Anatoly Dymarsky
email: dymarsky@ias.edu

Stanislav Kuperstein
email: stanislav.kuperstein@cea.fr

Non-supersymmetric Conifold

We find a new family of non-supersymmetric numerical solutions of IIB supergravity which are dual to the N = 1 cascading "conifold" theory perturbed by certain combinations of relevant single trace and marginal double trace operators with non infinitesimal couplings. The SUSY is broken but the resulting ground states, and their gravity duals, remain stable, at least perturbatively. Despite the complicated field theory dynamics the gravity solutions have a simple structure. They feature the Ricci-flat non-Kähler metric on the deformed conifold and the imaginary self-dual three-form flux accompanied by a constant dilaton.

Introduction

The problem of finding a holographic description for QCD is an outstanding challenge. Even in the strongly coupled regime, when the dual background is expected to be weakly curved and potentially can be described by supergravity, to find the corresponding gravity background is beyond our reach. Partially this can be attributed to the lack of supersymmetry. So far supersymmetry was the main vehicle to find new solutions and to guarantee their stability. Because of supersymmetry the dual solutions had a simple and elegant structure, but what we could learn about the dual field theories was bounded to supersymmetric dynamics.

In this paper we want to make a step in the direction of finding a gravity background dual to a non-supersymmetric field theory. As we are aiming at a confining theory the deformed conifold geometry of the Klebanov-Strassler (KS) background [START_REF] Klebanov | Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities[END_REF] seems a natural starting point. The KS geometry admits a global SU (2)×SU (2) symmetry, which, although redundant from the QCD point of view, drastically simplifies the story on the gravity side. We would have to maintain this symmetry throughout the paper.

At the next step we would like to break supersymmetry by means of perturbing theory by relevant operators, say, by giving mass to gauginos. This will break supersymmetry while preserve the global SU (2) × SU [START_REF] Kuperstein | Analytic non-supersymmetric background dual of a confining gauge theory and the corresponding plane wave theory of hadrons[END_REF]. The corresponding gravity solution is straightforward to find when the gaugino mass is infinitesimally small and SUSY is softly broken [START_REF] Kuperstein | Analytic non-supersymmetric background dual of a confining gauge theory and the corresponding plane wave theory of hadrons[END_REF][START_REF] Kuperstein | Non-supersymmetric deformation of the Klebanov-Strassler model and the related plane wave theory[END_REF][START_REF] Dymarsky | On gravity dual of a metastable vacuum in Klebanov-Strassler theory[END_REF]. Conceptually to find the corresponding solutions when the masses are large is also straightforward. Because the global SU (2) × SU (2) symmetry is preserved the corresponding 10d IIB supergravity background can be described by the so-called Papadopoulos-Tseytlin (PT) ansatz [START_REF] Papadopoulos | Complex geometry of conifolds and 5-brane wrapped on 2-sphere[END_REF] that involves only ten functions of radial variable. The supergravity equations of motion reduce to ten coupled non-linear second order ODEs which one would need to solve imposing proper boundary conditions in the IR and the UV.

In practice such a solution can be found only numerically but even in this case it is a formidable task. 1 To make the problem more manageable one needs to employ some trick which would drastically reduce the complexity of the system. For instance this was successfully done in [START_REF] Butti | The baryonic branch of Klebanov-Strassler solution: A supersymmetric family of SU(3) structure backgrounds[END_REF] where a new family of gravity solutions dual to the baryonic branch of field theory was found. Thanks to supersymmetry the problem reduced just to two coupled first order ODEs with simple boundary conditions while other eight functions from the PT ansatz are algebraically dependent on the first two. Later these solutions were generalized by use of a solution generating technique based on string theory dualities [START_REF] Gaillard | The warped, resolved, deformed conifold gets flavoured[END_REF][START_REF] Caceres | Heating up the Baryonic Branch with U-duality: A Unified picture of conifold black holes[END_REF][START_REF] Elander | Towards multi-scale dynamics on the baryonic branch of Klebanov-Strassler[END_REF]. The resulting solutions are still attractively simple but develop various singularities. Typically these solutions have singular metric near the tip. And whenever solution is IR regular there is a deviation from the asymptotically AdS behavior in UV signaling the presence of the irrelevant dimension eight operator. These solutions are of no use for us as we aim to find a regular non-supersymmetric background. Moreover we want to preserve the decoupling limit and therefore we do not allow perturbations by irrelevant operators.

A good idea would be to preserve the simplicity of the supersymmetric solution while breaking supersymmetry explicitly. One example of such a trick in the context of D7-brane embedded into a supersymmetric background was developed in [START_REF] Dymarsky | Chiral Symmetry Breaking with non-SUSY D7-branes in ISD backgrounds[END_REF]. It was shown there that in a special case of constant dilaton background the supersymmetry conditions for D7 can be partially relaxed. Then the supersymmetry will be completely broken but the resulting solution for D7 will maintain the simplicity of a fully supersymmetric one. It is interesting to note that such a trick fails if the D7brane is embedded into a more complicated supersymmetric background with running dilaton [START_REF] Dymarsky | Flavor brane on the baryonic branch of moduli space[END_REF]. In our case the situation is similar. Our trick is based on the following observation [START_REF] Grana | Supersymmetric three-form flux perturbations on AdS(5)[END_REF][START_REF] Grana | Gauge / gravity duals with holomorphic dilaton[END_REF][START_REF] Giddings | Hierarchies from fluxes in string compactifications[END_REF]. The simplest version of the bulk supersymmetry condition requires the RR four form to be related to the warp-factor C 4 = h -1 Vol(R 3,1 ) and the three-form flux to be a (2, 1) primitive form. Furthermore, in the absence of D-branes the dilaton is constant while the unwarped 6d metric is Kähler and Ricciflat. A remarkable observation is that if the three-form flux is an arbitrary imaginary self-dual (ISD) form (not necessarily (2, 1) primitive) this is enough for the fluxes to decouple from the metric/dilaton equations. Without sources the dilaton will have to be constant and one would end up with a Ricci-flat unwarped 6d metric which is not necessarily Kähler. A non-Kähler metric and a generic ISD flux are both breaking supersymmetry. We will denote this type of solution with the ISD three-form flux, a constant dilaton and a Ricci-flat 6d metric as the GKP (for Giddings, Kachru, Polchinski [START_REF] Giddings | Hierarchies from fluxes in string compactifications[END_REF]) background. These solutions are significantly simpler than the generic ones as the fluxes completely decouple from the equations for metric.

Relaxing the SUSY condition only for the three-form flux while keeping the metric intact in the context of the KS background does not yield any new non-singular solution [START_REF] Kuperstein | Analytic non-supersymmetric background dual of a confining gauge theory and the corresponding plane wave theory of hadrons[END_REF]. Therefore we are motivated to find a new Ricci-flat, presumably non-Kähler, metric on the deformed conifold. Below we argue that the conventional Kähler Ricci-flat metric on the deformed conifod [START_REF] Candelas | Comments on Conifolds[END_REF] -the unwarped metric of the KS solution -can be generalized to a one-parametric family of IR-regular Ricci-flat metrics which asymptote to the conventional metric in the UV. These new non-Kähler metrics can be used to construct novel gravity backgrounds which are regular everywhere and approach the KS solution in the UV. Therefore these backgrounds must be dual to the original N = 1 cascading "conifold" theory perturbed by some particular combinations of SUSY-breaking relevant and/or marginal operators. We do not expect these particular combinations of couplings to be special in any way from the field theory point of view. Quite the opposite, there must be a larger space of SUSY-breaking relevant deformations which lead to well-defined theories with stable vacua.

To illustrate the point it is convenient to parameterize the family of new backgrounds which we find below, by U -the vev of the bottom component of the U (1) baryon multiplet. This is because the supersymmetric solution for infinitesimally small U is of the GKP type [START_REF] Gubser | Symmetry breaking and axionic strings in the warped deformed conifold[END_REF]. Hence U is a good coordinate to parametrize the new one-dimensional family of the GKP backgrounds, at least locally near U = 0. Value of U will specify the location of the new non-SUSY vacuum on the baryonic branch of the original SUSY theory. Let us start with the original SUSY theory in a vacuum on the baryonic branch with some non-trivial value of U . For finite U the dual gravity background is quite complicated: it is a SU (3)-structure solution with running dilaton etc. [START_REF] Butti | The baryonic branch of Klebanov-Strassler solution: A supersymmetric family of SU(3) structure backgrounds[END_REF]. Then we start turning on SUSY-breaking relevant and marginal couplings such that the vacuum value of U stays intact. Overall there is a large space of such combinations of couplings and the corresponding theories which we denote by M U . Since SUSY is broken the corresponding gravity backgrounds presumably are even more complicated than the original point of our journey -the SU (3) structure solution of [START_REF] Butti | The baryonic branch of Klebanov-Strassler solution: A supersymmetric family of SU(3) structure backgrounds[END_REF] with the given U . Yet, we claim, there is at least one particular point in M U such that the gravity dual admits the simple GKP structureit has an ISD flux, a constant dilaton and a Ricci-flat metric. These are the theories, and their gravity duals, we study in this paper. The schematic picture is shown in Fig. 1. This paper is organized as follows. In the next section we numerically construct a one dimensional family of gravity backgrounds that corresponds to the blue line in fig. 1. We construct the novel SU (2) × SU (2) invariant Ricci-flat non-Kähler metrics on the deformed conifold and then dress them up by the appropriate fluxes and warping. We define M U as a space of combinations of relevant/marginal couplingsand the corresponding theories -such that the bottom component of the U (1) baryon multiplet has the vacuum value U . Schematically we denote x to be the coordinates on M U (we do not specify the dimension of M U ) such that x = 0 corresponds to the unperturbed N = 1 theory. Red dashed line corresponds to the motion in the space of theories starting from the unperturbed N = 1 theory such that vev U stays constant. Most of these theories have complicated gravity dual. At a specific point the gravity dual accidently becomes simpler -it is of GKP type. The one-dimensional family of such theories forms the thin blue line. These are the theories we study.

Then in Section 3 we discuss stability of the resulting solutions and their meaning from the dual field theory point of view. We outline our results in Section 4.

2 Supergravity solutions

Ricci-flat metric on the deformed conifold

The starting point of our journey is the observation that the Gubser-Herzog-Klebanov (GHK) solution that corresponds to the vacua of the original N = 1 theory with the infinitesimally small U has the GKP structure [START_REF] Gubser | Symmetry breaking and axionic strings in the warped deformed conifold[END_REF][START_REF] Gubser | Variations on the warped deformed conifold[END_REF]:

• Constant dilaton, e Φ = g s ;
• Imaginary self-dual (ISD) three-form flux, iG 3 = 6 G 3 , where 6 is the 6d Hodge dual and

G 3 ≡ F 3 + i gs H 3 ; • RR four-form C 4 = h -1 Vol(R 3,1
), where h is the warp factor:

-2

s ds 2 10 = h -1/2 dx µ dx µ + h 1/2 ds 2 M 6 ;
(2.1)

• Ricci-flat 6d unwarped metric on M 6 .

Beyond linear order the baryonic branch solutions of [START_REF] Butti | The baryonic branch of Klebanov-Strassler solution: A supersymmetric family of SU(3) structure backgrounds[END_REF] do not preserve these properties. A natural question then is wether it is possible to continue the GHK solution beyond linear order such that the Ricci-flatness of the unwarped metric and other nice GKP properties of the solutions are preserved. We will argue that this is indeed possible and there is a one-dimensional family of such backgrounds which form the blue line in Fig 1 touching the horizontal line, the baryonic branch of the SUSY theory, at the origin.

More broadly we can pose a question of finding non-singular Ricci-flat deformations of the conventional Kähler metric on the deformed conifold. As an extra condition we will require the new metric to approach the original Kähler metric in the UV. For simplicity we also preserve the SU (2) × SU (2) symmetry. The most general metric compatible with these symmetries is given by the PT ansatz and depends on four functions of the radial variable τ [5]2 :

-4/3 ds 2 6 = 2 3 e -8p+3q dτ 2 + g 2 5 + e 2p+3q cosh(y) e z (e We refer the reader to [START_REF] Papadopoulos | Complex geometry of conifolds and 5-brane wrapped on 2-sphere[END_REF] for the definitions of the angular forms e i , i , and g 5 . For the KS solution one finds z(τ ) = 0 and:

e 10p(τ ) = K 3 (τ ) sinh(τ ) , e 6q(τ ) = 3 1/2 8 K 4/5 (τ ) sinh 8/5 (τ ) , e y(τ ) = tanh τ 2 ,
where

K(τ ) ≡ (sinh(2τ ) -2τ ) 1/3 2 1/3 sinh(τ ) . (2.3)
This 6d metric is a solution of the 10d supergravity EOMs from [START_REF] Papadopoulos | Complex geometry of conifolds and 5-brane wrapped on 2-sphere[END_REF], together with zero three-form fluxes and Φ = const, provided we redefine some of the metric fields in the following manner (the expression for A PT ensures that the warp factor is one, h(τ ) = 1)3 : e x , e p , e 2A PT = 2 -1/2 3 -1/4 e 2p+3q , 3 1/4 e p-q , 2 -1/2 3 -3/4 e 5q here , e g = e z cosh(y)

, a = e z tanh(y) .

(

2.4)

There is a Z 2 symmetry I that exchanges the two two-spheres of the conifold (and also changes sign of the three-forms). In the coordinates z i satisfying 4 i z 2 i = 2 it acts by z 4 → -z 4 . This symmetry leaves the variables p(τ ), q(τ ) and y(τ ) invariant and flips the sign of z(τ ). In the z = 0 case the geometry is I invariant as in [START_REF] Klebanov | Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities[END_REF] (see also [START_REF] Kuperstein | Analytic non-supersymmetric background dual of a confining gauge theory and the corresponding plane wave theory of hadrons[END_REF], [START_REF] Pando Zayas | 3-branes on spaces with R x S**2 x S**3 topology[END_REF]).

For the metric (2.2) to be Ricci-flat the functions p(τ ), q(τ ), y(τ ) and z(τ ) have to satisfy four second order ODEs and a first order constraint (the zero energy condition). Hence the full space of SU (2)×SU (2) invariant Ricci-flat metrics on conifold is sevendimensional. Obviously the shift τ → τ + const is a symmetry which we can fix by choosing the tip of the geometry to be at τ = 0. Another simple parameter is the overall re-scaling of the metric given by q → q+const. Alternatively, this shift controls the size of the three-sphere at the tip of the conifold which can be identified with the deformation parameter . Thus we are left with only five non-trivial parameters and .

The counting above agrees with the linearized analysis in the vicinity of the singular conifold z 2 i = 0. The warped product of singular conifold and the Minkowski space is a gravity dual to a particular CFT [START_REF] Klebanov | Superconformal field theory on threebranes at a Calabi-Yau singularity[END_REF]. Therefore all small Ricci-flat perturbations of the unwarped metric are in one-to-one correspondence with couplings and vevs of certain operators in field theory. There are three such operators in this case [START_REF] Ceresole | Spectrum of type IIB supergravity on AdS(5) x T**11: Predictions on N=1 SCFT's[END_REF][START_REF] Ceresole | K K spectroscopy of type IIB supergravity on AdS(5) x T**11[END_REF][START_REF] Baumann | D3-brane Potentials from Fluxes in AdS/CFT[END_REF]:

1. The gaugino bilinear λ 1 λ 1 -λ 2 λ 2 , 2. The bottom component of (W 2 W 2 ) + , 3. The bottom component of the U (1) baryon multiplet U ∼ Tr(|A 2 | -|B 2 |) .
We will denote these operators by their dimensions: 3, 6 and 2 respectively. The first two operators are even, while the last one is odd under I. Three operators correspond to six small Ricci-flat perturbations of metric on the singular conifold. This is not in contradiction with our previous finding because in the singular conifold case the shift of τ = ln(r 3 / 2 ) and an overall rescaling of conic metric dr 2 + r 2 ds 2 T 1,1 coincide. Hence we are left with six independent linear modes that correspond to the three operators in the dual field theory.

The analysis of small Ricci-flat perturbations around the deformed conifold should give similar results simply because in the UV region the deformed conifold approaches the singular one, up to 1/r corrections. Indeed there are six infinitesimal Ricci flat perturbations, two I odd and four I even. The I odd perturbations correspond to the coupling and the vev of U . The four I even perturbations correspond to the couplings and vevs of the operators of dimension 6 and 3. Because there are two different dimension three operators in this theory -two combinations of gauginos λ 1 λ 1 ± λ 2 λ 2 -which mix in the deformed conifold case (but not in the singular conifold case) we will not be able to explicitly identify the operator responsible for this metric perturbation. In what follows we will simply denote this operator as λλ and call the corresponding coupling the gaugino mass m λλ . Four I even modes plus two I odd modes give six in total, but this is not in contradiction with five non-trivial parameters we found above. This is because one of the six modes is actually the shift of τ i.e. the τ -derivative of the background solution.

These six infinitesimal Ricci-flat perturbations around the conventional conifold metric have been found explicitly. Unfortunately the full description is quite technical. We briefly mention it below but a detailed knowledge of these modes is not necessary to follow our logic. An uninterested reader can skip until the next paragraph. The I-odd Ricci-flat perturbations were found by GHK in [START_REF] Gubser | Symmetry breaking and axionic strings in the warped deformed conifold[END_REF]. The I-even ones can be found using the formalism to study linear I even SU (2) × SU (2) invariant perturbations around the KS solution proposed by Borokhov and Gubser [START_REF] Borokhov | Non-supersymmetric deformations of the dual of a confining gauge theory[END_REF] and later developed in [START_REF] Bena | On the Existence of Meta-stable Vacua in Klebanov-Strassler[END_REF]. The full space of solutions is parameterized by sixteen constants X 1 , . . . , X 8 , Y 1 , . . . , Y 8 . To find the Ricci-flat perturbations of the conifold metric one can impose that full background is of the GKP type, i.e. the dilaton is constant, three-form flux is ISD etc. Then ignoring an overall rescaling one recovers exactly four modes. In the notations of [START_REF] Dymarsky | On gravity dual of a metastable vacuum in Klebanov-Strassler theory[END_REF] these modes are parameterized by

X 3 , X 4 , Y 2 , Y 3 while X 2 = -2X 3 /3, Y 1 = -5Y 2 /3
and all other X i and Y i are zero. Only three modes are truly non-trivial as the mode Y 2 = const, Y 3 = X 3 = X 4 = 0 is the derivative of the background with respect to τ . It can be attributed to the vev of an operator of dimension 3 in a sense that the linear mode asymptotes to 1/r 3 at infinity. Then X 4 corresponds to the coupling m λλ while X 3 , Y 3 correspond to the coupling and the vev of the dimension six operator correspondingly.

Since we know the linear Ricci-flat perturbations explicitly we can analyze their behavior in the UV and IR. There is only one UV divergent mode which corresponds to turning on the coupling of the irrelevant dimension six operator in the dual field theory. To keep the UV asymptotic the same as in the conventional case and hence the field theory well defined in the UV we want to keep this mode "turned off". In the IR there are two regular modes. One is I odd and corresponds to the infinitesimal motion along the baryonic branch. The other is the I even mode that turns on the dimension six coupling -the same mode we discussed above. So there is no Ricci-flat IR and UV regular I invariant infinitesimal deformation of the conventional deformed conifold metric. To find a new Ricci-flat background one has to consider the I breaking modes as well.

The two IR regular linearized modes can be extended into two-parameter family of IR regular Ricci-flat metrics on the conifold. With vanishing fluxes, constant dilaton and h = 1 the IIB supergravity action of [START_REF] Papadopoulos | Complex geometry of conifolds and 5-brane wrapped on 2-sphere[END_REF] reduces to the 6d Hilbert-Einstein action, and so we can read from it the four Ricci flatness equations. We relegated these equations to Appendix A. The two-parameter family of the IR regular solutions is given 4 by:

e 10p(τ ) = 2 3 τ + - 4 45 + ζ 2 τ 3 + . . . , 1 q(τ ) = 15 4 τ + - 7 8 + 45 26 • ζ 2 τ 3 + . . . , e y(τ ) = 1 2 τ + - 1 24 - 15 52 • ζ 2 - 1 2 • ζ 2 1 τ 3 + . . . , z(τ ) = ζ 1 τ 2 + - 7 15 • ζ 1 - 4 5 • ζ 3 1 - 15 13 • ζ 2 ζ 1 τ 3 + . . . . (2.5)
We write here only the result for q(τ ) since adding constant to q(τ ) corresponds to the overall rescaling of the metric (2.2). This rescaling can be in turn absorbed in a redefinition of the deformation parameter . Fixing of in the new solution is an important issue and we will specifically address it in the end of this section. We introduced here two parameters ζ 1 and ζ 2 such that the I symmetry flips the sign of ζ 1 while keeping ζ 2 invariant. The point (ζ 1 , ζ 2 ) = (0, 0) is the conventional deformed conifold metric, i.e. the unwarped metric of the KS solution, while the infinitesimal ζ 2 and ζ 1 correspond to "turning on" the dimension six mode and the GHK mode respectively. Notice that both ζ 1 and ζ 2 appear in the sub-leading order in p(τ ), q(τ ) and y(τ ) and so the solution is indeed regular at τ = 0 for any ζ 2 and ζ 1 exactly as the original KS background.

Let us stress again that we solve the full-nonlinear Ricci-flatness equations and ζ 2 , ζ 1 are not assumed to be infinitesimally small in (2.5).

Regularity in the UV provides only one constraint on ζ 1,2 and therefore we expect to find a one-parameter family of non-singular Ricci-flat metrics on the deformed conifold, which approach the conventional metric at large radius. The boundary conditions at τ = ∞ should coincide with the deformed conifold values (2.3):

e 10p(τ ) ≈ 1 , 1 q(τ ) ≈ 9 2
, e y(τ ) ≈ 1 , z(τ ) ≈ 0 .

(2.6)

Figure 3: The four lines show numerical solutions for e 10p(τ ) , 1/ q(τ ), e y(τ ) and z(τ ) for (ζ 1 , ζ 2 ) = (0.193(9), 0.009) (the last point on the previous plot).

In Appendix A we study the subleading terms and show that the modes of dimension 2, 3 and 6 indeed appear in the large τ expansion as expected.

Near the origin the leading perturbation (see the z(τ ) expansion in (2.3)) is proportional to ζ 1 (meaning it is I odd) and therefore at least for small ζ 2 and ζ 1 is it convenient to parameterize the family of new Ricci-flat metrics by ζ 1 :

ζ 2 (ζ 1 ) ∼ ζ 2 1 + O(ζ 4 1 ) , (2.7)
where the ζ 3 1 term is ruled out because of the I-parity. We found the function ζ 2 = ζ 2 (ζ 1 ) numerically using the shooting technique and confirm a generic prediction that the expansion (2.7) starts with ζ 2 1 (see Figure 2). We also present a numerical solution for the functions e 10p(τ ) , 1/ q(τ ), e y(τ ) , z(τ ) for some particular values of ζ 1 and ζ 2 , see Figure 3. Finally, on Figure 4 we compare our numerical results for e y(τ ) with its analytical conventional (KS) counterpart of [START_REF] Candelas | Comments on Conifolds[END_REF].

It would be interesting to know how far one can move along this new family of solutions and if there is any limit on the resulting value of U . These questions would require careful numerical studies which we leave for the future. 9), 0.009), while the red dashed line corresponds to e y 0 (τ ) = tanh( τ 2 ) of the conventional deformed conifold of [START_REF] Candelas | Comments on Conifolds[END_REF]. Notice that red line converges to the asymptotic faster than the blue line. This is so because for large τ the function 1 -e y(τ ) goes like e -τ for the the deformed conifold, and as e -τ /3 for our solution.

Ricci-flat metric on deformed conifold: a summary

The construction of the novel family of Ricci-flat metrics on deformed conifold presented above constitutes a mathematical result which could be of interest in its own right, with no connection to the ten dimensional supergravity or dual gauge theory. To make this aspect of our work more accessible to reader less familiar with the holographic context we briefly outline here main steps of section 2.1. Our goal was to construct novel Ricci-flat metrics on the deformed conifold -the complex three-fold described by the equation 4 i z 2 i = 2 . We construct the Ricci-flat metric by help of the ansatz (2.2). This ansatz is by no means general. In particular it explicitly preserves the SU (2) × SU (2) symmetry. Since all functions p, q, y, z depend only on the radial coordinate τ the Ricci-flatness condition R µν = 0 reduces to a set of coupled ordinary differential equations (A.1). Our ansatz (2.2) is invariant under the shift τ → τ + const. We choose this freedom to set τ = 0 as the origin -the tip of he conifold. That would be reflected in the appropriate behavior of the metric there i.e. particular boundary conditions for p, q, y, z at τ = 0. We find a two-parameter family of solutions regular at τ = 0, i.e. satisfying these boundary conditions, by ex-panding in power series in τ for small τ . The result is given by (2.5). Two integration constants ζ 1,2 are free parameters at this point. We want the metric to approach the usual conical form at large radius τ → ∞. Here by usual we mean the canonical form of the metric on the cone over T 1,1 . There is one Ricci-flat metric on the deformed conifold, found by Candelas and de la Ossa [START_REF] Candelas | Comments on Conifolds[END_REF], which does just that. This particular metric is captured by our ansatz. At the next step we study the behavior of linear fluctuations around this metric, all within our ansatz of choice. The system of linear equations reveals that there is only one mode which tends to grow at large τ . All other modes actually vanish in this limit. On general grounds we expect that by choosing appropriate infinitesimal ζ 1 , ζ 2 one can fine-tune the behavior of the metric near τ → 0 such that the growing linear mode will not be present at large τ . Hence the behavior at large τ would be the same as in the solution of [START_REF] Candelas | Comments on Conifolds[END_REF], as well as the linear analysis around it: there is still only one growing mode at large τ . In this way we find that there must be a line in the ζ 1 , ζ 2 plane such that for each point on this line the resulting metric has the same asymptotic behavior at large τ . These metrics are Ricci-flat by construction, as well as regular everywhere including appropriate behavior at the tip τ = 0 and at infinity τ → ∞.

Adding fluxes and warping

In the previous subsection we outlined the way to find a regular Ricci-flat metric on the deformed conifold such that it approaches the conventional metric at large radius. The next step is to supplement such a metric with a smooth ISD three-form flux which would approach the KS asymptotic in the UV. Below we give an argument why this is possible for any metric found in the previous section.

We start with the PT ansatz for the three-form flux which automatically includes M units of RR flux through the three sphere at the tip and write down the ISD condition:

ḟ = (cosh(y) cosh(z) + sinh(y)) 2 -F • cosh(2y) + sinh(2y) cosh(z) k = cosh 2 (y) sinh 2 (z) + F • cosh(2y) -sinh(2y) cosh(z) Ḟ = 1 2 (k -f ) (2.8) χ = -cosh(y) sinh(z) (cosh(y) cosh(z) + sinh(y)) -2F • sinh(y) .
These functions are related to the functions used in [START_REF] Papadopoulos | Complex geometry of conifolds and 5-brane wrapped on 2-sphere[END_REF] in the following way:

(h 1 , h 2 , b, χ) PT = (-P g s (k + f ), -P g s (k -f ), 2F -1, 2P χ) here , (2.9)
where P is a constant proportional to M , P = -M 2 s /4. It is easy to verify that for e y(τ ) = tanh(τ /2) and z(τ ) = 0 this system of equations reduces to the one discussed originally in [START_REF] Klebanov | Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities[END_REF].

Naïvely the solutions are parameterized by four integration constants. One of the constants is a shift of χ → χ + const which corresponds to a gauge transformation B → B + const • dg 5 and hence is unphysical. Another constant is a shift (f, k) → (f + b, k + b) for a constant b which corresponds to a "large gauge transformation" B → B + b • w 2 , where w 2 is a Betti form on the base of the conifold. This shift introduces extra Page D3-brane charge to the system. This symmetry will be fixed by the requirement that the Page D3-charge vanishes at the origin, namely that B ∧ F 3 = 0 at τ = 0. Therefore there are only two non-trivial parameters left.

In the UV the metric and fluxes should approach the KS values. Linear analysis around the KS background shows that apart from the simultaneous shift of f and k there are indeed two modes [START_REF] Kuperstein | Analytic non-supersymmetric background dual of a confining gauge theory and the corresponding plane wave theory of hadrons[END_REF]: one is UV regular and the other is UV singular diverging as e τ . The former is related to the ∆ = 3 operator, while the latter is the (0, 3) flux dual to the ∆ = 7 operator d 2 θ(W 2 W 2 ) + . Eventually we would like to "turn off" this irrelevant operator "killing" one of the two free parameters.

It seems like we get a one-parameter family of UV-regular solutions of the ISD equations (2.8). This is not the end of the story, however, as we have yet to address the IR regularity. Differentiating f (τ ) and k(τ ) we can arrive at a second order PDE for F (τ ) only. Using (2.5) we see that, exactly as in the KS case, near τ = 0 the two solutions of this equation are τ -1 and τ 2 . Plugging the latter into the equations for f (τ ) and k(τ ) one finds that these functions have the same regular behavior in the IR like in [START_REF] Klebanov | Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities[END_REF]. This in turn implies that |F 2 3 | = |H 2 3 | does not diverge at τ = 0 and so the background is IR regular. To be more specific, requiring that f (τ ) and k(τ ) vanish at τ = 0 we derive from (2.8) and (2.5) that there is a one dimensional family of IR regular ISD fluxes parametrized by ζ 3 : We now have to find ζ 3 so that the divergent ∆ = 7 mode is "turned off" in the UV. Then for large τ we have F (τ ) ≈ 1/2, while both f (τ ) and k(τ ) go like τ /2, reproducing the UV behavior of the KS solution. Employing the shooting technique once again we found that ζ 3 = 0.186 [START_REF] Gaillard | The warped, resolved, deformed conifold gets flavoured[END_REF] for ζ 1 = 0.193(9) (the last point in Figure 2). The corresponding numeric solution for F (τ ) is shown in Figure 5. With F (τ ) at hand, we can easily calculate the remaining three functions. In Figure 6 we provide a plot for χ(τ ). As a matter of convenience, we fixed the integration constant so that χ(τ ) vanishes for large τ , see Figure 6.

F (τ ) ≈ (1 + ζ 3 ) • τ 2 12 , f (τ ) ≈ τ 3 12 , k(τ ) ≈ (1 + ζ 3 ) • τ 3 , χ ≈ - 1 6 ζ 1 (4 + ζ 3 ) • τ 2 . ( 2
To complete our discussion of the ten dimensional supergravity background we need to accompany our solution by the warp factor of the 10d metric (2.1). The equation for h(τ ):

h(τ ) = √ 3 8 α ∞ τ dx e -4p(x)-6q(x) (1 -F (x))f (x) + F (x)k(x) , (2.11)
where:

α ≡ 4 g s M 2 s 2 • -8/3 . (2.12)
This equation immediately reveals that h(τ ) is regular5 at the tip and approaches the The last touch is to fix the deformation parameter (as we have already mentioned below (2.5) this is equivalent to adding a constant to the field q(τ )).

Since is the only dimensional parameter of the solution we can only talk about fixing it in the context of comparing for two different solutions. We want to fix (ζ 1 ) such that the corresponding family of gravity backgrounds would correspond to the same dual field theory perturbed by some relevant operators in the UV. In particular, this would require running of the Maxwell D3-charge 6 Q D3 with the scale to be the same at large radius r for all ζ 1 . A convenient reference point is ζ 1 = 0, i.e. the KS solution. Thus, to match the KS theory in the UV we require:

Q D3 (r c ) = Q KS D3 (r c ) (2.13)
at a sufficiently large cutoff radius r c . Here the Maxwell D3-charge is (see, for example, vanishes at infinity exactly like in the unperturbed KS background. 6 To be more precise, one has to compare the running of the Page charges, since unlike the Maxwell charge it is quantized and so has the proper interpretation as the gravity dual of the cascading gauge theory rank. It turns out, however, that the matching of the Maxwell charges yields exactly the same result for .

[4]):

Q D3 (τ ) = g s π M 2 ((1 -F (τ ))f (τ ) + F (τ )k(τ )) . (2.14)
It seems that this expression is completely independent of . Notice, however, that τ = ln(r 3 / 2 ) and so requiring that the two Maxwell charges in (2.13) coincide for the same r = r c (but different τ 's) we can read from (2.13) the ratio between the KS parameter KS and the parameter of the perturbed theory, . For instance, for the theory corresponding to the last point on Figure 2 (the one with ζ 1 = 0.193( 9)) we have Q D3 (τ = 11.866( 2)) = Q KS D3 (τ = 12) which implies / KS = 1.069. It is worth emphasizing here that for large enough r c (or equivalently τ ) the value of / KS is not sensitive to the variation of the cut-off, since asymptotically Q D3 (τ ) ∼ τ + const.

A more detailed discussion of the condition (2.13) and the resulting constraint on in the context of SUSY and non-SUSY vacua of the KS theory can be found in [START_REF] Dymarsky | On gravity dual of a metastable vacuum in Klebanov-Strassler theory[END_REF].

Dual field theory and stability

In this section we discuss the field theories dual to our gravity backgrounds. We suggest that these non-supersymmetric theories correspond to the KS theory perturbed by both a single and a double-trace operators. We also argue that, at least for large M , these theories are long-lived both perturbatively and non-perturbatively.

The dual field theory interpretation

Once we outlined the way to find the gravity backgrounds we want to analyze their meaning from the dual field theory point of view. As we require the backgrounds to approach the KS solution in the UV the resulting field theory must be the original cascading N = 1 "conifold" theory perturbed by some relevant and/or marginal operators. In the single trace sector these are the I-odd bottom component of the U (1) baryon multiplet U and the gaugino bilinears. It is easy to see that the marginal top components of W 2 ± are not turned on: the value of dilaton and running of the Maxwell D3-charge are the same for large r as in the KS case. (In any way adding top components of W 2 ± would only result in the renormalization of gauge couplings). Hence the solutions in question are dual to the vacua described by the N = 1 KS theory perturbed by some, not necessary small, combination of the gaugino masses m λ 1 λ 1 ± m λ 2 λ 2 and the operator U . We will not be able to identify the particular combination of m λ 1 λ 1 ± m λ 2 λ 2 that is getting turned on (see our discussion in the previous section) and in what follows will refer to this combinations simply as m λλ . The vacuum solutions also acquire a non-trivial vev of U , as well as vevs of other relevant and irrelevant operators.

For most operators it is straightforward to distinguish the vev from the coupling as the two modes have different powers of 1/r where the radial coordinate r = 2/3 e τ /3 .

(

The operator U , however, has dimension two and therefore the two modes mix. The corresponding wave-function, which for large radius can be defined through:

δds 2 = h(r)(e 2 1 + e 2 2 -2 1 -2 2 )Ψ U (3.2)
has the following asymptotic [START_REF] Gubser | Symmetry breaking and axionic strings in the warped deformed conifold[END_REF]:

Ψ U = β ln r r 0 + α . (3.3)
Here r 0 is some IR scale which we choose such that, in the absence of multi-trace deformations, the coefficient β corresponds to the vev of U while α corresponds to its coupling. In general r 0 may depend on various couplings of the theory, such as m λλ . In the KS case, when all relevant couplings (including m λλ ), are turned off, r 0 = 2/3 e 1/3 [START_REF] Gubser | Symmetry breaking and axionic strings in the warped deformed conifold[END_REF].

We do not know r 0 (m λλ ), although this can be established in principle. Therefore even if we find our solutions numerically with a very high precision, we will not know α(ζ 1 ). 7 Nevertheless, on general grounds we expect:

α ∼ ζ 3 1 + O(ζ 5 1 ) , β ∼ ζ 1 + O(ζ 3 1 ) . (3.4)
This scaling is easy to explain. β corresponds to the vev of U which, by definition of ζ 1 , scales as ζ 1 . Moreover at the linear order in ζ 1 the solution coincides with the GHK background, i.e. has zero coupling α. Since α is odd under I symmetry, as well 7 In practice the main difficulty here is to find the solutions numerically with enough precision to distinguish small α which appears only in the ζ 3 1 order from the large logarithmic contribution β ln(r/r 0 ) ∼ ζ 1 τ .

as ζ 1 , it can only start with a cubic term. Let us demonstrate that the same scaling is compatible with (3.3). From the fact that m λλ is I-even we find that:

m λλ ∼ ζ 2 1 + O(ζ 4 1 ) . (3.5)
Now, we generally expect r 0 to be m λλ -dependent such that ∂r 0 /∂m λλ = 0. The change in r 0 can be reabsorbed into α which indeed indicates that α ∼ ζ 3 1 . A non-zero value of α implies that besides the gaugino mass the dual field theory is perturbed by the dimension two operator U . Definitely this can not be the full story: the theory perturbed by the potential V = aU does not have a vacuum at small U . We find a consistent interpretation only at the next order in U -at least for small ζ 1 the gravity solutions in question can be interpreted as a dual to the theory perturbed by the marginal double-trace8 operator cU 2 /2. The coupling c will alter the boundary conditions in the UV and hence the dual field theory interpretation of α. More precisely, when the field theory perturbed by a potential:

V = aU + c U 2 2 , (3.6)
the boundary condition for Ψ U becomes [START_REF] Witten | Multitrace operators, boundary conditions, and AdS / CFT correspondence[END_REF][START_REF] Berkooz | Double trace' deformations, boundary conditions and space-time singularities[END_REF]:

α = a + cβ . (3.7)
When c = 0 we get the usual relation α = a, i.e. in this case α is the coupling of U . Since the theory with the V = aU potential has no vacuum state there should be no IR regular solution with α = a = 0. In the general c = 0 case the resulting field theory has a unique vacuum at U = -a/c. The corresponding gravity dual background should have α and β such that they satisfy (3.7) and the geometry is regular in the IR. Since the IR regularity requires α = 0, we see that β = -a/c, in full agreement with β = U and dV /dU = 0.

It is important to note that the gravity solution itself can not distinguish between different combinations of a, c which result in the same vev of U . 9 In fact the same gravity solution admits interpretation as both being dual to a field theory with and without the double trace deformation (see [START_REF] Hertog | Towards a big crunch dual[END_REF][START_REF] Hertog | Holographic description of AdS cosmologies[END_REF] and [START_REF] Maldacena | Vacuum decay into Anti de Sitter space[END_REF] correspondingly). Using this freedom we would like to interpret our solutions as the gravity dual of the field theory perturbed by certain single trace relevant operators and also the marginal double-trace operator U 2 . Our logic here was to show that it is a consistent interpretation when in the dual field theory a ∼ ζ 1 and c ∼ ζ 2 1 . Strictly speaking, the consideration above and the equivalence between vacuum value of U in the field theory and in the bulk is only valid as long as the theory with a = c = 0 has a flat direction for VEV of U . This is indeed true for infinitesimal ζ 1 as it it can be shown (using the charges under R-symmetry) that the gaugino masses can only contribute to the potential for U at the subleading order m 2 λλ ∼ ζ 4 1 . But we believe that our interpretation -that the solutions we found are dual to the KS theory perturbed by gaugino and the potential V = aU + cU 2 /2 -should be correct beyond the infinitesimal order in ζ 1 . In this case the gaugino masses induce a potential for U (on top of V = aU + cU 2 /2 which we add "by hands" in the UV) and the calculation of vacuum value of U is obscured on the field theory side. Similarly on the supergravity side (3.7) still holds but we do not know which combination of α, β corresponds to the IR regular background. Thus we can not easily compare field theory and gravity to prove our point. At the same time the fact that we included all relevant/marginal operators (allowed by symmetries) into consideration and the fact that this interpretation works for small ζ 1 , suggests that the same interpretation should be correct beyond the infinitesimal order in ζ 1 .

Stability

The field theory interpretation suggested above ensures the perturbative stability for infinitesimally small ζ 1 as long as c > 0 (see (3.6)). There are many field theories dual to the same gravity background and we choose the one with positive c: for any theory with a, c dual to a given background the theory with -a, -c is dual to the same background as well. Indeed when α = 0 this flipping will not modify β, as one can see from (3.7), and so the gravity solution will be the same.

On the gravity side the stability would follow from the analysis of small perturbations around the classical solutions in question with the boundary conditions: δα = c δβ .

(3.8)

Clearly the sign of the perturbation mass squared will be sensitive to the sign of c. More specifically, in solving the equations for massive modes, the sign of the 4d dimensional mass squared will be sensitive to the relative sign of δα and δβ, which in turn is fixed by c.

By interpreting the gravity solutions as being dual to the field theory perturbed by the double-trace operator we ensured perturbative stability, at least for the solutions with infinitesimally small ζ 1 . For larger ζ 1 we would need to calculate the mass-squre m 2 U of small perturbations of U to check that is it positive. For infinitesimally small ζ 1 , m 2 U ∼ c and the next correction appears in the ζ 4 1 order

m 2 U ∼ ζ 2 1 + O(ζ 4 1 ) . (3.9)
Since the coefficient in front of the first term is positive, m 2 U is positive at least for some finite range of ζ 1 1. Hence the vacuum states, and the dual gravity backgrounds, are perturbatively stable for ζ 1 1.

Besides the perturbative stability, one may also worry about possible tunneling between the 2M vacua of the original KS theory with different phases of Λ 3 . When ζ 1 is small, i.e. when m λλ is small, one can estimate the shift of vacuum energy in each of those vacua as [START_REF] Kuperstein | Analytic non-supersymmetric background dual of a confining gauge theory and the corresponding plane wave theory of hadrons[END_REF]:

V M Re m λλ Λ 3 . (3.10)
Our solutions correspond to m λλ aligned with Λ 3 -they have the opposite phase such that their product is real. This is so since we have not included in our setup the mode λ(τ ) mentioned briefly in footnote 2 and its counterpart in the fluxes. It remains unclear, though, whether m λλ Λ 3 is positive or not. Hence our solutions may be unstable non-perturbatively.

Let us show that in any case the theories in question are long-lived, if we do not depart too far from the KS solution. At the leading order the tension of the domain wall separating various vacua remains the same as in the supersymmetric case [START_REF] Dymarsky | On the moduli space of the cascading SU(M+p) x SU(p) gauge theory[END_REF]: is smaller than the decay rate for tunneling into the true vacuum in one leap. The latter is suppressed by: e -S bubble ∼ e -M (Λ/m λλ ) 3 .

T n = M |Λ|
(3.13)

Because of large M for m λλ Λ i.e. ζ 1 1 the solutions in question are metastable.

This analysis of stability may be too conservative. In fact it may be that the signs of m λλ and Λ 3 are aligned such that the solutions we find correspond to the true vacuum.

Outline

We constructed a one-dimensional family of IIB supergravity solutions which are dual to the N = 1 gauge theory perturbed by some combination of relevant single trace and marginal double-trace operators. Thus the family of solutions corresponds to a line in the space of couplings g i (ζ 1 ). The origin of the line g i (0) = 0 corresponds to the original N = 1 theory [START_REF] Klebanov | Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities[END_REF]. Except for the small region near the origin ζ 1 → 0 the couplings are not infinitesimally small and the corresponding operators explicitly break supersymmetry. Hence our solutions are gravity duals for non-supersymmetric field theories.

The original N = 1 theory has only one massless mode which acquires a positive mass, at least for the theories within a range ζ 1 1. Therefore the solutions in question are perturbatively stable. There is a possible non-perturbative instability associated with tunneling into a vacuum with Λ 3 new = -Λ 3 . But at least for ζ 1 1 the decay rate is exponentially suppressed.

The geometry of the solutions is extremely simple. The unwarped metric on the deformed conifold is Ricci-flat but not Kähler for ζ 1 = 0. The three-form fluxes are ISD and the dilaton is constant. There is also a warp-factor which has a similar behavior to its KS counterpart. The solutions are regular and all fields approach their KS values in UV, up to 1/r corrections.

The apparent simplicity of the solutions makes them a natural arena to study strongly coupled dynamics of the non-SUSY confining gauge theories. Similarly to the KS background, which was a playground for numerous phenomenological models, our solutions can be used for model-building with an advantage that the resulting models are not supersymmetric. Because of the GKP structure of the solutions the probe D3-brane in such backgrounds experiences no force. Hence our solutions can be a natural starting place for the models of stringy inflation based on the dynamics of a mobile D3-brane [START_REF] Kachru | De Sitter vacua in string theory[END_REF][START_REF] Kachru | Towards inflation in string theory[END_REF][START_REF] Baumann | A Delicate Universe[END_REF][START_REF] Baumann | Towards an Explicit Model of D-brane Inflation[END_REF][START_REF] Baumann | Holographic Systematics of D-brane Inflation[END_REF][START_REF] Baumann | D3-brane Potentials from Fluxes in AdS/CFT[END_REF]. In a similar way the GKP structure makes it possible to use the trick of [START_REF] Dymarsky | Chiral Symmetry Breaking with non-SUSY D7-branes in ISD backgrounds[END_REF] in its original form and embed the U-shaped D7-branes inside the conifold. Eventually one arrives at a holographic model of baryonic matter with the SUSY broken explicitly by the relevant operators in the gauge sector, and not only by the flavor sector. Moreover because of the KS asymptotic at infinity the resulting baryons will develop a realistic attractive potential at least in some range of parameters [START_REF] Dymarsky | Attractive Holographic Baryons[END_REF]. Finally, it would be interesting to understand the appearance of the flat moduli associated with the mobile D3-branes from the point of view of the dual non-supersymmetric field theory (see [START_REF] Klebanov | Membranes with Topological Charge and AdS 4 /CF T 3 Correspondence[END_REF] for a similar situation in a 3d dimensional theory).

  Figure1: We define M U as a space of combinations of relevant/marginal couplingsand the corresponding theories -such that the bottom component of the U (1) baryon multiplet has the vacuum value U . Schematically we denote x to be the coordinates on M U (we do not specify the dimension of M U ) such that x = 0 corresponds to the unperturbed N = 1 theory. Red dashed line corresponds to the motion in the space of theories starting from the unperturbed N = 1 theory such that vev U stays constant. Most of these theories have complicated gravity dual. At a specific point the gravity dual accidently becomes simpler -it is of GKP type. The one-dimensional family of such theories forms the thin blue line. These are the theories we study.

Figure 2 :

 2 Figure 2: The red points show the values of ζ 1 and ζ 2 for which the Ricci-flat metric is regular in both the IR and the UV. Notice that the points near the origin lie on the parabola (the blue line). The deviation of the points with larger ζ 1 from the parabola most likely indicates the contribution of the ζ 4 1 term in (2.7).

Figure 4 :

 4 Figure 4: The blue solid line show the numerical result for e y(τ ) with (ζ 1 , ζ 2 ) = (0.193(9), 0.009), while the red dashed line corresponds to e y 0 (τ ) = tanh( τ 2 ) of the conventional deformed conifold of[START_REF] Candelas | Comments on Conifolds[END_REF]. Notice that red line converges to the asymptotic faster than the blue line. This is so because for large τ the function 1 -e y(τ ) goes like e -τ for the the deformed conifold, and as e -τ /3 for our solution.

. 10 )

 10 In particular ζ 1 , ζ 3 = 0 for the KS background.

Figure 5 :

 5 Figure 5: The solid blue and the dashed red lines correspond to the numeric solution for F (τ ) with ζ 1 = 0.193(9) and the same function in the KS geometry, respectively.

Figure 6 :

 6 Figure 6: The numerical solution for χ(τ ) with ζ 1 = 0.193(9).

3 1 -

 1 e iπn/M , n = 1, . . . , 2M .(3.11)When m λλ is small enough we can use the thin wall approximation and estimate the size of a new vacuum bubble by minimizing the following effective action:S bubble ∼ ∆V R 4 -T ∆n R 3 . (3.12)For the adjacent vacua with ∆n = 1 we have T ∼ |Λ| 3 while ∆V is 1/M suppressed. For a one leap transition into the true vacuum ∆n = M we get T ∼ M Λ 3 and ∆V ∼ M m λλ Λ 3 . The decay rate for tunneling into a new vacuum with Λ 3 → Λ 3 e iπ/M

This question is successfully attacked in[START_REF] Bennett | The Non-SUSY Baryonic Branch: Soft Supersymmetry Breaking of N=1 Gauge Theories[END_REF].

It is possible to introduce another fifth function λ(τ ) by substituting ψ → ψ + λ while keeping dψ intact. Obviously the constant part of λ is a pure gauge associated with the action of U (1) R . Introducing λ will not lead to any new solutions as the Ricci-flatness requires λ = 0.

Our z(τ ) coincides with the one used in[START_REF] Gubser | Symmetry breaking and axionic strings in the warped deformed conifold[END_REF] only at the linear order. Also, for z = 0 our notations reduce to those of[START_REF] Kuperstein | Analytic non-supersymmetric background dual of a confining gauge theory and the corresponding plane wave theory of hadrons[END_REF]. This is the reason for using p in the new notations.

We choose the field parametrization here (e 10p(τ ) instead of p(τ ), q-1 (τ ) instead of q(τ ) etc) so that all the expressions will be regular both at the origin and for large τ .

We fixed the integration constant in the equation for h(τ ) by requiring that the warp function

The dimension of the double trace operator U 2 is twice the dimension of U up to 1/N corrections.

We thank I. Klebanov and J. Maldacena for discussing this point.

Acknowledgments

We thank N. Halmagyi for collaboration at the initial stages of this project and G. Giecold and C. Nuñez for reading the manuscript. We are also grateful to I. Klebanov, Z. Komargodski, J. Maldacena, N. Seiberg, M. Shifman and A. Yung for numerous discussions. A.D. thanks the theory group at Institut de Physique Théorique, CEA-Saclay and High Energy Group at Cornell University for hospitality. The research of A.D. was supported by the DOE grant DE-FG02-90ER40542, by Monell Foundation, and in part by the grant RFBR 07-02-00878 and the Grant for Support of Scientific Schools NSh-3035.2008.2. The work of S.K. was supported in part by the ERC Starting Independent Researcher Grant 240210 -String-QCD-BH. S.K. would like to thank Institute for Advanced Study for hospitality and the European Commission Marie Curie Fellowship (under the contract IEF-2008-237488) for the financial support during the US visit.

Appendix A A. The Ricci flatness equations

Here we summarize the Ricci-flatness equations for the 6d metric (2.2) p + ṗ(4 ṗ + 6 q) = -2 15 e -20p 1 + 2 cosh 2 y sinh 2 z (A. The solutions of these EOM are also subject to the zero energy (zero Hamiltonian) condition following from the action. This condition can be used to find q(τ ), and so we are left only with the equations for p(τ ), y(τ ) and z(τ ).

To study the large τ behavior we have to expand the equations linearly around (2.4). The equations for δy(τ ) and δz(τ ) are both homogenous. The δz(τ ) equation is solved by e -2τ /3 and τ e -2τ /3 , as expected for the the ∆ = 2 operator. The two solutions for δy(τ ) are e -τ and e -τ /3 corresponding to ∆ = 3. Finally, the homogenous part of the equation for δp(τ ) gives two modes, e -2τ and e 2τ /3 , implying that the mode has dimension ∆ = 6.


Alain Nadaï
email: nadai@centre-cired.fr

Olivier Labussière
email: olivier.labussiere@umrpacte.fr

Social acceptance: beyond criticism and critical, a call for experimental ontology

Keywords: social acceptance, renewable energy technologies, interferences, co-production, experimental ontology

This chapter looks back at our own research practice about the social acceptance of renewable energy technologies (RET) and reflects on the articulation between our practice and the way in which we conceptualise social acceptance.

We do so based on our own work about French onshore wind power development over the past fifteen years. We examine the factors that influenced the course of our research work, we retrospectively distinguish between three ways in which we have addressed acceptance issues, and discuss this evolution in relation with the academic field of social acceptance of RET.

This allows us to reflect on the dynamics of this academic field and to conclude that a different relation and engagement of social science researchers with the fieldwork is called for.

Introduction

Research on the social acceptance of renewable energy technologies (RET) started in the nineties, with successive 'waves' of research problematising and critiquing the notion [START_REF] Batel | Research on the social acceptance of renewable energy technologies: Past, present and future[END_REF]. While early works uncritically examined this issue with a view to promote acceptance -most often under the heading of the NIMBY notion -social sciences have then challenged the relevance of NIMBY. They argued that the notion lacks consideration for the potential of the 'social' to evolve, because it locates the potential for change in the technologies and the potential for resistance in the 'social' [START_REF] Shove | Social Theory and Climate Change[END_REF][START_REF] Batel | Social acceptance of low carbon energy and associated infrastructures: A critical discussion[END_REF]. As a matter of fact, all developments in RET technologies confirm this criticism : new collectives and new relations emerge with these developments and witness of the potential of the 'social' to accompany, if not to take an active part in, these changes. A major issue for the protagonists in these processes -and an actual source for resistance -is how to express the problematic consequences of these developments so as to make these consequences exist as a shared problem, and be addressed (Labussière & Nadaï, 2018).

As suggested in the framing of this part of the book, any research work endorses a specific way of representing the issues that arise in relation with RET development, and it is then interesting to better understand the articulation between research practices and ways of conceptualising social acceptance. This chapter ventures in this direction by looking back at our own research practice in analysing French onshore wind power development over the past fifteen years. Our research work is embedded in what we may call, referring to Bruno Latour, a politics of things (Latour, 2005a[START_REF] Latour | Reassembling the Social-An Introduction to Actor-Network-Theory[END_REF]. Much of our effort has been targeted at understanding the opposition to RET in relation to the politics that underlie their developments by following, describing and acknowledging the way in which many entitieshumans but also non humans -are embarked and transformed in the process of developing these technologies. In what follows, we try to reflect on the articulation of this research practice with the conceptualisation of social acceptance, to posit our successive problematisations of social acceptance within the academic field, and to a certain extent, to reflect on the representation of the dynamics of this academic field.

We first examine the factors that have influenced the course of our research work ( §.1), we retrospectively distinguish between three ways in which we have addressed acceptance issues, which we successively present ( §.2 to 4) before discussing this evolution in relation with the academic field of social acceptance of RET ( §.5).

This reflection points at a reformulation of the notion of social acceptance in order to address the web of interferences 1 at play in energy transitions processes. It also points out that 1 The notion of "interference" brings the emphasis on emerging relational realms induced by energy transition processes.

We distinguish between four notions : "interaction" and "interference" , "relationship" and "interrelationship". To put it shortly, the notion of interaction pertains to analytical strands -such as system analysis -which presuppose the existence of distinct entities and analyse the set of relations between these given entities. Different from this, the notion of interference points out situations in which the parts do not necessarily precede their being part of a common reality. In such situations, the relations are constitutive of the parts, which ontologies then depends on the set of relations at work. Importantly, the becoming of interferences is highly uncertain, as they can evolve towards reciprocal modes of relationships acknowledging the consequences of the process for all the parties -which we term "interrelationship" -or towards ill-related and sometimes conflictual reality.

pretending to do so challenges our position as social scientists in taking part into the field, the temporality and the reciprocity of fieldwork. Hence, challenging the notion of social acceptance is not enough. Shifting to interferences is just but a point of departure towards more engaged research practices.

Analytical threads

Different factors may influence the way in which we, as researchers in social science, construct our research questions and programs and, henceforth, frame our conception of Our research program on French wind power started with the analysis of the first French parliamentary debate (Nadaï, 2007a) concerning this development and the rising opposition to it, five years after the adoption of national feed in tariffs for renewable energies. We then followed during ten years the evolution of this policy. We emphasized an inherited centralisation of energy policy and infrastructures, an early economic policy framing -feed in tariff being its pivotal instrument -, a reliance on private wind power developers, and a defiance of national politicians towards the capacity of local authorities to regulate the spatial development of renewable energies (Labussière & Nadaï, 2015). It was only at the end of the last decade -marked with a major concertation process about environmental issues, the Issues of social acceptance of renewable energy projects have thus been framed in different ways by French policy makers -through economic incentives, local planning, regional planning, participative financing.

As researchers in social science, who were partly funded by public policy programs, we faced these evolutions both as issues and political framing. While most of the calls for research under which we got financing left us pretty free of our analytical framing, the academic requirement to regularly -hence, swiftly -publish has probably been the most impacting requirement on our explorations, as it imposed a pace that did not necessarily allow time for deep critical reflexivity. Such reflexivity had to be gained and constructed through time, along opportunities for publishing, and with the hint of academic networks, and fieldwork encounters. Opportunities for publishing often had to be seized in the flow and came as moments of guided re-elaboration of already undertaken case studies, as crystallization of understandings. Academic and theoretical affinities have been decisive, if only because they steered encounters.

As far as our past publications about French wind power development witness it, our academic affinity has been with what we could name relational approaches -i.e. approaches that do not take individualities or entities as given and stable, but rather as in-the-making through processes and relations. This meant to follow, describe and acknowledge the way in which processes of developing renewable energies embarked entities -humans and non humans -and changed ways of relating with / among them. While this resumes an enduring line of concern in our analyses of RET development, the theoretical strands we conveyed in the analysis have evolved. Part of them have been key references along the way, such as [START_REF] Akrich | La construction d'un système socio-technique. Esquisse pour une anthropologie des techniques[END_REF], [START_REF] Latour | Reassembling the Social-An Introduction to Actor-Network-Theory[END_REF] or [START_REF] Simondon | Du mode d'existence des objets techniques[END_REF] relational approach to technology. Others have being called for in specific works or on specific objects, such as: non representational approaches to analyse the spatial/ landscape planning of RET [START_REF] Deleuze | Différence et répétition[END_REF] -map/tracing, sign ; affect/ethos) ; and pragmatist sociology to analyse the democratic dimension of energy transition processes (Dewey, 1946 -public, problematicness;[START_REF] Marres | The Issues Deserve More Credit: Pragmatist Contributions to the Study of Public Involvement in Controversy[END_REF]Marres, , 2012 -issue -issue making, material participation).

In the end, this line of works converged with critical strands to both social acceptance (challenging the idea that opposition to RET should be reduced/ overcome -Batel, 2020)and the energy transition (challenging the democratic dimension of energy transition processes - Labussière & Nadaï, 2018), as well as with a co-productionist agenda according to which the social and the sociotechnical are jointly produced (Chilvers & Kearnes, 2016).

However, the many factors influencing our analyses and the particular organisation and pacing of academic production, make it impossible to delineate clear chronological sequences in the work : publication do not necessarily follow fieldworks in time; occasions for publication (e.g. thematic issues) may influence the framing of case studies; fieldwork encounters can impact the framing of older case studies in new publications … Rather, we distinguish between three threads of analyses, respectively interested in: i/ the role of planning / permitting procedures and landscape policy in wind power development; ii/ the way in which different entities, humans or non humans, are acknowledged (or not) in the development of renewable energy projects; iii/ the way in which resources, space and time are constructed in energy transition processes, resulting in certain political framing of -and a democratic deficit in -energy transition processes.

While this is based on reviewing our own work, it of course partly echoes results that have been pointed out in social acceptance research with different frames and terms [START_REF] Ellis | The social acceptance of wind energy: Where we stand and the path ahead[END_REF]. The purpose in this chapter is to both point out the factors having influenced our research practices and the articulation between these practices and our evolving conception of social acceptance.

In what follows, we successively present each of these threads and the way in which they frame tensions around wind power development ( §.2 to §.4), before discussing the ways in which social acceptance has been (implicitly) thematised and displaced in our work as well as the relation of our work to the academic field of sociological analysis of social acceptance.

Thread 1: Acceptance as a consequence of spatial planning (representationsmilieu)

This line of analysis is interested in the role that spatial planning plays in the emergence of tensions around wind power development. It has been initiated in the early phase of our research work, partly because of our background -one of us had a background as a geographer with an interest in the role of aesthetics and forms in spatial planning [START_REF] Labussière | Eléments pour une prospective du milieu. L'enjeu du sensible en aménagement[END_REF], the other as a landscape architect with a recent interest in STS and the emergence of form in landscape projects [START_REF] Nadaï | Site ou l'émergence d'un paysage[END_REF]) -partly because the development of French wind power was triggering landscape, heritage and planning issues (Cf. §. 1).

Early field encounters were decisive in steering our attention to such issues. In 2007, the French Ministry of Culture had gathered in Paris, for a crisis meeting, all of its local branches for heritage protection. Wind turbines threat on heritage landscape, because of their far reaching visual presence in the countryside, had become a key issue. The administrative officer in charge of the protection of the Chartres Cathedral (Eure et Loir, France) presented a bright red urchin-like scheme figuring the dramatic State failure to protect the cathedral (a national heritage, at the centre of the urchin) from the visual interferences of wind turbines (located as dots in radiating red cones). The dotted red radiating scheme was materialising both the visual governmentality of French landscape (cones radiating for the cathedral, to be protected from visual intrusion) and its failure to govern (the decentralised presence of wind turbines in the yet unqualified landscape of the countryside). This setting was so telling that it triggered, for us, an in-depth, theoretical and empirical, inquiry into the making / unmaking of what we ended up naming a 'State landscape' -a formal, visual and centralized administrative tradition of landscapes protection, that was challenged by the emergence of wind power (Nadaï & Labussière, 2015). Why and how this was so, became a research concern. A parallel fieldwork, in the Aveyron department [START_REF] Nadaï | Wind power planning in France (Aveyron): from State regulation to local experimentation[END_REF], also witnessed the failure of normative planning processes. Administrative top down sieve map planning in South Aveyron had steered wind power developers towards areas judged less sensitive as to their heritage or environmental dimensions. This raised local concern about what was perceived as a spatial / environmental disqualification, if not injustice, and triggered the emergence of a local opposition to wind power in the concerned areas.

Working at the crossroad between critical geography, landscape studies and STS, we followed a 'constructive' approach [START_REF] Latour | The Promises of Constructivism, in Chasing Technoscience[END_REF] attentive to the way in which landscape/space was represented and circulated (graphic forms, material supports) in spatial planning. The focus on the relation between the material (un)making of landscape in planning and the rise of local opposition to wind power, was a way to point out factors behind opposition to wind power ('bad' planning). It also shed light on the relation of co-emergence between the technology and its human-non human 'milieu' [START_REF] Simondon | Du mode d'existence des objets techniques[END_REF][START_REF] Akrich | La construction d'un système socio-technique. Esquisse pour une anthropologie des techniques[END_REF]. This consideration was progressively sharpened by analysing a case of successful wind power planning (Narbonnaise, South West France) [START_REF] Nadaï | Playing with the Line, Channelling Multiplicity -Wind Power Planning in the Narbonnaise (France, Aude)[END_REF], which had allowed for a rebound in wind power development, after a local moratorium had been put on this development because of local tensions. The case study proved that, symmetrically, a potential for change and wind power development could emerge from planning processes that were attentive to the milieu (rather than to a predefined norm). This analytical line was further amplified by comparing cases studies and emphasizing the importance of weaving together current socio spatial configurations with inherited ones [START_REF] Labussière | Unexpected Wind Power 'Potentials': The Art of Planning with Inherited SocioGeographical Configurations (France)[END_REF].

Along this thread, the underlying narrative is that social acceptance, even if not explicitly flagged in these analyses, resulted from reciprocal alignment between the technology and its milieu through attentive enough landscape planning processes or RET policy frameworks.

While dialoguing within an academic strand interested in the role of spatial planning [START_REF] Strachan | Learning from Wind Power: Governance, Societal and Policy Perspectives on Sustainable Energy[END_REF], space and place (Devine-Wright & Howes, 2010) for wind power development -and thus pertaining to a "criticism" of social acceptance [START_REF] Batel | Research on the social acceptance of renewable energy technologies: Past, present and future[END_REF] -this thread of analysis, was also feeding a co-productionist agenda (hence 'critical'), be it only because of its reliance on Simondon notion of milieu.

Thread 2: Acceptance as part of the construction of a relational potential (actants -milieu)

The second line of analyses has been more directly interested in the fate of those which/who are embarked in the processes of wind power development. Rather than looking at the work of (mis) alignment through certain mediations -such as planning and landscape representations -the analysis aimed at targeting more directly those humans or non humans that were affected and contributed to the (non) development of specific wind power projects: birds in one case study, villagers in the other.

Once again, this line of analysis has been strongly suggested by field encounters. In 2006, as we were analysing the innovative wind power planning process in the Narbonnaise, we interviewed a local bird watching NGO, which had just undertaken field experiments with a wind power developer. The project was located in a European migratory corridor at the French-Spain border on the Mediterranean shore (Languedoc-Roussillon, France). It involved a collaboration between the local branch of the French bird protection NGO (LPO) and a wind power developer [START_REF] Nadaï | Birds, turbines and the making of wind power landscape in South France (Aude)[END_REF] in order to adapt the siting of an existing wind farm (repowering) to bird migration. Existing wind turbines were turned into a type of lab-scape: bird watchers hided behind the turbines, observed and followed individual birds in their crossing through the existing wind farm, gauging their individual cognitive/strategic ability in flying through it. Both the reshuffling of bird classifications (according to crossing ability rather than to statutory protection) and the drawing of individual bird trajectories allowed for the mapping of migratory micro-corridors, which paved the way for wind farm siting proposals judged compatible with bird migration. By focusing on individual stories so as to capture birds' intelligence, this experiment endowed birds with new capacities and opened a new potential for sharing the wind. In this process, entities were requalified and performed differently: birds became more skilled (wind farm compatible) and the wind farm became more compatible with migrating birds. By focusing on human -non human relations (affect, animal charisma), the analysis brought forth the ontological dimension of the process and its potential to give rise to renewed compatibility between wind power development and bird protection (human-non human assemblage).

The second case study followed the progressive structuring of a network of opponents to a local wind power project in a village located in the Parisian basin (2003-2009) [START_REF] Nadaï | Landscape commons, following wind power fault lines: The case of Seine-et-Marne (France)[END_REF]. The analysis traced issues as they emerged in relation with the project and its interferences with shared landscape resources (a local 'plain') that were collectively managed (land consolidation association) and used at the village level (Sunday walk). Relying on pragmatic sociology of issue making and public formation (Dewey, Marres), it followed a collective of opponents in the making. It showed the ways through which part of these villagers, which at the outset were not opponents to wind power per se or to the project, progressively became opposed to the way in which it was developed. We showed that the incapacity of permitting and policy institutions to acknowledge the collective trouble raised by the project (unravelling of village life, divisions within families, stalling of social activities …) mirrored a narrow framing on renewable energy resources, as wind speed and private land. This case study thus pointed at a failed wind power potential: opposition to wind power resulted from a lack of attention to relations, and especially to the collective dimension of certain resources.

In both case studies, the focus was on the construction (or lack thereof) of specific relations.

Both case studies foreground an ontological dimension of wind power development processes: identities, roles and capacities to influence the course of RET project development are distributed by and through these processes. The case studies flag relational potentials for change in the (un) making. Acceptance (issue) is thus conceived as the outcome of the construction (or lack thereof) of this relational potential. As much as thread 1, this thread is both 'critic' and 'critical' to social acceptance. Yet, it is much more explicit in its ontological lessons, because it makes clear the transformations that entities undergo through RET development processes and the related potentials for contribution (or not) to these processes they are given (or not).

In so doing, such a perspective echoes analyses in the field of participation, which emphasize the construction of publics by participatory settings [START_REF] Lezaun | Consulting citizens: technologies of elicitation and the mobility of publics[END_REF][START_REF] Laurent | Technologies of Democracy. Experiments and Demonstrations[END_REF].

Thread 3: Acceptance (issue) as an expression ('problematicness') of interferences

The third thread of analysis, more recent, has built on the former two. Building on pragmatic thinking, in particular that of John [START_REF] Dewey | The Public and its Problems. An Essay in Political Inquiry[END_REF] about 'publics' as political moments and collectives, and its extension to issue making and material participation [START_REF] Marres | The Issues Deserve More Credit: Pragmatist Contributions to the Study of Public Involvement in Controversy[END_REF][START_REF] Marres | Material Participation. Technology, the Environment and Everyday Publics[END_REF], the idea was to focus on the relationships between all the entities convened (intentionally or unintentionally) in energy transition processes and explore the transformative (or disruptive) scope of the corresponding relational assemblages. Building on a large set of case studies, we proposed to distinguish between transition trajectories and the processes that underlie them and condition their deployment (Labussière & Nadaï, 2018).

"Trajectories" are devised in advance, through actors and categories of action that have been formed and stabilized -often involving modelling exercises and technology based visions in the field of the energy transition. Different from this, "processes" refer to a multiplicity of entities that participate in the construction of these trajectories, sometimes at the level of RET projects, without necessarily having control over them.

For instance, the development of a new wind farm may cause discontinuities in experiences (e.g. impeded animal migrations, fragmented agricultural communities, distant co-visibilities and changed perception of heritage elements) which, if not perceived and dealt with, give rise to potential inequalities. Building on thread 2, the analysis thus approaches the transition as a period of "ontological trouble" according to which not all the entities involved in the processes are called upon to "transition" with the same opportunities for being taken into account (Cf. the contrast between the Narbonnaise and Seine et Marne case studies). These entities find different existences in energy transition processes according to the degrees to which they are concerned, impacted, involved, acknowledged, and redefined with or without their consent. Following Noortje Marres, such an approach to potential endows the collective qualification of problems ('problematicness') with a central role in the formation of material 'publics' and of their capacity for action.

Based on this observation, the challenge is no longer to operationalize trajectories in order to reach predefined potentials. It is to acknowledge and assess the "interferences" that these trajectories generate, which in turn allows for the analysis of the (non) emergence of shared transition potentials. The question of "interferences" thus becomes the focal point of the inquiry and the concern for symmetry -i.e. acknowledging all the parties convened into the process, addressing them with the same analytical lens, and giving the same status to failed and successful options -is the analytical key to understanding the democratic dimension of the process.

Under such an approach, acceptance (issue) endorses a broader and deeper meaning as it is the expression ('problematicness') of interferences or, said differently, the expression of a democratic potential (deficit) in the conduct of the energy transition [START_REF] Bell | Re-visiting the 'Social Gap': Public Opinion and Relations of Power in the Local Politics of Wind Energy[END_REF]Labussière & Nadaï, 2018). Most importantly, however, while acceptance still relates to alignment and ontological work, turning to interferences broadens the scope of the analysis .

Indeed, analysing interferences challenges as well the construction of resources, space and time in energy transition processes, because this construction underlies the potential for participation and change.

This approach also has two major implications for thinking about the energy transition.

Firstly, it recalls us that energy is always more than just a matter of energy ; it questions just as much our relationships with biodiversity, landscapes, or spatial planning, and through them the handling of new tensions between development, justice and the environment. Secondly, relationships -which are at the heart of pragmatic inquiry -constitute a critical resource for the transition in that their collective specification and qualification makes it possible to support solutions that bring about social and environmental justice. Such conclusions echo the co-productionist agenda and its recent call for a just energy transition [START_REF] Jasanoff | Just transitions: A humble approach to global energy futures[END_REF].

Discussion: interference as hint for different encounters

This chapter proposed to undertake a reflective exercise about our preconception of social acceptance by considering our research practice about the development of wind power and energy transition processes over the past fifteen years.

Attempting to trace back the ways in which we framed the tensions around RET development in France, we have shown that several factors influenced it -including: our backgrounds, our academic networks and, importantly, fieldwork encounters. As one of the authors has written about, the field has a potential to resist our research framing and programming, and the capacity to entice us to explore new tracks [START_REF] Labussière | Le terrain ? C'est ce qui résiste. Réflexion sur la portée cognitive de l'expérience sensible en géographie[END_REF]. As far as we are concerned, several encounters have been decisive in the way we evolved our questioning about wind power development.

Important also has been the process of production of research papers, often resulting from academic network opportunities, which enticed us to re-interpret case studies along specific concerns, and makes it difficult to ex-post delineate a clear temporal succession in our research work. We thus distinguished between 'threads of works' mirroring the way in which we felt our reflection about the tensions raised by French RET development had progressed.

Each of these threads foregrounds a different articulation to which to relate social acceptance and its emergence (or lack thereof): representation and milieu, actants and milieu, interferences. Hence, each of these articulations points to a different problematisation of social acceptance: acceptance (issue) as a result of reciprocal alignment between the technology and its milieu through attentive spatial planning processes; acceptance (issue) as the outcome of the construction (or lack thereof) of a relational potential; acceptance (issue)

as the expression ('problematicness') of interferences.

As we have shown, each thread is embedded in a definite analytical strand, at the same time that it is both 'a criticism' and 'critical' to the notion of social acceptance. In the end, even if the 'critical' perspective has been said to challenge the relevance of RET development (because of its consequences) when the 'criticism' pointed out factors of resistance to it [START_REF] Batel | Research on the social acceptance of renewable energy technologies: Past, present and future[END_REF], it seems difficult to clearly sort out these perspectives here. The reason for this is that factors of opposition ('criticism' perspective) most often result from the ontological consequences of RET development, so that, at least in our research work, pointing out these factors challenges RET development by shedding light on its social and political consequences (discriminations, injustice and lack of democracy). Again, this may be singular to our research practice, and relate to our embedding in an academic field (STS, critical geography) that challenges ontologies (hence, the premise of individuality). As emphasized by [START_REF] Batel | Research on the social acceptance of renewable energy technologies: Past, present and future[END_REF], the 'criticism' approach to social acceptance, has otherwise significantly been developed by (social, environmental) psychology, a discipline that works within this premise.

Each thread in our research work echoes works and results pertaining to the academic field of social acceptance. This especially is the case if we include in the definition of this academic field, the works concerned with public participation and the coproduction agenda (Chilvers, Laurent, Jasanoff, Marres). The co-production agenda, though, has several scales of interpretation and operationalization: it points to both the co-production of orders (social and technological) and objects (technologies, individualities). As far as the above witnesses, our research work has mostly proceeded from the latter, through detailed case studies, in order to point to the former ; we did build from Laurent and Marres framing, but not from Chilvers or Jasanoff ones.

Reformulating acceptance issues as the expression of interferences in energy transition processes, however, should not be regarded as a closure within this agenda. As far as we are concerned with the urgency of environmental issues, following interferences calls for enlarging the requirement for symmetry so as to include ourselves (the observer).

Interferences are not only among and for the others. They cannot be formulated in terms of distant options or alternatives. Following their making calls for attending to the multiple arenas in which ontologies and their politics are constructed and at play, including that of the social scientists themselves -a point that Noortje Marres has made very clear with the notion of experimental ontology (2013). Thus, emphasizing the pragmatic heritage of the coproduction agenda, carries with it a methodological displacement that goes beyond the critical perspective and a new departure point in how we address these issues (Cf. Figure 1). The point is no longer to challenge the relevance of RET development by only shedding light on its underlying politics and consequences ('critical perspective'). It also is -by including ourselves as sources and ends of interferences -to reflexively engage with the field and be challenged by it as social scientists, because this paves the way to enlarged experimentations and ontologies, and thus to new possibilities.

Figure 1 : Overall displacement from 'pre-conception' to 'academic engagement'

Research implication Interferences Approaches to acceptance

From pre-conception The 'technology' and the 'social' pre-exist to their "interactions" ; Pretending to follow interferences, challenges our position as social scientists, the method and the material devices we recourse to in taking part into the field, as well as the temporality and the reciprocity of fieldwork. While such a change has been partly thematised under the headings of material participation and experimental ontology [START_REF] Marres | Why political ontology must be experimentalized: on the ecoshowhome as a participation device[END_REF]Labussière & Nadaï, 2018, p. 22-30 for an application to energy transition issues) much remains to be done challenge and invigorate social sciences to contribute to the setting up of more just processes of energy transition. A challenge that is ahead of us, and to which some works already call us to engage with (Marres, Guggenheim & Wilkie, 2018;[START_REF] Segers | Tactics for tough issues[END_REF][START_REF] Wilkie | Designing and doing: Enacting energy-and-community[END_REF].

  people's relation to new energy infrastructures. In this section, we sum up the evolving French political context regarding the development of wind energy in France, under which we developed our research program since the year of 2005; we point to a few factors, which we think influenced the course of our research program ; and propose to sketch this program through three analytical threads.

  It was very much triggered by fieldwork encounters in Aveyron -meeting with overwhelmed opponentswhich pointed out a type of democratic deficit in the way RET was developed. Democratic deficit meant here that the parties affected by this development did not have adequate access to the arenas in which the decisions about this development were made. Moreover, both the Aveyron and the Seine-et-Marne case studies pointed at the relation between the way in which the energy resource was framed as naturally given (wind speed) and appropriable by anyone who could invest in / develop a technological artefact (turbine, solar panel…), regardless of the collective dimension of other resources engaged (land or landscape uses, effort spent in coordinating shared visions about where to develop these new energies …), and the rise of acceptance issues.

  about how to operationalize such a change. Interestingly, and echoing to the role of fieldwork in enticing methodological changes, recent research works about what has been called 'research fatigue' in the field of energy transition research come as a warning (Walsh et al., forthcoming). 'Research fatigue' points out situations in which actors in the field refuse to collaborate with social scientists because they no longer believe in the ongoing politics of 'the' energy transition (our experience in Aveyron), or because they do not see the value of the usual talk-leave-and-publish research sequence of social scientists, and probably because of both. Of course, 'research fatigue' is not just a symptom manifested by an external bodyfield but makes ourselves part of the problem. It opens a more general questioning about how we articulate both the field and a global knowledge economy. Hence, what 'research fatigue' also carries with it, is a call for a different relation and engagement of social science researchers with the field. Collectively experimenting ways to acknowledge interferences and support emerging publics (e.g. collaborative research, art/sciences experiments) may usefully