized without including any symmetry factors so that one has to count all possible contractions when evaluating the diagrams.

Diagrams

One enumerates now all three-loop vacuum diagrams. More precisely, such diagrams are all the diagrams contributing at order These diagrams consist in thirteen subdiagrams each for the diagrams of the upper line, and of seventeen, ten and eighteen subdiagrams for the bottom line, from left to right.

Finally, the last ve pure Liouville diagrams come from using four cubic vertices:

Note that the term ∼ ln AΛ 2 , although divergent, has a physical meaning. Indeed, once all other divergences cancelled by appropriate counterterms, one has W (3) [A] = W (3) [A] loops + W

(3) [A] ct = d 6 ln AΛ 2 + d 7 + O ln AΛ 2 AΛ 2 so that lim Λ→∞ Z[A] Z[A 0 ] 3-loop+ct = A A 0 d 6 , (2.2.10)
showing that d 6 is the three-loop plus counterterm, order 1 κ 4 , contribution to γ str .

Cancellation of the Λ 4 divergence Moreover, contrary to the preceding, somewhat naive power counting argument, one observes unexpected Λ 4 divergences appearing in the diagrams indicated in Tab. 2.2.

They appear through the following integrals:

J i j = dx dy K (t i , x, x) K(t j , y, y) K(t m , x, y) K(t n , x, y) , J i,j k = dx dy K (t i + t j , x, x) K(t k , y, y) K(t m , x, y) K(t n , x, y) , J i,j k,l = dx dy K (t i + t j , x, x) K(t k + t l , y, y) K(t m , x, y) K(t n , x , y) , (2.2.11)
where i, j, k, l, m and n are dierent. From (1.3.47) one gets the leading divergences

J i j ∼ Λ 4 α i α j J , J i,j k ∼ Λ 4 (α i + α j )α k J and J i,j k,l ∼ Λ 4 (α i + α j )(α k + α l ) J , (2.2.12)
with J = dx dy K(t m , x, y) K(t n , x, y). Thus these Λ 4 divergences come with three dierent structures in the α i . All these unwanted divergences are displayed in Tab. 2.2. When summing them up, all three structures (2.2.12) cancel and there is no net The true leading divergence contributing to the partition function at three loops is in

Λ 4 divergence ! Total J i j 9 -3 - 15
1 4 5 2 25 4 0 J i+j k -12 12
AΛ 2 ln AΛ 2 2
. As already emphasized, the main goal of this three-loop investigation is to analyse this leading divergence, check that it does not miraculously cancel between the diagrams and determine the structure of the required counterterms.

Out of the twenty-nine vacuum diagrams displayed in section 2.2.1, only the fourteen diagrams shown in Fig. 2.1 contribute to the leading divergence in AΛ 2 ln AΛ 2 2

. Note that all the diagrams with a single propagator between two vertices (i.e. one-particle reducible) do not contribute, as it was already the case for the two-loop computation. This is because there is no zero-mode and a single propagator connecting two parts of a vacuum diagram should carry only the zero-mode.

7

Consider again the ower diagram made from the sextic vertices, whose decomposition in subdiagrams was given in the previous subsection. Since only one vertex is involved, no integration has to be done to extract the divergences and it is the second simplest diagram to compute. (The simplest is the gure-eight diagram coming from the quartic 7 In at space, by momentum conservation, such a propagator would carry zero momentum. In our curved geometry the argument is more complicated and such one-particle reducible diagrams can still be non-vanishing, but using (1.3.44) one can show that they do not contribute to the present computation.

THREE-LOOP INVESTIGATION

74 measure vertex.) The rst subdiagram may be written in our regularization as:

I = - 4 5 (8π) 2 κ 4 dx K (t 1 , x, x) K (t 2 , x, x) ∆ x * - 5 6 R * K(t 3 , x, z) x=z = - 4 5 (8π) 2 κ 4 dx K (t 1 , x, x) K (t 2 , x, x) K(t 3 , x, x) + R * 6 K(t 3 , x, x) , (2.2.13)
The second subdiagram is slightly more complicated, because of the Laplacian acting on several propagators:

I = - 2 9 (8π) 2 κ 4 dx K (t 1 , x, x) ∆ x * K (t 2 , x, x) K(t 3 , x, z) z=x . (2.2.14)
The Laplacian term gives

∆ x * K (t 2 , x, x) K(t 3 , x, z) z=x = K (t 2 , x, x) ∆ x * K(t 3 , x, z) z=x + ∆ x * K (t 2 , x, x) K(t 3 , x, x) -2 g ij * ∂ x i K (t 2 , x, x) ∂ x j K(t 3 , x, z) z=x = K (t 2 , x, x) K(t 3 , x, x) + R * K(t 2 , x, x) K(t 3 , x, x) + ∆ x * K (t 2 , x, x) K(t 3 , x, x) -g ij * ∂ x i K (t 2 , x, x) ∂ x j K(t 3 , x , x) . (2.2.15) Inserting (2.2.15) into (2.2.14)
, and integrating the last term by parts leads to:

I = - 2 9 (8π) 2 κ 4 dx K (t 1 , x, x) K (t 2 , x, x) K(t 3 , x, x) + R * K (t 3 , x, x) K(t 2 , x, x) + 1 2 K (t 3 , x, x) ∆ x * K(t 2 , x, x) .
(2.2.16)

Similarly, the third and fth subdiagrams give

I = - 2 9 (8π) 2 κ 4 dx ∆ x * K(t 1 , x, z) K(t 2 , x, z) K(t 3 , x, z) z=x = - 2 3 (8π) 2 κ 4 dx K (t 1 , x, x) K (t 2 , x, x) K(t 3 , x, x) + R * K (t 3 , x, x) K(t 2 , x, x) - 1 4 K (t 3 , x, x) ∆ x * K(t 2 , x, x) , I = - 1 2 (8π) 2 κ 4 dx K(t 1 , x, x) ∆ x * K(t 2 , x, z) K(t 3 , x, z) z=x = - (8π) 2 κ 4 dx K (t 1 , x, x) K (t 2 , x, x) K(t 3 , x, x) + R * K (t 3 , x, x) K(t 2 , x, x) - 1 8 K (t 3 , x, x) ∆ x * K(t 2 , x, x) , (2.2.17
I = - 1 2 (8π) 2 κ 4 dx K(t 1 , x, x) K(t 2 , x, x)∆ x * K(t 3 , x, x) .
(2.2.18)

The overall contribution from the ower diagram is thus

I = 15 I + 9 I + 6 I + 3 I + 12 I = - (8π) 2 κ 4 dx K (t 1 , x, x) K (t 2 , x, x) 30 K(t 3 , x, x) + 20 R * K(t 3 , x, x) . (2.2.19)
(Note that the K K∆ K terms have cancelled.) The leading divergence of the second term is in ln AΛ 2 3

. These divergences will be discussed at the end of this subsection where it is shown that all ln AΛ 2 3 divergences cancel between the dierent diagrams.

The rst term on the right-hand-side of (2.2.19) contributes to the leading divergence, giving -30

πκ 4 AΛ 2 ln AΛ 2 2 ∞ 0 dα 3 ϕ(α 3 ) α 3
.

Leading divergence of the partition function per diagram

The previous computation already gives the contribution of the ower diagram to the leading divergence of the partition function:

I = AΛ 2 πκ 4 ln AΛ 2 2 - 30 α 1 + O(Λ 2 ln AΛ 2 ) . (2.2.20)
The only other diagram involving only one vertex is one of the measure gure-eight diagrams. It contributes

I = AΛ 2 πκ 4 ln AΛ 2 2 3 2 1 α 1 + O(Λ 2 ln AΛ 2 ) . ( 2

.2.21)

There are six diagrams built from two vertices that contribute to the leading singularity: , , , , and . The integrals to perform are similar to those previously done in order to compute the two-loop vacuum diagrams. It

THREE-LOOP INVESTIGATION

76

is rather straightforward to obtain:

I = AΛ 2 πκ 4 ln AΛ 2 2 18 α 1 + α 2 + O(Λ 2 ln AΛ 2 ) , I = AΛ 2 πκ 4 ln AΛ 2 2 18 α 1 + 9 α 1 + α 2 + O(Λ 2 ln AΛ 2 ) , I = AΛ 2 πκ 4 ln AΛ 2 2 24 α 1 + 48 α 1 + α 2 + O(Λ 2 ln AΛ 2 ) , I = AΛ 2 πκ 4 ln AΛ 2 2 - 2 α 1 + O(Λ 2 ln AΛ 2 ) , I = AΛ 2 πκ 4 ln AΛ 2 2 - 3 α 1 + O(Λ 2 ln AΛ 2 ) , I = AΛ 2 πκ 4 ln AΛ 2 2 - 15 α 1 + 12 α 1 + α 2 + O(Λ 2 ln AΛ 2 ) . (2.2.22)
As always, according to our regularization scheme (1.3.28), these expressions are to be understood as multiplied with the regulator functions i ϕ(α i ) and integrated over i ∞ 0 dα i . For instance I contributes as AΛ 2 πκ 4 ln AΛ 2 2 c , with c = 24

∞ 0 dα 1 ϕ(α 1 ) α 1 + 48 ∞ 0 dα 1 dα 2 ϕ(α 1 )ϕ(α 2 ) α 1 +α 2
being a number once the regularization function ϕ(α) is chosen. Note that the results for the diagrams involving the counterterm vertex, I in (2.2.22) and I in (2.2.23) below, do not depend on the free (two-loop) renormalization constants c φ and c R . In the next subsection, dedicated to the computation of the full contributions of the counterterms to all divergences, we will see indeed that the leading contribution involving c φ and c R is in AΛ 2 ln AΛ 2 .

When considering three vertices or more, computations become more technical. While for and it is easy to get:

I = AΛ 2 πκ 4 ln AΛ 2 2 7 2 1 α 1 + O(Λ 2 ln AΛ 2 ) , I = AΛ 2 πκ 4 ln AΛ 2 2 35 2 1 α 1 - 14 α 1 + α 2 + O(Λ 2 ln AΛ 2 ) , (2.2.23)
with and already, one stumbles over the same kind of technical diculties as those faced when computing the one-loop two-point Green's function at coinciding points (see the appendix of [START_REF] Bilal | 2D quantum gravity on compact Riemann surfaces and twoloop partition function: A rst principles approach[END_REF] and appendix C of this thesis). One of the integrals encountered in is for instance dx dy dz K(t 1 , x, z) K(t 2 , x, z) K(t 3 , y, z) K(t 4 , y, z) K(t 5 , x, y) .

Trouble comes from the fact that the three Ks in the integral force the three variables x, y and z to be all close to each other. For instance, integrating over y through the term K(t 4 , y, z) requires to Taylor expand

K(t 5 , x, y) = K(t 5 , x, z) + (y -z) i ∂ i z K(t 5 , x, z) + 1 2 (y -z) i (y -z) j ∂ i z ∂ j z K(t
= (-1) n n! (4π) 2 Λ 2 α 2 + α 5 n+1 ln AΛ 2 + O(Λ 2n+2 ) (2.2.27)
so that one can easily resum all the terms. Therefore, the previous integral contributes to the leading divergence by:

AΛ 2 (4π) 3 ln AΛ 2 2 ∞ n=0 (-1) n α n 4 (α 2 + α 5 ) n+1 = AΛ 2 (4π) 3 ln AΛ 2 2 1 α 2 + α 5 + α 4 .
(2.2.28)

Of course, this is valid for α 4 α 2 +α 5 < 1. However, the initial expression was symmetric under exchange of α 2 and α 4 (upon also exchanging α 1 and α 3 ). Hence, if α 4 α 2 +α 5 > 1 one simply exchanges the roles of α 2 and α 4 in the derivation (since now α 2 α 4 +α 5 < 1) and one gets the same result.

Considering carefully each integral, nally one gets 8 for and :

I = AΛ 2 πκ 4 ln AΛ 2 2 - 21 α 1 - 12 α 1 + α 2 - 24 α 1 + α 2 + α 3 + O(Λ 2 ln AΛ 2 ) , I = AΛ 2 πκ 4 ln AΛ 2 2 - 42 α 1 + α 2 - 46 α 1 + α 2 + α 3 + O(Λ 2 ln AΛ 2 ) .
(2.2.29)

One encounters similar problems for the diagrams with four vertices and .

Taylor expanding leads to series of divergent contributions. In addition to the series (2.2.28), one obtains

∞ n=0 ∞ m=0 n + m n (-1) n+m α n 1 α m 2 (α 3 + α 4 ) n+m+1 = 1 α 1 + α 2 + α 3 + α 4 .
(2.2.30)

More details on the integrals generating such series are given in the appendix C. Thus, one gets:

I = AΛ 2 πκ 4 ln AΛ 2 2 14 α 1 + α 2 + 16 α 1 + α 2 + α 3 + α 4 + O(Λ 2 ln AΛ 2 ) , I = AΛ 2 πκ 4 ln AΛ 2 2 16 α 1 + α 2 + α 3 + 8 α 1 + α 2 + α 3 + α 4 + O(Λ 2 ln AΛ 2 ) . ( 2
d 1 = 1 4πκ 4 - 26 α 1 + 132 α 1 + α 2 - 216 α 1 + α 2 + α 3 + 96 α 1 + α 2 + α 3 + α 4 .
(2.2.32)

Clearly, the leading divergence in AΛ 2 ln AΛ 2 2

is not vanishing and new counterterms will be required. They should be determined by ensuring that the one-loop three-point and four-point functions, as well as the two-loop two-point function be all nite. The computation of these one-loop n-point functions is beyond the scope of this thesis, but it is nevertheless already interesting to look at the possible counterterms one could consider and to calculate their contributions to the various divergences of the partition function.

This will be done in the next subsection.

8 Note again that the αi are to be multiplied with ϕ(αi) and integrated. This implies that any expression involving several αi can be symmetrized and that one can also rename the indices. In particular, the 1 α 2 +α 5 +α 4 in (2.2.28) has been rewritten as

1 α 1 +α 2 +α 3 .
Cancellation of the ln AΛ 2 3 divergence When the counterterm contributions to the three-loop partition function are computed in the next subsection, it will be shown that local counterterms with local coecients (i.e. not involving explicitly ln AΛ 2 ) cannot give contributions to the ln AΛ 2 3 divergence. Now, it is easy to see that such ln AΛ 2 3 divergences are present in individual three-loop diagrams. In particular, this was the case for the ower diagram, see (2.2.19) and the remarks that followed. The only way to ensure niteness of the partition function then is that these individual divergences cancel between the three-loop vacuum diagrams.

Among the twenty-nine diagrams, eight contribute to the ln AΛ

Counterterms

There are several types of counterterms one may add in the three-loop computation.

Cubic or quartic counterterms lead to diagrams similar to the ones generated by the cubic and quartic measure vertices (cf Tab. 2.1). One may also expand the coecients of the quadratic counterterms introduced in the two-loop computation (2.1.97)-(2.1.99),

(2.1.100), and consider their κ -4 contributions. Of course, only local counterterms will be introduced. This means, on the one hand, that the counterterms are polynomial in the Kähler eld φ with only nitely many derivatives acting on them, and, on the other hand, that the coecients of these counterterms are local expressions. In particular, a counterterm coecient involving the area e.g. through ln AΛ 2 is non-local. However, following the procedure applied in the previous section for the two-loop case, we do allow for counterterm coecients ∼ 1

A since they are already present in the measure action due to the absence of the zero-mode. In the previous section (see 2.1.5), we introduced the strong locality condition, i.e. absence of 1

A terms in the total action. Imposing such a condition at two loops led us to x one of the counterterms (namely c m ) precisely to the KPZ value. We also impose the strong locality condition at three loops, by xing the relevant coecients of the three-loop counterterms. Therefore, the only A-dependent (non-local) remaining terms involve a ln AΛ 2 # coecient.

THREE-LOOP INVESTIGATION 80

In this subsection, we will write out the counterterms contributing to the partition function at the same order as the three-loop diagrams, i.e. at order 1 κ 4 and give their contributions to W [A] = ln Z[A]. Since the divergences in AΛ 2 can always be absorbed in the cosmological constant they will be ignored in the following. Similarly, the nite area-independent contributions of the counterterms will not be spelt out.

Cubic counterterms

The new types of counterterms one may introduce are cubic and quartic ones. The allowed cubic counterterm action is

S c ct = 16π 3/2 κ 3 1 2 d 2 x √ g * f φ φ 2 (∆ * -R * ) φ + f R R * φ 3 + f m φ 3 (2.2.33)
where

f φ = f (1) φ , f R = f (1) R , (2.2.34) f m = f (1) m Λ 2 + f (2) m
A .

By dimensional analysis, the coecients f

(1) i and f

(2) i are dimensionless numbers. As already emphasized in the two-loop analysis (see section 2.1.3) they may depend on the regularization through the α i and are then to be integrated with the given ϕ(α i ), resulting in a number. But they do not depend on the cut-o Λ 2 . The action (2.2.33) contributes via the two two-loop diagrams and at the same order in κ -4

as the three-loop diagrams studied above.

We rst show that the glasses diagram gives no relevant contribution. It may be written as a sum of four subdiagrams. One gets:

I = 1 4 (8π) 2 κ 4 dx dy K(t 1 , x, x) K(t 2 , y, y) × f φ - d K(t , x, y) dt t=t 3 + f φ R * + 3 (f m + f R R * ) K(t 3 , x, y) + 3 R * (f m + f R R * ) K(t 3 , x, y) .
(2.2.35)

THREE-LOOP INVESTIGATION 81

Integrating and taking into account the absence of zero-modes leads to:

I = (4π) 2 κ 4 f φ dx G ζ ∆ * G ζ + 3 (f m + f R R * ) dx G ζ (x) G ζ (x) - 1 A dy G ζ (y) + 3 (f m + f R R * ) R * dx dy G ζ (x) K(t 3 , x, y) G ζ (y) . ( 2
I = (4π) 2 κ 4 f φ d 2 x √ g 0 G A 0 ζ ∆ 0 G A 0 ζ (2.2.37) + 3 A A 0 f m + f R R 0 d 2 x √ g 0 G A 0 ζ (x) G A 0 ζ (x) - 1 A 0 d 2 y g 0 (y) G A 0 ζ (y) + 3 A A 0 f m + f R R 0 R 0 d 2 xd 2 y g 0 (x)g 0 (y) G A 0 ζ (x) K 0 ( A 0 A t 3 , x, y) G A 0 ζ (y) .
The rst term is obviously independent of the area A and thus of no interest here. The only A dependence in the second line comes from the A A 0 f m term through f (1) m A 0 AΛ 2 . However, the parenthesis being A independent, this term can be included in the cosmological constant and is not signicant. The last term is slightly more subtle to handle because of the remaining K 0 ( A 0 A t 3 , x, y) term. For the non divergent counterterms f

(1) R and

f (2) m
A , the short-distance logarithmic singularity in K 0 ( A 0 A t 3 , x, y) being integrable, one may take the limit t 3 → ∞. Doing so leads to an A independent quantity. Finally, doing a nite expansion in x -y in the integral yields either A-independent or 1 Λ 2 -terms or terms that vanish exponentially as Λ → ∞. Thus, the remaining quadratically divergent counterterm Λ 2 f

(1) m only leads to terms nite or to be included in the cosmological constant. None of these terms is of any interest here. This glasses diagram thus gives no contribution to the pertinent divergences of the partition function (2.2.9). Note that diagrams with a single propagator joining two or three loops were already discarded from the diagrams contributing to the leading divergence in the previous subsection. + K (t 2 , x, y) K (t 3 , x, y)

+ 1 2 [(f φ + 3f R ) R * + 3f m ] K (t 1 , x, y) K (t 2 , x, y) K (t 3 , x, y) + R * 2 (f R R * + f m ) K (t 1 , x, y) K (t 2 , x, y) K (t 3 , x, y) .
This leads to the following divergences:

I = 1 κ 4 6f (1) m AΛ 2 ln AΛ 2 2 + 8 α 2 + α 3 f (1) φ + 12 f (1) m (G 0 -ln(α 2 + α 3 )) AΛ 2 ln AΛ 2 + 6 f (2) m + 4 f (1) φ + 6 f (1) R -6 α 3 f (1) m AR * ln AΛ 2 2 + 16 f (1) φ 3 -α 3 f (1) m AR * - α 2 α 3 α 2 + α 3 8 α 2 + α 3 f (1) φ + 12 f (1) m AR * -48π f (1) φ -2 α 3 f (1) m + 2 6 f (2) m + 4 f (1)
φ + 6 f (1) R -6 α 3 f (1) m AR * (G 0 -ln(α 2 + α 3 )) ln AΛ 2 (2.2.39)
where G 0 was dened in (2.1.54). The expression (2.2.39) is the full contribution from the cubic counterterms to the diverging part of the partition function.

Quartic counterterms

The quartic counterterm action is

S q ct = (8π) 2 κ 4 1 2 d 2 x √ g * q φ φ 3 (∆ * -R * ) φ + q φ φ 2 (∆ * -2R * ) φ 2 + q R R * φ 4 + q m φ 4 (2.2.40) with q φ = q (1) φ , q φ = q (1) φ , q R = q (1) R , (2.2.41) q m = q (1) m Λ 2 + q (2) m A .
Again, the coecients q (j) i may depend on the α k but not on the cuto Λ. This action gives a gure-eight diagram :

I = (8π) 2 κ 4 dx - 3 2 q φ K(t 1 , x, x) K (t 2 , x, x) + (q R R * + q m ) K (t 1 , x, x) K (t 2 , x, x) + q φ -2 K (t 1 , x, x) K (t 2 , x, x) + R * K (t 1 , x, x) K (t 2 , x, x) , (2.2.42)
which contributes as

I = 1 κ 4 -6 q (1) m AΛ 2 ln AΛ 2 2 -6 q (2) m + -4 q (1) φ + 6 q (1) R AR * ln AΛ 2 2 - 1 α 1 6 q (1)
φ + 8 q

(1) φ

+ 12 q (1) m (G 0 -ln α 1 ) AΛ 2 ln AΛ 2 - 7 6 AR * -4π 6 q (1)
φ + 8 q

(1)

φ -12 α 1 q (1) m + 2 6 q (2) m + -4 q (1) φ + 6 q (1) R AR * (G 0 -ln α 1 ) ln AΛ 2 (2.2.43)
with G 0 given in (2.1.54).

Quadratic two-loop counterterms

The quadratic counterterms (2.1.79) did contribute via one-loop diagrams to the two-loop partition function, but also via two-loop diagrams to the three-loop partition function as shown in the above computation. However, as always, the counterterm coecients get contributions at dierent orders in perturbation theory: we can add to c φ , c R and c m an additional piece

1 κ 2 c φ , 1 κ 2 c R and 1 κ 2 c m , so that c tot φ = c φ + 1 κ 2 c φ + O( 1 κ 4 ), etc.
Overall, the c are accompanied by a factor 1 κ 4 and they contribute via one-loop diagrams to the three-loop partition function. Thus we also add the following counterterm action

S quad' ct = 8π κ 4 d 2 x √ g * c φ 2 φ(∆ * -R * ) φ + c R 2 R * φ 2 + c m 2 φ 2 , (2.2.44)
where, again,

c φ = c (1) φ , c R = c (1)
R ,

(2.2.45)

c m = c (1) m Λ 2 + c (2) m
A .

The counterterm action (2.2.44) then provides a new one-loop diagram of order κ -4 :

I = - 1 2 8π κ 4 dx c φ K(t, x, x) + c R R * + c m K(t, x, x) (2.2.46)
leading to the following divergences:

I = 1 κ 4 -c (1) m AΛ 2 ln AΛ 2 -c (2) m + c (1)
R AR * ln AΛ 2 .

I = 1 κ 4 - 2 α 1 c φ AΛ 2 ln AΛ 2 -2 7 6 AR * -4π c φ ln AΛ 2 , I = 1 κ 4 - 10 α 1 + 8 α 1 + α 2 c φ AΛ 2 ln AΛ 2 + 4AR * c φ c R -2 7 6 AR * + 4π c φ + 4AR * c φ + c R 2π 3 -ln(α 1 + α 2 ) + ln α 1 -1 - 2α 1 α 2 (α 1 + α 2 ) 2 ln AΛ 2 , I = 1 κ 4 12 1 α 1 + 1 α 1 + α 2 c φ AΛ 2 ln AΛ 2 + 12AR * c φ + c R 2π ln AΛ 2 2 + 12AR * c φ + c R 2π 2 G 0 -
I = 1 κ 4 -8 1 α 1 + α 2 + 2 α 1 + α 2 + α 3 c φ AΛ 2 ln AΛ 2 -18 c φ + 7 c R π AR * ln AΛ 2 2 + -24 R 2 * c R dx dy K (t 1 , x, y) K (t 2 , x, y) -2 G 0 18 c φ + 7 c R π AR * + 4AR * c φ + c R 2π 1 + 2α 1 α 2 (α 1 + α 2 ) 2 + 3 ln(α 1 + α 2 ) + 6 c φ + c R 3π ln(α 1 + α 2 + α 3 ) + 4 c φ 36π -AR * 11 3 + 4 α 2 1 (α 1 + α 2 + α 3 ) 2 ln AΛ 2 .
(2.2.49)

None of these contains a AΛ 2 (ln AΛ 2 ) 2 divergence and this is why these nite counterterm coecients c φ and c R did not contribute to the previous computation of the leading threeloop divergence made in section 2.2.2.

THREE-LOOP INVESTIGATION 85

Total counterterm contribution to the partition function

W (3) [A] ct = 1 κ 4 Ω 1 AΛ 2 ln AΛ 2 2 + Ω 2 AΛ 2 ln AΛ 2 + Ω 3 ln AΛ 2 2 + Ω 4 ln AΛ 2 (2.2.50)
with

Ω 1 = 6 f (1) m -q (1)
m

Ω 2 = - 16 α 1 + α 2 + α 3 c φ + 1 α 1 + α 2 8 f (1)
φ + 12 c φ -

1 α 1 6 q (1) φ + 8 q (1) φ -c (1) m + 2 Ω 1 G 0 -12 f (1) m ln(α 1 + α 2 ) -q (1) m ln α 1 Ω 3 = Ω (a) 3 + Ω (b) 3 + Ω (c) 3 Ω (a) 3 = -6 q (2) m + 4 q (1) φ -6 q (1) R 8π (1 -h) Ω (b) 3 (α 1 ) = 6 f (2) m + 4 f (1)
φ + 6 c φ + c R 2π + f (1) R -α 1 f (1) m 8π (1 -h) Ω (c) 3 = -12 c φ + c R 3π 8π (1 -h) Ω 4 = 2 Ω 3 G 0 -2 Ω (a) 3 ln α 1 + Ω (b) 3 (α 1 ) ln(α 1 + α 2 ) + Ω (c) 3 ln(α 1 + α 2 + α 3 ) + 4π 1 - 7 3 (1 -h) 6 q
(1)

φ + 8 q

(1)

φ + 12 α 1 f (1) m -12 f (1) φ + 2 α 1 Ω 1 + 8π (1 -h) 26 3 c φ -f (1) φ + 12 α 2 1 α 1 + α 2 f (1) m - 8 α 1 α 2 (α 1 + α 2 ) 2 f (1) φ - 16 α 2 1 (α 1 + α 2 + α 3 ) 2 c φ + 48π c φ -c (2) m + 8π (1 -h) c (1) R + 32π (1 -h) c φ c R (2.2.51)
where G 0 was dened in (2.1.54). Once again, the divergences in AΛ 2 (which can always be absorbed in the cosmological constant) as well as the nite area-independent contributions of the counterterms (which drop when considering W (3) [A] -W (3) [A 0 ]) have been ignored here. This is the total contribution to the three-loop partition function of the counterterms that have not been previously xed by the order 1 κ 2 (two-loop) computation made in section 2.1.4. Requiring the one-loop two-point function to be nite and regulator independent xed c m and parts of c φ and c R . Thus, only their so-far undetermined regularization-independent parts c φ and c R have been included in (2.2.51).

One way to determine some of these counterterms is to compute the two-loop twopoint function (order 1 κ 4 ) and the one-loop three-point function (order 1 κ 3 ) and one- loop four-point function (order 1 κ 4 ) and to require them to be nite and regularization independent. Imposing niteness will completely determine certain combinations of the counterterm coecients, while imposing regularization independence of the nite terms will x certain other combinations up to constants.

The computations of the two-loop two-point function and of the one-loop three-point and four-point functions clearly are beyond the scope of this work. However, there are still interesting remarks that can be made without actually doing these computations.

One can rather easily determine the contributions of the counterterms to these n-point functions. This will tell us which combinations of the counterterm coecient would be xed by such computations. We will nd that the relevant combinations are indeed the same as those appearing in the Ω i of the three-loop partition function. Although expected, this is by no means obvious and constitutes a nice consistency check.

It is straightforward to see that the cubic and quartic counterterms contribute to the diverging parts of the three-and four-point functions as

CT div = - 48π 3/2 κ 3 f (1) m Λ 2 dx K (t 1 , a, x) K (t 2 , b, x) K (t 3 , c, x) , (2.2.52) CT div = -12 (8π) 2 κ 4 q (1) m Λ 2 dx K (t 1 , a, x) K (t 2 , b, x) K (t 3 , c, x) K (t 4 , d, x) .
Thus niteness of these functions xes both f

m and q

m and hence, Ω 1 . Finiteness of the two-point function at one loop (order 1 κ 2 ) was already imposed in section 2. 2.3 and resulted in the determination of c m to this order. Thus, only the two-loop (order 1 κ 4 ) part of the two-point function 9 G (3) (a, b) will be considered here. We nd that the contribution of the counterterms to its diverging part is CT

1 κ 4 , div = 8π κ 4 ρ 1 Λ 2 ln AΛ 2 + ρ 2 Λ 2 + ρ 3 ln AΛ 2 A dx K (t 1 , a, x) K (t 2 , b, x) + ρ 4 Λ 2 d 2 x √ g * G A 0 ζ (x) K (t 1 , a, x) K (t 2 , b, x) + ρ 5 ln AΛ 2 G (a, b) + 24π f (1) m Λ 2 H(a, b) , (2.2.53) with ρ 1 = 2 Ω 1 , ρ 2 = Ω 2 -6 3 f (1) m -2 q (1) m G 0 + ln A 0 µ 2 + γ , ρ 3 = 2 Ω 3 , ρ 4 = 24π 3 f (1) m -2 q (1) m , ρ 5 = -6 q (1)
φ + 8 q

(1)

φ + 12 α 1 f (1) m + 12 c φ -12 f (1) φ (2.2.54) and H(a, b) = R * dx dy K (t 1 , a, x) K (t 2 , x, y) K (t 3 , x, y) K (t 4 , y, b) + 1 2 dx K (t 1 , a, x) K (t 2 , b, x) K (t 3 , a, x) + K (t 3 , b, x) - 1 A dx dy 1 2 K (t 1 , x, y) K (t 2 , x, y) K (t 3 , a, x) + K (t 3 , y, b) + 2 K (t 1 , a, x) K (t 2 , x, y) K (t 3 , y, b) + R * dx dy K(t 1 , a, x) K(t 2 , x, b) K(t 3 , x, y) G ζ (y) + dx K(t 1 , a, x) + K(t 2 , x, b) × K(t 3 , a, b) G ζ (x) - 1 A dy K(t 3 , x, y) G ζ (y) . (2.2.55)
The contributions per diagrams are given in the appendix B.2. Finiteness of the twopoint function at order 1 κ 4 then xes all combinations ρ 1 , . . . ρ 5 and hence Ω 2 , Ω 3 and the combination 6 q

(1)

φ + 8 q (1) φ + 12 c φ -12 f (1) φ .
Thus, all the coecients Ω 1 , Ω 2 and Ω 3 of the diverging parts of the counterterm contributions to the partition function (2.2.50) are exactly determined by the requirement of the niteness of the two-loop two-point function and of the one-loop three-point and four-point functions ! Obviously, we expect this determination to be such that (2.2.50) precisely cancels the divergences of the genuine three-loop part of this partition function, as was indeed the case in section 2.1.5 for the two-loop computation.

Let us next discuss Ω 4 which is the counterterm contribution to the order 1

(b) 3 , c φ , c R , f (1) φ , c (2)
q (2) m , f (2)
m and c (2) m as q (2) m = -1 , f (2) m = - 4 3 , c (2
√ π κ 3 f (2) m + 4 3 φ 3 + 1 κ 4 c (2) m φ 2 + 8π 2 κ 4 q (2) m + 1 φ 4 = 0 . ( 2
(b) 3 , c φ , c R , f (1)
φ , and c

(1) R , that is to say four more than the two-loop free parameter in γ str : c R . In the next chapter, we will discuss how the nite part of the two-point function can partly x these remaining coecients.

Finally, as anticipated in section 2.2.2, none of the counterterms contributes to the ln AΛ 2 3 divergence. The only way to generate such divergences would be by introducing non-local counterterm coecients that already involve a factor of ln AΛ 2 . However, as repeatedly argued, such counterterms should be forbidden. Then, since there is no possible counterterm for a ln AΛ 2 3 divergence, such a divergence is required to cancel in the rst place between the three-loop vacuum diagrams. As shown above, this is indeed the case. Let us once more remark that the cancellation of this divergence is also required for background independence.

Discussion

The purpose of this section was to check if and which new counterterms are required at three loops. We have therefore computed the leading divergence of the three-loop partition function at xed area, cf (2.2.32). It does not vanish and thus genuine threeloop counterterms are required. As mentioned before, the two-loop computation already pointed to the insertion of new counterterms at three loops. Indeed, the counterterms inserted at two loops have a strong similarity with the measure terms at two loops, as was discussed in the previous section. Yet, at three loops, the measure action gives rise to cubic and quartic vertices unlike the two-loop counterterm vertices (cf Tab. 2.1). Therefore, one could have expected additional counterterms to be needed. This argument can be generalized to all orders, as the measure action gets additional structures at every order in the loop-expansion. If the counterterms are to be understood as a renormalization of the measure action, the latter itself coming from the regularization of the measure for the metrics, then new counterterms have to be introduced at every order in the perturbation series. On the other hand, what is really surprising and encouraging is that if one requires the counterterms to be local, in particular that no counterterm coecient with a ln AΛ 2 divergence is allowed, then all the divergences may be oset but the ln AΛ 2 3 divergence. However, as we showed, this divergence cancels out between the three-loop diagrams, meaning that local counterterms are enough to balance all the non-local divergences. Moreover, the required counterterm action has a structure similar to those of the measure action, supporting the understanding of counterterms as a renormalization of the measure.

Nevertheless, with no other way to discriminate the counterterms than to forbid (ln AΛ 2 )-like non-local terms, many new free parameters appear. At three loops, doing so gives rise to twelve new parameters. Imposing the divergences to vanish in the one-loop three-and four-point functions and in the two-loop two-point function xes two parameters and three combinations of the parameters. We found that with these parameters and combinations of parameters xed, the diverging part of the three-loop partition function is also completely xed with no additional adjustable parameter remaining. Obviously,

THREE-LOOP INVESTIGATION 89

as it was the case at two loops, we expect this to happen precisely in such a way that all divergences in the three-loop partition function cancel, except for the (ln AΛ 2 )-piece that yields the three-loop contribution to the string susceptibility. Indeed, this is the only coecient of the three-loop partition function which contains undetermined nite renormalization constants. To restrain the number of free parameter in this coecient, we argued that there are two dierent notions of locality of the counterterm coecients.

While coecients involving ln AΛ 2 were excluded, we did allow coecients proportional to 1 A since such non-local terms already appeared through the measure action. Introduc- ing such 1 A counterterms in precisely such a way as to cancel the corresponding 1 A terms in the measure action was referred to as strong locality condition. Imposing this condition xes one of the six free parameters in the contribution to the string susceptibility, leaving us with ve free renormalization constants on which the three-loop contribution to γ str depends. One of these free renormalization constants was already present in the two-loop string susceptibility, so that at three-loops, four new constants play a role.

Several additional requirements should be considered, such as the condition that neither the n-point functions nor the partition function should depend on the choice of regularization. In particular, the regularization function ϕ(α i ) satises ∞ 0 dα i ϕ(α i ) = 1 and certain regularity conditions at 0 and innity, but is otherwise arbitrary. Its choice should not impact any nal, physical result. This means that all the dependence in the α i must disappear in the end. Although important, this argument is not enough to fully determine the counterterms, in particular it cannot x any α-independent pieces. It will be discussed in the next chapter of this thesis.

Background independence does not seem to help xing the counterterms, since the background dependent quantities appear with the same coecients as the divergences and hence are automatically removed when the divergences are absorbed. Nevertheless, it could well be that some indirect criteria for background independence xes some or all of our free renormalization constants. Still, as no obvious criterion has been identied, we will proceed keeping track of the free renormalization constants.

Chapter 3

Towards non-conformal matter Qu'une chose soit dicile doit nous être une raison de plus pour l'entreprendre.

Rainer Maria Rilke in Lettres à un jeune poète.

In this chapter, following [START_REF] Bilal | 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions[END_REF], we consider a somewhat more general action including the Mabuchi action:

S grav [g 0 , g] = κ 2 8πε S L [g 0 , g] + β 2 ε S M [g 0 , g] + S measure [g 0 , g] , (3.0.1)
where ε is a loop counting parameter. As previously discussed in section 1.2, the Mabuchi action

S M [g 0 , g] = d 2 x √ g 0 2π(h -1)φ∆ 0 φ + 8π(1 -h) A 0 -R 0 φ + 4 A σe 2σ (
κ 2 8πε S L [g 0 , g] = κ 2 2 ε (1 -h) ln A A 0 + 1 2 d 2 x √ g * φ ε (A∆ * ) 2 (∆ * -R * ) φ ε + O(ε) , β 2 ε S M [g 0 , g] = 2β 2 ε ln A A 0 + 16π β 2 κ 2 d 2 x √ g * φ ε A∆ * (∆ * -R * ) φ ε + O(ε) , (3.0.3)
in the background metric g * of constant curvature R * and area A. From eq. (3.0.3), one directly reads the propagator for the eld φ ε G(x, y)

L+M = x| A∆ * (∆ * -R * ) A∆ * + 32π β 2 κ 2 -1
|y .

(3.0.4)

This propagator highly diers from the pure Liouville theory. Indeed, if we only consider the Liouville action, the term

β 2
κ 2 is vanishing and we can make the change of variable 2 φ = A∆ * φ ε .

(3.0.5)

1 We recall that AR * = 8π(1 -h). (3.0.8)

The coecients of the small t expansion of the heat kernel associated to the operator

D M * = ∆ * + T are now (see section 1.3.3) a M 0 (x, y) = 1 , a M 1 (x, y) = -T , a M 2 (x, y) = T 2 2
. . .

(3.0.9) and √ g * = 1. The regularized propagator K(t, x, y) is to be written in terms of these modied coecients. In particular, we have

K(t, x, x) = 1 4πt 1 -T + 4π A t + T 2 + 4π A Tt 2 + O(t 2 ) , K(t, x, x) = G ζ (x) - 1 4π ln µ 2 t + γ -T + 4π A t + O(t 2 ) .
(3.0.10)

In the section 3.1, we present the two-loop computation of the partition function in this Liouville and Mabuchi theory, while section 3.2 is dedicated to the three-loop study.

Thus, we expand the Mabuchi action around the metric g * of constant vanishing scalar curvature up to the three-loop order ε 2 :

β 2 ε S M [σ] = 2β 2 ε ln A A 0 + 32π A β 2 κ 2 d 2 x √ g * 1 2 φ 2 + √ 4πε 3 κ φ 3 + 4πε 3 κ 2 φ 4 (3.0.11) + 16(πε) 3/2 5 κ 3 φ 5 - (8πε) 2 κ 4 2 15 φ 6 + O(ε 3 ) .
In the following, we adopt the previous notation dx for d 2 x g * (x) .

= - √ 4πε κ ∆ * + 1 3 T , = - 4πε κ 2 4 3 (∆ * + T) , = - 4πε κ 2 1 2 (∆ * -2T) .
(3.1.1)

Diagrams

We get the same diagrams as in the previous study of the pure Liouville theory: the gure-eight diagram ii , the setting sun diagram i , the glasses diagram i i and the measure diagram a `i. In the following, we give the contributions from each diagram (when all its subdiagrams are summed up) and then their contributions to the partition function in terms of the divergences AΛ 2 ln AΛ 2 and ( ln AΛ 2 2 and in ln AΛ 2 which leads to the two-loop contribution for γ str . We drop the terms proportional to AΛ 2 since they will anyway be reabsorbed in the cosmological constant, and the constant terms since they vanish when considering ln Z

[A] Z[A 0 ] .
The gure-eight diagram First, we have the gure-eight diagram:

I ee= 4πε κ 2 dx -6 K(t 1 , x, x) K (t 2 , x, x) + 5 T K (t 1 , x, x) K (t 2 , x, x) , = ε κ 2 3 2π - AΛ 2 α 1 + AT + 4π ln AΛ 2 + 5 4π AT ln AΛ 2 2 + 2 G 0 -ln α 1 ln AΛ 2 (3.1.2)
with G 0 dened in the previous chapter in (2.1.54). Remember that this expression has to be understood with the implicit integration ∞ 0 dα 1 ϕ(α 1 ). We insist also on the fact that the coecients in each K and K are changed due to the new propagator (3.0.7) and not only the coecients in front of each integral. Note also that the leading logarithmic divergence ln AΛ 2 2 , which was vanishing when considering the summed contributions in the pure Liouville theory appears through this diagram. As already discussed in the previous chapters, this two-loop divergence has to vanish in order to ensure the background independence of our results, but also the possibility to renormalize through only local (or measure-like) counterterms.

TWO-LOOP PARTITION FUNCTION 93

The setting sun diagram Then, we have the setting sun diagram:

I e= 4πε κ 2 dx dy K(t 1 , x, y) -K(t 2 , x, y) d dt K(t, x, y) t=t 3 + 2 K (t 2 , x, y) K (t 3 , x, y) -4 T K (t 2 , x, y) K (t 3 , x, y) , = ε κ 2 1 π AΛ 2 α 1 + α 2 + AT 2 2α 1 α 2 (α 1 + α 2 ) 2 -1 -6π ln AΛ 2 - 5 4π AT ln AΛ 2 2 + 2 G 0 -ln(α 1 + α 2 ) ln AΛ 2 , (3.1.3)
whose contribution to the leading logarithmic divergence ln AΛ 2 2 is precisely the one required to cancel the contribution from the gure-eight diagram ii .

The glasses diagram

Similarly to the pure Liouville theory case, the glasses diagram:

I e e= 4πε κ 2 dx dy K(t 1 , x, x) K(t 2 , y, y) × - 1 2 d dt K(t, x, y) t=t 3 -2 T K (t 3 , x, y) + 2 T 2 K (t 3 , x , y)
I a `d = 4πε κ 2 dx K(t 1 , x, x) K(t 2 , x, x) = ε κ 2 1 4π AΛ 2 α 1 -AT -4π ln AΛ 2 . (3.1.5)

Total contribution to the two-loop partition function

Summing up the contributions (3.1.2)-(3.1.5) of these diagrams, we get the total contribution to the two-loop (order ε) partition function:

ln Z[A] L+M 2-loop = ε κ 2 1 2π 2 α 1 + α 2 - 5 2 α 1 AΛ 2 ln AΛ 2 (3.1.6) + -1 + 8 β 2 κ 2 3 -5 ln α 1 α 2 (α 1 + α 2 ) 2 + 4 α 1 α 2 (α 1 + α 2 ) 2 ln AΛ 2
We are only interested in the diverging and regulator-dependent part of G (2) (u, v). Thus, the result for G (2) (u, v) and the contributions per diagram given hereafter are written without their nite regulator independent parts. The computations themselves are very similar to those done in the pure Liouville theory case and those details are given in appendix B.1. Thus, we only give the results here. As already emphasized, the expressions (1.3.34) and (1.3.42) for K(t, x, y) (and thus for d dt K(t, x, y)) and K(t, x, y) are to be considered with the coecients (3.0.9).

G

(2)

u-O-v (u, v) = ε κ 2 G(u, v) 12 ln AΛ 2 -4 1 + 2 α 1 α 2 (α 1 + α 2 ) 2 + 3 ln(α 2 + α 3 ) + dx K(t 1 , u, x) K(t 2 , x, v) -20 T ln AΛ 2 + G 0 (x) -ln(α 3 + α 4 ) + 8Λ 2 α 3 + α 4 + T -4 + 8 α 1 α 2 (α 1 + α 2 ) 2 - 80π A , (3.1.10) G (2) u e v (u, v) = ε κ 2 -12 G(u, v) ln AΛ 2 -ln α 3 + dx K(t 1 , u, x) K(t 2 , x, v) 20 T ln AΛ 2 + G 0 (x) -ln α 3 -12 Λ 2 α 3 + 12 T + 48π A , (3.1.11) G (2) measure (u, v) = ε κ 2 dx K(t 1 , u, x) K(t 2 , x, v) 2 Λ 2 α 3 -2 T - 8π A , (3.1.12) G (2) ct (u, v) = ε κ 2 -8π c (1) φ G(u, v) -8π c (1)
T

T + c (1) m Λ 2 + c (2) m A dx K(t 1 , u, x) K(t 2 , x, v) , (3.1.13) with G 0 (x) = 4π G A 0 ζ (x) -γ -ln A 0 µ 2 . (3.1.14)
The diagram has neither diverging nor regulator dependent parts, and is therefore omitted in the following.

G (2) (u, v) = 4 ε κ 2 G(u, v) 3 2 ln α 1 α 2 (α 1 + α 2 ) 2 -1 - 2 α 1 α 2 (α 1 + α 2 ) 2 -2π c (1) φ + dx K(t 1 , u, x) K(t 2 , x, v) Λ 2 2 α 1 + α 2 - 5 2α 1 -2π c (1) m + T - 5 2 ln α 1 α 2 (α 1 + α 2 ) 2 + 3 2 + 2α 1 α 2 (α 1 + α 2 ) 2 -2π c (1) T - 10 π + 2π c (2) m
A .

(3.1.15)

Finiteness and regulator dependence partly x the counterterms. Moreover, imposing the strong locality condition also xes c

(2) m = -1 such that we get:

c φ = 1 2π 3 2 ln α 1 α 2 (α 1 + α 2 ) 2 -1 - 2α 1 α 2 (α 1 + α 2 ) 2 + c φ , c T = 1 2π - 5 2 ln α 1 α 2 (α 1 + α 2 ) 2 + 3 2 + 2α 1 α 2 (α 1 + α 2 ) 2 + c T , c m = Λ 2 2π 2 α 1 + α 2 - 5 2α 1 - 1 A . (3.1.16)
Note that both c φ and c m take the same value as in the case of the pure Liouville theory. This was expected, since the Mabuchi action comes with a coupling β 2 only present through the T coecient carried in the counterterm action by c T .

Two-loop partition function

The counterterms contribute to the partition function through the diagram :

I = - 4πε κ 2 dx c φ K(t, x, x) + (c T T + c m ) K(t, x, x) , (3.1.17) such that W (2) [A] M+L ≡ ln Z[A] M+L 2-loop = ε κ 2 1 2π 2 α 1 + α 2 - 5 2 α 1 -2π c (1) m AΛ 2 ln AΛ 2 + -1 -c (2) m + 8 β 2 κ 2 3 -5 ln α 1 α 2 (α 1 + α 2 ) 2 + 4 α 1 α 2 (α 1 + α 2 ) 2 -4π c (1) T ln AΛ 2 = -32π ε β 2 κ 4 c T (3.1.18)
with the values (3.1.16). In the pure Liouville theory, the two-loop contribution to γ str is proportional to (1 -h) (see (2.1.106)): Once more, at this order in the perturbation theory, we end up with two free nite and regularization independent constants, c φ and c T , with only one contributing to the string susceptibility.

γ (2) str = - 4 κ 2 c R (1 -h) . ( 3

Three-loop discussion

We present now the extension to the Liouville and Mabuchi theory on the torus of the results of the three-loop study carried on in the previous chapter for the pure Liouville theory. As already discussed, the leading divergence remains the same. Thus, we will only check in the following that the leading logarithmic divergence in ln AΛ 2 3 cancels between the diagrams. Indeed, this divergence which comes together with unwanted background dependent terms can not be absorbed by the addition of local (or non-local but measure-like) counterterms alone. Thus, this cancellation is a requirement for the consistency of our theory.

Then, we present the contributions of the counterterms to both the three-loop partition function and to the diverging and regulator dependent part of the two-loop two-point function (order ε 2 ). Indeed, the diverging part of the three-and four-point functions are unchanged and x uniquely both f

m and q

(1) m as dened in the previous chapter, thus, we only discuss the two-point function in the following.

First, let us write the genuine three-loop (i.e. order ε . Thus, for consistency, we computed its contribution in each of the 29 types of diagrams at three-loop (see the previous chapter). Indeed, although the coecients in the vertices and in the regularized propagator are modied in the Liouville plus Mabuchi theory on the torus, their structure remains the same and we can build exactly the same diagram than in the previous study. These diagrams contribute the same leading divergence but their subleading terms dier. However, once more, the miracle occurs and the ln AΛ 2 3 divergence cancels between the diagrams, as one can read in Tab. Let us stress that this cancellation was once more by no means obvious and that it is crucial for the consistency of our theory. Similarly to the pure Liouville theory, we expect this to happen for the leading logarithmic divergence at each order in ε.

Counterterm contributions

We introduce the same counterterm actions as in the previous chapter. Namely, inserting the undetermined counterterms from the two-loop computation, we have: S q ct = (8πε) 2 κ 4 1 2 dx q φ φ 3 (∆ * + T) φ + q φ φ 2 (∆ * + 2T) φ 2 + q T T φ 4 + q m φ 4 .

Once again, we include the ε κ 2 terms in the expansion of the quadratic counterterms.

These counterterms may be expanded as:

c φ = c (1) φ , c T = c (1)
T , c m = c

(1)

m Λ 2 + c (2) m A , f φ = f (1) φ , f T = f (1) T , f m = f (1) m Λ 2 + f (2) m A , q φ = q (1) φ , q φ = q (1) φ , q T = q (1)
T , q m = q (1)

m Λ 2 + q (2) m A , (3.2.3)
where we recall that c φ and c T are regulator independent constants.

Three-loop vacuum diagrams

The diagrams we have to consider are the following: , , , , , and . Indeed, the glasses-like diagrams only contribute to the nite part of the partition function or through terms proportional to AΛ 2 that would be included in the cosmological constant. The contributions per diagrams are given hereafter, up to these irrelevant terms. First, we have the measure-like diagrams:

I = - 4πε 2 κ 4 dx c T T + c m K(t, x, x) (3.2.4) = - ε 2 κ 4 c (1) m AΛ 2 ln AΛ 2 + c (2) m + c (1)
T AT ln AΛ q φ + 2 q φ K(t 1 , x, x) K (t 2 , x, x)

+ q φ T + 3 2 (q T T + q m ) K (t 1 , x, x) K (t 2 , x, x) , = -ε 2 κ 4 6 q (1) m AΛ 2 ln AΛ 2 2 + 1 α 1 6 q

(1)

φ + 8 q

(1) φ +12 q (1) m (G 0 -ln α 1 ) AΛ 2 ln AΛ 2 + 6 q (2) m + AT 4 q

(1)

φ + 6 q (1)
T ln AΛ 2 2 -AT + 4π 6 q

(1)

φ + 8 q

(1) φ -12 α 1 q (1) m -2 6 q (2) m + AT 4 q

(1)

φ + 6 q (1)
T (G 0 -ln α 1 ) ln AΛ where we dene C = c φ -c T 2 for the sake of brevity. , Ω (a) 3 = -6 q (2) m -AT 4 q

(1)

φ + 6 q (1) φ + 8 q

(1) φ + 12 α 1 f (1) m -12 f

(1)

φ + 2 α 1 Ω 1 -AT 28 3 c φ -8 f (1) φ + 12 α 2 1 α 1 + α 2 f (1) m - 8 α 1 α 2 (α 1 + α 2 ) 2 f (1) φ - 16 α 2 1 (α 1 + α 2 + α 3 ) 2 c φ -c (2) m + AT c (1)
T .

(3.2.14)

Of course, only the parts involving the coupling to the Mabuchi action β 2 , that is to say AT = 32π β 2 κ 2 are modied with respect to the pure Liouville theory. However, as for the pure Liouville theory, these counterterms for the Louville plus Mabuchi theory do not contribute to the leading logarithmic divergence at three loops ln AΛ 2 3

. Then, its magic cancellation through the diagrams (see Tab. (1)

φ + 8 q

(1)

φ +
φ + 176π c φ + 16πAT c φ c T , ρ 8 = 6 f (2) m + AT -f

φ + 6 f

T + 24 c φ -14 c T .

(3.2.20)

Imposing this structure to be regulator independent in the full two-loop two-point function result in xing ρ 7 to a constant c (ρ 6 = ρ 3 being already xed). This means that Ω (a) 4

is xed in terms of c, f

φ , c φ and c T . Injecting this into (3.2.14), we end up with four undetermined constants at three loops. This happens also in the pure Liouville theory on surfaces of any genus studied in the previous chapter, where the number of remaining free counterterms is also reduced to four. However, we still need new conditions to impose in order to completely x these parameters, and hence our theory.

Discussion

The path to a quantized theory of gravity has not been marked out yet. Consequently, several ways are currently explored. One of them relies on the continuous approach of the Liouville theory, which describes the coupling of any conformal eld theory to gravity in two dimensions. One object we can characterize in this approach is the partition function at xed area whose area dependence is described by a critical exponent called the string susceptibility γ str . The formalism of the functional integral appears to be a natural one to compute both the xed area partition function and the string susceptibility. However, one of the major diculties lies in the non-at integration measure over the space of the metrics. Moreover, in a theory of quantum gravity one has to sum over the topologies.

Nevertheless, under the simplifying assumption of a free-eld measure, a formula for γ str has been obtained in the end of the 1980s through the DDK argument. This formula, referred to as the KPZ formula since it has rst been conjectured by Khnizik, Polyakov and Zamolodchikov on the sphere, has since been found in agreement with several results

from the discrete methods.

The present work follows these footsteps. Taking advantage of recent advances in ecient multi-loop regularization methods on curved space, it has been possible to precisely regularize the complicated integration measure and thus to treat it without the strong hypothesis of a free-eld measure. Then, a rst-principles computation of the string susceptibility has been initiated at one loop, in a generalized Liouville theory including the Mabuchi action, which appears as a rst order correction to the Liouville theory if the matter is massive and hence non-confomal. The one-loop result for γ str is in agreement with the KPZ formula (in the case of conformal matter only). This thesis goes further with the full two-loop computation in to cases: rst in a pure Liouville theory on a surface of arbitrary genus, and then, in the special case of the torus, in a Liouville plus Mabuchi theory, thus investigating the coupling to non-conformal matter.

The non-trivial nature of the quantum gravity integration measure is revealed beyond one loop, starting at two loops. Then, it generates a measure action whose has to be added to the Liouville (and the Mabuchi) actions.

The results obtained from these computations present divergences, and thus one needs to renormalize them. The consistency of this approach is emphasized by the fact that the divergences that would require non-local counterterms, cancel among themselves. It is remarkable that those vanishing terms come together with background dependent structures which are, of course, undesirable in a theory of quantum gravity and which vanish together with the divergences. Consequently, the renormalization procedure is possible through only local counterterms, with the highlighted exception of the so-called "measure-like" counterterms, presenting the same dependence in the area than the measure action. These counterterms are thus understood as a renormalization of the complicated non-at quantum gravity integration measure. (C.2)

The only divergence investigated here is the one in AΛ 2 ln AΛ 2 2 , which cannot appear unless two Ks are present, since the logarithmic divergence comes from such terms (2.1.47). The terms without at least two Ks after doing the expansions are thus discarded in the following. Remembering that t = α Λ 2 , the previous integrals may then be rewritten as:

J (1) = dx K 3,4 (x,

Résumé

Le but de cette thèse est d'étudier la gravité quantique bidimensionnelle. Nous nous intéressons plus particulièrement aux approches dans le continu. Ces dernières reposent principalement sur l'action de Liouville qui décrit le couplage entre théorie conforme et gravité. Si cette action, bien connue, est très bien comprise, la mesure de l'intégrale fonctionnelle sur l'espace des métriques pose plus de problèmes. Toutefois, sous l'hypothèse simplificatrice d'une mesure de champ libre, la dépendance en l'aire de la fonction de partition de la gravité quantique en présence de matière conforme a pu être établie. Malgré l'hypothèse assez forte sur la mesure d'intégration, cette formule (dite KPZ), a été confirmée par des calculs issus de méthodes discrètes, et ce dans plusieurs cas particuliers.

Grâce à une nouvelle méthode de régularisation spectrale en espace courbe, cette mesure d'intégration a récemment pu être proprement définie. Dans cette thèse, un calcul perturbatif de la fonction de partition à aire fixée est mené, jusqu'à trois boucles, en considérant l'action de Liouville et des surfaces de Kähler de genre quelconque (qui coïncident avec l'ensemble des surfaces à deux dimensions). Des divergences apparaissant dans les calculs, il est nécessaire de renormaliser les actions. Cette renormalisation peut être interprétée comme une renormalisation de la mesure d'intégration. Nos résultats à deux boucles sont finis, indépendants de la régularisation et compatibles avec le résultat KPZ, mais dépendent d'un paramètre libre. L'étude à trois boucles suggère que la théorie reste renormalisable aux ordres supérieurs mais dépend de nouveaux paramètres à chaque ordre. Ces résultats ont été généralisé dans le cas du tore au couplage à de la matière non-conforme.

Mots Clés

Gravité quantique bidimensionnelle, théorie de Liouville, action de Mabuchi, susceptibilité de la corde, régularisation spectrale, renormalisation.

1 1. 1 .

 11 TWO-DIMENSIONAL QUANTUM GRAVITY 2 Consider two metrics g and g 0 of area A = d 2 x √ g and A 0 related in the conformal gauge by the conformal factor σ: g = e 2σ g 0 .

κ 2 .

 2 The left and right conformal weights of : e 2ασ :=: e 2α κ σ : are well-known. Requiring them to equal unity yields

  grav [g 0 , g] are appropriate local gravitational actions, it would be desirable to express them in terms of more geometric quantities like the conformal factor or the Kähler potential, as was the case for S (0) grav [g 0 , g] and S(1) grav [g 0 , g] with the Liouville, Mabuchi and Aubin-Yau actions. As previously stated, in the following of this thesis, both the Liouville action (for conformal matter) and the Mabuchi action are considered, the later being involved in the rst-order mass correction to the gravitational action.A writing of these S (r) grav [g 0 , g] in terms of the conformal factor or the Kähler potential could open a way for a continuation of the present study. To our knowledge however, there does not seem to exist any appropriate functional in the mathematical literature. Nevertheless, since the G (0) r are entirely determined in terms of the properties of the Riemann surface, they are purely geometric quantities.

1. 3 . 21 1. 3 . 2

 32132 REGULARIZATION OF THE PARTITION FUNCTION One-loop expansion of the partition function at xed areaThe semi-classical partition function at xed area is dominated by the minimum of the gravitational action at xed area. As previously stated in section 1.2.3 (see (1.2.29)), the classical saddle-points at xed area of both the Liouville (1.1.4) and the Mabuchi (1.2.26) actions are the constant curvature metrics of arbitrary area A. In terms of the background metric g 0 of constant curvature and of area A 0 , the saddle-point value of σ is σ cl = 1 2 ln A A 0

  .(1.3.16) is a generalization in agreement with(1.3.18).

( 1 . 3 . 28 )

 1328 If a given Feynman diagram (integral) I n contains n propagators, we can now dene its regularized version as

  our later computations in Chapter 2 and 3.

  n e -δ B S , δ B S = -Tr log(1 + M ) = -Tr M + O(ω 2 ) . (1.4.27) Since ψ n |δψ ∆ n = -ψ n |ω|ψ n , we get upon regularization

  independent terms only lead to irrelevant normalization factors and will drop out in 1.4. BACKGROUND INDEPENDENCE 36 the end upon computing Z[A]/Z[A 0 ]. Anticipating the discussion of the scaling of the Green's function in the next Chapter, G(x, z) does not depend of the area. Then, from eq. (1.2.23) and (1.4.29), and the three preceding formulae, we nd that

10 3 ζ 2 A

 32 Note that from(1.4.37), we expect the structure R G not to be background independent either.Chapter multi-loop rst principles analysisLe courage consiste à donner raison aux choses quand nous ne pouvons les changer.Marguerite Yourcenar in Alexis ou le Traité du Vain Combat.

1 κ 2

 12 which corresponds to the two-loop contributions to ln Z[A]. The regularized loopintegrals are then evaluated in subsection 2.1.2. It is remarkable that the individual (ln AΛ 2 ) 2 -divergences are always accompanied by certain structures that are not background independent. The fact that the (ln AΛ 2 ) 2 -divergences cancel is equivalent to the cancellation of these background non-invariant structures! However, the leading divergence AΛ 2 ln AΛ 2 is non-vanishing and we are led to introduce counterterms. This is done in the subsection 2.1.3. Note that the appearance of a non-local divergence should not be a surprise. In a local quantum eld theory, one-loop divergences always are local, but starting at two loops the divergences are not necessarily local, in particular also due to the so-called overlapping divergences. However, we know from the standard BPHZ proofs [114116] that they can always be cancelled by one-loop diagrams including local counterterm vertices (as well as tree diagrams including further local counterterm vertices). These counterterm vertices occurring in the one-loop diagrams are themselves determined from the cancellation of the divergences of the corresponding one-loop npoint functions. The same does happen here. This will be worked out in subsection 2.1.4.The subsection 2.1.4 is devoted to computing the one-loop two-point function of the Kähler eld and determining the necessary counterterms to make it nite and regulator independent. This computation is done on an arbitrary Riemann surface and is done consistently in the same spectral cuto regularization as the computation of the partition function. Note that this two-point function is closely related to the two-point function of the conformal factor < e 2σ(x) e 2σ(y) >. Most of the lengthy computational details

1 2

 1 ln(1 -φ) = σ -σ cl as the basic eld instead? One important point concerns the zero-mode. The absence of zero-mode is easy to implement for φ, while for σ -σ cl = -1 2 φ-1 4 φ 2 -1 8 φ 3 -. . . it results in a very complicated constraint.Taking this constraint properly into account is highly non-trivial and probably equivalent in diculty to working with the complicated actions (2.1.5) or (2.1.7).

  .1.13) with (∆ * -2R * ) acting on the two propagators connected to the bold part of the vertex (of course, upon integrating by parts, it does not matter whether one chooses the two 2 Recall that the considered metrics are of constant curvature at xed area. lines to the left or the two lines to the right), and

  connected by propagators that are G(x, y) = x|(∆ * -R * ) -1 |y .

  [A 0 ] instead. The coecients of ln A A 0 yield the contributions to γ str -3 and thus from (2.1.21) and (2.1.22)

  The two quartic vertices both give a gure-eight diagram ii with the four lines of a single vertex connected by two propagators. The cubic vertex gives two types of diagrams: the setting sun diagram i with two cubic vertices connected by three propagators, and the glasses diagram i i with the two vertices joined by a single propagator and the remaining two lines of each vertex connected by a propagator. Finally the measure vertex gives the measure (one-loop) diagram a `i with the two lines of the vertex connected by a single propagator. The gure-eight diagram: This diagram actually gets three contributions : one from the quartic vertex (2.1.14) and two from the dierent ways to contract the lines of the quartic vertex (2.1.13). Taking into account the dierent numbers of contractions in each case (3, 2 and 1) yields

ζ

  R (s, x, y) = ζ(s, x, y) s a 0 (x, y)e -2 (x,y)/4t

. 1 .

 1 53) and the denition (2.1.54) for G 0 . Before actually computing the counterterm coecients in the next subsection, let us discuss what are the desired values of the c φ , c R and c m . In particular, the non-local 4 This means that if any given c(αi) depends on any given number of αi, i = 1, . . . r, one should really think of it as c[ϕ] = ∞ 0 dα1 . . . dαrϕ(α1) . . . ϕ(αr)c(α1, . . . , αr).

1 )R

 1 = 0, as well as c

  are many dierent contributions to each diagram. The details of the computation are deferred to the appendix B.1 where the results for the three one-loop diagrams are given in (B.1.11), (B.1.15) and (B.1.21) and for the measure contribution in (B.1.22).

.1. 97 )

 97 These are exactly the desired values (2.1.82), (2.1.83) and (2.1.86). In particular, the value of c (2) m is exactly what is needed to cancel the divergent two-loop contributions in W [A] = ln Z[A] ! As explained above, the value (2.1.97) for c (2) m really means that

  c

  c φ , c R and c m are true (α i -independent, Λ-independent) constants. With these values of the counterterm coecients, eq. (2.1.96) reduces to

  The same is true for the terms in the second line of the right hand side. Cancelling these divergences would require a non-vanishing counterterm coecient c (1) φ , which was excluded by demanding niteness of the two-point function at u = v in the previous computation.

1 κ 4 . 12 .

 412 This involves genuine three-loop diagrams made from the Liouville vertices only, as well as two-loop and one-loop diagrams involving also the vertices from the measure or counterterm action. Combining all these vertices gives twenty-nine types of vacuum diagrams, each of them receiving contributions from subdiagrams. Fifteen of these diagrams come from pure Liouville contributions, nine involve the measure and six the two-loop counterterms . The decomposition of the diagrams is detailed hereafter. The sextic vertices give one diagram, the ower diagram, which may be written as the sum of ve subdiagrams: The weight factors in front of the dierent subdiagrams take into account the multiplicity of the diagram, including the symmetry factors and the contractions. Combining the quintic and cubic vertices yields two types of diagram: 2.2. THREE-LOOP INVESTIGATION 69 composed of respectively eight and ten subdiagrams. Using two quartic vertices gives two diagrams: made of respectively ve and eleven subdiagrams. Five types of diagrams are built by a quartic vertex and two cubic vertices:

Figure 2 . 1 :

 21 Figure 2.1: Relevant diagrams for the leading divergence in AΛ 2 ln AΛ 2 2

) 2 . 2 .

 22 THREE-LOOP INVESTIGATION 75 while one reads directly the fourth subdiagram

t=t 3

 3

( 2 . 2 . 47 )

 2247 Moreover, two parameters of the two-loop counterterms (2.1.80) are still unconstrained: c φ and c R . Although only c R appears in the two-loop partition function (2.1.104), both may contribute to the divergent part of the partition function at three loops, through the diagrams , , and . Their diverging contributions are displayed below:

m

  and the ρ i been xed, also Ω 1 , Ω 2 and Ω 3 are xed and we consider Ω .e. of Ω (b) 3 , c φ and c R (cf (2.2.51)). (Note that the second line in the expression of Ω 4 can be expressed though ρ 5 , c φ and Ω 1 .) Thus Ω 4 depends on the following six undetermined constants: Ω

8 √

 8 3.0.2) appears (together with the Aubin-Yau action) as the rst-order mass correction to the Liouville action. As previously stated in section 1.2.3 (see eq. (1.2.29)), the classical saddle-points at xed area of the Mabuchi action are also the constant curvature metrics of arbitrary area A 1 . In terms of the rescaled Kähler potential φ ε = κ πε φ, the one-loop expansions (1.3.13) of the Liouville and Mabuchi actions are rewritten as

2 β 2 κ 2 )

 22 In the previous chapter, ε was absorbed in κ or equivalently taken to one. 90 91 Thus, we have to consider the much simpler propagator G(x, y) L = x|(∆ * -R * ) -1 |y .(3.0.6) The full Liouville and Mabuchi theory is then expected to behave signicantly dierently from the pure Liouville theory. However, the writings of the vertices in terms of the propagator (3.0.4) is quite challenging and the computation of the two-loop contribution to the string susceptibility for this Liouville and Mabuchi theory on a surface of arbitrary genus is beyond the scope of this thesis. Thus, in this chapter we only present the much simpler case of the torus, where the constant scalar curvature R * = 8π(1-h) A vanishes. Then, the previous change of variable (3.0.5) stands and the propagator associated to the eld φ is simply G(x, y) L+M = x|(∆ * + 32π A -1 |y , ≡ x|(∆ * + T) -1 |y ,

( 3 . 1 . 4 )

 314 only contributes through constant or O(1/Λ 2 ) terms.The measure diagramFinally, we have the measure diagram, whose contribution to γ str and to the diverging part of the two-loop partition function (up to terms proportional to the cosmological constant) is simply:

3. 1 .

 1 TWO-LOOP PARTITION FUNCTION 96 Summing up eq. (3.1.10)-(3.1.13), we get the diverging and regulator dependent parts of the full one-loop two-point function:

3. 2 . 1

 21 Cancellation of the (ln AΛ 2 ) 3 divergence Similarly to what happened in the pure Liouville case on surfaces of any genus, the local or measure-like counterterms one can introduce do not contribute to the leading 3.2. THREE-LOOP DISCUSSION 98 logarithmic divergence ln AΛ 2 3

ln AΛ 2 3

 3 contributions per diagrams. The rst line corresponds to the Liouville and Mabuchi theory on the torus whereas the second line corresponds to the Liouville theory on surfaces of any genus.

c φ + ε κ 2 c+ ε κ 2 c m φ 2 f φ φ 2

 222 φ φ (∆ * + T) φ + c T + ε κ 2 c T T φ 2 (∆ * + T) φ + f T T φ 3 + f m φ 3 , (3.2.2)

t=t 3 + 2 + 6 fTW ( 3 ) [A] ct = ε 2 κ 4 Ω 1 1 , Ω 2 = Ω Liouville 2 ,

 326341122 K (t 2 , x, y) K (t 3 , x, y) , f(1) m (G 0 -ln(α 2 + α 3 )) AΛ 2 ln AΛ (2) m + AT -f + α 3 f (1) m (G 0 -ln(α 2 + α 3 )) ln AΛ 2 . (3.2.11)Total contribution of the counterterms to the two-loop partition function Summing the previous contributions (3.2.4)-(3.2.11), one gets: AΛ 2 ln AΛ 2 2 + Ω 2 AΛ 2 ln AΛ 2 + Ω 3 ln AΛ 2 2 + Ω 4 ln AΛ 2 ,

3 =

 3 AT (20 c φ -8 c T ) , Ω 4 = 2 Ω 3 G 0 + Ω (a) 4 + 48π c φ + 8π AT c φ c T , , Ω 1 ) ln(α 1 + α 2 ) + Ω (c) 3 ln(α 1 + α 2 + α 3 ) + (4π + AT) 6 q

3 . 1 ) 2 ,div = 8πε 2 κ 4 ρ 1 Λ 1 , ρ 2 = ρ Liouville 2 , ρ 3 = 2 Ω 3 , ρ 4 = ρ Liouville 4 , ρ 5 =

 31211232345 was indeed crucial for the consistency of our theory.Finiteness of the two-point functionAs previously stated, the diverging parts of the three-and four-point function are not changed with respect to the pure Liouville theory by the inclusion of the Mabuchi action, when studying the torus. Thus, in order to look at the freedom of the three-loop counterterms in (3.2.14), it is sucient to give their diverging contributions to the two-point function only. We have CTε 2 ln AΛ 2 + ρ 2 Λ 2 + ρ 3 ln AΛ 2 A dx K (t 1 , a, x) K (t 2 , b, x) + ρ 4 Λ 2 dx G A 0 ζ (x) K (t 1 , a, x) K (t 2 , b, x) + ρ 5 ln AΛ 2 K (t 1 , a, b) + 24π f (1) m Λ 2 H M (a, b) , -6 q

2 ,n = 8π κ 4 1 A dx ρ 6

 216 (3.2.17) Finiteness of the two-point function at order ε 2 then xes all the combinations ρ 1 , . . . ρ 5 . Namely, ρ 2 xes Ω 2 and ρ 3 xes Ω 3 , while Ω 1 is xed by the niteness of the three-and four-point functions. Thus, we do not have any freedom in the counterterm coecients of the divergences of the three-loop partition function. As it was the case in the two-loop study, we expect these combinations to be xed by the niteness of the n-point functions in such a way that they absorb the divergences of the partition function.If we consider now Ω 4 which contributes to the three-loop contribution to γ str , ρ 5 xes the second line in the coecient Ω (a) 4 . Moreover, the strong locality condition impose the values q (2) m = -1 , f(2) free parameters in Ω 4 (and thus in γ str : Ω (b) 3 , c φ , c T , f the nite part of the two-point function, and in particular the nite coecient ofdx K (t 1 , a, x) K (t 2 , b, x) and of dx G A 0 ζ (x) K (t 1 , a, x) K (t 2 , b, x) can help x one constant. Indeed, we have CT ε 4π G A 0 ζ (x) -γ -ln A 0 µ 2 + ρ 7 G (a, x) G (b, x) + ρ 8 4π A dx G ζ (x) -1 A dy G ζ (y) G (a,x) G (b, x) + ...

  They can be xed by the requirements of niteness and regularization independence and by imposing the so-called strong locality condition which imposes the area dependent terms in the measure 105B n (t a , t b , t c ; x) = dz K a (x, z) K b (x, z) d n dt n c K c (x, z) , C n,m (t a , t b , t c ; x) = dz K a (x, z) d n dt n b K b (x, z) d m dt m c K c (x, z) , D n (t a , t b , t c ; x) = dz K a (z, z) K b (x, z) d n dt n c K c (x, z) ,E n,m (t a , t b , t c ; x) = dz K a (z, z)

  2 A of S grav . Note that S M = 8π(1 -h)S AY + d 2 x

				√ g 0	4 A σe 2σ -R 0 φ . Eq. (1.2.21) relates the variations
	δσ and δφ as
	δσ =	δA 2A	-	A 4	∆δφ and δ	e 2σ A	= -	1 2	∆ 0 δφ .	(1.2.28)
	It is then straightforward to show that the variations of the Liouville, Mabuchi and
	Aubin-Yau actions are given by

is the geodesic distance between x and y in the metric g. One can show [53] that both denitions of G ζ are equivalent and dene the same quantity. The same relations hold between

  G ζ (x), G(x, y) and ζ(s, x, x). Note that G ζ (x) contains global information about the Riemann surface and cannot be expressed in terms of local quantities only.

							More precisely
	G ζ (y) = lim x→y	G(x, y) +	1 4π	ln	2 g (x, y)µ 2 4	+ 2γ	(1.2.42)
	where 2 g (x, y)

  These formulae would in principle allow us to compute δ K. However, we only need the asymptotic for large cuto. From eq. (1.3.47) and (1.4.13), we nd δ B K(t, x, x) ∼ 7 12π (∆ * ω -ωR * ) + O(t) and thus (1.4.21) becomes

  .29) we need to compute the variation of the Green's function at coinciding point δ B G ζ (x) and thus the variation δ B G(x, y) of the Green's function. From the equations (1.4.22) and (1.4.23) giving the transformations of the eigenvalues λ L n and eigenfunctions ψ n . One straightforwardly nds

  once regularized (see section1.3.3). Since K(t, x, x) = Λ 2 4πα + 7 24π R * -1 A + . . . does not depend 2 on x and since φ has no zero-mode, the rst term in the action drops out (cf eq. (1.3.39)). Thus, at two loops, S measure provides a quadratic vertex with a diverging coecient. This is very similar to a counterterm, as we will see later-on. One might wonder what is the advantage of rewriting the simple-looking Liouville action (2.1.2) in terms of φ or φ which has resulted in a complicated action (2.1.5)

or (2.1.7) with cubic and quartic (and higher) interactions that all involve derivatives.

  .1.34) It follows from (1.2.23) that ψ n and λ L n ≡ λ n scale as

  .1.37) It then straightforwardly follows from (2.1.36) together with (1.2.23) that all the integrals I r (t i ) as given in (2.1.30)-(2.1.33) satisfy

  The relation(2.1.39) that gives the area dependence of G A ζ will play a most important role below, since it is through this relation that the ln A A 0 terms appear in the logarithm of the partition function. Let us prove it. To begin with, G ζ (x) is dened in terms of the spectral ζ-function (without zero-mode) in (1.2.41) as

  .1.58) Since dx dy ≡ d 2 x g * (x) d 2 y g * (y) scales as A 2 and R 2 * ∼ 1 A 2 this term is obviously an A-independent constant, up to vanishing terms. All the other terms in (2.1.57) involve at least one K(t i , x, y) or d dt K(t, x, y).

								We
	write
		K(t, x, y) = K(t, x, y) -	e R * t A		(2.1.59)
	d dt	K(t, x, y) =	d dt	K(t, x, y) -	R * A	e R * t	(2.1.60)

  -ln(α 1 + α 2 ) + G 0 +O(ln AΛ 2 /Λ 2 ) . The relevant integrals that multiply terms involving either G ζ and/or ln µ 2 t 3 are all listed in the appendix A. Other integrals like d 2 ze -z 2 E 1 (z 2 t 3 t 1 )E 1 (z 2 t 3 t 2 ) only contribute nite constants that only depend on ratios of the α i . One gets

	2.1. TWO-LOOP LANDSCAPE			50
	(1.3.38).
				(2.1.62)
	ending up with an A-independent term. It is also straightforward to get
	e R * t 3 A	dx dy K(t 1 , x, y)K(t 2 , x, y) =	1 4π	ln AΛ 2 (2.1.63)
	We consider next R * dx dy K(t 1 , x, y) K(t 2 , x, y)K(t 3 , x, y). For this integral, one only
	needs to keep the leading term in the small t i expansion of K(t 3 , x, y) (1.3.45) and one
	can drop all terms O(t i ) or O(z j ) in the expansion of the K(t i , x, y) (1.3.45), or of	√ g *

t 3 A dx dy 4 K(t 1 , x, y)K(t 2 , x, y) + R * K(t 1 , x, y) K(t 2 , x, y)

+ c(α i ) + O (ln AΛ 2 ) 2 /Λ 2 .

(2.1.61)

The last two terms are easy to compute. Similarly as previously, one may indeed safely replace

R * A e R * t 3 dx dy K(t 1 , x, y) K(t 2 , x, y) = R * A dx dy ( G(x, y)) 2 + O (ln AΛ 2 ) 2 /Λ 2 ,

  expansion of K(t 1 , x, y). Doing the integrals over z results in

	2.1. TWO-LOOP LANDSCAPE				51
	short-distance
						and (1.3.45) to get
	g * (x) K(t 2 , x, y)K(t 3 , x, y) =	e -z 2 /4T (4π) 2 (t 2 + t 3 )T	1 +	7 6	R * (t 2 + t 3 ) + . . . , (2.1.66)
	where T = t 2 t 3 t 2 +t 3	. Thus, one has once again to keep the terms O(t 1 ) or O(z i z j ) in the

  AΛ 2 (ln AΛ 2 ) 2 and ∼ (ln AΛ 2 ) 2 are absent from W(2) [A] loops , as given in(2.1.75) and, hence, should also be absent from(2.1.81). It is easy to see that this implies

	2.1. TWO-LOOP LANDSCAPE	56
	terms ∼

  ,(2.1.84) where c ct (α i ) is some A-independent nite function of the α i . The terms in the second line are either ∼ A and can again be changed by adding an additional cosmological

constant counterterm, or irrelevant area-independent constants. If one adds (2.1.84) to (2.1.75), one gets

  .1.91) The counterterm coecients are now assumed to have the general form (2.1.80), without any further assumptions. Note that, while c φ and c R involve at most an ln AΛ 2 divergence, the coecient c m could have a Λ 2 ln AΛ 2 divergence. This means that, in order to compute the nite contributions, one should keep terms in K(t 1 , u, x) K(t 2 , x, v) that are O(1/Λ 2 ). (This is one of the reasons why all external propagators must also consistently be replaced by the regularized ones.) Thanks to(1.3.27), one directly reads:

  [START_REF] Hawking | Space-Time Foam[END_REF] , x, z) + ... As just explained, the terms + . . . contribute at the same order and cannot be dropped. Keeping only the terms that contribute to the leading singularity AΛ 2 ln AΛ2 2

									(2.2.24)
	When x, y and z are close, such that l 2 (x, y) ∼ l 2 (x, z) ∼ 1 Λ 2 , all terms in the expansion
	give contributions of the same order. One gets:
	dx dy dz K(t 1 , x, z) K(t 2 , x, z) K(t 3 , y, z) K(t 4 , y, z) K(t 5 , x, y)
	= dx dz K(t 1 , x, z) K(t 2 , x, z) K(t 3 + t 4 , z, z)
		× K(t 5 , x, z) -t 4 -	d K(t, x, z) dt	t=t 5	+ R * K(t 5 , x, z)
		+	t 2 4 2	d 2 K(t, x, z) dt 2	t=t 5	-2R *	d K(t, x, z) dt	t=t 5	+ R 2 * K(t 5 , x, z) + ...
	-	1 A	dx dy dz K(t 1 , x, z) K(t 2 , x, z) K(t 3 , y, z) K(t 5 , x, y) .	(2.2.25)
									, one has
	dx dy dz K(t 1 , x, z) K(t 2 , x, z) K(t 3 , y, z) K(t 4 , y, z) K(t 5 , x, y)
	= dx dz K(t 1 , x, z) K(t 2 , x, z) K(t 3 + t 4 , z, z)	∞ n=0	t n 4 n!	d n K(t, x, z) dt n	t=t 5	.
									(2.2.26)
	Furthermore,
	dx K(t 1 , x, z) K(t 2 , x, z)	d n K(t, x, z) dt n	t=t 5

  Since the glasses diagram has no divergence other than in AΛ 2 , the total contribution

	one could get from the counterterms to the three-loop partition function is given by
	summing (2.2.39), (2.2.43), (2.2.47), (2.2.48) and (2.2.49). Recalling AR * = 8π(1 -h),
	cf. (1.2.23), one gets:

  From eq. (2.2.1) and (3.0.11), we get the two-loop vertices (up to the rst order in ε) for the Liouville plus Mabuchi theory: a cubic vertex, two quartic vertices and a measure vertex. The measure vertex is unchanged (unless for the changes in K(t, x, x) due to the new a i 's), whereas the other vertices are modied such that:

	3.1. TWO-LOOP PARTITION FUNCTION	92
	3.1 Two-loop partition function

  Liouville contribution to γ str is vanishing on the torus and it is not surprising that our result(3.1.18) is simply proportional to β 2 , the coupling constant to the Mabuchi action. It is remarkable that in this (simplied) Liouville and Mabuchi theory on the torus, the requirement of niteness and regularization independence of the two-point function is once more sucient to ensure the niteness and regularization independence of the partition function (at least at two loops). Note that the counterterms we introduced are either local or non-local in 1/A but satisfying the strong locality condition that the global action including the counterterms, the Liouville and Mabuchi actions and the contribution from the measure does not contain any non-local term.

	3.2. THREE-LOOP DISCUSSION	97
	Then, the
		.1.19)

  2 ) vertices for our Liouville and Mabuchi theory. From eq. (2.2.1) and (3.0.11), one reads the vertices that appear in addition to (3.1.1):

	= -	(8πε) 2 κ 4	4 5	∆ * +	1 6	T	,			= -	(8πε) 2 κ 4	1 2	∆ * ,	= -	(8πε) 2 κ 4	2 9	∆ * ,
		= -	16(πε) 3/2 κ 3	∆ * +	1 5	T	,		= -	16(πε) 3/2 κ 3	2 3	∆ * ,	(3.2.1)
				=	(8πε) 2 κ 4	1 2	K(t, x, x) ,		=	16(πε) 3/2 κ 3	2 3	K(t, x, x) .

  AΛ 2 ln AΛ 2 + 2 AT + 4π c φ ln AΛ 2 , (c T T + c m ) K(t 1 , x, y) K(t 2 , x, y) + α 2 c φ AΛ 2 ln AΛ 2 + ln AΛ 2 8πAT c φ c T + 2 1 4 AT -4π c φ + 4AT (5 c φ -3 c T ) ln(α 1 + α 2 ) -ln α 1 + ( c φ -c T ) 1 + 2α 1 α 2 (α 1 + α 2 ) 2 ,where we used the values (3.1.16) for the counterterms c φ , c T and c m and kept only the parts involving one (or more) counterterm. Then, the gure-eight diagrams:+ 3 c φ K(t 1 , x, x) K (t 2 , x, y) K (t 3 , x, y)-5 c T T 2 K(t 1 , x, x) K (t 2 , x, y) K (t 3 , x, y) + α 2 c φ AΛ 2 ln AΛ 2 + (12 c T -20 c φ ) AT ln AΛ 2 2 + (12 c T -20 c φ ) AT 2 G 0 -ln α 1 -ln(α 1 + α 2 ) -24 AT + 4π c φ -80π T 2 c T dx dy K (t 1 , x, y) K (t 2 , x, y) ln AΛ 2 ,

	3.2. THREE-LOOP DISCUSSION	100
	and
	I	= -	(8πε) 2 κ 4	dx	3 2
						2 ,
	I = -	c φ 2	(8πε) 2 κ 4	dx K(t 1 , x, x) K(t 2 , x, y) K(t 3 , x, y)	(3.2.5)
	= c φ I = ε 2 κ 4 -2 α 1 1 2 (8πε) 2 κ 4 dx c φ (3.2.6)
	= α 1 I ε 2 κ 4 -10 α 1 + = (8πε) 2 κ 4 dx dy (-5 c + 8 3 2 c = ε 2 κ 4 12 1 α 1 1 + α 1 (3.2.7)

φ + 3 c T ) T K (t 1 , x, x) K (t 2 , x, y) K (t 3 , x, y) φ K(t 1 , x, x) -K (t 2 , x, y) d dt K (t, x

, y) t=t 3

+ K (t 2 , x, y) K (t 3 , x, y)

  2 . (3.2.8) with G 0 = 4π A 0 d 2 x g 0 (x) G A 0 ζ (x) -γ -ln A 0 µ 2 given in (2.1.54).Then, nally we have the two setting sun diagrams. As previously done for the other diagrams, the rst diagram can be written in terms of the regularized propagators and (thus in terms of the heat kernels) as:+ c φ K (t 1 , x, y) d K dt 3 K (t 3 , x, z) K (t 2 , x, y) K (t 4 , y, z) + 1 2 K (t 2 ,x, y) K (t 4 , y, z) ,

	I =	(8πε) 2 κ 4	dx dy dz 4 c (3.2.9)

T T 2 K (t 1 , x, y) K (t 2 , x, y) K (t 3 , x, z) K (t 4 , y, z) -c φ K(t 3 , x, z) + c T T K(t 3 , x, z) K(t 4 , y, z) × K (t 1 , x, y) K (t 2 , x, y) -K (t 1 , x, y) d K dt 2 (t 2 , x, y) + T K(t 1 , x, y) K(t 2 , x, y) C K (t 3 , x, z) K (t 4 , y, z) -c φ K (t 3 , x, z) d K dt 4 (t 4 , y, z) + K (t 1 , x, y) K (t 3 , x, z) 4 C T K (t 2 ,

x, y) K (t 4 , y, z) -c φ K (t 2 , x, y) K (t 4 , y, z)

  + α 2 + α 3 c φ AΛ 2 ln AΛ 2 + 30 c φ -14 c T AT ln AΛ 2 2 + ln AΛ 2 80 π T 2 c T dx dy K (t 1 , x, y) K (t 2 , x, y) + 2 G 0 30 c φ -14 c T AT -4AT c φ -c T 1 + 2α 1 α 2 (α 1 + α 2 ) 2 -(10 c φ -4 c T ) ln(α 1 + α 2 + α 3 ) -(5 c φ -3 c T ) ln(α 1 + α 2 ) + 4 c φ 36π + AT

	3.2. THREE-LOOP DISCUSSION			101
	Then, we get
	I =	ε 2 κ 4 -8	1 α 1 + α 2	+	2 α 1 8 3	+	4 α 2 1 (α 1 + α 2 + α 3 ) 2	,
									(3.2.10)
	and
	I =	(8πε) 2 κ 4	dx dy	-f φ T +	3 2	(f T

T + f m ) K (t 1 , x, y) K (t 2 , x, y) K (t 3 , x, y) + f φ K(t 1 , x, y) -1 2 K (t 2 , x, y) d dt K (t, x, y)

  12 α 1 f (1) m + 12 c φ -12 f M (a, b) = -2 T dx dy K (t 1 , a, x) K (t 2 , x, y) K (t 3 , x, y) K (t 4 , y, b) (t 1 , a, x) K (t 2 , b, x) K (t 3 , a, x) + K (t 3 , b, x) , x, y) K (t 2 , x, y) K (t 3 , a, x) + K (t 3 , y, b) + 2 K (t 1 , a, x) K (t 2 , x, y) K (t 3 , y, b) -2 T dx dy K(t 1 , a, x) K(t 2 , x, b) K(t 3 , x, y) G ζ (y) + dx K(t 1 , a, x) + K(t 2 , x, b) × K(t 3 , a, b) G ζ (x) -1 A dy K(t 3 , x, y) G ζ (y) .

	3.2. THREE-LOOP DISCUSSION	103
	and
	H + dx K -1 2 1 A dx dy	1 2	K (t 1
			(1) φ	= ρ Liouville 5	(3.2.16)

  x) (t 1 , t 2 , t 5 ; x) , J (2) = -dx K 3,4 (x, x) (t 1 , t 2 , t 5 ; x) , (t 4 + t 5 , t 3 , t 6 ; x) , + α 2 D 0 (t 4 + t 5 , t 3 , t 6 ; x) + J 1,2 4,5 , J (3) = dx K 1,2 (x, x) + α 5 D 0 (t 1 + t 2 , t 3 , t 6 ; x) ,

					∞ n=0 C 0,n ∞ t n 4 n! n=0 t n 4 n! B ∞ n=0	t n+1 4 (n + 1)!	C 1,n (t 1 , t 2 , t 5 ; x) ,
	J	(4) = J	(2) + dx	1 4π	Λ 2 α 3 + α 4	B 0 (t 1 , t 2 , t 5 ; x) ,
	J	(1) = dx K 3,4 (x, x)	∞ n=0	t n 4 n!	E 0,n (C.3)
	J	(2) = J	(1) + dx	1 4π	Λ 2 α 3 + α 4 D ∞ n=0 t n 4 n! D n+1 (t 1 , t 2 , t 5 ; x) ,
	J E n,m J (1) = dx K 1,2 (x, x) n,m≥0 t n 2 n! t m 5 m! (2) = J 1 Λ 2 (1) + 2 dx 4π α 1 n,m≥0 t n+1 2 (n + 1)! t m 5 m!	E n,m+1 (t 4 + t 5 , t 3 , t 6 ; x) + J	(3) ,
	J	(4) = J	(3) + dx	1 4π	α 4	Λ 2

n+1 (t 1 , t 2 , t 5 ; x) , J (3) = dx K 3,4 (x, x) -B 1 (t 1 , t 2 , t 5 ; x) + 0 (t 1 , t 2 , t 5 ; x) , J

(3)

= -dx K 3,4 (x, x)

As previously stated, the zero mode of φ is unphysical and has to be excluded. Thus we will only consider quantities with the zero mode excluded. This is indicated by a tilde such as in G(x, y).

Indeed, in this thesis, the operator Dg will be either A∆ * ∆ * -R * κ 2 32π A∆ * + β 2 or of the form ∆ * -R * + m 2 . In any case, if the zero mode of the Laplacian is excluded then λ D n > 0 for n > 0.

1.4. BACKGROUND INDEPENDENCE

The analogue of momentum space is the mode decomposition with respect to the eigenfunctions ψr(x) of ∆, replacing the plane waves e ipx . An important property is momentum conservation that follows from dxe i(p 1 +p 2 +p 3 ) ∼ δ(p1 + p2 + p3). On a curved manifold one then needs the Crst = dxψrψsψt, etc. While on the round sphere, where the ψr are the spherical harmonics, the Crst are the well-known Clebch-Gordan coecients, on the higher genus surfaces much less is known about the Crst and things are much more complicated.

Contrary to what we did at tree-level where we used G and K, resp Gϕ,Λ to denote the Green's function and its regularized version, here we just write G[START_REF] Penrose | Gravitational collapse and space-time singularities[END_REF] since we will always deal with the regularized two-point function.

Acknowledgements

My rst acknowledgements go to my supervisor Adel Bilal who rst introduced me to theoretical physics and then made me discover 2D quantum gravity and Liouville theories.

composed of eleven, thirteen, six, six and eighteen subdiagrams, from left to right and from top to bottom.

The measure and counterterm vertices contribute to fourteen diagrams. They may be classied according to the corresponding two-loop terminology: the gure-eight, the setting sun, the glasses and the measure diagrams. At the three-loop order, there are three gure-eight-like diagrams, three setting sun-like diagrams, ve glasses-like diagrams and nally three measure-like diagrams.

THREE-LOOP INVESTIGATION 71

From left to right, both the gure-eight and setting sun diagrams have respectively one, four and ve contributions. Concerning the glasses diagrams: the upper diagram has two contributions, and, from left to right, the diagrams in the second line have respectively four and three contributions, and the diagrams involving the counterterms six and four respectively. Finally, the measure diagram on the right gets two contributions whereas both diagrams involving the measure vertex have no other subdiagram.

Divergences of the partition function

Expected divergence structure All vacuum diagrams are dimensionless and can depend on A and Λ only through the dimensionless combination AΛ 2 . They contribute to various divergences to the partition function. Standard power counting shows that any loop-diagram has a supercial degree of divergence equal to 2. This means that divergences such as AΛ 2 ln AΛ 2 # are allowed.

To have a more precise idea of the leading divergence, consider a diagram with I internal lines and V vertices. Each internal line, that is to say each regularized propagator K, gives a logarithmic divergence, according to (2.1.47). Besides, each vertex, carrying a Laplacian, transforms such a propagator into the corresponding heat kernel K thanks to (1.3.24), leading to a quadratic divergence (1.3.47). Each vertex also implies an integration over the manifold. Due to the term e -l 2 /4t in the heat kernel (1.3.45), every integration contributes a factor t i ∼ 1 Λ 2 at most. (The subtraction of the zero-mode terms ∼ e R * t A does not change the nal conclusion.) For the last integration, however, all quantities to be integrated only depend on one point, hence no Gaussian integration can be performed and one just gets a factor of A. Putting everything together, the leading singularity of this L-loop vacuum diagram is

= ln AΛ 2 L-1 AΛ 2

(2.2.7) since I -V = L -1 for every diagram. Therefore, the leading divergence at three loops is AΛ 2 ln AΛ 2 2 . Note that the vertices not only contain a Laplacian but also terms

For V > 1 this is vanishing. This means that the subleading divergence with the largest power of logarithms is ln AΛ 2 L . Consequently, the expected divergences in W (3) [A] are

(2.2.9)

THREE-LOOP INVESTIGATION

where we did not include neither the constant terms nor those proportional to AΛ 2 . At this point, we should make a few remarks. First, the leading (non-vanishing) divergence in eq. (3.1.6),

is the same as in the pure Liouville theory, as it will be the case for the leading three-loop divergence in section 3.2. This is not surprising since the Mabuchi action (or its loop expansion), missing one Laplacian in comparison to the Liouville action, only modies the subleading terms, through the coecient T = 32π A β 2 κ 2 . We are then once more led to introduce counterterms in order to make this partition function nite and regulator independent.

Secondly, the (leading) logarithmic divergence ln AΛ again miraculously" cancel in the end was all but obvious in the beginning.

Counterterm contributions

We choose to only introduce local or non-local measure-like counterterms in 1/A. Thus, we consider the following action for the counterterms:

φ , c T = c

(1)

A .

(3.1.9)

In order to simplify the notation, we do not write the subscript L + M over the counterterms. To determine these counterterms, as previously we compute the full one-loop (order ε) two-point function G (2) (u, v) (at non coinciding points) and we require it to be both nite and regularization independent.

One-loop two-point function

As previously, we have to compute ve diagrams:

. action to cancel with the counterterms. We obtain an expression for the string susceptibility which depends on one free parameter at two loops and which is compatible with the KPZ formula. Although one of the counterterms involved in γ str is xed to its KPZ value by the strong locality condition, there seems to be no reason in our formalism to

x the remaining counterterm to its KPZ value.

Then, a three-loop computation has been performed. In particular, the leading divergence is non-vanishing and the renormalization procedure is once more required. Very similarly to the two-loop order, the only divergence which the introduction of local or measure-like counterterms could not cancel is vanishing. This is crucial for the consistency of our theory. Moreover, niteness of the n-point function automatically xes the counterterm coecients of the diverging part of the partition function. If one requires in addition its regularization independence, and if one imposes the strong locality condition, one ends up with four free independent constants in γ str , that is to say three more than at two loops.

As emphasized previously, the fact that only local counterterms are required to guarantee the niteness of the theory is due to the fact that some divergences vanish. This is far from obvious and may be due to a hidden symmetry of the theory. This symmetry is most probably also the reason why the niteness of the two-point function seems to automatically imply the niteness of the partition function. This has been shown at two loops and is hinted at three loops. Unravelling this hidden symmetry is thus the key to ensure renormalizability at all orders.

Appendix A

Some useful integrals

Here we list various integrals of the form

where d 2 z is the at measure: any non-trivial expansion of √ g is included in f . Then

One has also

and

where the E n are the exponential integral functions dened in (1.3.41) :

Of course, insertion of any odd number of components of z into any of these integrals gives a vanishing result, while insertions of an even number can be replaced according to the usual rules

and

One-and two-loop contributions to the two-point function

In this appendix are given the details of the computation of the one-loop and two-loop contributions (order 1/κ 2 and 1/κ 4 respectively) to the two-point function G (2) .

B.1 One-loop contributions to the two-point function

In this section we write computational details of the one-loop contributions to the two-

Recall that G (2) (u, v) always includes the two external propagators that, by consistency, are also regularized, i.e. replaced by K. Then, for the diagram i , due to the many ways the derivatives in the cubic vertex can act, one gets many dierent contributions.

They yield

where K K K K stands for K(t 1 , u, x) K(t 2 , x, y) K(t 3 , x, y) K(t 4 , y, v) etc. We now evaluate this for u = v. More precisely, since we work at nite cuto, we do not want 2 (u, v) to be as small as 1 Λ 2 and require 2 (u, v)

) is exponentially small and can always be dropped. Also

, y, u) we may replace the K by G since these integrals have nite limits as Λ → ∞. (The logarithmic short-distance singularity (ln µ 2 (y -u) 2 ) n is integrable for any integer n.)

Denoting by + . . . terms that are either O(ln AΛ 2 /Λ 2 ) or exponentially small as just explained, we nd

Combining everything, we get

where K K ≡ K(t 3 , x, y) K(t 4 , y, y) and similarly for the other terms. This simplies considerably since K(t i , y, y) does not depend on y and in K(t j , y, y)

The contribution to the Green's function then is

where KB K ≡ K(t 1 , u, x)B(x) K(t 2 , x, v) and similarly for the other terms. Inserting the expression for B and evaluating the integrals gives One nds, using the fact that K(t 2 , x, x) -G ζ (x) does not depend on x,

The three remaining terms in (B. 1.16) are straightforward to evaluate:

up to vanishing terms. Combining everything gives

The total one-loop contributions to G (2) (u, v) for u = v is the sum of the contributions computed here and of the one from the counterterms (2.1.93). It is given and discussed in section 2.1.4.

B.2 Two-loop contributions to the two-point function

Here we give the diverging contributions of the counterterm to the two-loop two-point function 1 G (3) (a, b) per diagrams 2 . We only consider the so far not xed counterterms, i.e. the genuine three-loop counterterms and the two-loop counterterms c φ and c R .

The diagrams contributing to G

ct (a, b) are the following:

. Since the one-loop two-point function G (2) (a, b) has already been renormalized (which led us to x the two-loop counterterms except for c φ and c R ), the summed contribution of , ,

should only contribute to the nite part of G (3) (a, b) and thus we will not give their contributions. The diagrams and do not have a diverging part and thus will also not be considered here. Then, we write

1 We consider the two-point function at non coinciding point i.e. a = b.

2 The computations are quite lengthy but straightforward and thus left to the motivated reader.

B.2. TWO-LOOP CONTRIBUTIONS TO THE TWO-POINT FUNCTION 115

with the understanding that we will only give the diverging contributions. We have rst for the cubic counterterms

Then, we also have for the cubic counterterms

B.2. TWO-LOOP CONTRIBUTIONS TO THE TWO-POINT FUNCTION 116

The unique diagram built from the quartic counterterms gives

φ + 8 q

(1) φ

For the c quadratic counterterms, we have

The two remaining diagrams only involves c φ and c R . They contribute . The details of the computation of these series are described hereafter. For the sake of brevity, notational short-cuts are dened: dx = d 2 x g * (x), K i (x, y) = K(t i , x, y) and K i,j (x, y) = K(t i + t j , x, y).

Among the series of diverging contributions appearing in these four diagrams, some may be straightforwardly computed by Taylor expanding the terms, as was done in section 2.2.2. These integrals are hereafter noted by J. Namely, they are:

= dx dy dz K 1 (z, z) K 2 (x, z) K 3 (x, y) K 4 (x, y) K 5 (y, z) ,

= dx dy dz dw K 1 (x, y) K 2 (x, y) K 3 (y, z) K 4 (z, w) K 5 (z, w) K 6 (x, w) ,

Note that J

was already computed in section 2.2.2. It is useful to dene: 117 where all the terms in O(AΛ 2 ln AΛ 2 ) are discarded. The fact that the α i (and thus t i ) are dummy variables that can be renamed and are symmetrized, has been used to simplify the writings of J

(2)

and J

. Finally, the term J 1,2 4,5 in J

is the term proportional to Λ 4 dened in (2.2.11). All contributions ∼ Λ 4 have been discussed in section 3.2 and are summarized in Tab. 2.2. At present we are only interested in the other types of divergences and thus we will simply drop the term J 1,2 4,5 in the following. We conjecture:

α a and α b being dummy variables, this may be rewritten as

These relations are veried by the above expressions, before considering the symmetries between the α i . Putting everything together and remembering once more that the α i are dummy variables, one gets: 1) ,

up to subleading divergences. One encounters also integrals such as

whose explicit computation requires to Taylor expand a product of two K or K. We denote such integrals by L. Integrating over z around x through the exponential term in K 4 (x, z), see (1.3.45), leads us to Taylor expand K 1 (z, z) d dt 5 K 5 (y, z) in (z -x) around x. After integration, one gets terms such as:

with r + s even. If s is odd, then the function K 2 (x, y) K 3 (x, y)∂ x j 1 . . . ∂ x js -d dt 5 K 5 (x, y) is odd and performing the integral over y kills the contribution: r and s have to be even. Since K(t, x, x) depends on x only through GA 0 ζ (x), see (2.1.47), for r ≥ 2, ∂ x i 1 . . . ∂ x ir K 1 (x, x) does not contribute to the leading divergence by neither a factor ln AΛ 2 nor Λ 2 . The diverging contributions may thus only come from the integral over y. However, applying s derivatives on d dt 5 K 5 (x, y) for any even s leads to terms similar to (-1)

1+ s 2 K 5 (x, y). Integrating over y one gets B 1+ s 2 (t 2 , t 3 , t 5 ; x) which only produces one of the two ln AΛ 2 of the leading divergence. The only terms contributing to AΛ 2 ln AΛ 2 2 are thus the terms with r = 0 and s even.

Thus, up to subleading divergences, the previous integral gives: 2) .

(C.13) 121

Similarly, one may compute

Since at least two Ks are needed to obtain the two ln AΛ 2 of the leading divergence, it is easy to compute

Indeed, integrating over w through K 3 (z, w) requires to Taylor expand K 1 (x, w) K 2 (y, w).

The only term in this expansion keeping the structure of the Ks and thus contributing to the leading divergence is the rst one: K 1 (x, z) K 2 (y, z). Thus,

Similarly, in

if any partial derivative acts on one of the two Ks through the Taylor expansion, the ln AΛ 2 2 are lost. Thus, 5) = dx dy dz dw -

with the same reasoning. In these three integrals, integrating over w implies to Taylor expand a product of two Ks. Since there are only three Ks in these integrals, this means that, if one wants to extract the leading divergence with two ln AΛ 2 , one shall only keep in the Taylor expansion the terms with the derivatives acting on one K. Up to subleading terms, one has

Integrating over x in L

L

(3) = 2J (1) , L (4) = J (1) + 2J (1) , L (5) = J (1) + J (1) .

The same idea is used to compute L

(3) = dx dy dz dw K 1 (x, z) K 2 (x, z) d dt 3 K 3 (y, z) K 4 (y, w) K 5 (y, w) d dt 6 K 6 (x, w) , L (6) = dx dy dz dw K 1 (x, y) K 2 (x, z) d dt 3 K 3 (y, z) K 4 (z, w) K 5 (y, w) d dt 6 K 6 (x, w) ,

the requirement is then to leave two of the four Ks without derivatives. Among such terms, one gets also subleading terms not considered here. In the end, one obtains: 6) = 4J (1) .

(C.24)

Abstract

Nowadays, two-dimensional quantum gravity can be studied in two different approaches, one involving discrete theories (triangulation, matrix model...), the other continuous ones, mainly based on the so called Liouville action which universally describes the coupling of any conformal field theory to gravity. While the Liouville action is relatively well understood, the appropriate functional integral measure is however rather complicated. Nevertheless, a formula for the area dependence of the quantum gravity partition function in the presence of conformal matter has been obtained, under the simplifying assumption of a free-field measure. Notwithstanding its non-rigorous derivation, this formula, often referred to as the KPZ formula, has since been verified in many instances and has scored many successes.

Recent developments of efficient multiloop regularization methods on curved spacetimes opened the way for a precise and welldefined perturbative computation of the fixedarea partition function in the Kähler formalism. In this work, a first-principles computation of the fixed-area partition function in the Liouville theory is performed, up to three loops. Among other things, the role of the non-trivial quantum gravity integration measure is highlighted. Renormalization is required and may be interpreted as a renormalization of the integration measure. This leads to a finite and regularization-independent result at two loops, that is more general than the KPZ result, although compatible. Finiteness and regularization-independence seem also possible at three loops. These results are generalized to the coupling to non-conformal matter on the torus.