Cercl É Baptiste

AND Émi Rhodes

Vincent Vargas

PROBABILISTIC CONSTRUCTION OF SIMPLY-LACED TODA CONFORMAL FIELD THEORIES

Following the 1984 seminal work of Belavin, Polyakov and Zamolodchikov on two-dimensional conformal field theories, Toda conformal field theories were introduced in the physics literature as a family of two-dimensional conformal field theories that enjoy, in addition to conformal symmetry, an extended level of symmetry usually referred to as W-symmetry or higher-spin symmetry. More precisely Toda conformal field theories provide a natural way to associate to a finite-dimensional simple and simply-laced Lie algebra a conformal field theory for which the algebra of symmetry contains the Virasoro algebra. In this document we use the path integral formulation of these models to provide a rigorous mathematical construction of simply-laced Toda conformal field theories based on probability theory. By doing so we recover expected properties of the theory such as the Weyl anomaly formula with respect to the change of background metric by a conformal factor and the existence of Seiberg bounds for the correlation functions.

1. Introduction 1.1. Toda Conformal Field Theories in the physics literature. In 1981, Polyakov presented in a pioneer work [START_REF] Polyakov | Quantum Geometry of bosonic strings[END_REF] a canonical way of defining the notion of random surface, usually called Liouville conformal field theory (LCFT hereafter), which is now considered to be an essential feature in the understanding of non-critical string theory and two-dimensional quantum gravity. A few years later Belavin, Polyakov and Zamolodchikov (BPZ) presented in their 1984 seminal work [START_REF] Belavin | Infinite conformal symmetry in twodimensional quantum field theory[END_REF] a systematic procedure to solve models which like LCFT possess certain conformal symmetries, now referred to as two-dimensional conformal field theories (CFTs in the sequel). The main input of their method was to exploit the constraints imposed by conformal symmetry through the study of the algebra of its generators, the Virasoro algebra, which in turn completely determines (up to the so-called structure constants) the main quantities of interest, namely the correlation functions of certain special operators, thanks to a recursive procedure dubbed the conformal bootstrap.

A natural question which appeared shortly after these developments was: what happens when the algebra of symmetry stricly contains the Virasoro algebra? In other words, do the same techniques apply when Virasoro symmetry is extended to feature an additional level of symmetry? Certain extensions of the Virasoro algebra, called W -algebras, have been first studied by Zamolodchikov in his work [36] where was presented the notion of higherspin symmetry, and following this work two-dimensional CFTs having this extended level of symmetry appeared in the physics literature in [START_REF] Fateev | The Models of Two-Dimensional Conformal Quantum Field Theory with Z(n) Symmetry[END_REF][START_REF] Fateev | Conformal quantum field theory models in two dimensions having Z3 symmetry[END_REF]. In addition to being an object of interest in itself, the study of a CFT having what is called W -symmetry is crucial in the understanding of W -strings, W -gravity theories or certain statistical physics systems (the articles [START_REF] Fateev | Parafermionic Currents in the Two-Dimensional Conformal Quantum Field Theory and Selfdual Critical Points in Z(n) Invariant Statistical Systems[END_REF] and [START_REF] Jimbo | Solvable lattice models related to the vector representation of classical simple Lie algebras[END_REF] provide explicit instances of such systems) and can be applied to the understanding of some Wess-Zumino-Novikov-Witten models (such a link is explained in [START_REF] Fehér | On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories[END_REF]). From the representation theory viewpoint the study of W-algebras has proved to be a seminal topic with numerous applications ranging from integrable hierarchies to the geometric Langlands program (see [START_REF] Arakawa | Representation theory of W-algebras and Higgs branch conjecture[END_REF] and the references therein).

Toda Conformal Field Theories (TCFTs hereafter) may be thought of as realizations of these algebras of symmetry since they are assumed to provide highest-weight representations of W -algebras. In this context, the primary fields, defined as random fields on a Riemann surface Σ with special conformal covariance properties, are called Vertex Operators and their correlations are defined as an average of the product of the fields (taken at different points of the surface Σ) with respect to the law of a random map called the Toda field. This construction can be made somehow explicit thanks to the fundamental fact that TCFTs admit a path integral formulation based on the following Lagrangian (1.1) S T,g (ϕ, g)

:= 1 4π Σ ∂ g ϕ(x), ∂ g ϕ(x) g + R g Q, ϕ(x) + 4π r i=1
µ i e γ e i ,ϕ(x) v g (dx),

where the Toda field ϕ is a map Σ → h and:

• g is a Riemannian metric on a Riemann surface Σ with associated scalar curvature R g , gradient ∂ g and volume form v g , • h is the Cartan subalgebra of some finite-dimensional simple and simply-laced complex Lie algebra g, equipped with its standard scalar product •, • and norm | • | (see Subsection 2.1.2),

• the e i , 1 i r are the simple roots of the Lie algebra g relative to h, • •, • g is the scalar product associated to the tangent space of h-valued functions defined on Σ, • the constants µ i (1 i r) are positive and are dubbed the cosmological constants, • γ > 0 is the coupling constant, • Q is the h-valued background charge.

In order to ensure conformal symmetry, the background charge is related to the coupling constant via the relation Q := (γ + 2 γ )ρ where ρ is the Weyl vector associated to the Cartan subalgebra h. Let us emphasize that one recovers LCFT when g is the Lie Algebra sl 2 (of 2 × 2 complex matrices with vanishing trace, in which case r = 1) and that the convention for γ in this paper differs by a scaling factor of √ 2 from the standard convention for LCFT in the probabilistic literature. The next section 2 is devoted to providing more details on these notations. At this stage it is worth noticing that this action functional has a relevant geometrical meaning in the context of W -geometry, introduced by Gervais and Matsuo in [START_REF] Gervais | W geometries[END_REF]. Indeed, in the semi-classical limit γ → 0 such that for all i, µ i = Λ γ 2 with fixed Λ > 0, the function which minimizes the Toda action (rescaled by γ) converges towards the solution u : Σ → h (provided that it exists and is unique) of the Toda equation :

(1.2) 2∆ g u = R g ρ + 4πΛ r i=1
e i e e i ,u on Σ.

This means that, in the quantum theory, the Toda field (rescaled by γ) will have a tendency to remain close to the (classical) solution of this h-valued equation. It is explained in [START_REF] Gervais | W geometries[END_REF] that when Σ is the two-dimensional sphere S 2 , sl n Toda equations (where one adds appropriate conical singularities) can in some sense be interpreted as compatibility equations for a meromorphic embedding of a two-sphere into a complex projective plane CP n , and therefore establishes a correspondence between solutions of the Toda equation (1.2) and certain meromorphic embeddings from CP 1 to CP n , a problem which somehow provides a generalization of the celebrated uniformisation of Riemann surfaces.

In the context of the quantum theory, the Toda field ϕ should be understood as a random map from a Riemannian surface (Σ, g) to h whose law can be defined via the expression (here F is some real-valued functional on a space of maps Σ → h):

(1.3) E g γ,µ [F (ϕ)] := F T,g 1 T,g
where the quantities that appear in this definition are defined by the (formal) path integral (1.4) F T,g := F (X)e -S T,g (X,g) DX and DX refers to a "uniform measure" on the space of square integrable h-valued maps defined on Σ; the normalization factor 1 T,g corresponds to the partition function e -S T,g (X,g) DX of TCFTs. In the sequel, we will focus on the case where Σ is the Riemann sphere S 2 (though the present framework can be extended to other topologies). In this case, the partition function is not defined (i.e. infinite) and one can only define quantities with insertion of a certain number of Vertex Operators, whose parameters obey the Seiberg bounds [START_REF] Seiberg | Notes on Quantum Liouville Theory and Quantum Gravity[END_REF] in order to ensure existence of the corresponding correlation function (see next subsection).

1.2. A probabilistic construction. Though an algebraic approach to CFTs was developped shortly after the BPZ paper (see the notion of Vertex Operator Algebra [START_REF] Borcherds | Vertex algebras, Kac-Moody algebras, and the Monster[END_REF][START_REF] Frenkel | Vertex Operator Algebras and the Monster[END_REF]), a probabilistic approach to conformal invariance was only developed recently following the introduction by Schramm [START_REF] Schramm | Scaling limits of loop-erased random walks and uniform spanning trees [mr1776084[END_REF] of random curves, called Schramm-Loewner Evolutions (SLEs), which describe (conjecturally at least) the interfaces of critical models of statistical physics (such as percolation or the Ising model). More recently, there has been a huge effort in probability theory to make sense of LCFT within the realm of random conformal geometry and the scaling limit of random planar maps (see [START_REF] Gall | Uniqueness and universality of the Brownian map[END_REF][START_REF] Miermont | The Brownian map is the scaling limit of uniform random plane quadrangulations[END_REF][START_REF] Gwynne | Existence and uniqueness of the Liouville quantum gravity metric for γ ∈ (0, 2)[END_REF][START_REF] Duplantier | Liouville quantum gravity as a mating of trees[END_REF][START_REF] Ding | Tightness of Liouville first passage percolation for γ ∈ (0, 2)[END_REF][START_REF] Dubédat | Weak LQG metrics and Liouville first passage percolation[END_REF]). Another approach, based on the path integral formulation of LCFT in the physics literature, was developed in [START_REF] David | Liouville Quantum Gravity on the Riemann Sphere[END_REF][START_REF] David | Liouville quantum gravity on complex tori[END_REF][START_REF] Huang | Liouville quantum gravity on the unit disk[END_REF][START_REF] Guillarmou | Polyakov's formulation of 2d bosonic string theory[END_REF] to give a rigorous probabilistic construction of the correlation functions of LCFT. This construction initiated a program [START_REF] Kupiainen | Local Conformal Structure of Liouville Quantum Gravity[END_REF] to lay the mathematical foundations of the conformal bootstrap procedure envisioned in physics by BPZ, namely that one can express the LCFT correlation functions in terms of representation theoretical special functions. The building blocks are an explicit formula for the three point correlation functions (or equivalently the structure constants) and a recursive procedure for the higher correlations. The explicit formula for the three-point structure constants discovered in the physics literature, the celebrated DOZZ formula, was recovered probabilistically in [START_REF] Kupiainen | Integrability of Liouville theory: proof of the DOZZ formula[END_REF] and a probabilistic justification of the conformal bootstrap formalism for the higher order correlations was provided recently in [START_REF] Guillarmou | Conformal bootstrap in Liouville Theory[END_REF].

Building on [START_REF] David | Liouville Quantum Gravity on the Riemann Sphere[END_REF], our goal here is to provide a probabilistic definition of TCFTs in the case where we consider the underlying Lie algebra to be a finite-dimensional simple and simplylaced complex Lie algebra, and when the manifold on which the theory is constructed is the (Riemann) sphere. To do so we follow the ideas developed in the case of LCFT (which corresponds to the sl 2 case) in [START_REF] David | Liouville Quantum Gravity on the Riemann Sphere[END_REF] and interpret the path integral formulation of the theory as a formal way of defining a measure on some functional space. More precisely we interpret the mapping defined via the path integral (1.4) as a measure F → F T,g on the Sobolev space with negative index H -1 (S 2 → h, g) (which we define in (2.4)); in order to construct this measure we introduce a probabilistic framework which involves two objects: the Gaussian Free Field (GFF) and the exponential of the GFF called Gaussian Multiplicative Chaos (GMC).

The presence of the GFF is related to the presence of the square gradient term in the Toda field action and has proven to be particularly relevant in the context of constructive conformal field theory. But as opposed to LCFT where only one GFF is involved in the construction, in TCFTs we have to consider several GFFs that are coupled in a way that is prescribed by the underlying structure of the Lie algebra. For this coupling to be mathematically meaningful we need to consider simple Lie algebras that are simply-laced, that is for which the underlying Cartan matrix (which is used as a covariance matrix in our construction) is symmetric. Under this assumption, the fields of TCFTs are well-defined but non-regular since they exist only in the sense of Schwartz distributions; therefore the exponential terms that appear in (1.1) are not well-defined objects. However, GMC theory provides a way of making sense of these terms as (random) Radon measures.

This interpretation allows to construct a regularized partition function by taking F = 1 in (1.4); however and similarly to the existence of Seiberg bounds in LCFT [START_REF] Seiberg | Notes on Quantum Liouville Theory and Quantum Gravity[END_REF] this partition function will not converge in relation with an obstruction of geometrical nature: the Gauss-Bonnet theorem entails that classical Toda equations (i.e the equations of motion associated to this action) cannot admit solutions on the Riemann sphere. This difficulty can be overcome by looking at special functionals F that admit Vertex Operators as factors; by adding these extra terms to the measure the partition function becomes the correlation function of Vertex Operators and is predicted to exist as long as some conditions on these operators are satisfied. To define these Vertex Operators one relies on a regularization of the Toda field and introduces the regularized Vertex Operator V α,ε (z), which is expressed for z on the Riemann sphere in terms of the Toda field ϕ and a weight α ∈ h: up to constant terms, V α,ε (z) is defined as ε |α| 2 /2 e α,ϕε where ϕ ε is the field ϕ smoothed up at scale ε (its definition will be made precise in Subsection 2.2.3). Our main result provides a necessary and sufficient condition that ensures the existence of the correlation functions (defined as limits of N k=1 V α k ,ε (z k ) T,g when the cut-off ε is sent to 0). Moreover, the correlations are indeed conformally covariant as predicted by CFT: Theorem 1.1. Let g be a finite-dimensional simple and simply-laced complex Lie algebra and assume that γ ∈ (0, √ 2). If g is any Riemannian metric in the conformal class of the standard round metric on the sphere ĝ, let N k=1 V α k ,ε (z k ) T,g be the regularized correlation function of the g-Toda theory. Then:

1. (Seiberg bounds): The limit

N k=1 V α k (z k ) T,g := lim ε→0 N k=1 V α k ,ε (z k ) T,g
exists and is non trivial if and only if the two following conditions hold for all i = 1, . . . , r:

• N k=1
α k -2Q, ω i > 0 where the (ω i ) 1 i r are the fundamental weights of h,

• for all 1 k N, α k , e i < Q, e i = γ + 2 γ . 2. (Conformal covariance): For any Möbius transform of the plane ψ

N k=1 V α k (ψ(z k )) T,g = N k=1 |ψ ′ (z k )| -2∆α k N k=1 V α k (z k ) T,g .
where the conformal weights are given by ∆ α j := α j 2 , Q -

α j 2 .

(Weyl anomaly):

For appropriate ϕ (more precisely ϕ ∈ C1 (R 2 ): see notations in Section 2) then

N k=1 V α k (z k ) T,e ϕ ĝ = e c T 96π S L (ϕ,ĝ) N k=1 V α k (z k ) T,ĝ
where S L is the Liouville functional (with vanishing cosmological constant)

S L (ϕ, ĝ) := S 2 |∂ ĝϕ| 2 ĝ + 2R ĝϕ dv ĝ,
and the central charge c T is given by c T = r + 6|Q| 2 .

The value of the central charge can be described explicitly in terms of the coupling constant γ and the underlying Lie algebra. Indeed, by the classification of finite-dimensional and simple complex Lie algebra we know that g, which is simply-laced, is necessarily isomorphic to sl n for n 2, so 2n for n 4, E 6 , E 7 or E 8 for which the central charge is explicit: see (2.18) or (3.6). Our main statement can be understood as a rigorous definition of the correlation functions of TCFTs, but also of the law of the Toda field ϕ (when we have fixed marked points (z, α) = (z k , α k ) 1 k N that satisfy the Seiberg bounds) by setting

(1.5) E (z,α) [F (ϕ)] := F N i=1 V α i (z i ) T,g N i=1 V α i (z i ) T,g
where F is any bounded and continuous functional on H -1 (S 2 → h, g) and

F N i=1 V α i (z i ) T,g is the limit of F N k=1 V α k ,ε (z k ) T,g
when the cut-off ε goes to 0 (this more general case can be handled similarly to the case F = 1).

Remark 1.2. The above construction can be generalized when one considers as underlying Lie algebra any simply-laced, semisimple and complex Lie algebra. Indeed, from the classification of semisimple Lie algebras, such a Lie algebra can be written as a direct sum of simple Lie algebras g = ⊕ p k=1 g k . Moreover a general property of TCFTs (which can be derived from the form of the Toda field action) is that for A, B two semisimple and simply-laced Lie algebras

(1.6) N k=1 V (α k ,β k ) (z k ) A⊕B T,g = N k=1 V α k (z k ) A T,g N k=1 V β k (z k ) B T,g ,
where with the notation g T,g we have stressed the dependence on the Lie Algebra g. This provides a way of constructing correlation functions for general finite-dimensional simplylaced and semisimple complex Lie algebras. The latter equation also implies that the central charges add up, in the sense that

(1.7) c T,A⊕B = c T,A + c T,B
where here again the notation c T,g stresses the dependence on the Lie Algebra g.

Background and notations

2.1. Some reminders on conformal geometry and Lie algebras.

2.1.1. Conformal geometry on the Riemann sphere. The sphere S 2 can be mapped by stereographic projection to the (compactified) plane (i.e. the Riemann sphere) which we view both as R 2 ∪ {∞} and C ∪ {∞}. We will work under this more convenient framework in the sequel.

Metrics on the Riemann sphere.

We will consider differentiable conformal metrics on the two-dimensional sphere S 2 ; they can can be identified via stereographic projection with metrics on the plane of the form g = e ϕ ĝ with ĝ is the standard round metric

(2.1) ĝ := 4 (1 + |x| 2 ) 2 |dx| 2 ,
and ϕ ∈ C1 (R 2 ) where, for k 0, Ck (R 2 ) stands for the space of functions ϕ : R 2 → R that are k-times differentiable with continuous derivatives as well as x → ϕ(1/x) in a neighbourhood of x = 0. The reader may check that the metric ĝ is the pushforward (via stereographic projection) of the standard metric on the Riemann sphere S 2 . We will thus work with such metrics g on the plane, for which we will denote by ∂ g the gradient, △ g the Laplace-Beltrami operator, R g = -△ g ln √ det g the Ricci scalar curvature and v g the volume form. If u, v ∈ R 2 , we denote by (u, v) g the inner product with respect to the metric g (| • | g stands for the associated norm). When no index is given, this means that the object has to be understood in terms of the usual Euclidean metric on the plane (i.e. ∂, △, R, v and (•, •)). Since the stereographic projection is an isometry, we already know that the spherical metric ĝ is such that R ĝ = 2 (its Gaussian curvature is 1) with total mass v ĝ (R 2 ) = 4π.

More generally, two metrics g and g ′ will be said to be conformally equivalent when

g = e ϕ g ′ for ϕ ∈ C1 (R 2 ). It is readily seen that these conditions imply that R 2 |∂ g ′ ϕ| 2 g ′ + 2R g ′ ϕ dv g ′ < ∞
as soon as g ′ is in the conformal class of the spherical metric-that is when g ′ = e ϕ ĝ where ϕ is as above. Furthermore, for ϕ ∈ C2 (R 2 ), the curvatures of two such metrics are related by the relation

(2.2) R g = e -ϕ R g ′ -∆ g ′ ϕ .
In what follows and for given metrics g and h ∈ C1 (R 2 ), we will denote by m g (h) the mean value of h in the metric g, that is the quantity

(2.3) m g (h) := 1 v g (R 2 ) R 2 h(x) v g (dx)
and work in the Sobolev space H 1 (R 2 , g), which is the closure of C ∞ c (R 2 ) with respect to the Hilbert-norm

(2.4) R 2 h(x) 2 v g (dx) + R 2 |∂ g h(x)| 2 g v g (dx).
The standard dual of H 1 (R 2 , g) will be denoted H -1 (R 2 , g). It may be useful to note that the Dirichlet energy is a conformal invariant, that is to say is independent of the metric within a given conformal class:

(2.5)

R 2 |∂ g ′ h(x)| 2 g ′ v g ′ (dx) = R 2 |∂ g h(x)| 2 g v g (dx).
Green kernels. Given a metric g on the Riemann sphere that is conformally equivalent to the spherical metric ĝ, we denote by G g the Green function of the problem

△ g u = -2π (f -m g (f )) on R 2 , R 2 u(x) v g (dx) = 0
where f belongs to the space L 2 (R 2 , g) and u is in H 1 (R 2 , g). This means that the solution u can be expressed as

(2.6) u = R 2 G g (•, x)f (x)v g (dx) =: G g f with m g (G g (x, •)) = 0 for all x ∈ R 2 .
The kernel G g has an explicit expression given by (see [4, Equation (2.9)])

(2.7) G g (x, y) = ln 1 |x -y| -m g ln 1 |x -•| -m g ln 1 |y -•| + θ g
where

θ g := 1 v g (R 2 ) 2 R 2 R 2 ln 1 |x -y| v g (dx)v g (dy).
For instance for the spherical metric this becomes

(2.8) G ĝ(x, y) = ln 1 |x -y| - 1 4 (ln ĝ(x) + ln ĝ(y)) + ln 2 - 1 2 .
Another well-known property of these Green functions (see [START_REF] David | Liouville Quantum Gravity on the Riemann Sphere[END_REF]Proposition 2.2] for instance) is that they are conformally covariant in the sense that: Lemma 2.1 (Conformal covariance). Let ψ be a Möbius transform of the Riemann sphere and g be a Riemannian metric conformally equivalent to the spherical one. Then

(2.9) G g ψ (x, y) = G g (ψ(x), ψ(y))
where

g ψ (z) = |ψ ′ (z)| 2 g(ψ(z))
is the pullback of the metric g by ψ.

Again let us register what happens for the spherical metric:

(2.10)

G ĝ(ψ(x), ψ(y)) = G ĝ(x, y) - 1 4 (φ(x) + φ(y))
where φ is such that e φ = ĝψ ĝ . 2.1.2. Lie algebras and the Toda field action. . Finite-dimensional simple Lie algebras. The special linear group SL n (C) = {A ∈ M n (C); det(A) = 1} can be endowed with the structure of a smooth manifold of (complex) dimension n 2 -1. When equipped with the group operation of matrix multiplication, it becomes a Lie group. Its tangent space at identity sl n consists of all n × n complex matrices with trace 0 and becomes a Lie algebra, called the special linear Lie algebra, when equipped with the usual Lie bracket

(2.11) [A, B] := AB -BA for A, B ∈ sl n .
The adjoint action ad is the map ad A (B) := [A, B] for A, B ∈ sl n . The sl n Lie algebra is one of the simplest instances of complex simple Lie algebras. The interested reader may find more details on the notion of Lie algebra for instance in the textbook [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF]; for the purpose of the present document we shed light on the fact that in the study of finite-dimensional complex simple Lie algebras there is a key subalgebra that naturally arises: the so-called Cartan subalgebra of the Lie algebra g, usually denoted h. In our context it can be defined as a maximal commutative subalgebra of g for which the adjoint action ad H is diagonalizable for each H ∈ h. Such Cartan algebras exist and are unique up to isomorphism: in the case of sl n , it can be realized as the set of diagonal matrices whose diagonal entries sum up to 0, hence can be identified with h n := {x = (x 1 , . . . , x n ) ∈ C n | n i=1 x i = 0}. When working with complex simple Lie algebras we are naturally led to introduce the roots of the Lie algebra g with respect to the Cartan subalgebra h; these are the linear functionals α acting on h such that the set {x ∈ g| ∀H ∈ h, [H, x] = α(H)x} is not restricted to {0}. Among these roots it is possible to find a basis (e i ) 1 i n of the space h * (of linear functionals on h) such that any root can be written as a linear combination of this basis using only integers with same sign: these are the so-called simple roots of g relative to h. This Cartan subalgebra also comes equipped with a scalar product, the so-called Killing form B, inherited from the Lie algebra g and which is defined by (2.12) B(x, y) := tr (ad x • ad y ) .

A convenient way to classify finite-dimensional and simple complex Lie algebras is to use their Cartan matrix. This r × r matrix (r is the rank of the simple Lie algebra, i.e. the dimension of h) is defined by

(2.13) A i,j := 2 B(e i , e j ) B(e i , e i ) •
The entries of this matrix are integral, equal to 2 on the diagonal and non-positive elsewhere; the matrix is invertible. For instance the sl n Cartan matrix is tridiagonal with 2 on the diagonal and -1 on the entries (i, j) with |i-j| = 1. In the sequel and instead of working with the Killing form B we will use the scalar product on h * defined on its basis by e i , e j := A i,j . These two scalar products only differ by a multiplicative constant, usually referred to as the Dynkin index of the adjoint representation, and which is equal to 2g, where g is the so-called dual Coxeter number.

It is very natural to introduce the basis of the fundamental weights (ω i ) 1 i r , that is the basis of h * dual to that of the simple roots:

(2.14)

ω i := r l=1 (A -1 ) i,l e l .
They are defined so that (δ ij is the Kronecker symbol)

(2.15) e ∨ i , ω j = δ ij , ω i , ω j = r l,l ′ =1 (A -1 ) i,l A l,l ′ (A -1 ) l ′ ,j = (A -1 ) i,j
where e ∨ i := 2 e i e i ,e i is the coroot associated to e i . The Weyl vector which is defined by

(2.16) ρ := r i=1 ω i
naturally enjoys the property that ρ, e ∨ i = 1 for all 1 i r. The squared norm of this vector can be expressed explicitly depending on the Lie algebra under consideration via the Freudenthal-de Vries strange formula for simple Lie algebras [16,

α, β = r i=1 α, ω i β, e ∨ i .
In the sequel we will work with simple Lie algebras whose Cartan matrices are symmetric (this is actually the definition of simply-laced simple Lie algebras); under this framework coroots and roots are equal, i.e. e ∨ i = e i . By the classification of finite-dimensional Lie algebras, the Lie algebra is then isomorphic to the special linear Lie algebra sl n (for n 2), the even-dimensional special orthogonal Lie algebra so 2n (for n 4) or one of the exceptional Lie algebras E 6 , E 7 and E 8 .

Toda field action. Given ϕ and φ two differentiable maps from R2 to h * 2 , ϕ = r i=1 ϕ i ω i and φ = r i=1 φ i ω i , let us set

(2.20) ∂ g ϕ, ∂ g φ g := r i,j=1 ω i , ω j (∂ g ϕ i , ∂ g φ j ) g .
Recall that we have defined the Toda field action S T,g in the metric g for the Lie algebra g by the expression

(2.21) S T,g (φ, g) := 1 4π R 2 ∂ g φ(x), ∂ g φ(x) g + R g Q, φ(x) + 4π r i=1 µ i e γ e i ,ϕ(x) v g (dx)
where

(2.22) Q := (γ + 2 γ )ρ
is the background charge, µ := (µ 1 > 0, • • • , µ r > 0) are the cosmological constants and γ > 0 is the coupling constant. In the sequel, we assume that γ satisfies the condition

(2.23) γ ∈ (0, √ 2).
The condition on γ is the optimal condition ensuring that the probabilistic construction makes sense3 : this will become clear later when connecting to GMC theory. From the definition of Q we already know that for all 1 i r,

Q, e i = γ + 2 γ and |Q| 2 := Q, Q = (γ + 2 γ ) 2 |ρ| 2 .
The path integral we aim to construct corresponds to a measure on a suitable space of maps φ : R 2 → h * formally corresponding to (2.24) e -S T,g (φ,g) Dφ.

As we will explain below, this can be achieved thanks to the introduction of the GFF.

2.2. Probabilistic interpretation of the path integral.

2.2.1.

Gaussian measure interpretation of the squared gradient. The Toda field action can be decomposed as a sum of different terms. The first one is the quadratic term

1 2π R 2 ∂ g ϕ(x), ∂ g ϕ(x) g v g (dx) = 1 2π R 2 ϕ(x), -△ g ϕ(x) g v g (dx) =: ϕ, -△ g ϕ g
which is reminiscent of a Gaussian measure. Indeed, the measure formally written as (2.25) e -1 2 ϕ,-△gϕ g Dϕ, when restricted to the space

Σ := {ϕ ∈ H -1 (R 2 → h * , g); R 2 ϕ(x) v g (dx) = 0}
where H -1 (R 2 → h * , g) is the set of h * -valued (generalized) functions with each component (with respect to a basis of h * ) in H -1 (R 2 , g), can be understood as the measure on a Gaussian space ( ϕ, -△ g f g ) f ∈H 1 with covariance kernel given by h, -△ g f g . In other words we are looking for a Gaussian field enjoying the property that

E [ ϕ, -△ g f g ϕ, -△ g h g ] = h, -△ g f g for f, g ∈ H 1 (R 2 → h * , g). When r = 1
, this is achieved by introducing the GFF X g with vanishing v g -mean on the sphere, that is a centered Gaussian random distribution with covariance kernel given by the Green function G g (see [START_REF] Dubédat | SLE and the Free Field: partition functions and couplings[END_REF][START_REF] Sheffield | Gaussian free field for mathematicians[END_REF] for more details on this object). An important feature of the GFF is that it is not defined pointwise but rather belongs to the distributional space H -1 (R 2 , g). For generic rank this is done by considering additional fields and setting (2.26)

X g := r i=1 X g i ω i ,
where X g 1 , . . . , X g r are r such GFFs with covariance structure given by (2.27)

E[X g i (x)X g j (y)] = A i,j G g (x, y). Remark 2.2. The reason why we must assume the underlying Lie algebra to be simply-laced is due to the fact that the Cartan matrix is symmetric only in that case, so that it is indeed possible to construct GFFs with covariance kernel given by (2.27).

To summarize we may wish to interpret the formal Gaussian measure (2.25) restricted to Σ as

Z(g) -1 F (ϕ)e -1 4π R 2 ∂g ϕ(x),∂g ϕ(x) g vg(dx) Dϕ = E F (X g )
for each continuous and bounded functional F on H -1 (R 2 → h * , g), where Z(g) stands for the total mass of the Gaussian integral (2.25)

"Z(g) := det(Σ) -1 2 ", where Σ is the covariance matrix (2.28)

Σ := -△ g Vol g (R 2 )
A with A the Cartan matrix of g and det(Σ) is given by a regularized determinant. We know how this factor varies when we consider a conformal change of metric: it is proved in [31, Equation (1.13)] that

(2.29) log Z(e ϕ g) = log Z(g) + r 96π R 2 |dϕ| 2 g + 2R g ϕ v g (dx).
As a consequence, up to a global factor, one has

(2.30) Z(e ϕ ĝ) = det(A) -1 2 e r 96π R 2 ( |dϕ| 2 ĝ +2R ĝ ϕ)vg(dx)
within the conformal class of the spherical metric ĝ. However in the above construction we do not take into account the fact that the GFF X g has zero mean in the metric g; to overcome this issue we will introduce so-called zero modes in the interpretation of the squared gradient term as a Gaussian measure. To do so we introduce the Lebesgue measure dc on h * by setting for each positive measurable function

F : h * → R (2.31) h * F (c) dc := det(A) 1 2 R r F r i=1 c i ω i dc 1 . . . dc r ,
where dc 1 , . . . , dc r stands for the Lebesgue measure with respect to each variable c i

4

. By doing so we are led to the following probabilistic interpretation of the formal (full) Gaussian measure (2.25)

(2.32) F (ϕ)e -1 2 ϕ,-△gϕ g Dϕ = Z(g) h * E F X g + c) dc
for each continuous and bounded functional F on H -1 (R 2 → h * , g) and g in the conformal class of the spherical metric.

Gaussian Multiplicative

Chaos interpretation of the exponential potential. It remains to treat the other terms that appear in the Toda field action (1.1):

1 4π R 2 R g (x) Q, ϕ(x) + 4π r i=1
µ i e γ e i ,ϕ(x) v g (dx).

The first term perfectly makes sense if we remember that the GFF has a meaning in the distributional sense. However the second term does not make sense because of the lack of regularity of the field. For it to be meaningful we need to make use of the notion of Gaussian Multiplicative Chaos (see [START_REF] Kahane | Sur le chaos multiplicatif[END_REF][START_REF] Rhodes | Gaussian multiplicative chaos and applications: A review[END_REF]) which relies on a proper renormalization of some regularization of the GFF.

Definition 2.3. Let η ε := 1 ε 2 η( • ε
) be a smooth mollifier. We define the regularized field X ε by considering the convolution approximation of X:

(2.33)

X ε := X * η ε .
Usually one considers the regularization given by averaging the field on circles with radius ε. The reasoning is exactly the same. Definition 2.4. Assume that γ < √ 2. From (2.38) and basics of GMC theory [START_REF] Kahane | Sur le chaos multiplicatif[END_REF][START_REF] Rhodes | Gaussian multiplicative chaos and applications: A review[END_REF], we know that the following convergence holds in probability in the space of Radon measures (equipped with the weak topology):

(2.34)

M γe i ,g (dx) := lim ε→0 e γe i ,X g ε (x) -1 2 E[ γe i ,X g ε (x) 2 ] v g (dx).
The random measure M γe i ,g (dx) is non trivial (i.e. different from 0) and is called the Gaussian Multiplicative Chaos (GMC) measure associated to the field γe i , X g .

More generally the GMC measure

M α,g (dx) := lim ε→0 ε |α| 2 2 e α,X g ε (x) v g (dx)
exists and is non trivial if and only if |α| 2 := α, α < 4.

Remark 2.5. The statement of [4, Proposition (2.5)] can be easily adapted to show that the regularized GFF thus defined has a variance which evolves as

E X ĝ i,ε (x)X ĝ j,ε (x) = A i,j -ln ε - 1 2 ln ĝ(x) + θ η + o(1)
when ε goes to 0, and where we have set

θ η := C C η(x)η(y) ln 1 |x-y| v(dx)v(dy) + ln 2 -1 2
(recall that v denotes the standard Lebesgue measure). As a consequence the GMC measure defined above and the limiting measure defined by

lim ε→0 ε γ 2 e γe i ,X ĝ ε (x)+ Q 2 ln ĝ- γe i θη 2 v(dx)
actually define the same random measure.

In the end we interpret the path integral of TCFTs in a probabilistic way by making the identification for F ∈ H -1 (R 2 → h * , g):

(2.35) Z(g) -1 Σ F (ϕ)e -S T,g (ϕ,g) Dϕ := lim ε→0 h * E F X g ε + Q 2 ln g + c) e -1 4π R 2 Rg(x) Q,X g ε (x)+c vg (dx)-r i=1 µ i e γ e i ,c M γe i ,g ε (C) dc,
when the limit exists and where g = e ϕ ĝ is in the conformal class of the spherical metric.

Vertex Operators.

There is a class of functionals F which play a key role in the study of TCFTs; usually referred to as Vertex Operators, computing their correlation functions is often one of the main issue in the study of two-dimensional CFTs. As we will see below, they enjoy a certain conformal covariance identity which is the starting point in the understanding of the theory.

In TCFTs these Vertex Operators formally correspond to taking F = e α,ϕ(z) for z ∈ R 2 and α ∈ h * but since such F are not defined on H -1 (R 2 → h * , g), one must use a regularization procedure in order to define their correlations. This motivates the following definition: Definition 2.6. For z ∈ C and α ∈ h * the regularized Vertex Operator V g α,ε (z) is defined by

(2.36) V g α,ε (z) := ε |α| 2 2 e α,X g ε (z)+ Q 2 ln g+c- αθη 2
where X g ε (z) is the field regularized as above. Similarly to the GMC measure, this regularized Vertex Operator has same limit when ε → 0 as the Wick exponential

(2.37) e α,X ĝ ε (x)+c -1 2 E[ α,X ĝ ε (x) 2 ] ĝ(x) α 2 ,Q-α 2 .
Indeed for α ∈ h * the variable α, X g ε is given by the following formula

α, X g ε = r i=1 α, ω i X g i,ε
and thus has the following covariance structure, for α, β ∈ h *

(2.38) E[ α, X g ε (x) β, X g ε (y) ] = α, β G g,ε (x, y) where G g,ε is the covariance kernel of the field X g ε .

General definitions and existence theorems

Having introduced the probabilistic tools allowing to translate in mathematical terms the path integral formulation of TCFTs, we are now ready to investigate the existence of the partition function and of the correlation functions.

3.1. TCFT measure and correlation functions. To start with, recall that we have given in (2.35) a meaning to the measure on the space H -1 (R 2 → h * , g) formally defined by the expression (2.24) by setting for F a positive measurable function on H -1 (R 2 → h * , g):

F T,g := Z(g) lim ε→0 h * E F X g ε + Q 2 ln g+c) e -1 4π R 2 Rg(x) Q,X g ε (x)+c vg (dx)-r i=1 µ i e γ e i ,c M γe i ,g ε (C) dc.
The mapping F → F T,g generates a measure on H -1 (R 2 → h * , g). It is worth noticing that this measure has infinite mass because of the c → -∞ behaviour of the integrand, but is non trivial as we will see below. Indeed, there is a special family of functions F that deserve special attention: these are the Vertex Operators which allow to define the correlations function of TCFTs. Fix an integer N 1 and N distinct points z 1 , . . . , z N ∈ C with respective associated weights α 1 , . . . , α N ∈ h * . The regularized correlation function is defined by the expression :

(3.1) V α 1 ,ε (z 1 ) • • • V α N ,ε (z N ) T,g := Z(g) h * E N k=1 V g ε,α k (z k ) e -1 4π R 2 Rg(x) Q,X g ε (x)+c vg (dx)-r i=1 µ i e γ c,e i M γe i ,g ε (C) dc.
The correlation function is then set to be the limit when ε → 0 of the above regularization:

(3.2) V α 1 (z 1 ) • • • V α N (z N ) T,g := lim ε→0 V α 1 ,ε (z 1 ) • • • V α N ,ε (z N ) T,g .
The next subsection is devoted to the study of the convergence of this correlation function.

Existence of the correlation function. The form of the correlation function (3.1)

is not really convenient when it comes to investigating its convergence as ε → 0. To obtain a reformulation of the correlation functions, we introduce the random measures

(3.3) Z γe i (z,α) (dx) := e γ N j=1 α j ,e i G ĝ (x,z j ) M γe i ,ĝ (dx)
and we define (3.4) s := N j=1 α j -2Q γ as well as, for all i, s i :=

N j=1 α j -2Q, ω i γ •
The expression of the Vertex Operator as a Wick exponential (2.37) allows us to interpret the product in the regularized correlation function as a Girsanov transform (see Theorem 6.3), and has the effect of shifting the law of the GFF X g . This reformulation is essential to prove the following result:

Theorem 3.1. Existence and non triviality of the correlation function

V α 1 (z 1 ) • • • V α N (z N ) T,g
do not depend on the background metric g in the conformal class of the spherical metric. Furthermore:

1. (Seiberg bounds): The correlation function

V α 1 (z 1 ) • • • V α N (z N ) T,g
exists and is non trivial if and only if the two following conditions hold for all i = 1, . . . , r:

• s i > 0, • for all 1 k N, α k , e i < Q, e i = γ + 2 γ .

(Conformal covariance):

Let ψ be a Möbius transform of the plane. Then

V α 1 (ψ(z 1 )) • • • V α N (ψ(z N )) T,g = N k=1 |ψ ′ (z k )| -2∆α k V α 1 (z 1 ) • • • V α N (z N ) T,g .
where the conformal weights are given by ∆ α j := α j 2 , Q -

α j 2 .

(Weyl anomaly):

If ϕ ∈ C1 (R 2 ) then we have the following relation

V α 1 (z 1 ) • • • V α N (z N ) T,e ϕ ĝ = e c T 96π S L (ϕ,ĝ) V α 1 (z 1 ) • • • V α N (z N ) T,ĝ
where S L is the Liouville functional

S L (ϕ, ĝ) := R 2 |∂ ĝϕ| 2 ĝ + 2R ĝϕ dv ĝ ,
and the central charge c T is given by c T = r + 6|Q| 2 .

(GMC representation):

In the particular case where g = ĝ is the round metric, one gets the following explicit expression for the correlation function

(3.5) V α 1 (z 1 ) • • • V α N (z N ) T,ĝ = r i=1 Γ(s i )µ -s i i γ N k=1 ĝ(z k ) ∆α k e k<l α k ,α l G ĝ (z k ,z l ) E r i=1 Z γe i (z,α) (C) -s i .
The items 2 and 3 above characterize the theory as a CFT with central charge c T , where c T is given for g simple by:

(3.6) g c T,g sl n n -1 + (n-1)n(n+1) 2 q 2 so 2n n + (n -1)n(2n -1)q 2 E 6 6 + 468q 2 E 7 7 + 1197q 2 E 8 8 + 3720q 2
where q := γ + 2 γ .

Proof. To start with, and anticipating on the conformal anomaly formula, we can assume that we work with the spherical metric ĝ, which is such that, since X ĝ has zero mean value in the metric ĝ,

1 4π C Q + X ĝ, c R ĝ(x)v ĝ (dx) = 2 Q, c
. As a consequence the expression of the Vertex operators V α k ,ε (z k ) as Wick exponentials (2.37) allow to interpret them as Girsanov weights that have the effect of shifitng the law of the GFF by a additive term G ĝ,ε (•, z k ) where G ĝ,ε is a mollified version of G ĝ (see Theorem 6.3). This allows to rewrite the regularized correlation function as

N k=1 ĝ(z k ) ∆α k e k<l α k ,α l G ĝ,ε (z k ,z l ) R r e r i=1 s i c i E e -r i=1 µ i e γ c,e i Z γe i (z,α),ε (C) dc 1 ...dc r where Z γe i (z,α),ε (dx) := e γ N k=1 α k ,e i G ĝ,ε (x,z j ) M γe i ,ĝ (dx)
. Now, if one of the s i is non-positive, then the whole integral can be lower-bounded by

R r e r i=1 s i c i e -r i=1 µ i e γ c,e i M dc 1 ...dc r P ∀1 i r, Z γe i (z,α),ε (C) M = +∞
where M > 0 is taken so that P ∀1 i r, Z γe i (z,α),ε (C) M > 0 (to see why this is possible note that G ĝ,ε (x, •) is bounded over C and apply Lemma 4.1 with all the α taken equal to zero). Therefore the ε-regularized partition function is infinite if one of the s i is non-positive. Conversely if these s i are all positive and using Lemma 4.1 we can make the change of variable y i = µ i e γc i Z γe i (z,α),ε (C) in the integral so that we are left with

r i=1 Γ(s i )µ -s i i γ N k=1 ĝ(z k ) ∆α k e k<l α k ,α l G ĝ,ε (z k ,z l ) E r i=1 Z γe i (z,α),ε (C) -s i .
To conclude for the first and fourth item it remains to show that:

• If for all 1 i r, and 1 j N, α j , e i < Q, e i , then

lim ε→0 E r i=1 Z γe i (z,α),ε (C) -s i = E r i=1 Z γe i (z,α) (C) -s i > 0.
• If for some 1 i r and 1 k N, α k , e i Q, e i , then

lim ε→0 E r i=1 Z γe i (z,α),ε (C) -s i = 0.
Let us assume that for all 1 i r, and 1 k N, α k , e i < Q, e i . Then we know from Lemma 4.1 that the family of random variables

r i=1 Z γe i (z,α),ε (C) -s i ε 0
have (uniformly bounded in ε) positive moments of all orders. Thus we can write that

E r i=1 Z γe i (z,α),ε (C) -s i - r i=1 Z γe i (z,α) (C) -s i E Z γe 1 (z,α),ε (C) -s 1 -Z γe 1 (z,α) (C) -s 1 r i=2 Z γe i (z,α),ε (C) -s i + E r i=2 Z γe i (z,α),ε (C) -s i - r i=2 Z γe i (z,α) (C) -s i Z γe 1 (z,α) (C) -s 1 E Z γe 1 (z,α),ε (C) -s 1 -Z γe 1 (z,α) (C) -s 1 p 1 p E r i=2 Z γe i (z,α),ε (C) -qs i 1 q + E r i=2 Z γe i (z,α),ε (C) -s i - r i=2 Z γe i (z,α) (C) -s i p 1 p E Z γe 1 (z,α) (C) -qs 1 1 q
where we have used Hölder inequality with some p = q q-1 > 1. Therefore we can proceed by induction on r so that the only point to check is lim

ε→0 E Z γe 1 (z,α),ε (C) -s 1 -Z γe 1 (z,α) (C) -s 1 p 1 p = 0.
This fact has already been proved by the authors in [4, Lemma 3.3]. For the second bullet point, let us introduce the set

P := {i = 1, . . . , r |∃1 k N, α i -Q, e i 0}
and assume that it is non-empty. Then we can write that, for positive 1 p i and 1 q summing to one,

E r i=1 Z γe i (z,α),ε (C) -s i i∈P E Z γe i (z,α),ε (C) -p i s i 1 p i E i ∈P Z γe i (z,α),ε (C) -qs i 1 q
.

Then we have already seen that the second expectation in the right-hand-side had a finite limit as ε → 0 thanks to the results of Lemma 4.1. Conversely standard results of the GMC theory (see again [START_REF] David | Liouville Quantum Gravity on the Riemann Sphere[END_REF]Lemma 3.3] for instance) imply that for any i ∈ P,

lim ε→0 E Z γe i (z,α),ε (C) -p i s i 1 p i = 0.
This proves the first item in the statement of Theorem 3.1.

We now turn to the proofs of the second and third items. Let us start with the third item. By making a standard change of variable in the zero mode c, and using the fact that X gm ĝ(X g ) and X ĝ have same law, we can assume that X is a GFF with vanishing mean with respect to the round metric. Since g = e ϕ ĝ, we have that

V α 1 ,ε (z 1 ) • • • V α N ,ε (z N ) g = Z(g) N k=1 e α k ,Q 2 ϕ(x k ) × lim ε→0 h * e γ s,c E k j=1 V ĝ ε,α j (z j )e -1 4π C Rg Q,X (x)vg (dx)-r i=1 µ i e γc i C e γ 2 ϕ(x) 2 V ĝ ε,e i (x)vg (dx) dc
where this time regularization is done with respect to the round metric, and V is the vertex operator without constant mode. Recall that for all i, Q, e i = γ + 2 γ ; therefore:

E C R g (y) Q, X (y)v g (dy) e i , X (x) = Q, e i C R g (y)G ĝ(x, y)v g (dy) = (γ + 2 γ )2π (ϕ(x) -m ĝ (ϕ))
where we have used that R g (y)v g (dy) = (-∆ ĝ ϕ(y) + 2)v ĝ(dy) (here in the weak sense since ϕ ∈ C1 (R 2 )) and the definition of the Green function G ĝ. More generally when α ∈ h * :

E 1 4π C R g (y) Q, X (y)v g (dy) α, X (x) = Q, α 2 (ϕ(x) -m ĝ (ϕ)) .
Next we want to consider the exponential of this term C R g (y) Q, X (y)v g (dy) as a Girsanov transform. Its variance is given by

E 1 4π C R g (x) Q, X (x)v g (dx) 2 = 1 16π 2 C×C R g (x)R g (y)E[ Q, X (x) Q, X (y)]v g (dx)v g (dy) = 1 16π 2 |Q| 2 C×C R g (x)R g (y)G ĝ(x, y)v g (dx)v g (dy) = 1 16π 2 |Q| 2 C R g (x) C R g (y)G ĝ(x, y)v g (dy) v g (dx) = 1 8π |Q| 2 C R g (x) (ϕ(x) -m ĝ (ϕ)) v g (dx) = 1 8π |Q| 2 C (-∆ ĝ ϕ(x) + 2) (ϕ(x) -m ĝ(ϕ)) v ĝ (dx) = 1 8π |Q| 2 C |∂ ĝ ϕ| 2 v ĝ(dx)
and as seen above it has the effect of shifting the law of X by an additional factor ) Vε,e i (x)vg (dx) dc.

Q 2 (ϕ(x) -m ĝ(ϕ)). Therefore V α 1 (z 1 ) • • • V α N (z N ) g = Z(g)e |Q| 2 16π C |∂ ĝ ϕ| 2 v ĝ × lim ε→0 h * e γ s,c E N k=1 V ε,α k (z k )e -r i=1 µ i e γc i C e ϕ(x)(γ 2 |e i | 2 4 - γ Q,e i 2
Since for all i we have

|γe i | 2 4 -γ Q,e i 2 = -1, by the change of variable c i → c i -Q, e i 2 m ĝ(ϕ) we get that V α 1 (z 1 ) • • • V α N (z N ) g = Z(g)e |Q| 2 16π C |∂ ĝ ϕ| 2 v ĝ + |Q| 2 4π C ϕv ĝ × lim ε→0 h * E N k=1 V ĝ ε,α k (z k )e -r i=1 µ i C V ĝ ε,e i (x)v ĝ (dx) dc,
whence the result, by using the expression (2.30) for the regularized determinant Z(g).

For the second item, we see that according to our proof of the first item, F (ϕ ĝ) N k=1 V α k (z k ) , where F is continuous and bounded on H -1 (S 2 → h * , ĝ), is actually given by

N k=1 ĝ(z k ) ∆α k e k<l α k ,α l G ĝ (z k ,z l ) h * e s,c E F X ĝ + Q 2 ln ĝ + c + N k=1 α k G ĝ(•, z k ) e -r i=1 µ i e γ c,e i Z γe i (z,α) (C) dc.
Now since ψ is a conformal map we know that the Riemannian metric ψ * ĝ (that we have denoted ĝψ before) is in the conformal class of ĝ; as a consequence the free fields X ĝψm ĝ(X ĝψ ) and X ĝ have same law. Moreover from Lemma 2.1 we know that X ĝψ has same law as X ĝ • ψ. As a consequence and using (2.10) we see that the laws of

X ĝ + N k=1 α k G ĝ(•, z k ) and X ĝ + N k=1 α k G ĝ(•, z k ) • ψ + 1 4 N k=1 α k (φ + φ(z k )) -m ĝ(X ĝ • ψ)
are actually the same. Therefore shifting the zero mode c by the additional constant

m ĝ(X ψ * ĝ) -1 4 N k=1 α k φ(z k ) -1 4 sm ĝ (φ)
and collecting up terms using (2.10) yields:

N k=1 ĝψ (z k ) ∆α k e k<l α k ,α l G ĝ (ψ(z k ),ψ(z l )) h * e s,c E e s,m ĝ (X ψ * ĝ ) -m ĝ (φ) s,s 4
F Φ ĝ • ψ + Q ln |ψ ′ | + 1 4 s(φ -m ĝ(φ)) e -r i=1 µ i e γ c,e i Z γe i (z,α) (C) dc,
where Φ ĝ is a shorthand for

X ĝ + Q 2 ln ĝ + c + N k=1 α k G ĝ(•, z k ).
To conclude the proof it remains to apply the same reasoning as in the proof of the third item to see that the exponential term e s,m ĝ (X ψ * ĝ ) -m ĝ (φ) s,s 4 will shift the law of Φ ĝ • ψ by - 1 4 s(φm ĝ(φ)). In the proof of the conformal covariance property we have shown a slightly more general result. Indeed we have proved that under the Seiberg bounds and for any Möbius transform of the plane, the following was true for any continuous and bounded map F on H -1 (S 2 → h * , ĝ):

(3.7) F (ϕ • ψ + Q ln |ψ ′ |) N k=1 V α k (ψ(z k )) g = N k=1 |ψ ′ (z k )| -2∆α k F (ϕ) N k=1 V α k (z k ) g .
This statement is usually referred to as the conformal covariance of the Toda field.

The conditional volume measures

We wish to extend here the validity of the probabilistic representation (3.5). The point is that the explicit expression (3.5) allows us to isolate the constraints s i > 0 in the product of Γ functions. This term can obviously be analytically removed. The question is then to determine whether the expectation in (3.5) makes sense beyond the range of parameters permitted by the Seiberg bounds. So we claim Lemma 4.1 (Extended Seiberg bounds). The bound

(4.1) E r i=1 (Z γe i (z,α) (R 2 )) -s i < ∞ holds if and only if for all i = 1, • • • , r one has (4.2) -s i < 2 γ 2 ∧ min k=1,...,N 1 γ Q -α k , e i
Proof. We suppose that condition (4.2) holds. Let us consider the families of indices

P := {i = 1, . . . , r | s i 0} and N := {i = 1, . . . , r | s i < 0}. Choose p > 1 such that for all i ∈ N , -ps i < 2 γ 2 ∧ min k=1,...,N 2
γ Qα k , e i and fix the conjugate exponent q > 1 such that 1 p + 1 q = 1. By Hölder inequality we can write that

E r i=1 (Z γe i (z,α) (R 2 )) -s i E i∈P (Z γe i (z,α) (R 2 )) -ps i 1/p E i∈N (Z γe i (z,α) (R 2 )) -qs i 1/q .
The product running over i ∈ P is finite because GMC admits negative moments of all order (see [START_REF] Rhodes | Gaussian multiplicative chaos and applications: A review[END_REF]Theorem 2.12]). For the product running over i ∈ N , we use Corollary 6.2 in appendix as well as the relation (2.38), which shows that the GFFs γe i , X g and γe j , X g , for i = j, are negatively correlated since γe i , γe j = γ 2 A i,j (recall that A is the Cartan matrix) and all off-diagonal elements of A are nonpositive. Hence

(4.3) E i∈N (Z γe i (z,α) (R 2 )) -qs i i∈N E (Z γe i (z,α) (R 2 )) -qs i .
We conclude that each expectation in the product above is finite because [START_REF] David | Liouville Quantum Gravity on the Riemann Sphere[END_REF]Lemma A.1] ensures that it is indeed the case provided that -ps i < 2 γ 2 ∧ min k=1,...,N 1 γ Qα k , e i for all i, which we assumed to hold.

Conversely, assume that the expectation (4.1) is finite. By Corollary 6.2 in the appendix applied to the function

G(x 1 , . . . , x d ) = i∈N x -s i i i∈P
x -s i i ,

to r = |N |, and to the GFFs ( γe i , X g ) i=1,...,r we deduce

E r i=1 (Z γe i (z,α) (R 2 )) -s i E i∈P (Z γe i (z,α) (R 2 )) -s i E i∈N (Z γe i (z,α) (R 2 )) -s i .
Since GMC admits negative moments of all order [33, Theorem 2.12], the first expectation in the right-hand side above is a finite constant C > 0. This implies that the second expectation is finite too. From now on, we fix i 0 ∈ N and j ∈ {1, . . . , N}. Without loss of generality and for the sake of simplicity, we may assume that z j = 0. Then we can choose δ > 0 such that min j ′ =j |z ′ j | > 10 × δ and we can choose non empty balls (B i ) i =i 0 ,i∈N all of them at distance at least 10 × δ > 0 from each other and all of them at distance at least 10 × δ from all the z j 's. Set B i 0 := B(0, δ). Obviously we have

E i∈N (Z γe i (z,α) (R 2 )) -s i E i∈N (Z γe i (z,α) (B i )) -s i .
Consider the mean value of the field Y := 1 2πi |x|=δ X g (x) dx x . A simple check of covariances shows that the law of the field X g -Y is the independent sum of the field X h g -which coincides with X g -Y outside of B(0, 2δ) and corresponds inside B(0, 2δ) to the harmonic extension (component by component) of the field X g -Y restricted to the boundary ∂B(0, 2δ)-plus the Dirichlet field X d defined by (4.4)

X d = r i=1 ω i X d,i ,
where (X g d,1 , . . . , X g d,r ) is a family of centered correlated Dirichlet GFFs inside B(0, 2δ) with covariance structure given by

E[X g d,i (z)X g d,j (z ′ )] = A i,j G d (z, z ′ )
and G d (z, z ′ ) stands for the Dirichlet Green function inside B(0, 2δ). From now on we will write Z γe i ,X g (z,α)

(d 2 x) instead of Z γe i (z,α) (d 2 x
) to indicate in the notations the dependence on the underlying Gaussian field. This means that, generally speaking, we will write

Z γe i ,X (z,α) for Z γe i ,X (z,α) (d 2 x) := lim ε→0 e N j=1 α j ,γe i G ĝ (x,z j ) ε |γe i | 2 2 e γe i ,Xε(x) v g (dx)
where X ε stands for the ε-regularization of the field X in the metric g. So we can write

E i∈N (Z γe i ,X g (z,α) (B i )) -s i = E e -i∈N s i γe i ,Y i∈N (Z γe i ,X g -Y (z,α) (B i )) -s i .
Now we use the Girsanov transform to remover the factor e -i∈N s i γe i ,Y , which we renormalize by its variance. We do not need to compute explicitly the variance, nor the covariance of Y with X g -Y . Indeed, the variance is bounded as well as the covariance of Y with X g -Y .

This entails the existence of some constant C > 0 such that

E e -i∈N s i γe i ,Y i∈N (Z γe i ,X g -Y (z,α) (B i )) -s i CE i∈N (Z γe i ,X g -Y (z,α) (B i )) -s i .
Using the decomposition of the law of X g -Y = X d + X g h and independence of X d and X g h , we get

E i∈N (Z γe i ,X g -Y (z,α) (B i )) -s i E (Z γe i 0 ,X d (z,α) (B i 0 )) -s i 0 E e min x∈B i 0 X g h (x) i =i 0 ,i∈N (Z γe i ,X g -Y (z,α) (B i )) -s i .
This implies that both expectations in the right-hand side are finite (they are obviously nonzero). Finiteness of the first expectation above entails that -s i 0 < 2 γ 2 ∧ 1 γ Qα j , e i (see [START_REF] David | Liouville Quantum Gravity on the Riemann Sphere[END_REF]Lemma A.1]). Since the argument is valid for all i 0 ∈ N and all j, this yields the result.

Perspectives

TCFTs provide natural extensions of LCFT with a higher level of symmetry in addition to the Weyl anomaly which encodes the local conformal structure. In this document we have constructed TCFTs but we have not really shed light on where this W-symmetry does appear in the model and how useful it can be. We review below some interesting questions related to this observation. 5.1. W-symmetry and local conformal structure of Toda theories.

5.1.1.

Characterizing the sl 3 Lie algbera using W-symmetry. Let us consider the first nontrivial extension of the Liouville theory, i.e. when one works with two negatively correlated GFFs-like in the construction of the sl 3 Toda theory. Then we can create a one-parameter family of QFTs by changing the covariance between these two GFFs, Lie algebras arising only for some special values of this parameter, that is when the covariance matrix coincides with the Cartan matrix of some Lie algebra. More explicitly one can assume to be working with a pair of GFFs whose covariance matrix is given by

E[X g i (x)X g j (y)] = A i,j G g (x, y),
but where A = 2 c c 2 with c ∈ (-2, 2) would no longer be the Cartan matrix of some semisimple Lie algebra (except for the values c = 0, -1 corresponding to sl 2 ⊕ sl 2 and sl 3 Lie algebras). Interestingly these theories also enjoy conformal invariance thus define CFTs, but are not supposed to enjoy higher-spin symmetry. In this context it seems natural to wonder what is so specific about the theories defined via the sl 3 Lie algebra, and how can one see where W-symmetry does appear ? These questions are being investigated in a work in progress. More precisely it is expected that the existence of higher-spin currents that are holomorphic is granted if and only if the covariance matrix of the GFFs is given by the Green kernel times the Cartan matrix of the sl 3 Lie algebra.

5.1.2.

Local conformal structure of Toda Field Theories. In LCFT, the Weyl anomaly (combined with diffeomorphism invariance of the theory) is in some sense equivalent to the existence of a holomorphic current: the stress-energy tensor T (z). The expression of this tensor can be obtained by formally derivating the correlation function with respect to the metric

T µ,ν (z)V α 1 (z 1 ) • • • V α k (z N ) g := 4π ∂ ∂g µ,ν V α 1 (z 1 ) • • • V α k (z N ) g ,
and then setting T := T z,z + c 12 t where t is explicit and depends on the background metric g. The first Ward identity (5.1)

T (z 0 ) N k=1 V α k (z k ) = N k=1 ∆ α k (z 0 -z k ) 2 + N k=1 ∂ z k z 0 -z k N k=1 V α k (z k )
and the asymptotic behaviour of the stress-energy tensor T (z) ∼ 1 z 4 near infinity (which comes from the fact that we have conformally mapped the sphere to the plane) usually ensure conformal covariance of the model. In a similar way TCFTs feature higher-spin currents W (k) (z) which should encode higher-spin symmetry via equations that take the same form as the Ward identity. As an application of our formalism it should be possible to check that these identities hold (at least in the simplest sl 3 case) and study the properties of integrability provided by the W-symmetry. However the structure of W -algebras is much more complicated that the Virasoro one (for instance higher-spin tensors feature higher derivatives and the commutation relations of their modes are no longer linear) and the tools used to prove integrability of Liouville theory come with additional complications. However there is still some hope of actually solving these models: for instance it is predicted in the physics literature that one can find a differential equation for some four-point correlation functions, which would allow to derive the value of certain three-point correlation functions. See [START_REF] Fateev | On differential equation on four-point correlation function in the Conformal Toda Field Theory[END_REF]Equation (14)] for a precise statement. These questions will be addressed in an upcoming series of work by Y. Huang and one of the author. The probabilistic framework should in particular allow to write down an explicit expression for the descendants of the primary fields V α (x), expression which was left unidentified in the physics literature. 5.2. The semi-classical limit and Toda equations in W-geometry. Let us comment here on the geometrical signification of TCFTs. Their path integral formulation rely on the action functional (1.1) that corresponds to the quantization (i.e. where we have introduced a coupling constant and considered an appropriate renormalization of the field and cosmological constants) of the action

S c T (ϕ, g) := 1 4π Σ ∂ g ϕ(x), ∂ g ϕ(x) g + 2R g ρ, ϕ(x) + 2Λ n-1 i=1 e e i ,ϕ(x) v g (dx),
whose critical point is given by the solution of the Toda equation (1.2). Such a critical point exists and is unique as soon as Σ admits a metric for which R g is negative and constant; when the surface has the topology of the sphere or the torus one may need the field to have certain logarithmic singularities in order for such a problem to admit a unique solution.

In the simplest case where n -1 = 1 the equation corresponds to the Liouville equation that describes (the conformal factor of) Riemannian metrics with constant negative curvature equal to -Λ within the conformal class of (Σ, g). In general, such an interpretation remains possible but instead of working in the setup of conformal geometry the good framework to consider is the one of W-geometry. The interested reader may find more details on W -geometries in the work of Gervais and Matsuo [START_REF] Gervais | W geometries[END_REF]. One of its important features is that, in a way similar to the fact that W -algebras admit the Virasoro algebra as a subalgebra, W -geometries in some sense contain two-dimensional Riemannian geometry.

From the quantum theory viewpoint, since we have constructed TCFTs via a quantization of this classical model, it is natural to expect that when the coupling constant γ (which characterizes the level of randomness) is taken to zero-and under appropriate renormalizations-we recover the solution of the classical Toda equation (1.2) (when it exists and is unique). Such a result has been recently proved by H. Lacoin and the last two authors in [START_REF] Lacoin | The semiclassical limit of Liouville conformal field theory[END_REF] for LCFT; it should be reasonable to expect that the result extends for the general sl n TCFTs on the sphere.

Appendix

In the appendix, we gather rather general and classical results on Gaussian vectors: first comparison lemmas and then a statement of the Girsanov theorem. Lemma 6.1. Let F be some smooth function defined on (R n ) d with at most polynomial growth at infinity for F as well as for its derivatives up to order 2. Assume that for (x 1 , • • • , x d ) ∈ (R n ) d (where x i = (x i 1 , • • • , x i n )) the following inequalities hold:

for all i = j and k, k ′ ∂ 2 F ∂x i k ∂x j k ′ 0.

Let X := (X 1 , • • • , X d ) and X := ( X 1 , • • • , X d ) be two centered Gaussian vectors in (R n ) d such that 1) for all i = j and k, k

′ E[X i k X j k ′ ] E[ X i k X j k ′ ].
2) for all i, X i as the same law as X i .

Then the following inequality holds:

E[F (X 1 , • • • , X d )] E[F ( X 1 , • • • , X d )].
Proof. For t ∈ [0, 1], we set X t = √ tX + √ 1t X, where X and X are independent, and

G(t) = E[F (X t )].
By using Gaussian integration by parts, we get the following relation

G ′ (t) = 1 2 d i=1 n k=1 E ∂F ∂x i k (X t )( 1 √ t X i k - 1 √ 1 -t X i k ) = 1 2 d i=1 n k=1 d i ′ =1 n k ′ =1 E ∂ 2 F ∂x i k ∂x i ′ k ′ (X t ) E ( √ tX i ′ k ′ + √ 1 -t X i ′ k ′ )( 1 √ t X i k - 1 √ 1 -t X i k ) = 1 2 d i=1 n k=1 d i ′ =i n k ′ =1 E ∂ 2 F ∂x i k ∂x i ′ k ′ (X t ) E[X i ′ k ′ X i k ] -E[ X i ′ k ′ X i k ] 0.
Therefore G(1) G(0). Corollary 6.2. Let G be some smooth function defined on (R + ) d with at most polynomial growth at infinity for G as well as for its derivatives up to order 2 and consider a partition P 1 , . . . , P m of the set {1, . . . , d}. Assume that for (x 1 , • • • , x d ) ∈ R d + , the following inequality holds for all r, r ′ ∈ {1, . . . , m} with r = r ′ , all i ∈ P r and all j ∈ P r ′ ∂ 2 G ∂x i ∂x j 0.

Further assume that X 1 , • • • , X d is a family of continuous centered Gaussian fields respectively defined over domains D i ⊂ R n (for i = 1, . . . , d) such that for all r, r ′ ∈ {1, . . . , m} with r = r ′ , all i ∈ P r and all j ∈ P r ′ ∀x ∈ D i , ∀x ′ ∈ D j , E[X i (x)X j (x ′ )] 0.

Let X = ( X 1 , • • • , X d ) be another family of continuous centered Gaussian fields such that: 1) for all r = 1, . . . , m, ( X i ) i∈Pr has same distribution as (X i ) i∈Pr .

2) the families ( X i ) i∈P 1 , . . . , ( X i ) i∈Pm are independent. Eventually, let f 1 , . . . , f d be a family of positive functions each of which respectively defined on D i . For i = 1, . . . , d, we set

M i := D i e X i (x)-1 2 E[X i (x) 2 ] f i (x) dx and M i := D i e X i (x)-1 2 E[X i (x) 2 ] f i (x) dx.
Then the following inequality holds

E[G(M 1 , • • • , M d )] E[G( M 1 , • • • , M d )].
Proof. Up to a discretization of the fields, it suffices to apply Lemma 6.1 with

F (x 1 , . . . , x d ) := G k 1 p 1 k 1 e γX 1 k 1 -γ 2 2 E[(X k 1 ) 2 ] , • • • , k d p d k d e γX d k d -γ 2 2 E[(X d k d ) 2 ]
for some nonnegative numbers p i k i obtained by discretizing f i over D i . After these two comparison lemmas we recall the statement of the Girsanov theorem. It can be adapted in a way similar to the above proof via a regularization procedure in order to fit to the GFF we have considered throughout the present document:

Theorem 6 . 3 ( 2 F

 632 Girsanov theorem). Let D be a subdomain of C and (X(x)) x∈D := (X 1 (x), • • • , X n-1 (x)) x∈D be a family of smooth centered Gaussian field; also consider Z any Gaussian variable belonging to the L 2 closure of the subspace spanned by (X(x)) x∈D . Then, for any bounded functional F over the space of continuous functions one has thatE e Z- E [ Z 2 ] (X(x)) x∈D = E F (X(x) + E [ZX(x)]) x∈D .[36] A. B. Zamolodchikov. Infinite additional symmetries in two-dimensional conformal quantum field theory. Theoretical and Mathematical Physics, 65(3):1205-1213, December 1985. Email address: baptiste.cercle@universite-paris-saclay.fr Laboratoire de Mathématiques d'Orsay, Bâtiment 307. Faculté des Sciences d'Orsay, Université Paris-Saclay. F-91405 Orsay Cedex, France Email address: remi.rhodes@univ-amu.fr Aix-Marseille University, Institut de Mathématiques de Marseille (I2M), and Institut Universitaire de France (IUF) Email address: vargas@math.univ-paris13.fr LAGA Université Sorbonne Paris Nord, CNRS, UMR 7539, F-93430, Villetaneuse, France

  Equation (47.11)] 1 , 620 for the exceptional Lie algebras E 6 , E 7 and E 8 . To finish this quick introductory part on simple Lie algebras let us mention the following duality relation between two vectors α, β ∈ h

	(2.17)		|ρ| 2 =	g dim g 12	.
	For finite-dimensional simple complex Lie algebras this quantity is given by
	(2.18)	(n -1)n(n + 1) 12	for sl n ,	(n -1)n(2n -1) 6	for so 2n
	and 78, 399 2

*

(2.

[START_REF] Guillarmou | Polyakov's formulation of 2d bosonic string theory[END_REF]

This equation differs from the one in[START_REF] Freudenthal | Linear lie groups[END_REF] by a multiplicative factor

2g. This is due to our normalization convention for the scalar product •, • on h * .

The scalar field ϕ being studied in TCFTs usually has values in h. To keep notations simple we adopt the convention that ϕ actually takes values in the space of roots h * . This identification is possible thanks to the Riesz representation theorem.

Recall that here our convention on γ is different from the standard convention for LCFT in the probabilistic literature by a factor of √ 2.

The prefactor det(A) 1 2 in the equation comes from the fact that the basis (ω i ) 1 i r is not orthonormal.


Benoît Pelopidas
email: benoit.pelopidas@sciencespo.fr

Keywords: nuclear weapons, luck, scholarly responsibility, collapse, existential threats

This contribution argues that the concept of protean power opens a space to think about the limits of control and knowledge about catastrophic possibilities such as nuclear war.

To do so, it offers the first distinctive definition of nuclear luck, which has long been acknowledged by policy and military leaders but remains unaccounted for in scholarship. It further shows that the nuclear realm is defined by two key unknowables. However, it argues that protean power perpetuates a survivability bias which has characterized scholarship so far, before suggesting ways to overcome that bias and modify scholarly ethos to acknowledge such catastrophic possibilities.

The power that the madmen hold is power of an order that the sane alone know that they are not sane enough to use. But the madmen do not want us to know that this power is too absolute, too godlike, to be placed in any human hands: for the madmen dandle the infernal machine jauntily in their laps and their hands eagerly tremble to push the button.

Lewis Mumford, 'Gentlemen you are mad', Saturday Review of Literature, 2 March 1946, 5-6 Protean Power is a long-awaited contribution to International Relations (IR) theory, which makes three salutary interventions in the field. The authors first challenge assumptions of controllability and predictability, even when they are unacknowledged or denied. Conceptually, modes of reduction of uncertainty to risk lie at the heart of the problem they identify and tackle in the book: for too long in mainstream IR, power has been only treated as 'control power' conceived in a probabilistic world of calculable risk and bounded possibilities. 1 Instead, Lucia Seybert and Peter Katzenstein introduce 'protean power' as a way of engaging with uncertainty, as a productive force and as an effect of interactions among actors. Second, they note that IR scholarship tends to overlook imagined futures and invite imagination as a faculty shaping horizons for action. 2 In other words, protean power incorporates imagined futures as constitutive of our horizons of possible action in the world. Third, it challenges the founding inter-national dimension of the discipline of IR which singles out the problem of international anarchy and opens a space for rethinking the planetary and the global in a historical perspective. In doing so, Lucia Seybert and Peter Katzenstein and their co-authors modify the requirements of a scholarly ethos. 3 This welcome call appears in a very particular context of increasingly possible existential disaster and terrible track record of social sciences to anticipate large scale collapses and disasters.

Earth system scientists have identified nine planetary boundaries within which humanity can safely live. Attached to each of them is a process which may affect the habitability of Earth: climate change, ocean acidification, stratospheric ozone depletion, fresh water use, deforestation and other land system change, biodiversity loss, chemical pollution, particle pollution in the atmosphere, and biogeochemical flows. In 2009, these scientists had asserted that we had crossed three of those boundaries (CO 2 emissions as the essential proxy for climate change, biodiversity loss, and biogeochemical flows). In a 2015 update published in Science, they argued that a fourth boundary had been crossed (disruption of the cycle of phosphorus). 4 In 2017, 15,000 scientists from 184 countries published a second 'warning to humanity' in Bioscience, 25 years after the Union of Concerned Scientists' original call to curtail rampant environmental destruction that was likely to affect the prospects of humankind's survival on planet Earth. 5 In this letter, they show that freshwater resources per capita have been rapidly declining and acidification of the oceans has been increasing to a concerning degree between 1960 and 2016. 6 In 2019, the International Science-Policy Platform on Biodiversity and Ecosystem Services (IPBES), an independent body established by United Nations member states, released its first assessment regarding biodiversity since 2005. In it, scientists argue that around 1 million animal and plant species are now threatened with extinction, many within decades, more than ever before in human history. 7 Beyond these planetary boundaries, our condition when it comes to the possibility of thermonuclear war has been deteriorating as well. In January 2020, the Bulletin of the Atomic Scientists set the doomsday clock at '100 seconds to midnight', which is the closest we have ever been to annihilation in their power within IR have avoided that flaw for a long time. From a conventional IR standpoint, they may appear as political theorists outside of IR. See Harrington 2016 and Chaloupka 1992, 130-31. 2 Seybert and Katzenstein 2018, 12. 3 They write that 'the purpose of "power analysis" is to think about responsibility' (Katzenstein and Seybert 2018c, 292). assessment. 8 The Trump and Putin administrations have pulled out of the Treaty on Intermediate-Range Nuclear Forces. The prospects of negotiations to extend the so-called New START Treaty signed in 2010 and due to expire in 2021 are slim. The point is not to say that the cancellation of arms control treaties will open the floodgates to a new nuclear arms race but rather that all existing nuclear weapons states are already committed to 'modernize' their arsenals so that they could keep them until 2080, which is almost as long as nuclear weapons have existed on the planet. For most nuclear weapons states, the period between now and 2080 is much longer than the period between now and their acquisition of nuclear weapons. The omnicidal potential of one single thermonuclear war, either accidental, inadvertent, or deliberate, which would not require the use of all the 13,000 nuclear weapons present on the planet today, as well as other forms of collapse or tipping points that would make parts of the planet uninhabitable, invites us to interrogate one specific category of events: the unexpected, unprecedented, and unrepeatable. 9 Here, unrepeatable means that it cannot be tolerated more than once because there may simply not be enough people left to make the event equally meaningful a second time.

On the other hand, over the last 30 years, the world has experienced a series of collapses of all kinds which were meant to be predicted by social sciences and were not: the collapse of the Soviet bloc, which had been a structuring component of international politics since the end of World War II, was not anticipated by IR at a time when prediction was one of its key ambitions and the 2008 financial crisis was not considered possible either by mainstream economics or by criminology applied to the sector. 10 It has even been convincingly argued that mainstream economics produced blindness to climate change. 11 If one considers the Anthropocene (or capitalocene or eurocene) as an era defined by multiple self-inflicted disasters in the making, IR is extremely ill at ease with it: those who engage with it within IR disagree about almost everything except the need to move beyond IR as a discipline and modify its concept of power. 12 In such a context, it is interesting that the set of case studies which is offered in the book does not include a case of catastrophic existential possibility for humanity. I deliberately do not call it risk, contrary to the commonly accepted notion, because that would be tantamount to falling for the very confusion and obfuscation between uncertainty and risk which is the main intellectual target of the book. Such a possibility would certainly be uncertain and unexpected and, as such, fall under the remit of the volume. This is all the more surprising that two interesting chapters 8 I fully realize the problems associated with taking the 'doomsday clock' as an accurate assessment of the danger of nuclear war. One crucial problem is the inability of the clock to account for short-term danger. For instance, the Cuban Missile Crisis, one of the most dangerous events in the history of the nuclear age, took place at a time of declining risk according to the doomsday clock. 9 Social sciences usually think about frequent events and protean power does not entirely escape that bias (Clarke 2008, 669). 12 Booth 2007, 2, 396-98 as an exception with his notion of 'the great reckoning'; Burke and Fishel 2016, Burke and Fishel 2019, Chandler et al. 2018, Deudney 2018, Fishel and Burke. 2018, Grove 2018, Taylor 2018. are devoted to climate issues and nuclear weapons, but only from the perspective of international negotiations and power dynamics within them.

This will be the focus of this intervention. How does the concept of protean power help us understand catastrophic existential possibilities? Could the concept be mobilized in further studies of such cases? 13 Does our knowledge about catastrophic existential possibilities help us understand better the contribution of protean power and its limits? What does that tell us about scholarly responsibility? To do so, I will focus on one catastrophic existential possibility that would only happen once, in a short amount of time, and which would already be intolerable: global thermonuclear war. 14 Not controlling and not knowing: protean power as an antidote to overconfidence?

The concept of protean power opens up a space to understand inadvertent and accidental outcomes, the role of 'luck' in the non-catastrophic outcome of nuclear crises so far, as well as the paths towards and dangers of overconfidence in the way we scholars relate to them. 15 The role of luck has been evoked by political as well military leaders in the USA and the Soviet Union, including Robert McNamara, Secretary of Defense during the 1962 Cuban Missile Crisis, Dean Acheson, special envoy of President Kennedy dispatched to France at the time, Nikolaï S. Leonov, head of Cuban affairs in the KGB at the same time, and Gerard C. Smith, chief US delegate to the Strategic Arms Limitation Talks. Among post-Cold War figures, one can cite former head of the US Strategic Air Command and Strategic Command (1991-94) General George Lee Butler and US Secretary of Defense William Perry. 16 Surprisingly, scholars of nuclear politics at best mention it in passing but only few actually take it seriously and try to account for it. 17 When they do, it is mostly through either epistemic inconsistencies or practical inconsistencies. In other words, after acknowledging limits of safety they then either reduce the realm of what was possible to what is measurable, or reintroduce the idea of manageability 13 Katzenstein and Seybert 2018c, 268, invite further case studies. In his introduction to this symposium, Jacques Hymans engages briefly with the nuclear weapons realm by connecting protean power to the revolutionary decision of starting a nuclear weapon programme. Hymans, this symposium. Here, we engage with other revolutionary nuclear possibilities whether or not they connect to an isolated decision that can be identified. For an open-minded effort at outlining catastrophic existential possibilities and paths to human extinction, see Leslie 1998, 3-14. 15 Katzenstein and Seybert 2018c, 278. 16 Acheson 1969, 76; Butler 2016, 172; Smith 1996, viii; Perry 2015, ch. 1. Robert S. McNamara: 'In the end, we lucked out. It was luck that prevented nuclear war'. Nikolaï S. Leonov: 'One mistake at the wrong time in October 1962, and all could have been lost. I can hardly believe we are here today, talking about this. It is almost as if some divine intervention occurred to help us save ourselves, but with this proviso: we must never get that close again. Next time, we would not be so lucky, as you put it'. Cited in Blanton and Blight 2002, 7. 17 For mentions of the role of luck or good fortune in the noncatastrophic outcome of nuclear events in security studies scholarship, see, inter alia, Booth 2007, 406, Sagan 1993, 31, 154, Rendall 2007, 530, Lebovic 2013, xii, and Leitenberg 2018, 249. For early attempts at grasping luck in the nuclear weapons realm, see Pelopidas 2015a, 14-17; 2015b, 173-75; 2017. as the inevitable outcome of policy-relevant research. 18 These problems in the field of security studies are well pointed out in chapter 2. 19 In mainstream nuclear scholarship, power is too often reduced to control power and the environment is accounted for in terms of risk. And risk thinking is desire for control and faith in that control. 20 This reduction makes the role of luck and of any factor that would not fall within the realm of practices of control unthinkable. This is a consequential problem insofar as one can think about multiple paths towards desired nuclear outcomes that cannot be reduced to the practice of control power. Those outcomes can happen independently from any control practice; in spite of and not because of practices of control; because of the failure of control practices. This heuristic potential of protean power as a category appears, when Lucia Seybert and Peter Katzenstein write that: 'it's about being in the right place, at the right time'. 21 Those are exactly the words of Colonel Stanislav Petrov in reaction to the praise he received for being 'the man who saved the world'. Petrov disobeyed when he waited too long to report what he saw on his radar screen on that night of 26 September 1983. With protean power, scholars now have a category to direct their empirical investigation of the limits of nuclear controllability. Against the retrospective illusions of safety and control, protean power opens a space that forces analysts to take the possibility of uncontrolled escalation to global thermonuclear war seriously. It reminds us of the fundamental material vulnerability that has been ours since the coupling of thermonuclear weapons with intercontinental ballistic missiles. Such openings are important because they allow us to grasp the enormity of the bets on the future that underpin the massive nuclear weapons modernization programmes which are currently ongoing in all nuclear weapons states as acceptance of continued vulnerability for almost as long as the nuclear age.

Most interestingly, the concept of protean power opens up a space for thinking about the possibility of thermonuclear war and makes certain scholarly practices of denying this possibility more difficult. 22 The authors' rejection of the goal of more accurate predictions 23 points to the power effects of not knowing in the age of global nuclear vulnerability, which connects to the fruitful suggestion to remobilize the imagination and its constitutive effects on the present. 24 Indeed, instead of 'assuming away the unknown' 25 as we too often do, we should acknowledge the fundamental unknowability of the time and effects of the first nuclear war as a defining feature of our condition and age. Bernard Brodie is frequently quoted as summarizing the nuclear condition as follows: 'Everything about the atomic bomb is overshadowed by the twin facts that it exists and that its destructive power is fantastically great'.26 18 Pelopidas 2017, 248-51. 19 Katzenstein and Seybert 2018b, 42-47. 20 See also the chapter by Mendelsohn 2018, 207. 21 Seybert and Katzenstein 2018, 15. 22 Protean power goes against retrospective illusions of understanding and validity and the law of small numbers. See Kahneman 2011, chs. 10, 19, and 20. 23 Katzenstein and Seybert 2018b, 56. 24 Seybert and Katzenstein 2018, 12, Katzenstein and Seybert 2018b, 44, 47; 2018c, 292-97. 25 Katzenstein and Seybert 2018a, xii. I would contend that there is another set of twin facts. They are two known unknowables: we do not know, cannot know and will never know in advance when exactly nuclear war will happen and whether humankind will survive it. 27 This is all the more important as the nuclear discourse often ignores the first unknowable (the question of when) and mischaracterizes the second one (the question of survivability) as a knowable entity. The latter mischaracterization produces two opposite biases: a survivability bias and an extinction bias, both premised on an overstatement of what can be known for sure, neglecting the role of imagined futures, value judgments and memories of the past in the construction of such knowledge. On the one hand, the survivability bias is an explicit or implicit conviction that we can know in advance all the effects of a nuclear war and this knowledge should lead us to realize the species will survive. The 'limited nuclear war' tradition, from Herman Kahn and Albert Wohlstetter to Kier Lieber and Daryl Press through Colin Gray and Keith Payne, has argued that most explicitly. The labelling of nuclear war and of the end as 'unthinkable', which is frequent in the field, also suggests this survivability bias. 28 It starts from the known effects of individual explosions but neglects possibilities of unanticipated escalation, overreaction, and climate effects, which open a space of radical uncertainty and unknowability about the prospect of survival. On the other end of the spectrum, we find an extinction bias. The extinction bias is a claim to know that once nuclear war has started, it will be waged until extinction. It could derive from folk belief that any nuclear explosion means the end of the world or from the claim that the amount of devastation caused by the first strikes alone will trigger a drive for revenge that will prevent any form of further restraint. 29 Stances in the nuclear weapons policy debate may be aware of the unknowability of the survivability question but tactically operate as though we knew and therefore fall in the survivability or extinction bias in practice, if not in theory.

Extinction and civilizational collapse as boundary possibilities for protean power?

While the critique of control power opens a space for the possibility of luck and accidents, the binary approach between control power and protean power, worlds 27 This fundamental unknowability has been with us since the early years of the nuclear age and is well captured in the exchange that took place at the US Strategic Air Command headquarters in Omaha, Nebraska on 12 December 1960 after RAND analyst William Kaufman was giving a counterforce briefing. Opposing what he saw as restraint, General Thomas Power, then head of SAC, interrupted to say: 'At the end of the war, if there are two Americans and one Russian, we win!' Kaufman snapped back: 'you'd better be sure that they're a man and a woman' (Kaplan 1983, 246). This is not the case at all for the problems described above regarding planetary boundaries. See for instance Oreskes and Conway 2010. An important research programme is currently trying to go beyond Kaufman's intuition to identify the minimum viable population size. I do not enter this discussion here.

28

One has to acknowledge that the label unthinkable was most famously thematized by Hermann Kahn, an early theorist of limited nuclear war, who wanted to stigmatize the opposite tradition (Kahn 1962). German philosopher Gunther Anders was deeply concerned about human's inability to grasp the nuclear condition and the possibility of self-inflicted extermination. He called this problem 'inverted utopianism' (Anders 1962, 496-97).

of risk and worlds of uncertainty, seems to overlook the second known unknowable mentioned above and the possibility of extinction and/or civilizational collapse.

Protean power suffers from a survivability bias. Let's take three of its definitions: 'Protean power responds to and deepens unanticipated change and is often a response to crises that catch everyone by surprise'; 'the core of protean power lies in agility and the circulation of power potentialities'; 'Protean power rests in the ability to find channels of possibility where established means of control fail'. 30 In all those definitions, there is a world after. This may be most explicit in the following definition: '[Protean power] arises either through direct relations between actors or indirectly in the follow-on effects that reconfigures complex systems'. 31 In his chapter on the revolutions in rights, Christian Reus-Smit illustrates this point as he engages with the notion of collapse. He writes that: 'today's global system of sovereign states is the product of several waves of imperial collapse […] Post-1945 decolonization was the most momentous of these waves: not only did multiple empires collapse but so too did the institution of empire'. But he adds 'after the war, most imperial powers reasserted their commitments to empire' so that the phenomenon he describes ends up looking more like a process of reconfiguration and change of political form than like a collapse. 32 The mobilization of the concept of 'resilience' operates according to the same assumption of survivability. 33 Similarly, if I am not mistaken, the word 'survival' appears only four times in the book, either describing a political strategy which succeeded or a threat, and the notion of extinction is not mentioned at all. 34 In that respect, the path to extinction exceeds the dichotomy between control power and protean power. Indeed, control power will have failed and produced such massive, fast, and irrepressible forms of violence that there will not be channels of possibility left. The implicit certainty that control and protean power cover the whole spectrum of power makes omnicidal nuclear war unthinkable a priori and does not take the second unknowable seriously. It is written as though we knew that we would survive or as though we should act as if we will. This survivability bias is characteristic of the discipline of social sciences and, surprisingly, much less present in the general public. 35 It still implicitly maintains that existing power structures are compatible with and possibly responsible for the survival of our civilization and species.

My invitation to readers and students of protean power would be to adopt its effort at mobilizing imagined futures in a way that makes luck and accidents 30 Katzenstein and Seybert 2018a, xii, xv; 2018c, 274. 31 Seybert and Katzenstein 2018, 10. 32 Reus-Smit 2018, 60. 33 Katzenstein and Seybert 2018c, 296. 34 Katzenstein and Seybert 2018a, 38, 100, 119, and 191. 35 In a survey conducted in June 2018 among 7000 citizens of France, the UK, Belgium, Germany, Italy, the Netherlands, Poland, Turkey, and Sweden aged 18-50 and representative in terms of age, gender, and level of education, we found that when asked, 'if a nuclear war happens, which of the following would be the most likely consequence?' 61.8% of them respond either 'it would be the end of humankind' or 'it would not be the end of humankind but it would be the end of civilization as we know it' and only 11.4% respond: 'I don't think a nuclear war will ever happen'. This survey was funded by the French National Research Agency (ANR) under the VULPAN project.

thinkable and extend it one step further to the possibilities of civilization collapse and extinction which, so far, remain beyond the remit of our field. In other words, avoid a survivability bias.

Scholarly responsibility at the end of the world(s) The unknowability of the timing of nuclear war and its effects makes it possible that thermonuclear war will end the species before this paper goes to print but also that we avoid it until the death of the sun. In this context, claims of knowing and scoping possible nuclear futures appear as the ultimate acts of power. 36 As suggested by the authors, scholarly responsibility no longer lies in predicting; expert power does. As scholars and educators, we are uniquely placed to recreate expert accountability and expose what the available bets on the future are, instead of reproducing overconfidence in our survivability. This can be done on behalf of our fellowcommunity members, whatever the scale of the community, future, past, or present.

Exercising this responsibility would start with identifying the sources of overconfidence in the controllability and predictability of those futures as well as relabelling the claims of knowing the future as bets. 37 Such bets are underpinned by lessons of the past, imagined futures, specific meanings embedded in the categories used, sociotechnical assemblages and value choices. Picturing them as certainties is certainly inadequate because it underestimates how limited and contested existing knowledge about past nuclear crises is, it obfuscates the value choices on which those supposed certainties are based and it assumes that the effects of the policies preferred by those experts will work perfectly and that we can know that for sure in advance. In doing so, it removes space for political judgment, deliberation, and choice without taking responsibility for such a move.

Protean power, its critique of control power and predictability and invocation of the constitutive effects of imagined futures help us think about the possibility of unprecedented accidents and push back against institutional claims of control over the present and the future. 38 It also helps us avoid the temptation of presentism, which commonly distorts our thinking about responsibility.

This effort of imagination should move one step further and take seriously the possibility of collapses and ends of humanity as well as radical change of power structures along the way. Indeed, protean power does not seem to have yet overcome that survivability bias which has been present in the field. It connects to a discourse of resilience which simply negates the possibility of an end and makes the condition of vulnerability unproblematic. 39 This effort of imagination invites us to re-engage with normativity and utopia.40 Asserting a form of scholarly responsibility requires us to reflexively assert the 36 In the 1960s, for instance, a series of intellectual moves have entrenched the idea that nuclear weapons were 'here to stay'. See Pelopidas 2020. Another instance would be the deployment of the rhetorical veto player of NATO as a 'nuclear alliance' since 2010. See Egeland 2020. 37 Pelopidas 2015b. 38 For an important contribution in that direction in the field of nuclear studies, see Sylvest and van Munster 2016, ch. 5. fundamental normative underpinning of scholarship associated with existential dangers as opposed to convoluted obfuscations in the name of detachment and objectivity. Moreover, in the nuclear weapons realm, given the potential for devastation of this technology, the radical unacceptability of catastrophic failure of nuclear deterrence and the leap into the future that modernization programmes are asking us to make, we cannot escape utopia: we can either bet on nuclear weapons technology never failing or only failing in non-catastrophic ways in the next seven decades or on radical measures of nuclear disarmament and the possibility to invent new ones down to levels under the nuclear winter threshold, before nuclear war happens. 41 There is no third alternative. While often portrayed as a repetition of the past 70 years of successful nuclear control, the current bet on another 70 years of absence of unintended nuclear explosion should be called a technological utopia given that such a record of control of nuclear weapons technology has not been achieved. This becomes clear as soon as one takes lucky cases seriously and realizes that the public record of such cases is incomplete. The second utopia would mean dismantling a smaller number of nuclear warheads than we have during the 10 years between 1986 and 1996 but, as we get to low numbers, challenges of trust and stability may intensify to the point that pessimists call it utopian. 42 As a result, there is no non-utopian course of action if utopian means unusually challenging or unprecedented. 43 In that respect, protean power would be an apt concept to study the ongoing attempts at inventing nuclear disarmament. 44 Thinking and making space for the possibility of an end is not impossible: it is exactly what the futurologists managed to do after World War II even though the discomforts of doing so introduced negotiations away from it. 45 From this standpoint, we would construct timescapes in the shadow of possible ends in which we could re-allocate responsibilities. 46 Only then will we be able to assess whether 'the power that the madmen hold is power of an order that the sane alone know that they are not sane enough to use'.

encouragement. Kjølv Egeland, Thomas Fraise, and Sanne Verschuren gave detailed feedback which helped me improve this essay. It also benefitted from in depth conversations with Debak Das, Daniel Deudney, Martin Hellman, Jan Ruzicka, N.A.J. Taylor, Nick Ritchie, the Global Systemic Risk research cluster at Princeton University as well as the two reviewers for International Theory. Lyndon Burford deserves special thanks for helping me articulate the typology of lucky cases.

10

  Lebow 1994, Katzenstein and Nelson 2013, Blyth and Matthijs 2017, de Maillard 2011, chs. 9 and 10. 11Pottier 2016.

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additional scientists have signed the document since its publication in October 2017 (http://scientistswarning.forestry.oregonstate.edu/additional-signatories). Last consulted on 4 May 2019. 6 Ripple et al. 2017. 7 https://www.ipbes.net/news/Media-Release-Global-Assessment. 460 Benoît Pelopidas terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1752971920000299 Downloaded from https://www.cambridge.org/core. IP address: 86.225.231.8, on 30 Nov 2020 at 15:01:24, subject to the Cambridge Core

terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1752971920000299 Downloaded from https://www.cambridge.org/core. IP address: 86.225.231.8, on 30 Nov 2020 at 15:01:24, subject to the Cambridge Core

Benoît Pelopidas terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1752971920000299 Downloaded from https://www.cambridge.org/core. IP address: 86.225.231.8, on 30 Nov 2020 at 15:01:24, subject to the Cambridge Core

 Brodie 1946, 52.

Rose McDermott and her colleagues offer very interesting insights from psychology about the role of revenge in guaranteeing nuclear retaliation (McDermott et al. 2017).

Pelopidas and Weldes 2014.

This expands onPelopidas 2016, 331-33.

Acknowledgements. This essay has been made possible by funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 759707), NUCLEAR. I want to thank the Nuclear Knowledges team for its constant support and