POLYAKOV's FORMULATION of 2D BOSONIC STRING THEORY

POLYAKOV’S FORMULATION OF 2d BOSONIC STRING THEORY Colin Guillarmou, Rémi Rhodes, Vincent Vargas

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Colin Guillarmou, Rémi Rhodes, Vincent Vargas. POLYAKOV’S FORMULATION OF 2d BOSONIC STRING THEORY. Publications mathematiques de l’ IHES, 2019, 130 (1), pp.111-185. hal-01587093

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. POLYAKOV’S FORMULATION OF 2d BOSONIC STRING THEORY

COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

Abstract. Using probabilistic methods, we first define Liouville quantum field theory on Riemann surfaces of genus g > 2 and show that it is a conformal field theory. We use the partition function of Liouville quantum field theory to give a mathematical sense to Polyakov’s partition function of noncritical bosonic string theory [Po] (also called 2d bosonic string theory) and to Liouville quantum gravity. More specifically, we show the convergence of Polyakov’s partition function over the moduli space of Riemann surfaces in genus g > 2 in the case of D 6 1 boson. This is done by performing a careful analysis of the behaviour of the partition function at the boundary of moduli space. An essential feature of our approach is that it is probabilistic and non perturbative. The interest of our result is twofold. First, to the best of our knowledge, this is the first mathematical result about convergence of string theories. Second, our construction describes conjecturally the scaling limit of higher genus random planar maps weighted by Conformal Field Theories: we make precise conjectures about this statement at the end of the paper.

1. Introduction

In physics, string theory or more generally Euclidean 2d Quantum Gravity (LQG) is an attempt to quantize the Einstein-Hilbert functional coupled to matter fields (matter is re- placed by the free bosonic string in the case of string theory). The problematic can be briefly summarized as follows. First of all, a quantum field theory on a surface M can be viewed as a way to define a Sg(φ) measure e− Dφ over an infinite dimensional space E of fields φ living over M (typically φ are sections of some bundles over M), where Dφ is a “uniform measure” and S : E R is g → a functional on E called the action, depending on a background Riemannian metric g on M. The total mass of the measure

Sg (φ) Z(g) := e− Dφ (1.1) ZE is called the partition function. Defining the n-points correlation functions amounts to taking n points x ,...,x M and weights α ,...,α R and to defining 1 n ∈ 1 n ∈ n P αiφ(xi) Sg(φ) Z(g; (x1, α1),..., (xn, αn)) := e i=1 e− Dφ, ZE at least if the fields φ are functions on M. A conformal field theory (CFT in short) on a surface is a quantum field theory which possesses certain conformal symmetries. More specifically, the partition function Z(g) of a CFT satisfies the diffeormorphism invariance Z(ψ∗g) = Z(g) for all smooth diffeomorphism 1 2 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

ψ : M M and a so-called conformal anomaly of the following form: for all ω C (M) → ∈ ∞ ω c 2 Z(e g)= Z(g)exp ( dω g + 2Kgω)dvg (1.2) 96π M | | Z where c R is called the central charge of the theory, K is the scalar curvature of g and dv ∈ g g the volume form. The n-points correlation functions should also satisfy similar types of conformal anomalies and diffeormorphism invariance (see (4.2) and (4.4)). Usually, it is difficult to give a mathematical sense to (1.1) because the measure Dφ, which is formally the Lebesgue measure on an infinite dimensional space, does not exist mathematically. Hence, CFT’s are mostly studied using axiomatic and algebraic methods, or perturbative methods (formal stationary phase type expansions): see for example [FMS, Ga].

Liouville Quantum Field Theory. The first part of our work is to construct Liouville quantum field theory (LQFT in short) on Riemann surface of genus g > 2 and to show that this is a CFT. We use probabilistic methods to give a mathematical sense to the path integral (1.1), when Sg(φ) = SL(g, φ) is the classical Liouville action, a natural convex functional coming from the theory of uniformisation of Riemann surfaces that we describe now. Given a two dimensional connected compact Riemannian manifold (M, g) without boundary, we define the Liouville functional on C1 maps ϕ : M R by → 1 S (g, ϕ) := dϕ 2 + QK ϕ + 4πµeγϕ dv (1.3) L 4π | |g g g ZM 2 where Q,µ,γ > 0 are parameters to be discussed later. If Q = γ , finding the minimizer u γu of this functional allows one to uniformize (M, g). Indeed, the metric g′ = e g has constant 2 scalar curvature K ′ = 2πµγ and it is the unique such metric in the conformal class of g. g − The quantization of the Liouville action is precisey LQFT: one wants to make sense of the following finite measure on some appropriate functional space Σ (to be defined later) made up of (generalized) functions ϕ : M R → S (g,ϕ) F Π (g, F ) := F (ϕ)e− L Dϕ (1.4) 7→ γ,µ ZΣ where Dϕ stands for the “formal uniform measure” on Σ. Up to renormalizing this measure by its total mass, this formalism describes the law of some random (generalized) function ϕ on Σ, which stands for the (log-)conformal factor of a random metric of the form eγϕg on M. In 2 physics, LQFT is known to be CFT with central charge cL := 1 + 6Q continuously ranging in [25, ) for the particular values ∞ 2 γ γ ]0, 2], Q = + . (1.5) ∈ γ 2 Of course, this description is purely formal and giving a mathematical description of this picture is a longstanding problem, which goes back to the work of Polyakov [Po]. The rigorous construction of such an object has been carried out recently in [DKRV] in genus 0, [DRV] in genus 1 (see also [HRV] for the case of the disk). Let us also mention that Duplantier- Miller-Sheffield have developed in [DMS] a theory that lies at the “boundary” of LQFT: with the help of the Gaussian Free Field, their approach gives sense in the case of the sphere, disk or plane to some form of 2-point correlation function of LQFT and the corresponding LQG AND BOSONIC 2d STRING THEORY 3 equivalence class of random measures (defined up to a non trivial subgroup of biholomorphic transformations of the domain). Yet, another approach by Takhtajan-Teo [TaTe] was to develop a perturbative approach (a semiclassical Liouville theory in the so-called background field formalism approach): in this non-probabilistic approach, LQFT is expanded as a formal power series in γ around the minimum of the action (1.3) and the parameter Q in the action is given 2 by its value in classical Liouville theory Q = γ . We consider the genus g > 2 case and give a mathematical (and non perturbative) definition to (1.4). To explain the result, we need to summarize the construction. On a compact surface M with genus g > 2, we fix a smooth metric g on M and define for s R the Sobolev space s s/2 2 ∈ H (M) := (1+∆g)− (L (M)) of order s with scalar product defined using the metric g. Using the theory of Gaussian free field (GFF in short), we show that for each s> 0 there is a measure on H s(M) which is independent of the choice of metric g in the conformal class [g], and P′ − which represents the following Gaussian formal measure defined for each F L1(H s(M), ) ∈ − P′ by 1 2 R ϕ dvg 1 F (ϕ)e− 4π M |∇ |g Dϕ := F (ϕ)d ′(ϕ) (1.6) det (∆ ) P Z ′ g Z where det′(∆g) is the regularized determinant ofp the Laplacian, defined as in [RaSi]. The method to do this is to consider a probability space (Ω, , P) and a sequence (a ) of i.i.d. real F j j Gaussians (0, 1) and to consider the random variable (called GFF) N ϕj Xg = √2π aj (1.7) > λj jX1 s p with values in H− (M), if (ϕj)j > 0 is an orthonormal basis of eigenfunctions of ∆g with 1 eigenvalues (λj)j > 0 (and with λ0 = 0). The covariance of Xg is the Green function of 2π ∆g s s and there is a probability mesure on H0− (M) := u H− (M); u, 1 = 0 so that the law P s { ∈ h i } of Xg is given by and for each φ H0 (M), Xg, φ is a random variable on Ω with zero mean P 1 ∈ s h s i R and variance 2π ∆g− φ, φ . Then H− (M)= H0− (M) and we define ′ as the pushforward h i s⊕ P of the measure dc under the mapping (X, c) H− (M) R c+X, where dc is the uniform P⊗ ∈ 0 × 7→ Lebesgue measure in R. The formal equality (1.6) is an analogy with the finite dimensional setting. The next tool needed to the construction is Gaussian multiplicative chaos theory γ γX introduced by Kahane [Ka], which allows us to define the random measure g := e g dv on G g M for 0 < γ 6 2 when Xg is the GFF. This is done by using a renormalisation procedure. We can then define the quantity which plays the role of the formal integral (1.4) as follows: for F : H s(M) R (with s> 0) a bounded continuous functional, we set − → 1/2 Πγ,µ(g, F ) :=(det′(∆g)/Volg(M))− (1.8) E Q γc γ F (c + Xg)exp Kg(c + Xg)dvg µe g (M) dc × R − 4π M − G Z h Z i and call it the functional integral of LQFT (when F = 1 this is the partition function). Our first result is that this quantity is finite and satisfies diffeomorphism invariance and a certain γ 2 conformal anomaly when Q = 2 + γ . γ 2 6 Theorem 1.1 (LQFT is a CFT). Let Q = 2 + γ with γ 2 and g be a smooth metric on M. For each bounded continuous functional F : H s(M) R (with s> 0) and each ω C (M), − → ∈ ∞ 4 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

ω Πγ,µ(e g, F ) is finite and satisfies the following conformal anomaly: 2 ω Q 1 + 6Q 2 Πγ,µ(e g, F ) = Πγ,µ(g, F ( 2 ω)) exp ( dω g + 2Kgω)dvg . · − 96π M | | Z Let g be any metric on M and ψ : M M be an orientation preserving diffeomorphism, then → we have for each bounded measurable F : H s(M) R with s> 0 − → Π (ψ∗g, F ) = Π (g, F ( ψ)). γ,µ γ,µ · ◦ As a quantum field theory, the other objects of importance for LQFT are the correlation functions. In Section 4.3, we define the n-points correlation functions with vertex operators α Xg(x ) e i i where αi are weights and xi M some points, and we show their conformal anomaly ∈ n αiϕ(xi) required to be a CFT. This amounts somehow to taking F (ϕ) = i=1 e in (1.8), but it again requires renormalisation since ϕ lives in H s(M) with s> 0. At this level, the construc- − Q tion follows the method initiated by [DKRV] on the sphere. We notice that for the sphere, only the n-points correlation functions for n > 3 are well defined, while here the partition function is already well-defined.

Liouville Quantum Gravity and Polyakov partition function. Our next result is the main part of the paper and consists in giving a sense to the Liouville quantum gravity (LQG in short) partition function following the work of Polyakov [Po]. We stress that, even though the objects comes from theoretical physics, our result and proof is purely mathematical and can be viewed, from the perspective of a mathematician, as a a way to understand the behaviour of some natural interesting function near the boundary of moduli space, namely the LQFT partition function. Given a connected closed surface M with genus g > 2, quantizing the coupling of the gravitational ﬁeld with matter ﬁelds amounts to making sense of the formal integral (partition function),

SEH(g) SM(g,φm) Z = e− e− Dgφm Dg (1.9) ZR Z where the measure Dg runs over the space of Riemannian structures on M (the space of R S (g,φm) metrics g modulo diffeomorphisms), the functional integral for matter fields e− M Dgφm stands for the quantization of an action φ S (g, φ ) over an infinite dimensional space of m 7→ M m R fields describing matter, and SEH is the Einstein-Hilbert action 1 S (g)= K dv + µ Vol (M), (1.10) EH 2κ g g 0 g ZM where κ is the Einstein constant, µ R is the cosmological constant. The measure Dg repre- 0 ∈ sents the formal Riemannian measure associated to the L2 metric on the space of Riemannian metrics (its reduction to ). There are several possible choices for the matter fields and we R shall focus on the choice described in Polyakov [Po]. Giving a mathematical definition to the functional integral (1.9) has been a real challenge, and Polyakov [Po] suggested a decomposition of this integral in the case of bosonic string theory with D free bosons. In that case, S (g,φm) ZM (g) := e− M Dgφm is the partition function of a CFT with central charge cM = D, and is mathematically given by a certain power of the determinant of the Laplacian. The ar- R gument of Polyakov [Po] (see also D’Hoker-Phong [DhPh]) for defining (1.9) was based on the LQG AND BOSONIC 2d STRING THEORY 5 observation that each metric g on M can be decomposed as ω g = ψ∗(e gτ ) (1.11) where ω C∞(M), ψ is a diffeomorphism and (gτ )τ g is a family of hyperbolic metrics on ∈ ∈M M parameterizing the moduli space of genus-g surfaces. We recall that is the space of Mg Mg equivalence classes of conformal structures: it is a 6g 6 dimensional orbifold equipped with − a natural metric, called the Weil-Petersson metric, whose volume form denoted dτ has finite volume. In this way, the space of Riemannian structures is identified to the product of moduli R space with the Weyl group C (M) acting on metrics by (ϕ, g) eϕg. Applying the change Mg ∞ 7→ of variables (1.11) in the formal integral (1.9) produces a Jacobian (called Ghost determinant) taking into account the quotient of the space of metrics by the space of diffeomorphisms of M. The Ghost determinant turns out to be the partition function of a CFT with central charge c = 26 and Polyakov noticed that at the specific value D = 26 the conformal anomaly ghost − of ZM cancels out that of the Ghost term, giving rise to a Weyl invariant partition function

Z = ZM (gτ )ZGhost(gτ ) det Jgτ dτ. (1.12) Z g M p called critical string theory1. Concretely, this was further discussed by D’Hoker-Phong [DhPh] who wrote D det(P P ) 1/2 g∗ g det′(∆g) − 2 ZGhost(g)= ,ZM(g)= C (1.13) det Jg Volg(M) for some constant C, where the determinants are defined by zeta regularizations, Pg is a first- order elliptic operator mapping 1-forms to trace-free symmetric 2-tensors, and Jg is the Gram matrix of a fixed basis of ker Pg∗ (see Section 5.1 further details). Then Belavin-Knizhnik [BeKn] and Wolpert [Wo2] proved that the integral (1.12) with D = 26 diverges at the boundary of (the compactification of) moduli space, a problematic fact in order to establish well-posedness of the partition function for critical D = 26 (bosonic) strings. Noncritical string theories are not formulated within the critical dimension D = 26, yet they are Weyl invariant. The idea, emerging once again from the paper [Po], is that for D = 26 the 6 integral (1.9) possesses one further degree of freedom to be integrated over corresponding to ω the Weyl factor e in (1.11). For D 6 1, hence cM 6 1, Polyakov argued that integrating this factor requires using Liouville quantum field theory. In other words, applying once again the change of variables (1.11) to (1.9) yields

Z = ZM (gτ )ZGhost(gτ )Πγ,µ(gτ , 1) det Jgτ dτ. (1.14) Z g M p where Πγ,µ(gτ , 1) is the partition function of LQFT in the background metric gτ . As explained above, the partition function Πγ,µ(gτ , 1) depends on two parameters γ and µ (some of which are related in some speciﬁc way to cM). Later, Polyakov’s argument was generalized by David and Distler-Kawai [Da, DiKa] to CFT type matter ﬁeld theories with central charge cM 6 1 (thus including the case D = 1). In that CFT context, the integral (1.14) is often called partition function of Liouville quantum gravity. This approach has an important consequence related to

1The term ”critical” refers in fact to the critical dimension D = 26 needed to get a Weyl invariant theory without quantizing the Weyl factor eω in (1.11). 6 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS string theory as it paves the way to a rigorous construction of noncritical bosonic string theory 2 provided one can make sense of (1.14). This is the main purpose of this paper. The importance of this theory is discussed in great details in [Polch, section 5.1] or [Kleb] for instance.

Once having deﬁned the partition function Πγ,µ(g, 1) of LQFT, we introduce the partition function for LQG

ZLQG := ZM(gτ )ZGhost(gτ )Πγ,µ(gτ , 1) det Jgτ dτ (1.15) Z g M p where we choose for the matter partition function

cM det′∆g − 2 ZM(g)= , (1.16) Volg(M) the ghost determinant ZGhost(gτ ) being deﬁned by (1.13) and dτ the Weil-Petersson measure. Notice that (1.16) is nothing but the partition function (1.13) for D = cM free bosons extended to all possible values D 6 1. The parameters in (1.8) are tuned in such a way that the global conformal anomaly of the product

ZM(gτ )ZGhost(gτ )Πγ,µ(gτ , 1) vanishes, hence ensuring Weyl invariance of the whole theory (1.15). In view of Theorem 1.1, this gives the relation c 26 + 1 + 6Q2 = 0, M − hence determining the value of γ (encoded by Q) in terms of the central charge of the matter ﬁelds. We refer to Section 5.1 for more explanations. The main result of this paper is the following: Theorem 1.2 (Convergence of the partition function). For surfaces of genus g > 2, the integral deﬁning the partition function Z of (1.15) converges for γ ]0, 2], that is for c 6 1. LQG ∈ M

The integral defining ZLQG in the case cM = 0 corresponds to the case of pure gravity (i.e. no matter), c = 2 to uniform spanning trees and c = 1 (equivalently D = 1) to noncritical M − M strings (or D = 2 string theory). As far as we know, this is the first proof of convergence of string theory on hyperbolic surfaces with fixed genus. Using this Theorem, we can now see the metric g on M as a random variable with law ruled by the partition function (1.15). The Riemannian volume and modulus of this random metric are called the quantum gravity volume form (LQG volume form) and quantum gravity modulus, see Theorem 5.1. Furthermore we formulate conjectures relating the LQG volume form to the scaling limit of random planar maps in the case of pure gravity cM = 0, hence providing the scaling limit of the model studied in [Mi], or to the scaling limit of random planar maps weighted by the discrete Gaussian free field in the case cM = 1, see Section 5.5. Though we do not explicitly write a conjecture, we further mention here that the limit of random planar maps with fixed topology weighted by uniform spanning trees corresponds to c = 2. M − 2Noncritical bosonic string theory is sometimes referred to as critical D = 2 string theory, by opposition to the critical D = 26 string theory. The two dimensions correspond to one dimension for the embedding into R and one dimension for the Weyl factor: in other words the Weyl factor ω in (1.11) plays the role of a hidden dimension, see the explanations in [Polch] page 121. LQG AND BOSONIC 2d STRING THEORY 7

The main input of our paper is the proof of Theorem 1.2. We need to analyse the integrands in (1.15) near the boundary of moduli space and show that we can control them. The moduli space g can be viewed as a 6g 6 dimensional non-compact orbifold of hyperbolic metrics M − on M, that can be compactified in such a way that its boundary corresponds to pinching closed geodesics. Hyperbolic metrics on M corresponding to points in ∂ g are complete M hyperbolic surfaces with cusps and finite volume. The estimates of Wolpert [Wo2] describe the parts involving the ghost and matter terms. The heart of our work is to analyse the behaviour of Gaussian multiplicative chaos under degeneracies of the hyperbolic surfaces: this is rather involved since there are in general small eigenvalues of Laplacian tending to 0 and the covariance of the GFF (i.e. the Green function) is thus diverging. There is yet a huge conceptual gap between the cases cM < 1 and cM = 1. Roughly, the reason is the following: 2 c π (1 M ) − 3ℓj − 2 the product ZM(gτ )ZGhost(gτ ) is at leading order comparable to j e , where the product runs over pinched geodesics with lengths ℓ 0 when approaching the boundary j → Q of the (compactification of) moduli space – see subsection 2.3 for more precise statements– 2 1 π ( ) − 3ℓj − 2 γ γ whereas Πγ,µ(gτ , 1) is comparable to j e F ( g (M)) where F ( g (M)) is an explicit × G γ G functional expectation of the Gaussian multiplicative chaos g (M). Hence, for cM < 1, we Q γ G prove soft estimates on the functional F ( g (M)) that are enough to get an exponential decay G of the product Z (g )Z (g )Π (g , 1) at the boundary of g and thus integrability with M τ Ghost τ γ,µ τ M respect to the Weil-Petersson measure. In the case cM = 1, the leading exponential behaviors cancel out exactly so that the analysis must determine the polynomial corrections behind the leading exponential behavior, rendering the computations more much intricate. In order to analyse the mass of the Gaussian multiplicative chaos measure in these degenerating regions, we need to prove very sharp estimates (that, as far as we know, are new) on the Green function and on the eigenfunctions associated to the small eigenvalues in the pinched necks of the surface, as functions of the moduli space parameters τ when τ approaches the boundary ∂ g. Roughly M speaking, the crucial observation is that the GFF behaves like two independent Brownian motions in the variable transverse to the closed geodesic being pinched, and this allows us to translate the problem in terms of explicit functionals of Brownian motion. For (and only for) cM = 1, we show that the pinching produces an extra rate of decay of Πγ,µ(gτ , 1) as we approach ∂ g, implying the convergence. M To conclude this introduction, we point out that an interesting different approach to define path integrals for random Kähler metrics on surfaces was introduced recently by Ferrari- Klevtsov-Zelditch [FKZ, KlZe], but the link with our work is not established rigorously.

Acknowledgements: C.G. is partially funded by grants ANR-13-BS01-0007-01 and ANR-13- JS01-0006. R.R. and V.V. are partially funded by grants ANR-JCJC-Liouville. We would like to thank A. Bilal, I. Klebanov, F. Rochon, L. Takhtajan for useful conversations. We ﬁnally wish to thank B. Bluy.

2. Geometric background and Green functions

2.1. Uniformisation of compact surfaces of genus g > 2. Let M be a compact surface of genus g > 2 and let g be a smooth Riemannian metric. Recall that Gauss-Bonnet tells us 8 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS that

Kgdvg = 4πχ(M) (2.1) ZM where χ(M) = (2 2g) is the Euler characteristic, K the scalar curvature of g and dv the − g g Riemannian measure. The uniformisation theorem says that in the conformal class ϕ [g] := e g; ϕ C∞(M) { ∈ } of g, there exists a unique metric g = eϕ0 g of scalar curvature K = 2. For a metricg ˆ = eϕg, 0 g0 − one has the relation ϕ Kgˆ = e− (∆gϕ + Kg) where ∆g = d∗d is the non-negative Laplacian (here d is exterior derivative and d∗ its adjoint). Finding ϕ0 is achieved by minimizing the following functional

+ 1 2 ϕ F : C∞(M) R , F (ϕ) := ( dϕ + K ϕ + 2e )dv → 2 | |g g g ZM and taking ϕ0 to be the function where F (ϕ) is minimum at ϕ = ϕ0. We will embed this functional into a more general one, depending on three parameters, called Liouville functional: let γ,Q,µ > 0 and deﬁne 1 S (g, ϕ) := dϕ 2 + QK ϕ + 4πµeγϕ dv . (2.2) L 4π | |g g g ZM 2 1 ω When Q = 2/γ and πµγ = 1, we can write SL(g, ϕ)= 2γ2π F (γϕ). In fact, ifg ˆ = e g for some ϕ, the functional SL satisﬁes the relation

ω 1 1 Q 2 Q 2 SL g,ˆ ϕ = SL(g, ϕ)+ 2 dω g Kgω + Q ϕ∆gω dvg − γ 4π M γ − γ | | − γ − γ Z and in particular if Q = 2/γ it satisﬁes

ω 1 2 SL g,ˆ ϕ = SL(g, ϕ) 2 ( dω g + 2Kgω)dvg, (2.3) − γ − 4πγ M | | Z which is sometimes called conformal anomaly: changing the conformal factor of the metric entails a variation of the functional proportional to the Liouville functional. Similar properties will be shared by the quantum version of the Liouville theory, which fall under the scope of Conformal Field Theory (CFT for short, more later). At this stage let us just mention that we will show that the value of Q for the quantum Liouville theory to possess a conformal anomaly has to be adjusted to take into account quantum eﬀects. More precisely we will have in the quantum theory 2 γ Q = + . (2.4) γ 2

2.2. Hyperbolic surfaces, Teichmüller space and Moduli space. Let M be a surface of genus g > 2. The set of smooth metrics on M is a Fréchet manifold denoted by Met(M) and 2 contained in the Fréchet space of smooth symmetric tensors C∞(M,S M) of order 2. This space has a natural L2 metric given by

h , h := h , h dv (2.5) h 1 2i h 1 2ig g ZM LQG AND BOSONIC 2d STRING THEORY 9 where h , h T Met(M) = C (M,S2M) and , is the usual scalar product on endo- 1 2 ∈ g ∞ h· ·ig morphisms of T M when we identify symmetric 2-tensors with endomorphisms of T M through the metric g. A metric with Gaussian curvature 1 will be called hyperbolic, we denote by − Met 1(M) the set of such metrics on M. The group (M) of smooth diffeomorphisms acts − D smoothly and properly on Met(M) and on Met 1(M) by pull-back φ.g := φ∗g, moreover it − acts by isometries with respect to the metric (2.5). The subgroup (M) (M) of elements D0 ⊂ D contained in the connected component of the Identity also acts properly and smoothly and Mod(M) := (M)/ (M) is a discrete subgroup called mapping class group or moduli group. D D0 The Fréchet space C (M) acts on Met(M) by conformal multiplication (ϕ, g) eϕg. The or- ∞ 7→ bits of this action are called conformal classes and the conformal class of a metric g is denoted by [g]. The Teichmüller space of M is defined by

(M) := Met 1(M)/ 0(M). T − D By taking slices transverse to the action of (M), we can put a structure of smooth manifold D0 with real dimension 6g 6, it is topologically a ball, and its tangent space at a metric g − (representing a class in (M)) can be identified naturally with the space of divergence-free T trace-free tensors with respect to g by choosing appropriately the slice. Teichmüller space has a complex structure and is equipped with a natural Kähler metric called the Weil-Petersson metric, which is defined by h , h := htf , htf dv h 1 2iWP h 1 2 ig g ZM if h , h T (M) and htf = h 1 Tr (h )g denotes the trace-free part. The Weil-Petersson 1 2 ∈ gT i i − 2 g i metric is not the metric induced by (2.5) after quotienting by (M) but it is rather induced by D0 the L2 metric on almost complex structures, when we identify almost complex structures with metrics of constant curvature. We refer for to the book of Tromba [Tr] for more details about this approach of Teichmüller theory, the Weil-Petersson metric is discussed in Section 2.6 there.

The group Mod(M) acts properly discontinuously on (M) by isometries of the Weil- T Petersson metric, but the action is not free and there are elements of finite order. The quotient (M) := (M)/Mod(M) is a Riemannian orbifold called the moduli space of M, its orbifold M T singularities corresponding to hyperbolic metrics admitting isometries. Since (M), Mod(M) T and (M) actually depend only on the genus g of M, we shall denote them g, Modg and M T g. The manifold g is open but can be compactified into g, the locus of the compactifi- M M M cation is a divisor D g and the Weil-Petersson distance is complete on that space. Since ⊂ M we will need to understand the behavior of certain quantities on the moduli space, we now recall its geometry near the divisor D and we shall follow the description given by Wolpert ([Wo1, Wo2, Wo3]) for this compactification. On a surface M of genus g, there is a unique geodesic in each free homotopy class, and we call a partition of M a collection of 3g 3 sim- − ple closed curves γ1, . . . , γ3g 3 which are not null-homotopic and not mutually homotopic. { − } If g Met 1(M), there is a unique simple geodesic homotopic to each γj and we obtain a ∈ − decomposition of (M, g) into 2g 2 hyperbolic pants (a pant is a topological sphere with 3 − disk removed, equipped with a hyperbolic metric and with totally geodesic boundary). A subpartition of M is a collection of n simple curves γ , . . . , γ which are not null-homotopic p { 1 np } 10 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS and not mutually homotopic, with n 6 3g 3; they disconnect the surface into surfaces with p − boundary. A surface (M , g ) in ∂ g is a surface with nodes: M is the interior of a compact 0 0 M 0 surface M with np simple curves γ1, . . . , γnp removed and g0 is a complete hyperbolic metric with finite volume on M , the metric in a collar neighborhood [ 1, 1] (R/Z) of each γ 0 − ρ × θ j (with γ = ρ = 0 ) being j { } dρ2 g = + ρ2dθ2. 0 ρ2 R+ Notice that these corresponds to a pair of hyperbolic cusps, each one isometric to ( t 2 2t 2 t × (R/Z) , dt + e dθ ) by setting x = e . Now there is a neighborhood U of g in g θ − ± − g0 0 M represented by hyperbolic metrics g on M for some parameter (s,τ) C3g 3 m Cm near s,τ 0 ∈ − − × 0, with g which are smooth metrics on M when τ = 0 and complete smooth metrics with s,τ 6 hyperbolic cusps on M0 when τ = 0, and g0,0 = g0. Moreover, the metrics gs,τ are continuous with respect to (s,τ) on compact sets of M0 for (s,τ) near 0 (in the C∞ topology), they are given in the fixed collar neighborhood [ 1, 1] (R/Z) of γ by − ρ × θ j 2 ϕs,τ dρ 2 2 2 gs,τ = e 2 2 + (εj + ρ )dθ (2.6) εj + ρ with ε := 2π2/ log τ and ϕ C (M) satisfies j | | j|| s,τ ∈ ∞ eϕs,τ 1 0 as (s,τ) 0 − → → 0 in C norm (and in fact in C∞ on compact sets of M0). Here we notice that the metric dρ2 2 2 2 ε2+ρ2 + (εj + ρ )dθ has curvature 1 in the collar and is isometric to j − [ t ,t ] (R/Z) , dt2 + ε2 cosh(t)2dθ2 (2.7) − j j t × θ j by setting εj sinh(t)= ρ and εj sinh(tj) = 1. The geodesic γj(gs,t) for gs,t homotopic to γj has length ℓ (g )= ε (1 + o(1)) as (s,τ) 0 j s,τ j | |→ and is contained in a neighborhood [ cε , cε ] R/Z of the collar near γ for some c > 0 − j j × j independent of ε (or equivalently in t [ c, c]). Using eϕs,τ 1 < δ for some small δ > 0, the j ∈ − | − | set B = m M; d (γ (g ),m) > log ℓ (g ) is contained in a compact set of M uniform j { ∈ g j s,τ | j s,τ |} 0 in (s,τ) where the metrics depend continously on (s,τ). We can then use geodesic normal coordinates with respect to g around γ (g ) and the collar (g )= M B is isometric to j s,τ Cj s,τ \ j [ log ℓ (g ), log ℓ (g )] (R/Z) , dt2 + ℓ (g )2 cosh(t)2dθ2. (2.8) − j s,τ j s,τ t × θ j s,τ To summarize, the geometry is uniformly bounded outside r (g) for metrics g in a small ∪j=1Cj neighborhood U of g in g. g0 0 M The loops (γj)j define a subpartition. The open strata of D correspond to subpartitions up to equivalence by elements in Modg. For each β > 0, the set of metrics in g such that all M geodesics have length larger than β is a compact subset of g called the β-thick part. The M β-thin part of g is the complement of the β-thick part. By Lemma 6.1 of [Wo2], there exists M a constant β > 0 so that the β-thin part of g is covered by a finite set of neighborhoods M U(SPj), j = 1,...,np where SPj denote some subpartitions of M and U(SPj) denote the set of surfaces in Teichmüller space (up to Modg equivalence) for which the geodesics in the homotopy LQG AND BOSONIC 2d STRING THEORY 11 class of curves of SPj have length less than β and the other ones have length bounded below by β/2. Each U(SPj) is a neighborhood of a strata of D. For each pants decomposition of the surface (with genus g), one has associated coordinates τ = (ℓ1,...,ℓ3g 3,θ1,...,θ3g 3) where ℓj are the lengths of the simple closed geodesics bound- − − ing the pair of pants and θ [0, 2π) are the twist angles (see [Wo1]). The Weil-Petersson j ∈ volume form is given in these coordinates by

3g 3 − dτ := Cg ℓjdθjdℓj (2.9) jY=1 for some constant Cg > 0 depending only on the genus.

2.3. Determinant of Laplacians. For a Riemannian metric g on a connected oriented compact surface M, the non-negative Laplacian ∆g = d∗d has discrete spectrum Sp(∆g) = (λj)j N0 ∈ with λ = 0 and λ + . We can deﬁne the determinant of ∆ by 0 j → ∞ g det′(∆ ) = exp( ∂ ζ(s) ) g − s |s=0 s where ζ(s) := j∞=1 λj− is the spectral zeta function of ∆g, which admits a meromorphic continuation from Re(s) 1 to s C and is holomorphic at s = 0. We recall that ifg ˆ = eϕg P ≫ ∈ for some ϕ C (M), one has the so-called Polyakov formula (see [OPS, eq. (1.13)]) ∈ ∞ det′(∆ ) det′(∆ ) 1 log gˆ = log g ( dϕ 2 + 2K ϕ)dv (2.10) Vol (M) Vol (M) − 48π | |g g g gˆ g ZM where Kg is the scalar curvature of g as above. It is interesting to compare with the conformal anomaly (2.3) of the Liouville action SL. To compute det′(∆g), it thus suﬃces to know it for an element in the conformal class, and by the uniformisation theorem we can choose a metric g of scalar curvature 2 (or equivalently Gaussian curvature 1) if M has genus g > 2. Such − − hyperbolic surface can be realized as a quotient Γ H2 of the hyperbolic half-plane \ 2 2 dz H := z C;Im(z) > 0 with metric gH2 = | | { ∈ } (Im(z))2 by a discrete co-compact subgroup Γ PSL (R) with no torsion. In each free homotopy class ⊂ 2 on M =Γ H2, there is a unique closed geodesic, and we can form the Selberg zeta function \

∞ (s+k)ℓ(γ) Z (s)= (1 e− ), Re(s) > 1 g − γ k=0 Y∈P Y where denotes the set of primitive closed geodesics of (M, g) Γ H2 and ℓ(γ) are their P ≃ \ lengths (recall that primitive closed geodesics are oriented closed geodesics that are not iterates of another closed geodesic). By the work of Selberg, The function Zg(s) admits an analytic continuation to s C and it is proved by D’Hoker-Phong [DhPh] that ∈ (2g 2)C det′∆g = Zg′ (1)e − (2.11) where C is an explicit universal constant (see also Sarnak [Sa]). The behavior of Zg′ (1) near the boundary of g is studied by Wolpert [Wo2]: there exists Cg > 0 a constant depending M 12 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS only on the genus such that for all g g ∈ M

π2 π2 np np − 3ℓj (g) − 3ℓj (g) 1 e e Cg− λk(g) 6 Zg′ (1) 6 Cg λk(g) (2.12) ℓj(g) ℓj(g) jY=1 λk(gY)<1/4 jY=1 λk(gY)<1/4 where λk(g) are the eigenvalues of ∆g and ℓj(g) are the lengths of closed geodesics with length less than ε> 0 for some small ﬁxed ε> 0.

There is another operator which appears in the work of Polyakov [Po] and whose determinant is important in 2D quantum gravity. Let Pg be the differential operator mapping differential 1-forms on M to symmetric trace-free 2-tensors, defined by

P ω := 2 gω Tr ( gω)g. g S∇ − g S∇ If g is Levi-Civita connection for g, Tr denotes the trace with respect to g and denotes ∇ g S the orthogonal projection on symmetric 2-tensors. The kernel of Pg is the space of conformal Killing vector fields, which is thus trivial in genus g > 2. Its adjoint Pg∗ is given by Pg∗u := δ (u) = Tr ( gu) and called the divergence operator on symmetric trace-free 2-tensors. Its g − g ∇ kernel has real dimension 6g 6 and is conformally invariant. We denote by (φ1,...,φ6g 6) a − − fixed basis of ker P and by J the matrix (J ) = φ , φ The operator P P is an elliptic g∗ g g ij h i jig g∗ g positive self-adjoint second order differential operator acting on 1-forms, and we can define its determinant by

∞ s det(P ∗P ) = exp( ∂ ζ (s) ), ζ (s)= µ− g g − s 1 |s=0 1 j Xj=1 where µj > 0 are the non-zero eigenvalues of Pg∗Pg. The conformal anomaly for this operator is proved by Alvarez [Al, Eq. 4.27] and reads

det(P ∗Pgˆ) det(P ∗Pg) 13 log gˆ = log g ( dϕ 2 + 2K ϕ)dv (2.13) det J det J − 24π | |g g g gˆ g ZM ifg ˆ = eϕg. By [DhPh], one has for (M, g) a hyperbolic surface realized as Γ H2 \ 1 (2g 2)C′ det(Pg∗Pg) 2 = Zg(2)e − (2.14) for some universal constant C′, and Zg(s) is again the Selberg zeta function. The behavior of Z (2) near the boundary of g is also studied by Wolpert [Wo2]: there exists Cg > 0 a g M constant depending only on the genus such that for all g g ∈ M

π2 π2 np np e− 3ℓj (g) e− 3ℓj (g) 1 6 6 Cg− 3 Zg(2) Cg 3 (2.15) ℓj (g) ℓj (g) jY=1 jY=1 where the ℓj(g) are the lengths of closed geodesics with length less than ε> 0 for some small ﬁxed ε> 0. LQG AND BOSONIC 2d STRING THEORY 13

2.4. Green function and resolvent of Laplacian. Each compact Riemannian surface (M, g) has a (non-negative) Laplace operator ∆g = d∗d and a Green function Gg deﬁned to be 2 2 the integral kernel of the resolvent operator Rg : L (M) L (M) satisfying ∆gRg = Id Π0 2 → − where Π0 is the orthogonal projection in L (M, dvg) on ker ∆g (the constants). By integral kernel, we mean that for each f L2(M) ∈

Rgf(x)= Gg(x,x′)f(x′)dvg(x′). ZM It is well-known (see for example [Pa]) that the hyperbolic space H2 also has a family of Green functions (here dH2 (z, z′) denotes the hyperbolic distance between z, z′)

GH2 (λ; z, z′)= F (dH2 (z, z′)), λ D(0, 1/4) C (2.16) λ ∈ ⊂ so that F (r) is a holomorphic function of λ for r (0, ) satisfying λ ∈ ∞ 1 1 F (r) log(r) as r 0, F (r)= log(r)+ m(r2) (2.17) λ ∼−2π → 0 −2π with m being a smooth functions on [0, ), and GH2 (s) satisﬁes ∞ (∆H2 λ)GH2 (λ; , z′)= δ ′ − · z where δz′ denotes the Dirac mass at z′; in other words, GH2 (λ) is the Schwartz kernel of the 1 2 2 operator (∆H2 λ) on L (H ). To obtain the Green function G (x,x ), it suﬃces to know − − g ′ it for g hyperbolic (ie. g has constant Gaussian curvature 1) since for any other conformal − metricg ˆ = eϕg, we have that

G (x,x′)= G (x,x′)+ α u(x) u(x′), gˆ g − − 1 1 (2.18) with α = G , 1 1 , u(x) := G (x,y)dv (y). Vol (M)2 h g ⊗ igˆ Vol (M) g gˆ gˆ gˆ ZM ϕ This follows from an easy computation and the identity ∆gˆ = e− ∆g. Let us then assume that g is hyperbolic. We have

Lemma 2.1. If g is a hyperbolic metric on the surface M, the Green function Gg(x,x′) for ∆g has the following form near the diagonal 1 G (x,x′)= log(d (x,x′)) + m (x,x′) (2.19) g −2π g g for some smooth function m on M M. Near each point x M, there are isothermal g × 0 ∈ coordinates z so that g = dz 2/Im(z)2 and near x | | 0 1 G (z, z′)= log z z′ + F (z, z′) g −2π | − | with F smooth. Finally, if gˆ is any metric conformal to g, (2.19) holds with gˆ replacing g but with mgˆ continuous.

Proof. Near each point x0 M, there is an isometry from a geodesic ball Bg(x0, ε) for g to ∈ 2 2 the hyperbolic ball BH2 (0, ε) in H (0 denotes the center of H viewed as the unit disk), which provides in particular some local complex coordinates z near x so that g = eϕ dz 2 in the ball 0 | | Bg(x0, ε) for some function ϕ. In these coordinates,

log d (x,x′) = log dH2 (z, z′) = log z z′ + L(z, z′) with L smooth (2.20) g | − | 14 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS where d denote the distance for the metric g. Near any given point x M, one has g ′ ∈ (∆ λ)F (d ( ,x′)) δ ′ C∞(B (x′, ε)) (2.21) g − λ g · − x ∈ g where Bg(x′, ε) is a geodesic ball of center x′ and radius ε > 0 small. Denote by Rg(λ) = (∆ λ) 1 the resolvent of ∆ for λ / Sp(∆ ). By the spectral theorem, at λ = λ with g − − g ∈ g 0 λ Sp(∆ ) we have the Laurent expansion 0 ∈ g Π R (λ)= λ0 + R (λ )+ ((λ λ )), λ λ g λ λ g 0 O − 0 → 0 − 0 for some bounded operator R (λ ) and Π being the orthogonal projector on ker(∆ λ ). g 0 λ0 g − 0 Thus we obtain (∆ λ )R (λ )=Id Π g − 0 g 0 − λ0 and by elliptic regularity and (2.21), the Schwartz kernel G (λ; x,x ) of R (λ) for λ / Sp(∆ ) g ′ g ∈ g satisﬁes for dg(x,x′) < ε with ε> 0 small enough

Gg(λ; x,x′)= Fλ(dg(x,x′)) + Eg(λ; x,x′) (2.22) with Eg some smooth function on M M depending meromorphically of λ. At λ = 0 we deduce × ϕ(x)/2 (2.19). The part aboutg ˆ just follows from (2.18) and the fact that dgˆ(x,x′)= e dg(x,x′)+ (d (x,x )2) as x x. O g ′ ′ → The function x m (x,x) is often called the Robin constant at x. Notice that if we view 7→ g the hyperbolic metric g as an element representing a point of g and if ψ Modg, then we T ∈ have the modular invariance

Gψ∗g(λ; x,x′)= Gg(λ; ψ(x), ψ(x′)), mψ∗g(x,x′)= mg(ψ(x), ψ(x′)). (2.23)

We shall need to describe the Green function Gg when the metric g approaches the boundary of the compactiﬁcation of g. It turns out that positive small eigenvalues of ∆ appear M g sometime when g approaches a point in ∂ g: Schoen-Wolpert-Yau [SWY] proved that there M exist two positive constants α1, α2 depending only on the genus g so that the n-th positive eigenvalue λn(g) of ∆g satisfy

α1Ln(M, g) 6 λn(g) 6 α2Ln(M, g) if n 6 2g 2, and α1 6 λ2g 1 6 α2 − − where Ln(M, g) is the minimum (over subpartitions) of the sums of lengths of simple geodesics in subpartitions of M disconnecting M into n+1 connected components. Each metric g ∂ g 0 ∈ M is in a stratum corresponding to a subpartition SP containing np curves, with ns 6 np of these simple curves γ1, . . . , γns in the subpartition that disconnect the surface M into m+1 connected components. There is c0 > 0 depending on g0 such that for all δ > 0 small enough, there is a neighborhood U g of g such that for all g in the interior U := U g, there is at g0 ⊂ M 0 g0 g0 ∩ M most m positive eigenvalues less than δ and all other eigenvalues are bigger than c0. We call these eigenvalues the small eigenvalues of g near g0.

Proposition 2.2. Let (M , g ) ∂ g where M is a surface with nodes. For δ > 0 arbitrarily 0 0 ∈ M 0 small, take g in a suﬃciently small open neighborhood U of g in g so that the small g0 0 M LQG AND BOSONIC 2d STRING THEORY 15 eigenvalues of g in U = U g satisfy λ (g) 6 . . . 6 λ (g) 6 δ. The Green function G g0 g0 ∩ M 1 m g restricted to M can be written for g U as 0 ∈ g0

Πλj (g)(x,x′) G (x,x′)= + A (x,x′) (2.24) g λ (g) g 6 j λj (Xg) δ where Πλj (g) is the orthogonal projector on the corresponding eigenspace. In each compact set Ω of M , the map (g,x,x ) A (x,x ) is continuous on U (Ω2 ) if Ω2 := (Ω Ω) diag 0 ′ 7→ g ′ g0 × diag diag × \ and, near the diagonal of Ω Ω, one has × 1 A (x,x′)= log(d (x,x′)) + B (x,x′) g −2π g g with (g,x,x ) B (x,x ) C0(U Ω Ω). The Schwartz kernel m Π (x,x ) extends ′ 7→ g ′ ∈ g0 × × j=1 λj (g) ′ continuously to (g,x,x ) U Ω Ω with value at g = g ∂U the orthogonal projector ′ ∈ g0 × × ′ ∈ gP0 Π0(g′; x,x′) onto kerL2 ∆g′ .

Proof. After possibly splitting Ω in smaller pieces, we can assume that the radius of injectivity of all g U on Ω is bounded below by some uniform α> 0. Using the residue formula applied ∈ g0 to Rg(λ)/λ in a disk D(0, δ) of radius δ centered at λ = 0, one has

Πλ (g) 1 R (λ) R (0) j = g dλ. (2.25) g − λ (g) 2πi λ 6 j ∂D(0,δ) λj (Xg) δ Z and we denote by A (x,x ) the Schwartz kernel of 1 Rg(λ) dλ. Let Ω M be a small g ′ 2πi ∂D(0,δ) λ ′ ⊂ 0 neighborhood Ω of Ω so that the radius of injectivity of each g U is bounded below by ′ R ∈ g0 α/2. Let Lg(λ) be the operator on Ω′ with Schwartz kernel

Fλ(dg(x,x′)) where F is the function of (2.16). Take χ, χ C (M ) equal to 1 on Ω but with support λ ∈ c∞ 0 contained in Ω′, and such that χχ = χ. Then on Ω′ and on M0, we have e (∆ λ)χL (λ)χ = χ + [∆ , χ]L (λ)χ (2.26) ge− g g g Then multiplying (2.26) by χR (λ) on the left, we get g e e χR (λ)χ = χL (λ)χ χR (λ)[∆ , χ]L (λ)χ. g g − g g g The operators [∆ , χ]L (λ)χ have smooth kernel (we use that [∆ , χ] = 0 on supp(χ)), and g g e g extends continuously to g U since g extends continuously as a smooth metric to Ω and d ∈ g0 g on Ω Ω as well. Nowe we use the fact that for λ ∂D(0, δ), g Rg(λ)e extends continuously to × ∈ 7→ U as bounded operators Hk (M ) Hk (M ) for all k > 0 by a result of Schulze [Sc]: this g0 comp 0 → loc 0 implies that χR (λ)[∆ , χ]L (λ)χ extend continuously in g U as a family of bounded op- g g g ∈ g0 erators H k(M ) Hk (M ) for all k > 0, since [∆ , χ]L (λ)χ maps H k(M ) Hk(M ) − 0 → comp 0 g g − 0 → 0 uniformly in g Ug . Thuse the Schwartz kernels of the operators [∆g, χ]Lg(λ)χ extend as a ∈ 0 uniform family of continuous Schwartz kernels (when g eUg ). We then deduce that ∈ 0 e 1 χRg(λ)χ 1 χLg(λ)χ dλ = dλ + B′ 2πi λ 2πi λ g Z∂D(0,δ) Z∂D(0,δ) 16 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS where Bg′ is a family of operators, with Schwartz kernel Bg′ (x,x′) continuous as a function of (g,x,x ) U Ω Ω. Next, since by Cauchy formula 1 Fλ(z) dλ = F (z), we deduce ′ ∈ g0 × × 2πi ∂D(0,δ) λ 0 that R 1 χLg(λ)χ dλ (x,x′)= χ(x)χ(x′)F0(dg(x,x′)) 2πi ∂D(0,δ) λ Z and this Schwartz kernel has the desired property by using (2.17). This ends the proof of m (2.24). The proof of the fact that j=1 Πλj (g)(x,x′) converge to the projector onto the kernel of g′ as g g′ ∂U g0 is essentially the same as what we did (and even simpler) by applying → ∈ P 0 the residue formula to Rg(λ) in D(0, δ) instead of Rg(λ)/λ. The convergence in C norm 2 m is clear since convergence in L of j=1 Πλj (g) implies convergence in C∞ on Ω by elliptic regularity. P

2.5. Small eigenvalues and associated eigenvectors. In this section, we recall the asymptotics of the small positive eigenvalues λ (g) 6 . . . 6 λ (g) as g g approaches an element 1 m ∈ M g g by following Burger [Bu1, Bu2] and we will see that the proof of [Bu2] also gives 0 ∈ M an approximation of the projectors Πλj (g). Let (M0, g0) be a surface with nodes, with corresponding subpartition of the closed Riemann surface M given by simple curves γ1, . . . , γnp and

γ1, . . . , γns (with ns 6 np) are disconnecting M into m+1 connected components S1,...,Sm+1. For all δ > 0 small enough, there is a small neighborhood U g of g in g so that for g0 ⊂ M 0 M each g U = U g there are m small eigenvalues 0 < λ (g) 6 . . . 6 λ (g) 6 δ and ∈ g0 g0 ∩ M 1 m all others are larger than a constant c > 0 depending only on g . Each metric g U has a 0 0 ∈ g0 unique simple closed geodesic homotopic to γj for j 6 r, with length ℓj(g) 6 c1δ, while all other primitive closed geodesics have length bigger than c2 > 0, where c1, c2 are constants depending only on g0. Deﬁne the length Lij(g) := k E ℓk(g) where Eij = 1 6 k 6 ns; γk ∂Si ∂Sj . ∈ ij { ∈ ∩ } Let be the norm on Rm+1 given by || · ||g P m+1 a 2 = Vol (S )a2, with a = (a , . . . , a ) (2.27) || ||g g j j 1 m+1 Xj=1 m+1 and let Qg be the quadratic form on R given by

2 Qg(a)= (ai aj) Lij(g). (2.28) 6 6 − 1 i,jXm+1 Notice that Volg(Sj) are positive constants depending only on the topology of Sj (and not on g) by Gauss-Bonnet theorem. Then Burger [Bu2] showed the following estimate:

Theorem 2.3 (Burger 1990). If ν1(g) 6 . . . 6 νm(g) are the positive eigenvalues of Qg with respect to the norm on Rm+1, then there is C > 0 such that for all g U and each || · ||g ∈ g0 1 6 j 6 m νj(g) 1 νj(g) (1 Cδ 2 ) 6 λ (g) 6 (1 + Cδ log δ ). π − j π | |

Each simple small geodesic γj(g) of g (homotopic to γj) has a collar neighborhood (g)= x M; sinh(d (x, γ (g))) 6 1/ sinh(ℓ (g)) (2.29) Cj { ∈ g j j } LQG AND BOSONIC 2d STRING THEORY 17 and these collars are disjoints one from the other. The set M 6 (g) has m+1 connected \∪j ns Cj components S1′ ,...,Sm′ +1 that identify naturally with S1,...,Sm+1. One can deﬁne a map a Rm+1 f H1(M) (2.30) ∈ 7→ a ∈ by setting f (x)= a if x S and f being the unique harmonic function in (g) so that f a j ∈ j′ a Cj a is continuous on M. In [Bu2], Burger proved the following

Lemma 2.4 (Burger 90). There is C > 0 such that for all a Rm+1 and all g U ∈ ∈ g0 1 2 1 2 2 2 Q (a) 6 df 2 6 Q (a)(1 + Cδ), a (1 Cδ log δ ) 6 f 2 6 a . π g || a||L (M,g) π g || ||g − | | || a||L (M,g) || ||g

An estimate for λj(g) in terms of the pinched geodesics ℓk(g) is given by Schoen-Wolpert-Yau

[SWY]: let D be an n-disconnect, i.e a collection of closed simple geodesics γ1(g), . . . , γns (g) with respective lenghts ℓ1(g),...,ℓns (g) disconnecting M into n connected components, and ns deﬁne Ln(D, g) := j=1 ℓj(g). We set

P Ln(M, g) := min Ln(D, g) D n ∈D where is the set of all n-disconnects of M. Then, ordering the eigenavlues by increasing Dn order, it is proved in [SWY] that there is C > 1 depending only on the genus of M such that for each n 6 2g 2 − 1 C− Ln(M, g) 6 λn(g) 6 CLn(M, g).

As a consequence, in a neighborhood U g of a metric g ∂ g, we have the rough g0 ⊂ M 0 ∈ M estimate for all j 6 m and g U ∈ g0 1 λj(g) > C− ℓj(g) (2.31) where m + 1 is the number of connected components of the surface with cusps (M0, g0), ns is the number of pinched geodesics disconnecting the surface and ℓ1(g) 6 ℓ2(g) 6 . . . 6 ℓns (g) are the lengths of these separating geodesics ranked by increasing order.

Below, we take the convention that we repeat each eigenvalue according to its multiplicity, thus λj(g) can be equal to λj+1(g), and similarly for the νj(g).

Lemma 2.5. For each g U , let v = (4π(g 1)) 1 and v , . . . , v Rm+1 so that ∈ g0 0 − − 1 m ∈ (vi)i=0,...,m is an orthonormal basis of eigenvectors for Qg with vi associated to νi(g). There is C > 0 and L N such that for all g U , there exists an orthonormal basis ϕ ,...,ϕ of ∈ ∈ g0 1 m m ker(∆ λ (g)) satisfying ⊕j=1 g − j m 1 L 1 Πλj (g)(x,x′) fvj (x)fvj (x′) δ f ϕ 2 6 Cδ L , and = + || vj − j||L (M,g) λ (g) ν (g) O ν (g) 6 j j 1 λj (Xg) δ Xj=1 where the error term is in L norm on compact sets disjoints from (g). ∞ ∪jCj

Proof. To simplify notation we denote by fj the function fvj . We construct the basis ϕj by an 2 inductive process. Let (φj)j N0 be an orthonormal basis of L (M, g) of eigenvectors for ∆g, ie. ∈ 18 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

∆gφj = λj(g)φj . By Lemma 2.4, we have for k 6 m

2 ∞ 2 2 df 2 = λ (g) f , φ = Q (v )(1 + (δ log δ )) = ν (g) f 2 (1 + (δ log δ )) || k||L j h k ji g k O | | k || k||L O | | Xj=1 and by Theorem 2.3, this gives for each k = 1,...,m

∞ λj(g) 1 2 2 2 1 fk, φj = (δ fk L2 ). (2.32) λk(g) − h i O || || Xj=1 1 1 λ2 2 If 1 > δ 4 , we set ϕ := φ and i = 1. By (2.32) with k = 1, we get ∞ f , φ = (δ 4 ) λ1 1 1 1 j=2 k j | − | 1 h i O and thus f1 = φ1 + L2 (δ 8 ). Since fi,fj L2 = (δ log δ ) for i = j Pby Lemma 2.4, we get 1± O λ h i 1 O | | 6 f , ϕ = (δ 8 ) for all i> 1. If 2 1 6 δ 4 , we let i > 2 be the smallest integer such that i 1 λ1 1 h i O 1 | − | 1 λ λi +1 6 i 6 2i 1 2i1+1 6 for each i i1, λ 1 δ and λ 1 > δ , clearly i1 m since there are m small | i−1 − | | i1 − | eigenvalues. We deﬁne

i1 i1 j=1 f1, φj φj j=1 fi, φj φj ϕ1 = h i , and ϕi = h i for 1 6 i 6 i1. i1 f , φ φ i1 f , φ φ ||Pj=1h 1 ji j|| ||Pj=1h i ji j||

Then we construct ϕP2,...,ϕi1 by the Gram-Schmidte P orthonormalization process from ϕ2,..., ϕi1 . 1 1 λi +1 λi Since 1 1 > δ 2i1+1 and 1 1 = (δ 2i1 ), (2.32) tells us that for each i 6 i , λi λ1 1 | 1 − | | − | O e e i1 1 i +2 f = f , φ φ + 2 (δ 2 1 ) i h i j i j OL Xj=1 1 i +2 and thus ϕ = f + 2 (δ 2 1 ) for i = 1,...,i . Since f ,f 2 = δ + (δ log δ ) by Lemma i i OL 1 h i jiL ij O | | 2.4, we deduce that for i = 1,...,i1 1 e i +2 ϕ = f + 2 (δ 2 1 ). i i OL Now we prove the induction process in a way similar to the ﬁrst step. Suppose we have con- ℓ R 1 structed an orthornormal basis ϕ1,...,ϕℓ of j=1 φj so that ϕj = fj + L2 (δ L ) for some ⊕1 O L N and ℓ ℓ +1 and j 6 ℓ by the induction 1 1 ∈ λℓ+1 h i O 2 assumption. Then if 1 > δ L , (2.32) with k = ℓ + 1 gives ∞ fℓ+1, φj = (δ L ), | λℓ − | j=ℓ+2h i O thus if we set i = ℓ +1 and ℓ+1 P ℓ φℓ+1 j=1 φℓ+1, ϕj ϕj ϕℓ+1 = − h i φ ℓ φ , ϕ ϕ || ℓ+1 − Pj=1h ℓ+1 ji j || 1 1 P λℓ+1 we get ϕℓ+1 = fℓ+1+ L2 (δ 2L ) and we have increased the induction step by 1. If 1 6 δ L , O | λℓ − | λ 1 we let i 6 m be the smallest integer such that for all i = ℓ+1,...,i , i 1 6 δ L2i−ℓ−1 ℓ+1 ℓ+1 λi−1 1 | − | λiℓ+1+1 i −ℓ L2 ℓ+1 iℓ+1 ℓ and λ 1 > δ , and we will construct ϕℓ+1,...,ϕiℓ+1 . Let L′ = L2 − and | iℓ+1 − | deﬁne

iℓ+1 iℓ+1 j=ℓ+1 fℓ+1, φj φj j=ℓ+1 fi, φj φj ϕℓ+1 = h i , and ϕi = h i for ℓ + 1 6 i 6 iℓ+1. iℓ+1 iℓ+1 P fℓ+1, φj φj P fi, φj φj || j=ℓ+1h i || || j=ℓ+1h i || P e P LQG AND BOSONIC 2d STRING THEORY 19

Then we construct ϕℓ+2,...,ϕiℓ+1 by the Gram-Schmidt orthonormalization process from 1 λiℓ+1+1 L′ ϕℓ+2,..., ϕiℓ+1 . By induction assumption and λ 1 > δ , (2.32) tells us that for | iℓ+1 − | each i = ℓ + 1,...,iℓ+1, e e iℓ+1 1 ′ f = f , φ φ + 2 (δ 2L ) i h i ji j OL j=Xℓ+1 1 and thus ϕ = f + 2 (δ 2L′ ) for i = ℓ + 1,...,i . Since f ,f 2 = δ + (δ log δ ) by i i OL ℓ+1 h i jiL ij O | | Lemma 2.4, we deduce that for i = ℓ + 1,...,iℓ+1 e 1 ϕ = f + 2 (δ 2L′ ) i i OL and we have increased the induction step by i (ℓ + 1) > 1. This inductive construction ℓ+1 − produces a sequence of integers j0 = 1, j1 = i1, j2 = ii1+1,...,jN = m and N associated blocks E ,...E , with E = ϕ ,...,ϕ where the span of elements in E is the span of 1 N k { jk jk+1 } k φ ,...,φ . By construction we have { jk jk+1 } N jk+1 Πλ (x,x′) ϕ (x)ϕ (x ) j = j j ′ + (δ1/L) λ (g) λ (g) O 6 j j λj (Xg) δ Xk=0 jX=jk N jk+1 1 ϕ (x)ϕ (x ) δ L = j j ′ + νj(g) O ν1(g) Xk=0 jX=jk m 1 f (x)f (x ) δ L = j j ′ + νj(g) O ν1(g) Xj=1 1 δ L 2 for some L N large. Here we notice that the can be taken in L∞ norm since L ∈ O ν1(g) norms on eigenfunctions give directly uniform L∞norms on compact sets outside the collars (g). Cj 2.6. Example: the case of genus 2. For pedagogical purposes only, let us discuss more particularly the case of genus g = 2. In this case there can be only 3 simple curves in a partition and the maximal number of connected components separated by these curves is 2: either 1 curve separates M into two surfaces of genus 1 with 1 boundary component (Case 1) or two hyperbolic pants with 3 boundary components (Case 2). Consequently, the number of eigenvalues approaching 0 when we approach ∂ is m 1, 2 (including the eigenvalue M2 ∈ { } λ = 0), we call them λ0 = 0 and λ1(g) > 0 when m = 2.

In Case 1, take any partition SP1 by γ1, γ2, γ3 with γ1 being the only separating curve, and denote by ℓj(g) the length of the geodesic for g freely homotopic to γj. We have λ (g) c ℓ (g), as ℓ (g) 0 with g U(SP ) (2.33) 1 ∼ 1 1 1 → ∈ 1 where c1 > 0 depending only on g = 2.

In Case 2, take SP2 any partition where γ1, γ2, γ3 are separating simple curves and, if ℓj(g) is the length of the geodesic for g homotopic to γj, then by [Bu1], λ (g) c (ℓ (g)+ ℓ (g)+ ℓ (g)), as (ℓ (g),ℓ (g),ℓ (g)) 0 with g U(SP ) (2.34) 1 ∼ 2 1 2 3 1 2 3 → ∈ 2 20 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

dessin.pdf

γ2 γ3 γ2 γ1

γ 1 γ3 dessin-2.pdf dessin-2.pdf

Figure 1. On the left: Case 1. On the right: Case 2 for some c2 > 0 depending only on g = 2. In both cases, if g g with (M , g ) a surface with nodes, (M , g ) decomposes into → 0 0 0 0 0 two ﬁnite volume hyperbolic surfaces S1 and S2, with volume 2π (by Gauss-Bonnet), and by Proposition 2.2 1 Π (x,x′) (1l (x) 1l (x))(1l (x′) 1l (x′)) as g g (2.35) λ1(g) → 4π S1 − S2 S1 − S2 → 0 uniformly in (x,x ) on compact sets of M M . ′ 0 × 0 2.7. Green’s function near pinched geodesics. For later purpose, we will need a more detailed description of the Green’s function Gg than Proposition 2.2, in particular we shall need to know the behaviour of Gg near the pinched geodesics. A recent work of Albin-Rochon- Sher [ARS] gives a parametrix for ∆ λ when λ is outside the spectrum and when there is one g − pinched geodesic (or the geodesics are pinched at the same speed). Here we need to know the behaviour in all possible directions of approach of the boundary of g. Our estimates are done M in a way to be applied later for the study of the Gaussian multiplicative chaos measure in the cusp. The work of Melrose-Zhu [MeZh] gives a parametrix for some diﬀerent Green function.

Let (M0, g0) be a surface with nodes viewed as a surface with pairs of hyperbolic cusps, and let U be a local neighborhood of g in g made of hyperbolic metrics g as explained in g0 0 M s,t Section 2.2, and denote U = g U . For convenience, we remove the parameters (s,t) g0 M ∩ g0 and just write g for gs,t. As in Proposition 2.2, we set m Πλj (x,x′) 1 Rg(λ; x,x′) Ag(x,x′) := Gg(x,x′) = dλ (2.36) − λj 2πi ∂D(0,ε) λ Xj=1 Z where λj = λj(g) are the small eigenvalues tending to 0 as g approaches the boundary of moduli space, D(0, ε) is a small disk containing these eigenvalues (and only these ones) and Rg(λ; x,x′) is the integral kernel of the resolvent of ∆g. The hyperbolic surface (M, g) decomposes into M = S(g) np (g) where (g) are the ∪i=1 Cj Cj collars isometric to [ 1, 1] (R/Z) close to a given curve γ , where the metric g is given (in − ρ × θ j geodesic normal coordinates to the geodesic γj(g) homotopic to γj) by 2 dρ 2 2 2 gj = 2 2 + (ρ + ℓj )dθ , (2.37) ρ + ℓj LQG AND BOSONIC 2d STRING THEORY 21 where ℓj = ℓj(g) is the length of the geodesic γj(g) and S(g) is a compact manifold with boundary contained in a ﬁxed (independent of g) compact set of M0. This is also isometric to (by the coordinates change ρ = ℓj sinh(t)) [ d , d ] (R/Z) , g = dt2 + ℓ2 cosh2(t)dθ2, sinh(d ) = 1/ℓ . − j j t × θ j j j j The complete hyperbolic cylinder z eℓj z H2 is isometric to h 7→ i\ 2 R R Z dρ 2 2 2 j := ρ ( / )θ, gj = 2 2 + (ρ + ℓj )dθ . F × ρ + ℓj Notice that as ℓ 0, the Riemannian manifold ρ = 0 converges smoothly to two j → Fj \ { } disconnected surfaces (0, ) (R/Z) with metric dρ2/ρ2 + ρ2dθ2, which are isometric to two ∞ ρ × θ disjoint elementary quotients H2/ z z + 1 . The Laplacian ∆ is self-adjoint with spectrum h 7→ i gj [1/4, ) (its self-adjoint realisation is through Friedrichs extension on C ( )). It is invertible ∞ c∞ Fj on L2( ) and we can consider its resolvent R (λ) : L2( ) L2( ) which is holomorphic Fj gj Fj → Fj for λ / [1/4, ). This is studied in details for example in [Bo, Prop. 5.2] or [GuZw, Appendix]: ∈ ∞ writing λ = s(1 s) for s close to 1, we have − 2πik(θ θ′) Rgj (λ; ρ, θ, ρ′,θ′)= uk(s; ρ, ρ′)e − k Z X∈ for some explicit functions uk analytic in s for s close to 1. We denote by Ggj the Green function corresponding to λ = 0 (i.e. s = 1). We give a more explicit bound at λ = 0 in the following

Lemma 2.6. For ℓ 6 min( ρ , ρ ) 6 max( ρ , ρ ) 6 1 with ρρ > 0, the Green function G j | | | ′| | | | ′| ′ gj for the cylinder satisﬁes ρ ρ′ 2π arctan( ) arctan( ) +2πi(θ θ′) 1 ℓ ℓ ℓ G (ρ, θ, ρ′,θ′)= log 1 e− j | j − j | − gj − 2π − (2.38) 1 ρ ρ′ 1 ρ ρ′ + min( F ( |ℓ | ), F ( |ℓ | )) F ( |ℓ | )F ( |ℓ | )+ (1) ℓj j j − πℓj j j O du where F (x) := ∞ and the remainder is uniform. If ℓ 6 min( ρ , ρ ) 6 max( ρ , ρ ) 6 1 x 1+u2 j | | | ′| | | | ′| with ρ′ρ< 0, then R π 1 ρ ρ′ − 2ℓj Ggj (ρ, θ, ρ′,θ′)= F ( |ℓ | )F ( |ℓ | )+ (e ). (2.39) πℓj j j O If ρ [1/2, 1], ρ 6 1 and ρ ρ > δ for some δ > 0, then there is C depending only on δ | |∈ | ′| | − ′| so that

∂ G (ρ, θ, ρ′,θ′) 6 C. (2.40) | ρ gj | If χ C ( 1, 1) and λ [0, 1), then we have the pointwise estimate ∈ c∞ − ∈ 1 λ λ λ ρ− ρ− 1 ∞ Ggj (ρ, θ, ρ′,θ′)χ(ρ′)ρ′− dρ′dθ′ 6 2 χ L + − (2.41) R/Z 1 || || 1 λ λ Z Z− − ρ−λ 1 where by convention − = log ρ when λ = 0. Finally, the Robin mass of G satisﬁes λ | | || gj 1 2 1 π ρ 2 2 2 6 mgj (ρ, θ) 4 arctan( ℓ ) log( ρ + ℓj ) C. (2.42) − πℓj − j − 2π q

22 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

2 2 R R Z Proof. Let L be the operator L = (ρ + ℓj )∆gj acting on T := ρ ( / )θ with the measure 2 2 1 × 1 (ρ + ℓj )− dρdθ, it is symmetric on Cc∞(T ). Changing coordinates to t = ℓj− arctan(ρ/ℓj), L becomes the operator 2 2 π π L = ∂ ∂ on ( , )t (R/Z)θ − t − θ − 2ℓj 2ℓj × with the measure dtdθ. It is not self-adjoint but we can we consider the Friedrichs self-adjoint extension, which amounts to set Dirichlet conditions at t = π/2ℓj. It is clearly invertible ± 1 for each ℓj > 0 and the inverse can be computed using Fourier decomposition in θ. If L− is written under the form π 1 2ℓj (L− f)(t,θ)= G (t,θ,t′,θ′)f(t′,θ′)dθ′dt′ π L R/Z Z− 2ℓj Z for some Green kernel GL, then Ggj can be written as

′ arctan( ρ ) arctan( ρ ) ℓj ℓj Ggj (ρ, θ, ρ′,θ′)= GL , θ, ,θ′ . ℓj ℓj This is clear since the left-hand side maps C ( ) to L2( ) and is a right inverse for ∆ on c∞ Fj Fj gj C ( ). Now, computing G is quite simple: using the Fourier decomposition c∞ Fj L 1 2πikθ 1 L− f(t,θ)= e (Lk− fk)(t) k Z X∈ 2πikθ π π 2 where f(t,θ)= e fk(t) and Lk is the operator on Ij := ( , ) given by Lk = ∂ + k − 2ℓj 2ℓj − t 4π2k2 with Dirichlet condition at ∂I . For k = 0, straightforward Sturm-Liouville argument P j 6 gives the expression of the Green function for Lk: with k¯ = 2πk,

1 kt¯ k¯(t π/ℓ ) kt¯ ′ k¯(t′+π/ℓ ) j j ′ GLk (t,t′)= ¯ (e− e − )(e e− ) 1lt > t ¯ 2kπ/ℓj 2k(1 e− ) − − − kt¯ ′ k¯(t′ π/ℓ ) kt¯ k¯(t+π/ℓ ) + (e− e − j )(e e− j ) 1l ′ > − − t t k¯ t t′ 2πk/ℓ¯ kπ/ℓ¯ e− | − | e− j cosh(k¯(t t′)) e− j cosh(k¯(t + t′)) = + − − ¯ . 2k¯ k¯(1 e 2πk/ℓj ) − − π 2 If max( ρ , ρ′ ) 6 1, we have t 6 1+ (ℓ ) and same for t′ thus for ℓj small, | | | | | | 2ℓj − O j kπ/ℓ¯ k/¯ 2 2πk/ℓ¯ πk/ℓ¯ e− j cosh(k¯(t + t′)) e− e− j cosh(k¯(t t′)) e− j ¯ 6 , ¯ − 6 . k¯(1 e 2πk/ℓj ) k¯ k¯(1 e 2πk/ℓj ) k¯ − − − −

We then get k¯ t t′ +ik¯(θ θ′) ik¯(θ θ′) e− | − | − 1 2π( t t′ i(θ θ′)) GL (t,t′)e − = + (1) = log 1 e− | − |− − + (1) k 2k¯ O −2π − O k=0 k=0 X6 X6 when ℓ is small. Notice also that if ρ [ℓ , 1] and ρ [ℓ , 1] with ρ and ρ having diﬀerent j | |∈ j | ′|∈ j ′ sign, then π ¯ ′ ik(θ θ ) 6 − 2ℓj GLk (t(ρ),t′(ρ′))e − Ce . (2.43) k=0 X6

LQG AND BOSONIC 2d STRING THEORY 23

1 if t(ρ) = ℓj− arctan(ρ/ℓj) and similarly for t′(ρ′). Next for k = 0, the Green function is given by the expression 1 ℓj π GL0 (t,t′)= t t′ tt′ + , −2| − |− π 4ℓj thus we get

1 ρ ρ′ 1 π2 ρ ρ′ GL0 (t(ρ),t′(ρ′)) = arctan( ℓ ) arctan( ℓ ) + 4 arctan( ℓ ) arctan( ℓ ) . − 2ℓj j − j πℓj − j j du ∞ If F (x) := x 1+u2 , this can be rewritten as R 1 ρ ρ′ 1 ρ ρ′ 1 ρ ρ′ GL (ρ, ρ′)= F ( ) F ( ) + (F ( )+ F ( )) F ( )F ( ) 0 − 2ℓj ℓj − ℓj 2ℓj ℓj ℓj − πℓj ℓj ℓj 1 ρ ρ ′ 1 ρ ρ′ (2.44) min( F ( ), F ( )) F ( )F ( ), if ρ> 0, ρ′ > 0 ℓj ℓj ℓ j − πℓj ℓj ℓj = 1 ρ ρ′ ( F ( )F ( ), if ρ> 0, ρ′ < 0 πℓj ℓj − ℓj from which (2.38) and (2.39) follow. Now, we also see from the expressions of G above that if ρ [1/2, 1] and ρ ρ > δ for Lk | |∈ | − ′| some ﬁxed constant δ > 0, then

∂ G (ρ, θ, ρ′,θ′) 6 C | ρ gj | for some constant C depending only on δ but not on ℓj. du Next we prove (2.41). Let F (x)= ∞ and let c := χ ∞ . For ρ> 0, we write x 1+u2 || ||L 0 χ(ρ′) 1 Rρ ρ′ χ(ρ′) ∞ χ(ρ′) G (ρ, θ, ρ′,θ′) dx′ = F ( ) F ( ) dρ′ + G (ρ, θ, ρ′,θ′) dρ′ gj λ ℓj ℓj λ gj λ ρ′ ℓjπ 1 − ρ′ 0 ρ′ Z Z− Z 0 ρ 1 c ρ′ 1 c dρ′ dρ′ 6 F ( ) dρ′ + + c ℓj λ λ 1+λ πρ 1 − ρ′ ρ 0 ρ′ ρ ρ′ Z− Z Z ρ λ ρ λ 1 6 2c − + 2c − − 1 λ λ − where we used (2.44) in the second line and ρ 1 λ 0 − ρ′ dρ′ dρ′ ρ− 1 ρ′ dρ′ π 1 λ F ( ) 6 ℓ 6 ℓ − , F ( ) 6 ρ − ℓj λ j 1+λ j ℓj λ 1 − ρ′ ρ ρ′ λ ρ − ρ′ 2(1 λ) Z− Z Z− − for the third line. The same estimate works in the case ρ< 0. Using the estimates and expressions above, we also get as (ρ ,θ ) (ρ, θ) is small | ′ ′ − | 1 2 2 ℓj 2 π GL(t,θ,t′,θ′)= log( (t t′) + (θ θ′) ) t + + (1) + ( t t′ + θ θ′ ) −2π − − − π 4ℓj O O | − | | − | p where (1) is independent of all variables. We also have O 2 2 2 2 log(d (ρ, θ, ρ′,θ′)) = log( (t t ) + (θ θ ) ) + log( ρ + ℓ )+ ( t t′ + θ θ′ ) gj − ′ − ′ j O | − | | − | p q thus the Robin mass of Ggj satisﬁes (2.42).

Next we express R (λ)dλ/λ in terms of G . Let χ ,χ C (M) which are sup- ∂D(0,ε) g gj j j′ ∈ ∞ ported in (g), depending only on the variable ρ associated to the metric g, are equal to 1 in Cj R ρ 6 1/4 and such that χ = 1 on a neighborhood of supp(χ ). We will use the diﬀeomorphisms | | j′ j 24 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS between (g) and the subset ρ < 1 of to identify these sets (for notational simplicity we Cj | | Fj won’t input these diﬀeos below). Using that ∆ =∆ in (g), we have, with χ := χ g gj Cj j j (∆ λ) χ′ R (λ)χ = χ + [∆ ,χ′ ]R (λ)χ P g − j gj j g j gj j Xj Xj and thus

χj′ Rgj (λ)χj = Rg(λ)χ + Rg(λ) [∆g,χj′ ]Rgj (λ)χj . Xj Xj Similarly, we also have

χjRgj (λ)χj′ = χRg(λ)+ χjRgj (λ)[χj′ , ∆g]Rg(λ) Xj Xj and therefore (using also χ χ =0 if i = k) j′ k′ 6 χR (λ)χ = χ R (λ)χ χ R (λ)χ′ [∆ ,χ′ ]R (λ)χ g j gj j − j gj j g j gj j Xj Xj

+ χjRgj (λ)[∆g,χj′ ]Rg(λ) [χi′ , ∆g]Rgi (λ)χi Xj Xi = χ R (λ)χ K (λ)+ K (λ)R (λ)K (λ) j gj j − 1 2 g 3 Xj where K1(λ), K2(λ), K3(λ) are deﬁned by the equation. Remark that, in the right hand side, only the last term has poles (ﬁrst order) in D(0, ε). Therefore, using Cauchy formula

χRg(λ)χ dλ = χjRgj (0)χj K1(0) + K2(0)AgK3(0) ∂D(0,ε) 2πiλ − Z Xj K′ (0)Π K (0) K (0)Π K′ (0) − 2 0 3 − 2 0 3 m (K2(λk) K2(0)) (K3(λk) K3(0)) − Πλk K3(λk) K2(0)Πλk − − λk − λk Xk=1 where K (0) := ∂ K (λ) and A := Rg (λ) dλ. Let us analyse those terms more i′ λ i |λ=0 g ∂D(0,ε) 2πiλ carefully. R

Proposition 2.7. Let g ∂ g be a hyperbolic surface with cusps in the boundary of moduli 0 ∈ M space. Then there is a neighborhood U of g in g and C > 0 such that for all g U and g0 0 M ∈ g0 x,x (g) ′ ∈∪jCj χ(x)χ(x′)G (x,x′)= χ (x)G (x,x′)χ (x′)+ Q (x,x′) H (x)J (x′) J (x)H (x′) g j gj j g − 0 0 − 0 0 Xj m H (x) λ J (x) H (x ) λ J (x ) + k − k k k ′ − k k ′ √λk √λk Xk=1 with Q C (M M), H = χϕ , J C (M) satisfying g ∈ ∞ × k λk k ∈ ∞ (1 s ) Q ∞ 6 C, H ∞ 6 C, H (ρ, θ) 6 C ρ − − k , | g|L | 0|L | k | | | ρ (1 sk) 1 (1 sk) − − J0(ρ, θ) 6 C log ρ , Jk(ρ, θ) 6 C ρ − − + | | − | | | | || | | | | (1 sk) − LQG AND BOSONIC 2d STRING THEORY 25 with s (1 s )= λ (g) > 0 the small eigenvalues of ∆ converging to 0 as g approaches ∂ g k − k k g M and s = 1 λ (g)+ (λ (g)2). Here ρ = ℓ sinh(t) in (g) with t the signed distance to the k − k O k j Cj geodesic γj(g).

Proof. The integral kernel of χ R (0)χ is χ (x)χ (x )G (x,x ) in (g) with respect to j j gj j j j ′ gj ′ Cj the measure dv . This term has an explicit bound by using Lemma 2.6. The function Q (x,x ) gj P g ′ will be chosen to be the integral kernel of K (0) + K (0)A K (0), let us show this is smooth − 1 2 g 3 and uniformly bounded. The integral kernel of K1(0) is of the form

K1(0; x,x′)= χj(x) Ggj (x,y)χj′ (y)Pj(y)Ggj (y,x′)dvgj (y)χj(x′) ′ Xj ZCj in (g) with respect to the measure dv , where P is a smooth diﬀerential operator of order 1 Cj gj j supported in supp( χ ) (thus far from the γ (g)= ρ = 0 curve). Thus by (2.40) and (2.39), ∇ j j { } we directly get that for all x,x (g) with ρ(x) >ℓ and ρ(x ) >ℓ ′ ∈Cj | | j | ′ | j K (0; x,x′) 6 C, and if ρ(x)ρ(x′) < 0, K (0; x,x′) 6 Cℓ . | 1 | | 1 | j

By Proposition 2.2, the norm Ag L2(W ) L2(W ) is uniformly bounded if W := supp( χj′ ), || || → ∇ and by the same argument as for K1(0), K2(0) and K3(0) have smooth integral kernels that are uniformly bounded with respect to ℓ , thus there is C > 0 uniform so that for all x,x M j ′ ∈ (K (0)A K (0))(x,x′) 6 C. | 2 g 3 |

We can rewrite K2′ (0)Π0K3(0) by using that Π0 = c with c = 1/Volg(M):

(K2′ (0)Π0K3(0))(x,x′)=c χj(x)(∂λRgj (0)∆gχj′ )(x)χ(x′) Xj

=c χj(x)(Rgj (0)χj′ )(x)χ(x′)= H0(x′)J0(x) Xj where we used ∂λRgj (0)∆gj = Rgj (0), J0(x) := √c j χj(x) Ggj (x,x′)χj(x′)dvgj (x′) and Cj √ H0 := cχ are smooth functions on M. Similarly, weP get R

(K2(0)Π0K3′ (0))(x,x′)= H0(x)J0(x′). Now the bound (2.41) gives that J (ρ, θ) 6 C log ρ and H (x) 6 C for some uniform | 0 | | | || | 0 | C > 0. Using that (∆ λ )Π = Π (∆ λ ) = 0, we get g − k λk λk g − k (K2(λk) K2(0)) − Πλk K3(λk)= χjRgj (0)χj′ Πλk [χi′ , ∆g]Rgi (λk)χi λk Xj Xi

= χjRgj (0)χj′ Πλk χ. Xj Similarly, we also get

(K3(λk) K3(0)) K2(0)Πλk − = χjRgj (0)[∆g,χj′ ]Πλk χi′ Rgi (0)χi λk Xj Xi ′ =χΠλ1 χ′ Rg (0)χi λk χjRgj (0)χ′ Πλ χ′ Rgi (0)χi. i i − j k i Xi Xj Xi 26 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

Write Π (x,x )= ϕ (x)ϕ (x ) for some ϕ ker(∆ λ ) with L2 norm 1, then we get λk ′ λk λk ′ λk ∈ g − k (K2(λk) K2(0)) (K3(λk) K3(0)) − Πλk K3(λk)+ K2(0)Πλk − = λk λk H (x)J (x′)+ J (x)H (x′) λ J (x)J (x′) k k k k − k k k where H = χϕ , J (x) = χ (x) G (x,x )χ (x)ϕ (x )dv (x ). To conclude, we k λk k j j j (g) gj ′ j′ λk ′ gj ′ need some estimates on the eigenfunctionC ϕ associated to the small positive eigenvalue λ of P R λk k ∆g on M

Lemma 2.8. Let g ∂ be a surface with node with m + 1 connected components and 0 ∈ Mg denote by λ1, . . . λm the positive small eigenvalues. For each ε> 0, there is a neighborhood Ug0 of g0 and C > 0 such that for each g Ug0 , the following holds: if ϕλi is an eigenfunction for ∈ R λi which satisﬁes ϕλi ρ= 1 aij± < ε for some constants aij± in the collar j(g), then it | | ± − | ∈ C satisﬁes + s 1 ϕ (ρ, θ) = (a 1l +a− 1l ) ρ i− (1 + (ε)) + (ε) λi ij ρ>0 ij ρ<0 | | O O in the region ρ > Cℓ of the collar (g), where s (1 s ) = λ and s = 1 λ + (λ2) {| | j} Cj i − i i i − i O i when λ is small. In the region ρ < Cℓ , there is C > 0 such that i | | j ′ s 1 ϕ (ρ, θ) 6 C′ ρ i− . | λi | | |

Proof. To simplify notations, we remove the i, j indices from aij±, λi and si. We decompose ϕλ in Fourier modes in θ: there are b C ([ 1, 1]) so that k ∈ ∞ −

∞ 2πikθ ϕλ(ρ, θ)= bk(ρ)e k= X−∞ and the series converges uniformly. Since ϕλ can be supposed real-valued, b0 is real and b k = − bk. Moreover ak(t) := bk(ℓj sinh(t)) satisfies the ODE 2 2 2 4π k ∂t tanh(t)∂t + 2 2 λ ak(t) = 0, ak( tj)= ak± − − ℓj cosh(t) − ± if tj is defined by ℓj sinh(tj) = 1. We write a± := ak t= t = bk ρ= 1. First we make the k | ± j | ± following observation for each k = 0 : 6 + a (t) 6 max( a , a− ) < ε. (2.45) | k | | k | | k | + Indeed, assume that a achieves its maximum at T ( t ,t ) with a (T ) > max(a , a−), then k ∈ − j j k k k if ak(T ) > 0 4π2k2 ak′′(T )= λ 2 2 ak(T ) < 0 − − ℓj cosh(T ) when ℓj is smaller than a uniform constant. This contradicts that ak(T ) is a local maximum, + thus ak achieves its maximum at tj or its maximum is non-positive, in which case ak 6 0 ± + and a− 6 0. In both cases, a (T ) 6 max( a , a− ). The same argument works with the k | k | | k | | k | minimum and this shows (2.45). Next we analyze u0(t), and it is convenient for that to use s (0, 1) so that s(1 s)= λ (then s = 1 λ + (λ)2). There are 2 independent solutions of ∈ − − O LQG AND BOSONIC 2d STRING THEORY 27 the ODE with k = 0 (and no bounday condition), the first one v0 is odd in t, the other one u0 is even, they are given by [Bo, Chapter 5.1]

1 1+s 2 Γ( 2 s)Γ( 2 ) s 1+ s s 1 1 v0(t) =sign(t) − sinh(t) − F , , + s; − Γ(s 1 )Γ(1 s )2 | | 2 2 2 sinh(t)2 − 2 − 2 s 1 2 s 1 s 3 1 + sign(t) sinh(t) − F − , − , s; − , | | 2 2 2 − sinh(t)2 Γ( 1 s)Γ( s )2 s s + 1 1 1 u (t)= 2 − 2 sinh(t) sF , , + s; − 0 1 1 s 2 − 2 Γ(s )Γ( − ) | | 2 2 2 sinh(t) − 2 2 s 1 1 s s 3 1 + sinh(t) − F − , 1 , s; − | | 2 − 2 2 − sinh(t)2 where F (a, b, c; z) is the hypergeometric function, holomorphic in the variables a, b, c for a, b, c ∈ C in the half-plane C := c C, Re(c) > 0 if z ( , 0), it is smooth in z and for z < 0 + { ∈ } ∈ −∞ small F (a, b, c; z)=1+ ( z ) O | | where the remainder is uniform for a, b, c in compact sets of C+. In particular, there is C > 0 uniform in g so that for t >C, | | s 1 s s 1 s v (t) = sign(t) sinh(t) − + ( sinh t − ), u (t)= sinh(t) − + ( sinh t − ) 0 | | O | | 0 | | O | | where the remainder is uniform with respect to λ for λ> 0 small. We obtain

+ + (a0 + a0−) u0(t) (a0 a0−) v0(t) a0(t)= + − 2 u0(tj) 2 v0(tj) and we deduce that

+ s 1 s 1 s a (t) =(a 1l +a− 1l ) ℓ sinh(t) − + (ℓ − sinh(t) − ) 0 0 t>0 0 t<0 | j | O j | | s 1 s 2s 1 s s 1 We now use ℓ sinh(t)= ρ and ℓ − sinh(t) = ℓ − ρ 6 ε ρ if ρ > Cℓ with C large j j | |− j | |− | | − | | j enough (depending on ε). Next, consider the case ρ < Cℓ . We can also write u and v under the form (see [Bo, | | j 0 0 Chapter 5.5])

1+s 2 Γ( 2 ) 1+ s s 3 2 v0(t)= sinh(t)F , 1 , ; sinh(t) , Γ( 3 )Γ(s 1 ) 2 − 2 2 − 2 − 2 s 2 Γ( 2 ) s 1 s 1 2 u0(t)= F , − , ; sinh(t) Γ( 1 )Γ( 1 s) 2 2 2 − 2 2 − and this easily yields the desired estimate.

The proof is complete by noticing that the estimates on Hk, Jk follow fom this Lemma and (2.41), together with the fact that each ϕ is bounded uniformly in M (g) for g U λk \∪jCj ∈ g0 by Lemma 2.5. 28 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

3. Gaussian Free Field and Gaussian Multiplicative Chaos

In this section, we shall explain how to give a mathematical sense to the formal measure

S (g,ϕ) F F (ϕ)e− L Dϕ. (3.1) 7→ Z where SL(g, ϕ) is the Liouville functional defined in (2.2), g is a fixed metric on the surface M and ϕ varies among a certain space of functions so that eγϕg is parametrizing the conformal class [g] of g. This will allow us to define the partition function of Liouville Quantum Field Theory, and in fact ϕ will be a field, i.e. a random function or random distribution, that we will denote by Xg. The first step is to make sense of the part corresponding to the squared gradient term in SL(g, ϕ), i.e. the formal Gaussian measure

1 dϕ 2 F F (ϕ)e− 4π || ||L2 Dϕ. (3.2) 7→ Z Classically, we interpret the above field ϕ as a Gaussian Free Field (GFF in short): this is a Gaussian random variable taking values in some space of distributions in the sense of Schwartz. In particular, the field ϕ is not a fairly defined function and giving sense to the term eγϕ in (2.2) is thus not straightforward, but it can be done through the theory of Gaussian Multiplicative Chaos (GMC in short), which goes back to [Ka].

3.1. Gaussian Free Field. We describe the Gaussian Free Field (GFF) on a compact Rie- mannian surface (M, g) by using our previous description of the Green function. The definition of the GFF, as well as the definition of its partition function, can be carried out in a direct way ([Du1, She]). Yet, this path is maybe not as pedagogical as following the threads of ideas that have led physicists to our current knowledge of this object, which we try to summarize below in a rather heuristic way to end up with a mathematically sounded definition. As a warm up, let us quickly recall that the Gaussian measure

1 n/2 Ax,x (2π)− det(A)e− 2 h idx p on Rn when A is a positive definite symmetric operator is the law of the random variable X = n α ϕ / λ where (α ) are independent Gaussian random variables in (0, 1) (zero j=1 j j j j j N mean and variance 1), and (ϕ ) is an orthonormal basis of eigenvectors for A with eigenvalues P p j j λj > 0. As the GFF is an infinite dimensional Gaussian, it is natural to expect a construction through its projections onto finite dimensional subspaces, on which one can apply the construction described just above. For this, recall that the Laplacian ∆g has an orthonormal basis of real 2 valued eigenfunctions (ϕj )j N0 in L (M, g) with associated eigenvalues λj > 0. We set λ0 = 0 ∈ and ϕ0 to be constant. The Laplacian can thus be seen as a symmetric operator on an infinite dimensional space. Denote Hn the finite dimensional space spanned by the first n eigenfunctions n 2 n 2 (ϕ ) 6 6 of the Laplacian. Notice that for φ = α ϕ we have dφ = α λ . j 1 j n j=1 j j k kL2 j=1 j j P P e e LQG AND BOSONIC 2d STRING THEORY 29

Therefore the projection to Hn of the formal measure (3.2) is naturally understood as n n 1 2 1 2 4π dφ 2 (αj ) λj F (φ)e− k kL Dφ = F αjϕj e− 4π e dαj Rn ZHn Z j=1 j=1 X Y n n n 2 e 1/2 e αj n/2 − ϕj = (2π) λj F √2π αj e− 2 dαj Rn λ j=1 Z j=1 j j=1 Y X Y p for appropriate bounded measurable functionals F . The total mass of this measure is easily 1/2 n n − computed and equals (2π) j=1 λj . Furthermore, when renormalized by its total mass, this measure becomes thus a probability measure describing the law of the random function Q n ϕj Xn = √2π αj (3.3) λj Xj=1 where (α ) are independent Gaussian random variablesp with law (0, 1). j j N To obtain the description of the GFF, one has to take the limit n . It can be seen →s ∞ [Du1] that the sum (3.3) converges almost surely in the Sobolev space H− (M) for s> 0. The 1/2 total mass (2π)n n λ − diverges as n but this is not that much troublesome as j=1 j →∞ it is customary in physicsQ (and mathematics) to remove the diverging terms provided they are ”universal enough” (this procedure is called renormalization). Removing the diverging terms 1/2 1 − should give a limiting total mass equal to det′( 2π ∆g)/Volg(M) . So far, this is the picture the reader should have in mind to understand the construction of the GFF. Yet, for readers n who want to have more details, we stress that renormalizing the product j=1 λj turns out to be very troublesome and slight adaptations are necessary to recover the phenomenology Q explained above. The reader may consult the paper [BiFe] where these renormalization issues are discussed in further details.

The above formal discussion thus motivates the forthcoming deﬁnitions. The Green function G (x,x ) (with vanishing mean) is a distribution on M M which can be written as the series g ′ × (converging in the sense of distributions)

∞ ϕj(x)ϕj (x′) Gg(x,x′)= . λj Xj=1 Let (a ) be a sequence of i.i.d. real Gaussians (0, 1), deﬁned on some probability space j j N (Ω, , P), and deﬁne the Gaussian Free Field (GFF) with vanishing mean in the metric g by F ϕj Xg = √2π aj (3.4) > λj jX1 as a random variable with values in distributions onpM, i.e. almost surely X (M) (see g ∈ D′ [Du1] section 4.2 for instance). Notice that for each φ C (M), almost surely, we have ∈ ∞ ϕj ,φ E 2 Xg, φ = √2π j∞=1 aj h i which is a converging series of random variables as ( Xg, φ ) < h i √λj h i s 2 2 . In fact, if HP0− (M) is the kernel of the map X X, 1 L (dvg) on the L -based Sobolev ∞ s 7→ h i space H− (M) of order s R∗ , it is easy to see (see [Du1] again) that Xg makes sense as a − ∈ − 30 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

2 s random variable with values in L (Ω; H0− (M)) for all s> 0 by using the asymptotic counting function on the eigenvalues λ (i.e. Weyl law). If φ , φ C (M), we have the covariance j 1 2 ∈ ∞ value ∞ ϕj , φ1 ϕj, φ2 E[ Xg, φ1 . Xg, φ2 ] = 2π h ih i = 2π Gg, φ1 φ2 . h i h i λj h ⊗ i Xj=1 The covariance is the Green’s function when viewed as a distribution: if φ δ and φ δ ′ 1 → x 2 → x for x = x we have E( X , φ . X , φ ) 2πG (x,x ) and we write the formal equality for 6 ′ h g 1i h g 2i → g ′ the covariance of the GFF

E[Xg(x).Xg(x′)] = 2π Gg(x,x′).

Notice that the extra 2π factor serves to make the field Xg have exact logarithmic correlations s in view of Lemma 2.1. As in [She, Theorem 2.3], there is a probability mesure on H− (M) P 0 (for some natural σ-algebra) so that the law of X is given by and for each φ Hs(M), g P ∈ X , φ is a random variable on Ω with zero mean and variance 2π R (0)φ, φ . The mesure h g i h g i represents the Gaussian mesure (3.2) (times the det(∆ ) term) on the space of functions P g orthogonal to constants, thus to define (3.2) on the whole H s(M) space, we shall consider p − the tensor product dc where dc is the Lebesgue mesure on R viewed as the 1-dimensional P⊗ vector space of constant functions on M: in other words, we use the isomorphism s s H− (M) R H− (M), (X, c) X + c. 0 × → 7→ to define the mesure on H s(M) as the image of dc by this map. This mesure gives a P′ − P ⊗ proper sense to the Gaussian mesure (3.2). We have

Lemma 3.1. The mesure on H s(M) obtained by tensorizing the GFF mesure by dc is P′ − P conformally invariant in the sense that it does not depend on the conformal representative in a conformal class [g].

ω s Proof. Letg ˆ = e g for some ω C∞(M). Notice that H0− (M) depends on g, we thus denote s ∈ it H− (M, g) and we denote , the distribution pairing on M or M M induced by the 0 h· ·ig × measure dvg. First we claim that the probability law obtained from Xˆg := Xg cgˆ(Xg) is the ω − same as that of Xgˆ, if cgˆ(Xg) := Xg, 1 gˆ/Volgˆ(M) = Xg, e g/Volgˆ(M). The random field h i s h i Xˆ satisfies Xˆ , 1 = 0 and is thus in the space H− (M, gˆ), moreover E[ Xˆ , φ ] = 0 for all g h g igˆ 0 h g igˆ φ C (M). The covariance of Xˆ is given by ∈ ∞ g 2 E[ Xˆ , φ Xˆ , φ ]= G , φ φ + (Vol (M))− G , 1 1 φ , 1 φ , 1 h g 1igˆh g 2igˆ h g 1 ⊗ 2igˆ gˆ h g ⊗ igˆh 1 igˆh 2 igˆ 1 (Vol (M))− ( G , 1 φ 1, φ + G , φ 1 1, φ ) − gˆ h g ⊗ 2igˆh 1igˆ h g 1 ⊗ igˆh 2igˆ = G + α1 1 u 1 1 u, φ φ h g ⊗ − ⊗ − ⊗ 1 ⊗ 2igˆ where α = (Vol (M)) 2 G , 1 1 , u(x) = G (x,y)dv (y)/Vol (M). We recognize that gˆ − h g ⊗ igˆ M g gˆ gˆ this kernel is just the Green function forg ˆ (this is easily checked) against φ φ , showing R 1 ⊗ 2 that the correlation of Xˆg is that of Xgˆ. Since both random fields are Gaussian, we deduce that the law of Xˆ and X are the same and thus for F L1(H s(M), ), g gˆ ∈ − P′

E[F (Xgˆ + c)]dc = E[F (Xg cgˆ(Xg)+ c)]dc = E[F (Xg + c)]dc R R − R Z Z Z by making a change of variables in c. LQG AND BOSONIC 2d STRING THEORY 31

Finally, in view of the discussion above, the measure F E(F ) on H s(M) represents the 7→ − formal measure 1 2 1 4π RM dϕ gdvg det′( 2π ∆g) e− | | Dϕ (3.5)

q 1 χ(M) (1 ) 1 2 6 and using (2.10), we can write det′( 2π ∆g) = (2π) − det′(∆g). q p 3.2. Gaussian multiplicative chaos. To deﬁne quantities like eγX for some γ R we will use ∈ a renormalization procedure after regularization of the ﬁeld Xg. We describe the construction for g hyperbolic and we shall remark that in fact the construction works as well for any conformal metricg ˆ = eωg by using Lemma 2.1.

First, when ε > 0 is very small, we define a regularization Xg,ε of Xg by averaging on geodesic circles of radius ε > 0. Let x M and let (x, ε) be the geodesic circle of center n ∈ C n x and radius ε > 0, and let (fx,ε)n N C∞(M) be a sequence with fx,ε L1 = 1 which is ∈ ∈ || || given by f n = θn(d (x, )/ε) where θn(r) C ((0, 2)) non-negative supported near r = 1 x,ε g · ∈ c∞ such that f n dv is converging in (M) to the uniform probability measure µ on (x, ε) x,ε g D′ x,ε Cg as n (for ǫ small enough, the geodesic circles form a 1d-manifold and the trace of g along →∞ this manifold gives rise to a finite measure, which corresponds to the uniform measure after renormalization so as to have mass 1, it can also be defined in terms of 1-dimensional Hausdorff measure constructed with the volume form on M and restricted to this geodesic circle). Then we have the standard

Lemma 3.2. The random variable X ,f n converges to a random variable as n , which h g x,εi →∞ has a modiﬁcation X (x) with continuous sample paths with respect to (x, ε) M (0, ε ), g,ε ∈ × 0 with covariance

E[Xg,ε(x)Xg,ε(x′)] = 2π Gg(y,y′)dµx,ε(y)dµx′,ε(y′) Z and we have as ε 0 → E[X (x)2]= log(ε)+ W (x)+ o(1) (3.6) g,ε − g 1 where Wg is the smooth function on M given by Wg(x) = 2πmg(x,x)+ 2 log(2) if mg is the smooth function of Lemma 2.1.

n Proof. Let us ﬁx x, ε, then if Y := X ,f , it suﬃces to show that E(Y Y ′ ) has a limit n h g x,εi n n as (n,n ) to prove that Y is a Cauchy sequence in L2(Ω). Using Lemma 2.1 (and its ′ → ∞ n notation):

E n n′ (YnYn′ ) =2π Gg(y,y′)fx,ε(y)fx,ε(y′)dvg(y)dvg(y′) M M Z × n n′ = ( log(d (y,y′)) + 2πm (y,y′))θ (d (x,y))θ (d (x,y′))dv (y)dv (y′). − g g ε g ε g g g ZM M × (3.7) Clearly the term

n n′ mg(y,y′)θε (dg(x,y))θε (dg(x,y′))dvg(y)dvg(y′) M M Z × 32 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS is uniformly bounded in (n,n , ε) and, as (n,n ) , it converges, to ′ ′ →∞

mg(y,y′)dµx,ε(y)dµx,ε(y′) (x,ε) (x,ε) ZC ZC which in turn is smooth in x and converges, as ε 0, to m (x,x) uniformly in x. For ε > 0 → g small enough, we can use an isometry ψ between a small geodesic ball Bg(x, 3ε) of radius 3ε 2 and the ball BH2 (0, 3ε) in H viewed as the disk model, so that the integral (3.7) above reduces to an integral in B (x, 3ε) in both y,y . Using the coordinates z H2 induced by ψ and (2.20), g ′ ∈ we get

n n′ log(dg(y,y′))θε (dg(x,y))θε (dg(x,y′))dvg(y)dvg(y′)= M M Z × r i(α α′) r′ n r n′ r′ log tanh( 2 )e − tanh( 2 ) + L θ ( ε )θ ( ε )dαdα′ sinh(r) sinh(r′)drdr′. [0,1]2 [0,2π]2 | − | Z × where L = L(r, r , eiα, eiα) is continuous and L(0, 0, , ) = 0. The term involving L is clearly ′ · · uniformly bounded in (n,n ) and ε and converges just like for m above, and its limit as ε 0 ′ g → is 0. The part with the log term is also straightforward to deal with and is also uniformly bounded in (n,n′) for ﬁxed ε> 0 and we get

r i(α α′) r′ n r n′ r′ log tanh( 2 )e − tanh( 2 ) θ ( ε )θ ( ε )dαdα′ sinh(r) sinh(r′)drdr′ [0,1]2 [0,2π]2 | − | Z × ε 1 i(α α′) ε log tanh( 2 ) + 2 log e − 1 dαdα′ = log tanh( 2 ) . (n,n−→′) | | 4π 2 | − | | | →∞ Z[0,2π] We then have shown the convergence of X ,f n towards a random variable X (x) in L2(Ω). h g x,εi g,ε To show it has a modification X (x) that is sample continuous in (x, ε) M (0, ε ), it g,ε ∈ × 0 suffices to apply Kolmogorov multi-parameter continuity theorem exactly likee in the proof of 2 [DuSh, Prop. 3.1], we do not repeat the argument. The variance E(Xg,ε(x) ) is smooth in x and behaves like log(ε)+ 1 log(2) + 2πm (x,x)+ o(1) as ε 0, uniformly in x. − 2 g → Next from Lemma 3.2, we will be able to define the Gaussian Multiplicative Chaos (GMC) first considered by Kahane [Ka] in the eighties. The reader may also consult [DuSh, RhVa1, RoVa, Sha] on the topic (in particular, we recommend [Be] for the simplicity of the approach).

γ2 γ γXg,ε(x) Proposition 3.3. 1) Let γ > 0, the random measures g,ε := ε 2 e dvg(x) converge in G γ probability and weakly in the space of Radon measures towards a random measure g (dx). The γ G measure g (dx) is non zero if and only if γ (0, 2). G ∈ 2) One obtains a non trivial random measure that we will denote 2 in the case γ = 2 by Gg considering the limit in probability and in the sens of weak convergence of measures of the 2 γ 1/2 γ γX (x) family of random measures g,ε := ( ln ε) ε 2 e g,ε dv (x). G − g Proof. The proof is standard for convolution based regularizations of log-correlated Gaussian fields (first considered in [RoVa], see [Sha] for latest results) in the case γ < 2. The case γ =2is treated in [DRV, section 5] in the case of tori, relying on the result proved in [DRSV1, DRSV2] for GFF with Dirichlet boundary conditions. The same argument applies for general compact 2d-surfaces up to cosmetic modifications. LQG AND BOSONIC 2d STRING THEORY 33

Here we give a simple argument in the case γ < √2 for convenience of readers who are not familiar with GMC. Using the expression (3.6), it suﬃces to study the convergence of the measures 2 2 γ 2 γ γXg,ε(x) E[Xg,ε(x) ] Wg(x) e − 2 e 2 dvg(x). 2 γ W (x) Then by Fubini we directly get for each Borel set A M, with dσ = e 2 g dv (x) ⊂ g 2 γ 2 γ γXg,ε(x) E[Xg,ε(x) ] E[ (A)] = E[e − 2 ]dσ(x)= σ(A). Gg,ε ZA Using that there is C > 1 such that for all z C, z < 1 and ε> 0 small ∈ | | 2π 1/C + log( z + ε) 6 log z + εeiα dα 6 C + log( z + ε) | | | | 0 | | | | | | Z then the arguments in the proof of Lemma 3.2 and the expression (2.20) imply that there is C′ such that

1/C′ + log(d (x′,x)+ ε) 6 E[X (x)X ′ (x′)] 6 C′ + log(d (x′,x)+ ε) . (3.8) | g | g,ε g,ε | g | for all ε 6 ε, and all x,x M. In particular we get by using Fubini and the fact that X is a ′ ′ ∈ g Gaussian free ﬁeld 2 γ E 2 2 E γ 2 E γXg,ε(x) 2 [Xg,ε(x) ] [ g,ε(A) ]= e − dσ(x) G A h Z 2 i ′ γ 2 ′ 2 γ(Xg,ε(x)+Xg,ε(x )) (E[Xg,ε(x) ]+E[Xg,ε(x ) ]) =E e − 2 dσ(x)dσ(x′) A A h Z Z i 2 ′ γ E[Xg,ε(x)Xg,ε(x )] = e dσ(x)dσ(x′) ZA ZA 2 ′ which converges to eγ 2πGg (x,x )dσ(x)dσ(x ) < as ε 0 by using (3.8) and Lebesgue A A ′ ∞ → theorem - the condition γ2 < 2 appear here due to the log divergence of 2πG (x,x ) at x = x , R R g ′ ′ E γ γ 2 see Lemma 2.1. A similar argument and (3.8) also show that [( g,ε(A) g,ε′ (A)) ] 0 if γ G − G2 → (ε, ε′) 0, thus g,ε(A) is a Cauchy sequence, which therefore converges in L (Ω) to a random → G γ variable Z(A), of mean σ(A). By standard arguments, g,ε(dx) converges to a random measure γ γ G g satisfying E[ g (A)] = σ(A). The case γ [√2, 2) is more complicated and several methods G G ∈ have been proposed in the literature. We refer to [Be] for a simple argument.

In fact, the whole construction above is not so particular to choosing the hyperbolic metric: indeed it uses only the fact that the covariance of Xg is the Green function, the fact that near the diagonal 2πG (x,x ) = log d (x,x )+ F (x,x ) with F continuous, and finally the fact g ′ − g ′ ′ that in local isothermal coordinates z so that g = e2f(z) dz 2 | | log d (z, z′) = log z z′ + (1). g | − | O This allows to define a random measure γ just as above for any other metricg ˆ = eωg conformal Ggˆ to the hyperbolic metric g. For later purpose we will need to make the following observation. Ifg ˆ = eωg, define ˆ ˆn Xg,ε(x) := lim Xg, fx,ε gˆ (3.9) ε 0h i → for each x M where fˆn := θn(d (x, )/ε) with θn like above, so that fˆn dv converge as ∈ x,ε gˆ · x,ε gˆ n to the uniform probability measureµ ˆ on the geodesic circle (x, ε) of center x and →∞ x,ε Cgˆ 34 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS radius ε with respect tog ˆ. In isothermal coordinates at x so that z = 0 correspond to the point x and the metric is g = dz 2/Im(z)2, the circle (x, ε) is parametrized by | | Cgˆ 1 ω(z)+εhε(α) iα εe− 2 e , α [0, 2π] ∈ for some continuous function hε(α) uniformly bounded in ε. Then one has

E(Xˆg,ε(x)Xˆg,ε(x′)) = 2π Gg(y,y′)dµˆx,ε(y)dµˆx′,ε(y′) Z and by the arguments in the proof of Lemma 3.2, we have as ε 0 → E(Xˆ (x)2)= log(ε)+ W (x)+ 1 ω(x)+ o(1). (3.10) g,ε − g 2 Then by the arguments of Proposition 3.3, the random measure (add an extra push ( ln ε)1/2 − when γ = 2) 2 γ γ γXˆ (x) ˆ := ε 2 e g,ε dv (x) (3.11) Gg,ε gˆ γ converges weakly as ε 0 to some measure ˆg which satisﬁes → G 2 γ (1+ γ )ω(x) γ d ˆ (x)= e 4 d (x). (3.12) Gg Gg

4. Liouville Quantum field theory with fixed modulus

In this section we define Liouville Quantum Field Theory (LQFT) with fixed conformal class (also called modulus) and describe its main properties. It follows the approach of [DKRV] in the case of the Riemann sphere. Liouville Quantum Gravity (LQG) with fixed genus is a sum (“partition function”) over all possible metrics on a surface with fixed genus. The space of metrics splits into conformal classes and we want to decompose the partition function accordingly. Each conformal class has a unique hyperbolic metric, which plays the role of a background metric.

4.1. Axiomatic of CFT. Here we give a brief account of the axiomatic of Conformal Field Theories in order to motivate the forthcoming results. Our purpose will then be to construct the quantum Liouville theory and show that it satisﬁes this axiomatic. The reader is referred to [Ga] for more details related to this formalism. A CFT (on the surface (M, g)) is described by its partition function Z(g) as well as the correlation functions of the (spinless) primary ﬁelds (θi)i I denoted by ∈

Z(g,θi1 (x1),...,θin (xn)) where n > 1, i ,...,i I and x ,...,x are arbitrary points on M. Let us just roughly { 1 n} ∈ 1 n say that a CFT is supposed to give sense to ”random fields” defined on M, here the primary fields (θi)i, and the correlation functions can be thought of as the cumulants of these random fields. These correlation functions are supposed to satisfy the following conditions: Diffeomorphism covariance: for any orientation preserving diffeomorphism ψ • Z(g)= Z(ψ∗g) (4.1)

Z(g,θi1 (ψ(x1)),...,θin (ψ(xn))) = Z(ψ∗g,θi1 (x1),...,θin (xn)) (4.2) LQG AND BOSONIC 2d STRING THEORY 35

Conformal anomaly: for any smooth function ϕ on M • Z(eϕg) c ln = ( d ϕ 2 + 2K ϕ)dvol (4.3) Z(g) 96π | g |g g g ZM ϕ n Z(e g,θi1 (x1),...,θin (xn)) c 2 ln = ∆ik ϕ(xk)+ ( dgϕ g + 2Kgϕ)dvolg (4.4) Z(g,θi1 (x1),...,θin (xn)) − 96π M | | Xk=1 Z where the constant c is the so-called central charge of the CFT and the scalars (∆i)i I are ∈ called the conformal weights of the primary ﬁelds. One of the interesting feature of CFTs is their strong algebraic structure, which make them fall under the scope of techniques for integrable systems, leading to the possibility of obtaining exact expressions for the correlation functions, which is the main goal of a Quantum Field Theory.

4.2. The partition function of LQFT. The first step is to describe LQFT with fixed modulus. LQFT will describe the probability law of some random conformal factor, that is we consider the random metrics eγX g if g is a fixed metric and X is a random function. The law of X will be mathematically described by the Feynmann type measure (3.1). So, let g Met(M) ∈ be a fixed metric on M. The mathematical definition of the measure of LQFT (i.e. (3.1)) is the following. For F : H s(M) R (with s > 0) a bounded continuous functional, set for − → γ (0, 2] ∈ 1/2 Πγ,µ(g, F ) :=(det′(∆g)/Volg(M))− (4.5) E Q γc γ F (c + Xg)exp Kg(c + Xg)dvg µe g (M) dc. × R − 4π M − G Z h Z i This quantity, if it is finite, gives a mathematical sense to the formal integral

S (g,ϕ) F (ϕ)e− L Dϕ Z where SL(g, ϕ) is the Liouville action (2.2). The partition function is the total mass of this measure, i.e Πγ,µ(ˆg, 1).

Proposition 4.1. For g Met(M) and γ (0, 2], we have 0 < Π (g, 1) < + and the ∈ ∈ γ,µ ∞ mapping s F C (H− (M), R) Π (g, F ) ∈ b 7→ γ,µ defines a positive finite measure. When renormalized by its total mass, it describes the law s of a random variable living in H− (M) called the Liouville field. When g Met 1(M) is ∈ − hyperbolic, we further have

1/2 Qχ(M) Qχ(M) det′(∆g) Qχ(M) − 1 γ E γ γ Πγ,µ(g, 1) = γ− µ Γ( γ ) g (M) (4.6) Volg(M) − G h i where Γ(z) is the standard Euler Gamma function.

Proof. The proof of this proposition follows the same lines as in [DKRV, section 3.1]. We consider the case of a metric g Met 1(M), since the general case follows from that case, as ∈ − 36 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS is explained below in Proposition 4.4 for the correlations functions. In constant curvature, the Gauss-Bonnet theorem entails Q K (c + X )dv = Qcχ(M) 4π g g g ZM where χ(M) is the Euler characteristic of M and we get

1/2 det′(∆g) − Qcχ(M)E γc γ Πγ,µ(g, F )= e− exp µe g (M) dc. Volg(M) R − G Z h i After inverting expectation and integration, and using the change of variables y = µeγc γ (M), Ggˆ we get (4.6). Finiteness of this quantity is ensured by the fact that GMC has ﬁnite moments of negative orders as χ(M) < 0 - ﬁniteness of negative moments is proved for example in [RoVa, Proposition 3.6] for γ < 2 and in [DRSV2, ] in the case γ = 2.

4.2.1. Conformal anomaly and diﬀeomorphism invariance. Here we investigate the symmetries of the measure (4.5) and in particular how the partition function reacts to changes of background metrics. The following proposition is the quantum counterpart of (2.3).

γ 2 Proposition 4.2. (Conformal anomaly) Let Q = + with γ (0, 2] and g Met 1(M) 2 γ ∈ ∈ − be a hyperbolic metric on M. The partition function satisfies the following conformal anomaly: if gˆ = eωg for some ω C (M), we have ∈ ∞ 2 Q 1 + 6Q 2 Πγ,µ(ˆg, F ) = Πγ,µ(g, F ( 2 ω)) exp ( dω g + 2Kgω)dvg . · − 96π M | | Z Proof. We focus on the integral part in (4.5) (and hence let the determinant of Laplacian apart as its contribution is clear from (2.10)). First, by Lemma 3.1, we can replace Xgˆ by Xg in the expression defining Πγ,µ(ˆg, F ) and are thus left with considering the following quantity (with γ ˆg is the measure defined by (3.11)) G E Q γc ˆγ Aε := F (c + Xg)exp Kgˆ(c + Xg)dvgˆ µe g (M) dc. R − 4π M − G Z h Z i By (2.1), the term Qc K dv can be written as Qcχ(M) where χ(M) is the Euler − 4π M gˆ gˆ − characteristic. Define the Gaussian random variable R Q Q Y := K X dv = X , K eω . −4π gˆ g gˆ −4π h g gˆ ig ZM Let Rg(0) be the resolvent operator whose Schwartz kernel is Gg with respect to dvg. Since K eω =∆ ω + K , we compute, using that R (0)K = 0 (as K = 2), gˆ g g g g g − E[ X , K eω 2] =2π G , (∆ ω + K ) (∆ ω + K ) h g gˆ ig h g g g ⊗ g g ig ω,1 g 2 =2π ω h i , ∆gω 2 g = 2π dω dvg h − Volg(M) − i | |g ZM and similarly we have Q Q E[Y X ]= R (0)(K eω)= (ω c (ω)). g − 2 g gˆ − 2 − g LQG AND BOSONIC 2d STRING THEORY 37

ω,1 g if c (ω) := h i . Thus we get g Volg(M) Q2 Q 1 E[Y 2]= dω 2dv , E[Y X ]= (ω c (ω)). (4.7) 2 16π | |g g g − 2 − g ZM Therefore by applying Girsanov transform to the random variable Y , we can rewrite 1 E[Y 2] Qcχ(M) Q γQ E E γ(c+ 2 cg(ω)) 2 ω ˆγ A = e 2 − F (c + Xg + (Y Xg)) exp µe e− d g dc. R − M G Z h Z i γ 2 γQ ω γ γ With the help of the relation (3.12) and Q = + , we see that e− 2 d ˆg = g (M). Using 2 γ M G G (4.7), A can be written as R 2 Q 2 Q 16π dω 2 Qcχ(M) || ||Lg − E Q Q γ(c+ 2 cg(ω)) γ A = e F (c + Xg 2 ω + 2 cg(ω))) exp µe g (M) dc. R − − G Z h i It remains to make the change of variable c c Q c (ω) and we deduce that → − 2 g 2 Q 2 1 2 16π dω 2 Qcχ(M)+ Q χ(M)cg(ω) Q || ||Lg − 2 E γc γ A = e F (c + Xg ω)exp µe g (M) dc. R − 2 − G Z h i Since K = 2 and Vol (M)= 2πχ(M) we have g − g − Q 1 K (c + X )dv = Qcχ(M), c (ω)χ(M)= K ω dv −4π g g g − g 4π g g ZM ZM 2 6Q 2 Q R ( dω +2Kgω)dvg which shows that A = Π (g, F ( ω)) det′(∆ )/Vol (M)e 96π M | |g . Combin- γ,µ · − 2 g g ing with (2.10), the proof is complete. p

2 The constant cL := 1 + 6Q describing the conformal anomaly is called the central charge of the Liouville Theory. Since all the objects in the construction of the Gaussian Free Field and the Gaussian multiplicative chaos are geometric (deﬁned in a natural way from the metric), it is direct to get the following Diﬀeomorphism invariance:

Proposition 4.3. (Diﬀeomorphism invariance) Let g Met(M) be a metric on M and ∈ let ψ : M M be an orientation preserving diﬀeomorphism. Then we have for each bounded → measurable F : H s(M) R with s> 0 − → Π (ψ∗g, F ) = Π (g, F ( ψ)). γ,µ γ,µ · ◦ Proof. This follows directly from the fact that all the object considered in the construction of the measure are natural with respect to the metric g, thus invariant by isometries: more precisely, it follows from the identities

law G ∗ (x,y)= G (ψ(x), ψ(y)), K ∗ (x)= K (ψ(x)), X ∗ = X ψ. ψ g g ψ g g ψ g g ◦ which are obvious.

The two above results show that the axioms (4.1)+(4.3) are satisfied with central charge 2 cL =1+6Q . Yet we still have to define the primary fields and their correlation functions. This is the purpose of the next subsection. 38 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

4.3. The correlation functions. The correlation functions of LQFT can be thought of as the exponential moments eαϕ(x) of the random function ϕ, the law of which is ruled by the path integral (3.1), evaluated at some location x M with weight α. Yet, the field ϕ is not a s ∈ fairly defined function (it belongs to H− (M)) so that the construction requires some care. As before let g Met(M). We fix n points x ,...,x (n > 0) on M with respective ∈ 1 n associated weights α ,...,α R. We denote x = (x ,...,x ) and α = (α ,...,α ). The 1 n ∈ 1 n 1 n rigorous definition of the primary fields will require a regularization scheme. We introduce the following ε-regularized functional

x,α 1/2 Πγ,µ (g,F,ε) := det′(∆g)/Volg(M) − (4.8) Q E αi γc γ F (c + Xg) Vg,ε(xi) exp Kg(c + Xg)dvg µe g,ε(M) dc R − 4π − G Z i ZM h Y i where we have set, for ﬁxed α R and x M, ∈ ∈ 2 α α /2 α(c+Xg,ǫ(x)) Vg,ε(x)= ε e . Here the regularization is the one described in Lemma 3.2. Such quantities are called vertex α operators. Notice that Vg,ǫ also depends on the variable c but we have dropped this dependence in the notations. Then, the point is to determine whether the limit

x,α x,α Πγ,µ (g, F ) := lim Πγ,µ (g,F,ε) ε 0 → s exists and defines a non trivial functional on those mappings F : H− (M) R. If it does, the x,α x,α → quantity Πγ,µ (g) := Πγ,µ (g, 1) stands for the n-point correlation function of the primary fields α ϕ (e i )1 6 i 6 n respectively evaluated at (xi)1 6 i 6 n. Furthermore, another quantity of interest s is the probability law on H− (M) defined by the measure s x,α x,α F C (H− (M)) Π (g, F )/Π (g), ∈ b 7→ γ,µ γ,µ which describes the law of some formal ”random function” (it is in fact a distribution). We obtain a result similar to [DKRV] (done for the sphere).

Proposition 4.4. Let x = (x ,...,x ) M n and α = (α ,...,α ) Rn. Then for all 1 n ∈ 1 n ∈ bounded continuous functionals F : h H s(M) F (h) R with s> 0, the limit ∈ − → ∈ x,α x,α Πγ,µ (g, F ) := lim Πγ,µ (g,F,ε), ε 0 → x,α exists and is ﬁnite with Πγ,µ (g, 1) > 0, if and only if: α + 2Q(g 1) > 0, (4.9) i − Xi i, α < Q. (4.10) ∀ i In the case g Met 1(M), we have the following expression ∈ − P α +2Q(g−1) γ i i C(x) Pi αi 2Q(g 1) E γ e µ− − − Γ( i αi + 2Q(g 1)) g,x,α(M)− Πx,α(g)= − G γ,µ h i P det′(∆g)/Volg(M) p LQG AND BOSONIC 2d STRING THEORY 39 where Γ is Euler gamma function and, if Wg is the function appearing in Lemma 3.2,

γ (dx) := eγ Pi αi2πGg(xi,x) γ (dx), Gg,x,α Gg α2 (4.11) C(x) := i W (x ) + 2π α α G (x ,x ). 2 g i i j g i j Xi Xi

2 2 2 αi αi ˆ αi E ˆ 2 ˆ αi αic+ ϕ(xi)+ Wg(xi) αiXg,ε(xi) [Xg,ε(xi) ] Vg,ε(xi)= e 4 2 e − 2 (1 + o(1)) as ε 0, with the remainder being deterministic. Here we have used the notation Xˆ (x )= → g,ε i X , µˆ as before, whereµ ˆ is the uniform probability measure on the Riemannian circle h g xi,εi xi,ε (x , ε). Then applying Girsanov transform to Cgˆ i Q E ˆ αi γc ˆγ Aε := Vg,ε(xi) exp Kgˆ(c + Xg)dvgˆ µe g,ε(M) dc R − 4π − G Z i ZM h Y i we get

Cε(x) c(P αi Qχ(M)) Q γc Aε =e e i − E exp Xg, Kgˆ gˆ µe Zε dc (1 + o(1)) R − 4π h i − Z h i where

γ2 γ(Xˆg,ε+Hg,ε) Zε := ε 2 e dvgˆ ZM

Hg,ε(x) := 2παi Gg(y,x)dµˆxi,ε(y), gˆ(xi,ε) Xi ZC Q α2 C (x) := 2π α α G (x ,x ) K H dv + i (ϕ(x ) + 2W (x )). ε i j g i j − 4π gˆ g,ε gˆ 4 i g i i=j ZM i X6 X Notice that, since K dv = (∆ ϕ 2)dv , we have as ε 0 gˆ gˆ g − g → α2 Qα Q C (x) 2π α α G (x ,x )+ ( i i )ϕ(x )+ α c (ϕ)+ 1 α2W (x ). ε → i j g i j 4 − 2 i 2 i g 2 i g i i=j i i i X6 X X X By applying Girsanov transform just like in the proof of Proposition 4.2, we can get rid of the X , K term and this shifts the field Xˆ in Z by F (x)= Q (ϕ(x) c (ϕ)): h g gîg g,ε ε − 2 − g 2 Q 2 Cε(x)+ dϕ 16π || ||L2 c(P αi Qχ(M)) γc Aε =e g e i − E exp µe Zˆε dc (1 + o(1)) R − Z h i 2 γ γ(Xˆ +H +F ) where Zˆ := ε 2 e g,ε g,ε dv ; here we have denoted c (ϕ) = ϕ, 1 /Vol (M). By ε M gˆ g h ig g Lemma 3.2, H ∞ < thus by Proposition 3.3, we get that E[Zˆ ] < thus we can find || g,εˆ R||L ∞ ε ∞ 40 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

B > 0 such that P(Zˆε 6 B) > 0. We therefore get 0 γc c(P αi Qχ(M)) µe Aε > βε,x e i − − P(Zˆε 6 B) dc Z−∞ for some β x > 0, and this is infinite if α Qχ(M) 6 0. Then we assume (4.9). We also ε, i i − have as in (3.12) the relation P 2 γQ c (ϕ) γ γ(X +H ) γ Zˆ = e 2 g Z (1 + o(1)),Z := ε 2 e g,ε g,ε d . ε ε ε Gε ZM Making the change of variables c c Q c (ϕ), we obtain that A is equal to → − 2 g ε 2 2 2 αi Qαi Q 2 Q C(x)+P ( )ϕ(xi)+ dϕ + χ(M)cg(ϕ) i 4 − 2 16π || ||L2 2 c(P αi Qχ(M)) γc e g e i − E exp µe Zε dc R − Z h i times 1 + o(1) as ε 0, where C(x) is given by (4.11). In particular this implies (4.13) if we → can show that for the case ϕ = 0 the limit of Aε is finite. We now assume ϕ = 0, or equivalently γc we considerg ˆ = g the hyperbolic metric. We make the change of variable c = µe Zε in the c-integral defining Aε (recall that Zε > 0 almost surely), and we get

P α −Qχ(M) − Pi αi+Qχ(M) i i 1 C(x) i αi Qχ(M) γ A = γ− e µ γ Γ − E[Z− ]. ε γ ε P It remains to show that if αi

E s E s γHg γ lim [Zε ]= [Z0] (0, ),Z0 := e d (4.12) ε 0 ∈ ∞ Ggˆ → ZM s with Hg := limε 0 Hg,ε = 2π i αiGg(xi, ) in the H− (M) topology, and that if αi > Q for → · some i, then E[Zs] 0. But this part is only a local argument and therefore Lemma 3.3. ε → P of [DKRV] applies directly. The argument goes essentially as follows. We need to determine γ 1 whether the measure g (dx) integrates the singularity γα in the neighborhood of xi. G dg(x,xi) i Standard multifractal analysis shows that for any δ > 0 one can ﬁnd a constant Cδ such that γQ+δ γ sup r− g (Br(xi)) 6 Cδ r<1 G where Br(xi) stands for the geodesic ball of radius r centered at xi. This gives the condition αi

γc γ c P αi 2Q(1 g) µe x (M) e i − − E[e− Gg, ,α ] dc R Z γ is ﬁnite. As we have seen that g,x,α(M) is a well deﬁned non trivial random variable under the G condition (4.10), one may think of it as a macroscopic quantity and replace it by a constant quantity, say 1, so as to be left with the integral

γc c P αi 2Q(1 g) µe e i − − − dc, R Z which is straightforwardly seen to be converging if and only if (4.9) holds.

In fact the proof of the previous Proposition (adding a functional F does not change any- thing) also shows the LQG AND BOSONIC 2d STRING THEORY 41

Proposition 4.5. (Conformal anomaly and diﬀeomorphism invariance) Let x = (x1,...,xn) n n ω ∈ M and α = (α1,...,αn) R . Let g Met 1(M) be a hyperbolic metric and gˆ = e g for ∈ ∈ − some ω C (M). Then ∈ ∞ x,α 2 2 Πγ,µ (ˆg, F ) 1 + 6Q 2 αi Qαi log x,α Q = ( dω g + 2Kgω)dvg + ( )ω(xi) (4.13) Πγ,µ (g, F ( )) 96π M | | 4 − 2 ·− 2 Z Xi Let ψ : M M be an orientation preserving diﬀeomorphism. Then → x,α ψ(x),α Π (ψ∗g, F ) = Π (g, F ( ψ)). γ,µ γ,µ · ◦

5. Liouville Quantum gravity

5.1. The full partition function. The partition function of Liouville quantum gravity is a weighted integral over the moduli space of the Liouville quantum field theory coupled to a Conformal Field Theory (sometimes called matter field in physics in this context). The weight of each modulus is given by some explicit functional ZGhost(g) (this weight depends on the underlying surface (M, g)), called the ghost system in physics. This takes into account the factorization of the space of metrics by the action of the group of diffeomorphisms of the surface (as explained for example in [DhPh]). Let us first recall the physics heuristics that leads to the partition function, by following [Po, DhPh, DiKa, Da]; the following discussion is not mathematically rigorous but is rather a “state of the art” in physics literature. The partition function for (Euclidean) quantum gravity in 2D, coupled with matter, is

SEH(g) SM(g,φm) Z = e− e− Dgφm Dg ZR Z where = Met(M)/Diﬀ(M) is the space of Riemannian structures, the action S (g) = R EH µ Vol (M) is the Einstein-Hilbert action (or gravity action) with µ R the cosmological 0 g 0 ∈ constant and φm are matter ﬁelds with SM(g, φm) being the action for matter which depends in g in a conformally invariant way. Notice that, in comparison with (1.10), we got rid of the term M Kg dvg as it is a topological invariant in 2d because of the Gauss-Bonnet theorem: this is an important feature of 2d-quantum gravity. The quantity R

SM(g,φm) ZM(g) := e− Dgφm Z is supposed to be a CFT with central charge cM, Dgφm a formal measure depending on g and Dg is the formal Riemannian measure induced by the L2 Riemannian metric on Met(M) given by (2.5) (the group Diﬀ(M) acts by isometries on Met(M) thus the L2-metric on Met(M) descends to ). Each metric can be decomposed as g = ψ (eϕg ) where τ is a parameter on R ∗ τ moduli space g, g is a family of metrics representing moduli space and ψ Diﬀ(M), and M τ ∈ the formal measure Dg can be accordingly decomposed as

ϕ ϕ Dg = ZGhost(e gτ )De gτ ϕDτ 42 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS where ZGhost is the ghost determinant which comes from the Jacobian of the quotient of Met(M) by the group of diﬀeomorphism Diﬀ(M) (see for example [DhPh]), and given by

det(Pg∗Pg) 1/2 ZGhost(g)= det Jg where Pg, Jg are deﬁned in Section 2.3. The ghost determinant satisﬁes the conformal anomaly formula (4.3) with central charge

cghost = 26. (5.1) − Here Dτ is a measure on the slice of metrics gτ chosen to represent moduli space, whose value 1/2 is Dτ := (det Jgτ ) dτ with dτ being the Weil-Petersson volume form on the moduli space g ϕ M (somehow ZGhost(e gτ )Dτ is the quantity that makes invariant sense, as it does not depend on ϕ 2 the matrix Jg). The formal measure De gτ ϕ should be induced by the L Riemannian metric on metrics, which on the tangent space to the conformal orbit [g ] = eϕg ; ϕ C (M) is τ { τ ∈ ∞ } given by 2 2 ϕ ϕ f ϕ = ω e dv , f = ωe g T ϕ [g ]. || ||e gτ gτ τ ∈ e gτ τ ZM This measure depends non-linearly on ϕ and it is diﬃcult to “do the functionnal integral” for this measure, as written in [DiKa]. Therefore David and Distler-Kawai [Da, DiKa] made the “well-motivated” assumption that

SEH(g) SL(gτ ,ϕ) e− ZM(g)Dg = ZM(gτ )ZGhost(gτ )e− DτDgτ ϕ where SL(g, ϕ) is the Liouville action defined by (2.2) for some parameter Q,γ,µ to be chosen. A formal justification of this fact was written down later in [MaMi] and [DhKu]. Invariance of the theory by choice of slice gτ representing moduli space forces the partition function SL(gτ ,ϕ) 2 e− Dgτ ϕ to be a CFT with central charge cL =1+6Q such that the total conformal anomaly vanishes: R cghost + cM + cL = 0 and Q, γ need to be related by Q = γ/2 + 2/γ. Now we stop the physics parenthesis and come back to mathematics. For the matter field, we take the particular case the most studied in the physics literature, namely

cM/2 det′(∆g) − ZM(g) := (5.2) Volg(M) where cM is a constant in ( , 1]. Note that this has the central charge c by (2.10). Fur- −∞ M thermore, there are at least two important particular cases: pure gravity where cM = 0 and the 2d bosonic string in the case cM = 1. Because it is the critical situation os this approach, the latter case is especially interesting and raises serious additional diﬃculties. One could consider also other CFT partition functions provided that we get an expression explicit enough to determine how it behaves at the boundary of the moduli space. For each modulus τ g, we ∈ M can associate a hyperbolic metric gτ and we will denote by (gτ )τ the family of hyperbolic metrics representing the moduli space. By deﬁnition, the partition function of Liouville quantum gravity is given by the following formula:

Z := ZGhost(gτ ) ZM(gτ ) Πγ,µ(gτ ) Dτ (5.3) g × × ZM LQG AND BOSONIC 2d STRING THEORY 43

1/2 where Dτ := (det J ) dτ with dτ the Weil-Petersson volume form on the moduli space g, gτ M and Πγ,µ(g) is the partition function of the Liouville quantum ﬁeld theory. This can be reduced to

1/2 cM/2 Z = Cg det(Pg∗τ Pgτ ) det′(∆gτ )− Πγ,µ(gτ ) dτ (5.4) g × × ZM with Cg a constant depending only on the genus of M. Furthermore, as we have deﬁned LQFT only for γ ]0, 2[, this shows that the central charge cM of the matter ﬁeld must satisfy cM < 1 ∈ γ 2 and, combining with the expression Q = 2 + γ , we obtain another KPZ relation [KPZ]

√25 cM √1 cM γ = − − − , (5.5) √6 which ﬁxes the value of γ in terms of cM. Now, the main result of this section is the following

Theorem 5.1. If γ ]0, 2] and cM satisfies relation (5.5), the partition function Z given by ∈ (5.3) is finite. Hence it gives rise to a finite measure ν on Rad(M) g defined as follows: × M if (g ) is a family of hyperbolic metrics parametrizing the moduli space g, then τ τ M 1 det(Pg∗ Pgτ ) 2 γc γ τ E γc γ Qχ(M)c µe gτ (M) ν(F ) := F (e g (dz),τ)e− − G dτ dc R (det ∆ )cM+1 G τ Z g ′ gτ M × i for all continuous functionals F : Rad(M) g R. When renormalized by its total mass × M → P E Z = ν(1), it becomes a probability measure which we call (gτ )τ ,µ (with expectation (gτ )τ ,µ) and γc γ the couple (e g (dz),τ) becomes a random variable on Rad(M) g, which we denote by G τ × M ( ,R) and stands for the volume form of the space (called Liouville quantum gravity measure) Lγ and its modulus (called quantum modulus).

Furthermore, for all continuous functionals F : Rad(M) g R × M → E [F ( (dz),R)] (5.6) (gτ )τ ,µ Lγ 2Q(g 1) 1 γ Γ( − ) Q γ det(Pg∗τ Pgτ ) 2 gτ (dz) γ χ(M) = E F ξγ G ,τ (M) γ dτ 2Q(g−1) cM+1 γ gτ γ g (det′∆gτ ) gτ (M) G γZµ ZM G 2Q(g−1) γ 2Q(g−1) µ µx γ 1 where ξγ is a random variable with Gamma law of density 2Q(g−1) e− x − 1x > 0 and Γ( γ ) the random modulus R has density

(cM+1) 1 Q det′(∆g ) 2 χ(M) 2 τ − E γ γ det(Pg∗τ Pgτ ) gτ (M) Volgτ (M) G h i with respect to the dτ measure.

Let us make some comment on the above result. The LQG measure depends on the family of hyperbolic metrics (gτ )τ but this is not an issue since it enjoys the following invariance by reparametrization: if (ψτ )τ is a family of orientation preserving diﬀeomorphisms, we get the following equality for all τ

E ∗ [F ( γ ψτ ) R = τ]= E [F ( γ ) R = τ] (5.7) (ψτ gτ )τ L ◦ | (gτ )τ L | 44 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

5.2. Proof of Theorem 5.1 in the case γ = 2. Our purpose is to determine the behaviour of the partition function (5.3) of LQFT (for γ = 2) at the boundary of the moduli space with ZM(g) deﬁned by (5.2) and cM = 1. According to the relation (4.6), this amounts to showing that the integral 1 Q det(Pg∗ Pgτ ) 2 χ(M) τ E γ (M) γ dτ (5.8) cM+1 gτ g (det′∆gτ ) G ZM h i is ﬁnite. The singularities in this integral appear at the boundary of the moduli space, namely when the surface (M, g) gets close to a surface with nodes (M0, g0) by pinching np geodesics with respective lengths (ℓj)j on (M, g) (see section 2.5). According to the explicit bounds 1 ∗ det(Pgτ Pgτ ) 2 (2.12) and (2.15) for the product ′ cM+1 and the expression for the Weyl-Petersson (det ∆gτ ) measure (2.9), we can give an upper bound

np 3g 3 2 1 − C ℓj−′ λi− ℓj dℓjdθj (5.9) ′ jY=1 λiY<1/4 jY=1 1 ∗ det(Pgτ Pgτ ) 2 for the quantity ′ cM+1 dτ in the coordinate system (ℓj,θj)j=1,...,3g 3 associated to a (det ∆gτ ) − γ Qχ(M) pant decomposition. Hence it suﬃces to check the integrability of the expectation E g (M) γ G with respect to the measure (5.9) near (M0, g0). It turns out that the mass of the measure γ g (M) will become very large when g gets close to g0 in the pinched region of the surface G Qχ(M) γ (M, g), making the expectation E g (M) γ very small. We have to quantify the rate of G decay of this expectation to show integrability. Proposition 2.7 describes how the Green function behaves near the pinched geodesics. The purpose of what follows is to explain how these estimates transfer to the above expectation.

We assume (M0, g0) possesses m+1 connected components, and we consider a neighborhood of (M0, g0); there is a subpartition γ1, . . . , γnp of M corresponding to the pinched geodesics. { } np Let us denote by (S ) the connected components of M ( γ ′ ). Each γ ′ has a collar j j=1,...,m+1 \ ∪j′=1 j j neighborhood denoted ′ (see (2.29), for simplicity of notation we remove the g dependance Cj in ′ (g)). We denote by S the set obtained by removing from S all the collars Cj j′ j np

Sj′ := (Sj j′ ) ′ \C j\=1 m+1 np in such a way that M = S ′ . Furthermore, for each j = 1,...,m + 1, we deﬁne ∪j=1 j′ ∪j′=1 Cj I = j 1,...,n ; S ′ = the set of indices j such that the collar ′ encounters S . j { ′ ∈ p} j ∩Cj 6 ∅ ′ Cj j We deﬁne the following quantities for x> 0 and 1 6 j′ 6 np + ′ := ′ ρ > 0 −′ := ′ ρ 6 0 Cj Cj ∩ { } Cj Cj ∩ { } + ′ (x) := ′ x 6 ρ ′ (x)− := ′ ρ 6 x , Cj Cj ∩ { } Cj Cj ∩ { − } here ρ : ′ ′ [ 1, 1] is the function in the collars so that the metric is given by (2.37). ∪j Cj → − Let us denote by (ϕi)1 6 i 6 m the eigenfunctions associated to the small (non zero) eigenvalues LQG AND BOSONIC 2d STRING THEORY 45

1 (λ ) 6 6 and write the Green function G as Π +A . Consider X the Gaussian i 1 i m 1 6 i 6 m λi λi g g′ field with covariance 2πA , which is nothing but g P X , ϕ X = X (2π/λ )1/2 g i ϕ . (5.10) g′ g i h 1/2i i − 6 6 (2π) 1 Xi m Xg ,ϕi The finite sequence ( h i ) 6 6 is a sequence of i.i.d. standard Gaussian random variables, (2π)1/2 1 i m namely with law (0, 1), which we denote (a ) 6 6 . Furthermore, (a ) 6 6 and X are N i 1 i m i 1 i m g′ independent. In case the surface (M0, g0) is not disconnected, write Ag = G and Xg′ = Xg. We introduce the random measure: A M Borel set ∀ ⊂ 1/2 2 2X′ ′(A) := lim( ln ε) ε e g,ε dvg, G ε 0 − → ZA where Xg,ε′ is the regularization of Xg′ as in Lemma 3.2. Notice that the convergence in probability of this measure is ensured by the convergence in probability of the same measure involving the field Xg instead of Xg′ (both field coincide up to an additive continuous field so that convergence of one measure is equivalent to convergence of the other one). The main technical estimate we need in the proof is the following

Lemma 5.2. Let U g be a neighborhood of some metric with nodes g . For any j , δ > 0, 0 ⊂ M 0 ′ q > 0, λ [0, 1] there exists some constant C such that for all g U0, ℓ,ℓ′ > ℓj′ and A,B > 0 ∈ 1 ∈ ∀ and ψ : R R of class C such that ψ F ′ ( 1) = ψ F ′ (1) = 0 → ◦ j − ◦ j q 1 ψ F ′ − qλ q(1 λ) δ 2 E φe ◦ j d ′ 6 C A− B− − (ℓℓ′) 2 − exp C ψ′(r) dr . q 2π ′ G 1 6 r 6 | | h ZCj i Z | | min(ℓ,ℓ′)1−δ

′ ′ where the function φ is deﬁned on the collar j by φ(ρ)= A1 ′ (ℓ)+ + B1 ′ (ℓ′)− , and Fj (ρ) := C Cj Cj 2π ℓj′ arctan( ). The constant Cq depends on q and the mapping q [0, + [ Cq is locally ℓj′ ρ ∈ ∞ 7→ bounded.

The proof of this lemma is postponed to the end of this subsection. As a direct consequence we claim

Corollary 5.3. For any j , δ > 0 q > 0, there exists some constant C such that for all g U ′ ∈ 0 and ℓ,ℓ′ > ℓj′ and A,B > 0

+ q q 1/2 δ 1) E ′( ′ (ℓ) )− + E ′( ′ (ℓ)−)− 6 Cℓ − . G Cj G Cj +h i hq q/i 2 1/2 δ 2) E (A ′( ′ (ℓ) )+ B ′( ′ (ℓ′)−))− 6 C(AB)− (ℓℓ′) − . G Cj G Cj h i Now we complete the proof while considering two main situations: either the surface (M0, g0) is disconnected or not. Case (M , g ) is not disconnected: In that case the measure (5.9) can be estimated • 0 0 from above by the measure np 3g 3 1 − C ℓj−′ dℓjdθj. (5.11) ′ jY=1 jY=1 46 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

Obviously we have

np Qχ(M) Qχ(M) γ E (M) γ 6 E ′( ′ ) γ . Gg G Cj j′=1 X Now we observe that the cross covariances of the ﬁeld X in the various regions ′ are bounded g′ Cj by some uniform constant, i.e. sup sup ′ ′ sup E[X′ (x)X′ (y)] 6 C (see Propo- g U0 j1=j2 x ′ ,y ′ g g ∈ 6 ∈Cj1 ∈Cj2 | | sition (2.2)). Kahane’s inequality [Ka, Lemma 1] then tells us that, considering independent copies ( ′ ) 6 ′ 6 of , the latter expectation is less than (for some irrelevant constant C) Gj′ 1 j np G′ np Qχ(M) b γ CE ′ ′ ( ′ ) . Gj Cj j′=1 X b Then, we use the following elementary inequality valid for b1, . . . , bn > 0 and w1, . . . , wn > 0 n with j=1 wj = 1 and q > 0 n n P q wiq ( bi)− 6 b−i (5.12) Xi=1 Yi=1 Qχ(M) n p γnp to deduce that the above expectation is less than ′ E ′ ( ′ ) . Combining with j =1 Gj′ Cj Corollary 5.3, we obtain Q np b Qχ(M) γ γ 1 δ E (M) 6 C ℓ ′− Gg j j′=1 Y for some arbitrary δ > 0. This is enough to show integrability with respect to (5.11).

Case (M , g ) is disconnected: Recall that the eigenfunctions (ϕ ) converge • 0 0 i i=1,...,m uniformly on the compact subsets of each Sj respectively towards some ﬁxed value denoted v . From Lemma 2.8, we have the estimate in the region ρ > Cℓ ′ of the cusp ′ S ij | | j Cj ∩ j + ϕij 6 ϕi 6 ϕij− (5.13) where we have set

+ λ +C (v )λ2 ϕ :=v (1 (v )Cǫ) ρ − i S ij i Cǫ (5.14) ij ij − S ij | | − λ C (v )λ2 ϕ− :=v (1 + (v )Cǫ) ρ − i− S ij i + Cǫ (5.15) ij ij S ij | | for some constant C > 0; here we have denoted by the function x R (x) := sign(x). S ∈ 7→ S Restricting the integral to the cusp regions and then using (5.13), we have the estimate conditionally on the (ai)1 6 i 6 m

Qχ(M) Qχ(M) γ γ E (M) γ (a ) 6 6 6 E ( ′ ′ ) γ (a ) 6 6 Gg | i 1 i m Gg ∪j Cj | i 1 i m 1/2 Qχ(M) P 2(2π/λi) aiϕi =E e i d ′ γ (a ) 6 6 G | i 1 i m j′ Z j′ X C 1/2 S(ai) Qχ(M) P 2(2π/λi) aiϕ 6 E e i ij d ′ γ (a ) 6 6 , G | i 1 i m j,j′ Z j′ Sj X C ∩ LQG AND BOSONIC 2d STRING THEORY 47

(ai) + (ai) where ϕijS = ϕij if ai > 0 and ϕijS = ϕij− if ai < 0. Once again we can use Kahane’s convexity inequality to show that there exists a collection of mutually independent random measure ( ) 6 ′ 6 (and independent of the (a ) 6 6 ) such that, for some irrelevant G(′j′) 1 j np i 1 i m constant C > 0

1/2 S(ai) Qχ(M) P 2(2π/λi) aiϕ E e i ij d ′ γ (a ) 6 6 G | i 1 i m j,j′ Z j′ Sj X C ∩ 1/2 S(ai) Qχ(M) Pi 2(2π/λi) aiϕij γ 6 CE e d ′ ′ (a ) 6 6 . G(j ) | i 1 i m j,j′ Z j′ Sj X C ∩ Now we choose a collection of real-valued random variables (rj)1 6 j 6 m+1 that are non negative with j rj = 1 and measurable with respect to the family (ai)1 6 i 6 m. The precise choice of the family (r ) 6 6 will be made later when appropriate. Given those (r ) 6 6 , P j 1 j m+1 j 1 j m+1 ′ ′ we construct new weights (wj )1 6 j 6 np as follows. For each j = 1,...,m + 1, we define I+ = j 1,...,n ; S + = the set of indices j such that the collar + encounters j { ′ ∈ p} j ∩Cj′ 6 ∅ ′ Cj′ Sj and we define similarly Ij−. Also, for each j′ = 1,...,np, there exists a unique j such that + j′ Sj and we denote by j+′ that index. Similarly for j′ . Finally for j′ = 1,...,np, we define C ⊂ − r ′ r ′ + j+ j− + wj′ = + , wj−′ = + and wj′ = wj′ + wj−′ . (5.16) Ij′ + Ij−′ Ij′ + Ij−′ | + | | + | | − | | − | The first observation is that wj′ = 1. (5.17) ′ j =1X,...,np Indeed m+1 m+1 + wj′ = wj′ + wj−′ ′ j =1,...,np j=1 j′ I+ j=1 j′ I− X X X∈ j X X∈ j m+1 m+1 rj rj = + + + I + I− I + I− j=1 j′ I+ | j | | j | j=1 j′ I− | j | | j | X X∈ j X X∈ j m+1 = rj = 1. Xj=1 Relation (5.17) allows us to use inequality (5.12). Together with independence of the measures ( ) 6 ′ 6 conditionally on the (a ) 6 6 , this yields G(′j′) 1 j np i 1 i m Qχ(M) γ E (M) γ (a ) 6 6 Gg | i 1 i m np 1/2 S(ai) Qχ(M) P 2(2π/λi) aiϕ wj′ 6 C E e i ij d ′ γ (a ) 6 6 . G | i 1 i m j′=1 j Z j′ Sj Y X C ∩ Notice that the sum over j in the latter expression contains at most two non trivial terms as a cusp ′ possesses at most two non trivial intersections with the S ’s. More precisely Cj j 1/2 S(ai) 1/2 S(ai) 1/2 S(ai) P 2(2π/λi) aiϕ ′ P 2(2π/λi) aiϕ ′ P 2(2π/λi) aiϕ i ij i ij e i ij d ′ = e − d ′ + e + d ′. G − G + G j′ Sj ′ ′ Xj ZC ∩ ZCj ZCj 48 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

+ 1 1 Now introduce t ′ = F −′ (1) and t−′ = F −′ ( 1) and rewrite the above integrals as j j j j −

ψ F ′ φe ◦ j d ′ ′ G ZCj in view of applying Lemma 5.2, with ℓ = ℓ′ = ℓj′

S(a ) S(a ) 2(2π/λ )1/2a ϕ i (t+ ) 2(2π/λ )1/2a ϕ i (t− ) Pi i i ij′ j′ Pi i i ij′ j′ φ = e + 1 + + e − 1 − Cj′ Cj′ 1/2 (ai) 1 (ai) (ai) 1 (ai) + S S − S S + ψ = 2(2π/λi) ai (ϕij′ Fj−′ ϕij′ (tj−′ ))1 + (ϕij′ Fj−′ ϕij′ (tj′ ))1 . − ◦ − − Cj′ + ◦ − + Cj′ Xi w+ j′ Notice that ψ Fj′ (1) = ψ Fj′ ( 1) = 0. Then Lemma 5.2 with λ = gives the estimate ◦ ◦ − wj′

1/2 S(ai) Qχ(M) P 2(2π/λi) aiϕ wj′ E e i ij d ′) γ (a ) 6 6 G | i 1 i m j Z j′ Sj X C ∩ Qχ(M) S(a ) S(a ) 2(2π/λ )1/2a w−ϕ i (t− )+w+ϕ i (t+ ) Pi i i j′ ij′ j′ j′ ij′ j′ 2 γ − + 1 δ C Pi λiai 6 C e ℓj′− e . 2 To estimate the quantity 1 6 r 6 2π ψ′(r) dr appearing in the conclusion of Lemma 5.2, | | ℓ1−δ | | j′ we have used the chain ruleR formula for derivatives combined with the following elementary 1 ℓj′ 2π estimates for Fj−′ (r)= ℓ ′ r valid for all r [1, 1−δ ] and for some irrelevant constant c> 0 j ℓ ′ tan 2π ∈ j 1 1 c− 1 c c 1 c− 6 F −′ (r) 6 and 6 (F −′ )′(r) 6 . r j r − r2 j − r2 Combining with the expressions (5.14)+(5.15) we obtain the estimate

2 1/2 λ 1 2 ψ′(r) 6 C( λ a r i− ) , | | i i| | Xi yielding in turn, after integrating (and recalling that 0 < λi < 1/4),

2 1/2 2 ψ′(r) dr 6 C aiai′ (λiλi′ ) 6 C λiai 2π 1 6 r 6 | | ′ Z | | ℓ1−δ i,i i j′ X X for some global constant C (which may change along lines). Let us make a remark here. In the statement of Lemma 5.2, the exponent q is deterministic whereas here it is here random as a Qχ(M) measurable function of the wj′ (hence of the ai’s), namely q = wj′ γ . Yet conditionally on the ai’s the exponent can be seen as deterministic. We can thus apply Lemma 5.2. The resulting constant Cq is therefore a measurable function of the ai’s. Yet, because Cq is locally bounded as a function of q and because w ′ [0, 1] for all j , we can obtain an overall deterministic j ∈ ′ constant C in the above inequality by taking C = supq [0, Qχ(M) ] Cq. ∈ γ So far, we have established that

Qχ(M) 1/2 − S(ai) − + S(ai) + Qχ(M) ′ 2(2π/λ ) a w ϕ (t )+w ϕ (t ) 2 Pi,j i i j′ ij′ j′ j′ ij′ j′ γ γ C Pi λiai γ − + 1 δ E (M) 6 CE e e ℓ ′− . Gg j j′ h i Y LQG AND BOSONIC 2d STRING THEORY 49

We can convert the sums over j′ in the above expectation into sums of the type j j′ I± ∈ j (a ) S i and use the relation ϕij′ (tj±′ )= vij + (λi)+ (ǫ) for j′ Ij± (the entering thisP relationP ± O O ∈ O are uniform w.r.t. the ai’s as easily seen from the expressions (5.14) and (5.15)) to get for some C > 0 Qχ(M) E γ (M) γ (5.18) Gg |ai| Qχ(M) 2 C ( (ǫ)+ (λ )) 1/2 C λ a Pi i Pi,j 2(2π/λi) airj vij 1 δ 6 E Pi i i √λi O O γ C e e e ℓj′− . ′ h i Yj

To complete the proof we use the structure of the coefficients (vij)ij. Recall that the vectors v Rm+1 with components (v ) form an orthonormal family for the inner product i ∈ ij j (u, v) = j ujvjVolg0 (Sj) and the orthogonal complement of span(vi)i is the vector 1 with all components equal to 1. Now we define the precise values of the coefficients r involved in P j (5.18). Set V = max v and define ij | ij| r = R 1+ 1 (a )v Vol (S ) (5.19) j 2mV S i ij g0 j i X where R is a normalizing constant such that j rj = 1. Observe that 0 < rj < 1 for all j and with this choice P R i, r v = (a ). ∀ j ij 2mV S i Xj Now we plug this relation into the estimate (5.18) to get for some C > 0

|a | np Qχ(M) i 2 C Pi (1+ (ǫ)+ (λi))+C Pi λiai γ γ − √λi O O 1 δ E (M) 6 CE e ℓ ′− . Gg j j′=1 h i Y We can choose the neighborhood of (M , g ) in such a way that the term (ǫ)+ (λ ) is 0 0 |O O i | less than 1/2 uniformly with respect to i, (ai)i, in which case taking expectation of the above expression yields

|a | np m np Qχ(M) i 2 C Pi +C Pi λiai 1/2 γ γ − √λi 1 δ 1 δ E (M) 6 CE e ℓ ′− 6 C λ ℓ ′− . Gg j i j j′=1 i=1 j′=1 Y Y Y This establishes integrability with respect to the measure (5.9) by using the rough estimate (2.31) on the eigenvalues λj(g) in terms of the lengths ℓk(g).

5.3. Proof of Theorem 5.1 in the case γ < 2. The proof of the case γ < 2 is much simpler 1 ∗ det(Pgτ Pgτ ) 2 than the case γ = 2. The main reason is that the quantity ′ cM+1 dτ appearing in (det ∆gτ ) the integral (5.8) presents a more gentle behaviour at the boundary of the moduli space due to the lower central charge cM < 1. More precisely (2.12) and (2.15) now gives the following estimate for this term

3g 3 np π2(1−c ) 5−cM M cM+1 − 6ℓ − 2 − j′ − 2 C ℓj ℓj′ e λi dℓjdθj, (5.20) ′ jY=1 jY=1 λiY<1/4 Yj hence an additional exponential decay in comparison with the case cM = 1. 50 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

We stick to the notations introduced in section 2.4 and in the proof of Theorem 5.1 in the case γ = 2. We introduce (ai)1 6 i 6 m i.i.d. standard Gaussian random variables and consider the following Gaussian field: m ai Y (x)= fvi (x) √νi Xi=1 m+1 where the ν ’s are defined in Theorem 2.3 and f in (2.30), with v = (v ) 6 6 R i vi i ij 1 j m+1 ∈ from Lemma 2.5. Now, recall that on each Sj′ the field Y (x) is the constant random variable Y = m v ai . j i=1 ij √νi ByP combining Proposition 2.2 and Lemma 2.5, the Green function Gg is such that for all x,x 6 6 S ′ ∈∪1 j m+1 j′ m 1/L fvi (x)fvi (x′) Cδ Gg(x,x′) 6 + Ag(x,x′)+ νj(g) ν1 Xi=1 where C,L > 0 are global constants. Hence, if we introduce a standard normalized Gaussian variable Z and an independent Gaussian field Xg′ (living in the space of distributions) with covariance A , we get that for all x,x S g ′ ∈ 1 6 j 6 m+1 j′ E S E 1/L 1E 2 Gg(x,x′) 6 [Y (x)Y (x′)] + [Xg′ (x)Xg′ (x′)] + Cδ ν1− Z in such a way that Kahane’s convexity inequality [Ka] ensures that for all q > 0

γ q E ( 6 6 S′ )− Gg ∪1 j m+1 j m+1 2 2 γ 2 γ q γXg (x) E[Xg(x) ] Wg(x) − = E e − 2 e 2 dvg(x) S′ h Xj=1 Z j i m+1 2 2 2 2 γ 2 ′ γ ′ 2 1/L 1/2 γ 1/L γ q γY (x) E[Y (x) ]+X (x) E[X (x) ]+γ(Cδ /ν1) Z Cδ /ν1 Wg(x) − 6 E e − 2 g − 2 g − 2 e 2 dvg(x) S′ h Xj=1 Z j i 2 1/L m+1 2 2 2 2 γ Cδ γ 2 ′ γ ′ 2 γ q (q+q /2) 2 ν γY (x) E[Y (x) ]+γX (x) E[X (x) ] Wg(x) − 6 e 1 E e − 2 g − 2 g e 2 dvg(x) . S′ h Xj=1 Z j i Now, by Proposition 2.2 and Lemma 2.5, there is some global constant C > 0 such that for all j and x S we have ∈ j′ W (x) > E[Y (x)2] C (5.21) g − so that for some other global constant C > 0

1/L m+1 2 Cδ ′ γ ′ 2 q E γ q ν E γY (x)+γXg (x) 2 E[Xg(x) ] − g ( 1 6 j 6 m+1Sj′ )− 6 Ce 1 e − dvg(x) . G ∪ S′ j=1 Z j h X i Notice that

ai ai Volg(Sj)Yj = Volg(Sj)( vij )= ( Volg(Sj)vij) = 0 √νi √νi Xj Xj Xi Xi Xj LQG AND BOSONIC 2d STRING THEORY 51 since the vectors (v ) are orthogonal to 1 in the . norm (2.27). Hence there exists almost i i || ||g surely some j such that Yj > 0. Therefore, gathering the above considerations, we have

1/L m+1 2 Cδ ′ γ E ′ 2 q E γ m+1 q ν E γY (x)+γXg (x) 2 [Xg(x) ] − g ( j=1 Sj′ )− 6 Ce 1 e − dvg(x) G ∪ S′ h i h Xj=1 Z j i 1/L 2 Cδ ′ γ E ′ 2 q ν1 E γYj +γXg (x) 2 [Xg(x) ] − 6 Ce 1Yj > 0 e − dvg(x) S′ Xj h Z j i 1/L 2 Cδ ′ γ ′ 2 q ν γX (x) E[X (x) ] − 6 Ce 1 E e g − 2 g dvg(x) S′ Xj h Z j i Cδ1/L 6 Ce ν1 where in the last inequality we have used the fact that the covariance of Xg′ can be controlled independently of the size of the small eigenvalues (this is a consequence of Proposition 2.2 and Lemma 2.5). Combining with (5.20), this estimate shows integrability with respect to the measure (5.20) by recalling that ν1 > Cℓ1 for some some constant C > 0 and by choosing δ 2 1/L (1 cM)π such that Cδ < − 6 . Finally, it remains to identify the relation (5.6). Starting from the deﬁnition of ν, we have E [F ( (dz),R)] = ν(F )/ν(1) (gτ )τ ,µ Lγ 1 γ 1 det(Pg∗τ Pgτ ) 2 γc γ Qχ(M)c µeγc (M) = E F (e (dz),τ)e− − Ggτ dτdc. cM+1 gτ Z g R (det′∆gτ ) G ZM Z h i γc γ It suﬃces to make the change of variables y = e g (M) to get G τ E [F ( (dz),R)] (gτ )τ ,µ Lγ 1 γ 2Q(g−1) 2Q(g−1) 1 det(Pg∗τ Pgτ ) 2 gτ (dz) γ 1 µy = E F y G ,τ (M)− γ y γ − e− dy, cM+1 γ gτ γZ g (det′∆gτ ) gτ (M) G ZM h G i from which our claim follows.

5.4. Proof of Lemma 5.2. We will use the results in Lemma 2.42 and Proposition 2.7. So it is convenient to introduce the notations

+ 2π ℓj′ ±′ := j′ (ℓ) j′ (ℓ′)−, Fj′ (ρ) := arctan( ), Cj C ∪C ℓj′ ρ 2λ h(ρ) := ρ − i 1 + ln(1/ ρ ) . | | | | i X 1 We further denote by Fj−′ the inverse of the function Fj′ . Finally, in what follows, C stands for a generic irrelevant positive constant, the value of which may change along lines.

Proposition 2.7 ensures that the Green function Ag (which is the covariance of the field Xg′ iθ iθ defined in (5.10)) satisfies for x = ρe and x = ρ e in the collar ′ ′ ′ Cj 6 2 Ag(x,x′) Ggj′ (x,x′)+ C h(ρ)h(ρ′). (5.22) Otherwise stated, we have bounded the errors terms in Proposition 2.7 with the help of the function h. We stress that the function χ in Proposition 2.7 is worth 1 only for ρ 6 1/4 so | | 52 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS that the above bound is rigorously valid only for ρ 6 1/4. Yet in the following we assume | | that it is valid for ρ 6 1 for notational convenience because it does not change the validity of | | the argument.

Then we need to decompose the Gaussian field with covariance function Ggj′ according to its radial/angular parts. So we consider two independent centered Gaussian fields Xr and Xa r a defined on the collar ′ = [ 1, 1] (R Z) with respective covariance kernels G and G Cj − ρ × \ θ defined by

ℓj′ 2 r min(Fj′ ( ρ ), Fj′ ( ρ′ )) 2π2 Fj′ ( ρ )Fj′ ( ρ′ )+ C h(ρ)h(ρ′) if ρρ′ > 0 G (ρ, θ, ρ′,θ′) := | | | | − | | | | ℓj′ 2 ( 2 Fj′ ( ρ )Fj ( ρ′ )+ C h(ρ)h(ρ′) if ρρ′ 6 0 2π | | | | (5.23)

′ ′ a Fj′ (ρ) Fj′ (ρ ) +2iπ(θ θ ) G (ρ, θ, ρ′,θ′) := ln 1 e−| − | − 1 ρρ′ > 0 . (5.24) − − { }

r a The two quantities G and G are positive definite, hence the existence of such fields (pointwise for Xr and distributional for Xa). Combining the above two relations we get that for x = ρeiθ iθ and x = ρ e in the collar ′ ′ ′ Cj E E r r E a a E (0) (0) [Xg′ (x)Xg′ (x′)] 6 [X (x)X (x′)] + [X (x)X (x′)] + [(δ h(ρ))(δ h(ρ′))] (5.25) where δ(0) is a standard Gaussian random variable independent of everything. Hence we can use Kahane’s inequality [Ka, Lemma 1] to get the estimate (recall that mgj′ is the Robin constant defined in Lemma 2.42) for q > 0

q ψ F ′ − E φe ◦ j d ′ (5.26) G h Z (0) 2 r E r 2 4πmg q E ψ Fj′ 2Cδ h 2h 2X 2 [(X ) ] j′ a 6 Cq φ e ◦ e − e − e d g − ± G Z j′ C where Cq is an explicit constant such that ln Cq is quadratic in q. Here we have deﬁned the a random measure g as the limit in law as ǫ 0 (eventually up to some subsequence) of the G 1/2 2Xa 2E→[(Xa)2] family of random measures ( ln ε) e ε − ε dvg. From (2.42)+Proposition 2.7, we get 2 − 2 2 that 4πm 2E[X ] > 2 ln ρ 2C h(ρ) for ρ > ℓ ′ for some constant C > 0. Hence, gj′ − r | |− | | j (0) 2 r E r 2 4πmg 2Cδ h 2h +2X 2 [(X ) ] j′ a for ρ > ℓj′ , the measure e − − e d g is greater than the measure r| | G e2X eg(ρ)ρ 2 d a where we have set − Gg g(ρ) := 2Cδ(0)h(ρ)+4ln ρ 4C2h(ρ)2. (5.27) | |− Now that we have simpliﬁed the deterministic part of the measure we analyze the random part. For this, we write a path decomposition result for the process Xr

Lemma 5.4. Let us consider two standard Gaussian r.v. δ(1), δ(2) (0, 1) and two standard + ∼N Brownian bridges (Brρ )ρ [0,1] (Brρ−)ρ [0,1] , all of them mutually independent. We have the ∈ ∈ following equality in law in the sense of processes for ρ [ 1, 1] ∈ −

r + √ℓj′ /2 (1) (2) X (ρ)= 1 π/ ℓ ′ Br + 1 π/ ℓ ′ Br + F ′ ( ρ ) δ + Ch(ρ)δ . ρ>0 j ℓ ′ ρ<0 j −ℓ ′ π j { } j { } j | | 2 Fj′ ( ρ ) 2 Fj′ ( ρ ) p π | | p π | | LQG AND BOSONIC 2d STRING THEORY 53

Proof. Recall the covariance structure of the Brownian bridge E[Br+Br+]= ρ ρ ρρ . Hence ρ ρ′ ∧ ′ − ′ for arbitrary constants a, c > 0 and ρ< 1/c

+ + 2 2 2 E[(aBr )(aBr ′ )] = a cρ ρ′ a c ρρ′. cρ cρ ∧ − ℓ Adjusting the constants a, c to fit with the covariance function min(ρ, ρ ) j′ ρρ leads to ′ − π2 ′ ℓj′ a = π/ ℓj′ and c = π2 . One completes easily the proof of the claim by time changing with √ℓj′ /2 (1) (2) F ′ andp adding the covariance structure of the term F ′ ( ρ ) δ + Ch(ρ)δ . j π j | | ℓj 1 ′ 6 6 ′ Now, observe that if we restrict to those ℓj ρ 1 then π2 Fj ( ρ ) [0, 2 ] and the law of 1 | | | | ∈ the Brownian bridge Br on [0, 2 ] is absolutely continuous with respect to the law of Brownian B 2 + motion B with density 2e−| 1/2| . Using this relation with Br and Br− together with the law scaling relation for Brownian motion aBt/a2 = Bt for fixed a> 0, we deduce using (5.26) and Lemma 5.4 that q ψ F ′ − E φe ◦ j d ′ (5.28) G Z h + − q 2BF (ρ)+Θ(ρ) 2BF (ρ)+Θ(ρ) E ψ Fj′ j′ 2 a ψ Fj′ j′ 2 a − 6 Cq A e ◦ e ρ− d g + B e ◦ e ρ− d g , ′ (ℓ)+ G ′ (ℓ′)− G h ZCj ZCj i + where B ,B− are two independent Brownian motions, independent of everything, and the (1) (2) function Θ is defined by Θ(ρ) := 2ℓ ′ /πF ′ (ρ) δ + 2Ch( ρ )δ + g(ρ). Now we would like j j | | to get rid of the drift term Θ. The point is that the behaviour of Θ is rather tricky for those ρ p 1| δ| that are very close to (or less than) ℓ ′ whereas the contribution of the Θ(ρ), say for ρ > ℓ ′− j | | j for any δ > 0, turns out to be easily controlled. So we fix an arbitrary δ ]0, 1[ and remark ∈ that the expectation in the r.h.s. of (5.28) is less than the same expectation with integration ′1 δ 1 δ + restricted to ′ (ℓ ) and ′ (ℓ ) . Furthermore, we introduce the (random) function Y Cj − − Cj − ρ through the relation

Fj′ (ρ) ρ ]0, 1], Θ(ρ) = 2 Y du + Θ(1). (5.29) ∀ ∈ u ZFj′ (1) 1 δ 2π δ 1 We set κ(ℓ) := Fj′ (ℓ − )= arctan(ℓj′ ℓ − ). Then the Girsanov theorem tells us that, under ℓj′ the probability measure

′ ′ κ(ℓ) + κ(ℓ ) − 1 κ(ℓ) 2 1 κ(ℓ ) 2 R Yr dB +R Yr dB R Y dr R Y dr R dP, with R := e κ(1) r κ(1) r − 2 κ(1) r − 2 κ(1) r ,

1 δ + ′1 δ the processes ρ [1,ℓ − ] 2B and ρ [1,ℓ − ] 2B− have respectively the same ∈ 7→ Fj′ (ρ) ∈ 7→ Fj′ (ρ) 1 δ + 1 δ laws as the processes ρ [1,ℓ − ] 2B +Θ(ρ) Θ(1) and ρ [1,ℓ − ] 2B− +θ(ρ) ∈ 7→ Fj (ρ) − ∈ 7→ Fj (ρ) − Θ(1) under P. Therefore, using the Girsanov transform in the expectation (5.28), we get

q ψ F ′ − E φe ◦ j d ′ G Z h + − q 2BF (ρ) 2BF (ρ) E qΘ(1) ψ Fj′ j′ 2 a ψ Fj′ j′ 2 a − 6 4 Re− A e ◦ e ρ− d g + B e ◦ e ρ− d g ′1−δ ′ (ℓ1−δ)+ G (ℓ )− G h ZCj ZCj j′ i 2B+ 2B− pq 1 ψ F ′ F ′ (ρ) 2 a ψ F ′ F ′ (ρ) 2 a − p 6 4Eℓ,ℓ′ (q)E A e ◦ j e j ρ− d + B e ◦ j e j ρ− d . g ′ g ′ (ℓ1−δ )+ G ′ (ℓ 1−δ)− G h ZCj ZCj i 54 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS where we have used the H¨older inequality to get the last inequality with p,m,r > 1 and 1 1 1 p + m + r = 1 and set ′ ′ κ(ℓ) + κ(ℓ ) − 1 κ(ℓ) 2 1 κ(ℓ ) 2 m 1/m R Yu dBu +R Yu dBu R Yu du R Yu du qrΘ(1) 1/r Eℓ,ℓ′ (q) :=E e κ(1) κ(1) − 2 κ(1) − 2 κ(1) E[e− ]

2 2 ′ hm m κ(ℓ) 2 m m κ(ℓ ) 2 1/q i − R Yr dr+ − R Yr dr qrΘ(1) 1/r =E e 2 κ(1) 2 κ(1) E[e− ] . So we have got ridh of the drift term Θ(ρ) with thei Girsanov trick. The cost is the constant

E ′ (q) but an easy computation shows that sup ′ E ′ (q) < + for any q > 1. This ℓ,ℓ ℓ,ℓ > ℓj ℓ,ℓ ∞ computation is left to the reader but we give a brief convincing heuristic argument. The process Yu is deﬁned by (5.29). We have already explained in the proof of Theorem 5.1 that for small ℓ ′ and for ρ > ℓ ′ , F ′ (ρ) behaves like 1/ρ. Hence Y (ρ) is with good approximation given j | | j j by Θ (1/ρ)ρ 2. Then it is readily seen that Θ (1/ρ)ρ 2 is a sum of terms of the type − ′ − − ′ − cλi 1 n 1/2 1/ρ, ρ − − (ln ρ ) (for n = 0, 1, 2) or ℓj′ . It is then obvious to see that the square of | | | | κ(ℓ) every possible linear combination of such terms has its κ(1) -intergal bounded by constant independently of ℓ > ℓ ′ . j R We can use the same argument to explicitly determine the eﬀect of the drift term ψ F ′ . ◦ j The variance of the Girsanov transform to get rid of this term is less than

2 C ψ′(r) dr 1 6 r 6 2π/ min(ℓ1−δ ,ℓ′1−δ) | | Z | | 1 δ (here we have used the fact that κ(1) > 1 and κ(ℓ) 6 2π/ℓ − ). All in all, this entails that for arbitrary p> 1 there exists some constant Cp such that ′ δ−1 q 2π max(ℓ,ℓ ) ψ F ′ − 2 E φe ◦ j d ′ 6 Cp exp ψ′(r) dr (5.30) G 0 | | h Z Z 2B+ 2B− pq 1/p E Fj (ρ) 2 a Fj (ρ) 2 a − A e ρ− d g + B e ρ− d g , × (ℓ1−δ )+ G (ℓ′1−δ )− G h ZCj ZCj i which in turn less than 2π max(ℓ,ℓ′)δ−1 q 2 Cp(AB)− 2 exp ψ′(r) dr (5.31) 0 | | Z 1 2B+ pqλ 2B− pq(1 λ) F ′ (ρ) 2 a − F ′ (ρ) 2 a − − p E e j ρ− d e j ρ− d g ′ g × ′ (ℓ1−δ )+ G ′ (ℓ 1−δ )− G h ZCj ZCj i after using the elementary inequality (a + b) m 6 a λmb (1 λ)m for a, b > 0, λ [0, 1] and − − − − ∈ m> 0. It remains to evaluate the latter expectation. So we introduce the sets for n, k > 0 and ℓ> 0 A+(ℓ)= sup B+ B+ ]n,n + 1] n F ′ (u) F ′ (1) {u [ℓ1−δ,1] j − j ∈ } ∈ A−(ℓ′)= sup B− B− ]k, k + 1] k Fj′ (u) Fj′ (1) {u [ℓ′1−δ,1] − ∈ } ∈ as well as the stopping times + + + Tn = inf B B = n Tn− = inf B− B− ]n,n + 1] . {u ]0,1] Fj′ (u) − Fj′ (1) } {u > 0 Fj′ (u) − Fj′ (1) ∈ } ∈ LQG AND BOSONIC 2d STRING THEORY 55

+ Partitioning the probability space according to the events An (ℓ) and Ak−(ℓ′) and using sub- additivity of the mapping x R x1/p for p> 1, we get the estimate ∈ + 7→

2B+ pqλ 2B− pq(1 λ) 1/p E Fj′ (ρ) 2 a − Fj′ (ρ) 2 a − − e ρ− d g e ρ− d g 6 En,k 1−δ + G ′1−δ − G j′ (ℓ ) j′ (ℓ ) h ZC ZC i Xk,n where we have set

+ pqλ − pq(1 λ) 1 2BF (ρ) 2BF (ρ) p E j′ 2 a − j′ 2 a − − En,k := 1 + 1 − ′ e ρ− d g e ρ− d g . An (ℓ) Ak (ℓ ) ′ ′ (ℓ1−δ)+ G × ′ (ℓ 1−δ)− G h ZCj ZCj i

The idea is now the following: the ﬂuctuations of the Brownian motion B+ over an interval of + + + length 1 are of order 1. Hence over the interval n := [Fj′ (Tn ), Fj′ (Tn)+1], B is approximately I+ 1 + equal to n. Put in other words, the process B is worth n on the interval [F −′ (Fj′ (T )+ Fj′ (u) j n + 1), Tn ]. Same remark for B−. Hence En,k should be estimated by

1 1 (n+k)pqP + p P p En,k 6 e− (An ) (Ak−) pqλ pq(1 λ) 1 (5.32) E 2 a − 2 a − − p sup ρ− d g ρ− d g . × x,x′ ]ℓ ,1] ρ I+(x) G × ρ I−(x′) G ∈ j h Z{ ∈ } Z{ ∈ } i

+ 1 where for x ]0, 1], we denote I (x) := [Fj− (Fj (x) + 1),x] and for x′ [ 1, 0[, I−(x′) := 1 ∈ ∈ − [ x , F − (F (x ) + 1)]. We will conclude with the two following lemmas − ′ − j j ′ Lemma 5.5. For any q > 0, we have

q E 2 a − sup sup ρ− d g < + . ′ ℓ 6 1 x ]ℓ ′ ,1] ρ I+(x) G ∞ j ∈ j h Z{ ∈ } i

The same property for I (x ) and x [ 1, ℓ [. − ′ ′ ∈ − − j

Lemma 5.6. There is some constant C > 0 such that for any δ > 0, n, k > 0 and ℓ,ℓ′ > ℓj′

1 ′ 1 P + 2 (1 δ) P 2 (1 δ) (An (ℓ)) 6 Cnℓ − (Ak−(ℓ′)) 6 Ckℓ − .

Indeed, using H¨older inequality and Lemma 5.5 we see that the expectation involved in the r.h.s. of (5.32) is less than some constant independent of everything. Hence we get the bound (n+k)qpP + P (n+k)qp En,k 6 Ce− (An ) (Ak−). Lemma 5.6 and summability of the series n,k > 0 kne− complete the argument. P The only remaining detail to ﬁx is to show (5.32). This is an easy task as, using the independence of B+,B , a as well as the strong Markov property of the Brownian motion, we − Gg 56 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS have

+ − 2λpqBF (1) 2(1 λ)pqBF (1) 6 E − j′ − − j′ En,k 1A+(ℓ) A−(ℓ′)e n ∩ k h + + − − 1 2B 2B pqλ 2B 2B pq(1 λ) p Fj′ (ρ)− Fj′ (1) 2 a − Fj′ (ρ)− Fj′ (1) 2 a − − e ρ− d g e ρ− d g × + G × − G ZIn ZIn + − i 1 1 2λpqBF (1) 2(1 λ)pqBF (1) E − j′ − − j′ p pq(n+k) P + P p 6 [e ] e− ( (An (ℓ)) (Ak−(ℓ′))) + + − − s 1 s 1 pqλ minu∈I BF (u) B ′ pq(1 λ)minu∈I BF (u) B ′ E[e− n j′ − Fj j (1) ] p E[e− − n j′ − Fj j (1) ] p pqλ pq(1 λ) 1 E 2 a − 2 a − − p sup ρ− d g ρ− d g . x,x′ ]ℓ ′ ,1] ρ I+(x) G × ρ I−(x′) G ∈ j h Z{ ∈ } Z{ ∈ } i Standard estimates about the supremum of the Brownian motion over an interval of size 1 show + + − − pqλ min + B B pq(1 λ)min − B B that the E[e− u∈In Fj (u)− Fj (1) ]1/p and E[e− − u∈In Fj (u)− Fj (1) ]1/p are bounded by some constant independent of ℓ,ℓ′. Hence our claim.

Proof of Lemma 5.5. Recall that Gaussian multiplicative chaos at criticality (i.e. γ = 2) possesses moments of negative order (see [DRSV1, Prop. 5]). This entails that for any x > 0, q E 2 a − 1 x 6 ρ 6 1 ρ− d g < + . Then we observe that we have the relation Fj−′ (Fj′ (x)+ { } G ∞ 1)h>R Cℓj′ for some irrelevant i constant C and for all x>ℓj′ . Therefore, for some C and all x>ℓj′ q q E 2 a − E 2 2 1 a − ρ− d g 6 C (ρ + ℓj′ )− d g . I+(x) G I+(x) G h Z i h Z i 2 2 1 a To conclude, observe that the measure (ρ + ℓ ′ ) d is the pushforward of the measure j − Gg 2Xa(F −1(ρ)) 2E[Xa(F −1(ρ))2] e j′ − j′ dρ (implicitly understood as the limit of a regularized sequence) a 1 under the mapping ρ F ′ (ρ). The Gaussian random distribution ρ X (F −′ (ρ)) is sta- 7→ j 7→ j tionary and its law does not depend on ℓj′ in such a way that the process x R+ a −1 E a −1 2 ∈ 7→ x+1 2X (F ′ (ρ)) 2 [X (F ′ (ρ)) ] j − j ′ x e dρ is stationary (and its law does not depend on ℓj either) 2 2 1 a and has the same law as (ρ + ℓ ′ ) d , hence our claim. R I(x) j − Gg R Proof of Lemma 5.6. Recall the standard computation related to Brownian motion

1/2 P( sup Br 6 β) 6 βt− . r [0,t] ∈ Therefore

+ + 1 δ 1/2 P(A (ℓ)) 6 P sup B 6 n + 1 6 (n + 1)(F ′ (ℓ − ) F ′ (1))− . n F ′ (u) Fj (1) j j u [ℓ1−δ,1] j − − ∈ 1 ′ > > ′ We conclude by noticing that ℓj−′ arctan(ℓj /x) C/x for some constant C and all x ℓj . Same argument for Ak−(ℓ′). LQG AND BOSONIC 2d STRING THEORY 57

5.5. Relation with random planar maps. The purpose of this subsection is to write a precise mathematical conjecture relating LQG to the scaling limit of large planar maps. Following Polyakov’s work [Po], it was soon acknowledged by physicists that LQG should describe the scaling limit of discretized 2d quantum gravity given by ﬁnite triangulations of a given surface, eventually coupled with a model of statistical physics (often called matter ﬁeld in the physics language), see for example the classical textbook from physics [ADJ] for a review on this problem. We will describe two situations in what follows: pure gravity (no matter) or the bosonic string embedded in D = 1 dimension.

We consider a fixed family (gτ )τ of hyperbolic metrics on a compact surface M (without boundary) with genus g as previously and the associated Liouville measure under E [ ]. Lγ (gτ )τ ,µ · Let g be the set of triangulations with N faces with the topology of a surface of genus g. TN, Since these triangulations are seen up to orientation preserving homeorphisms, there are only a finite number of such triangulations. We equip T g with a standard metric structure h ∈TN, T where each triangle is given volume a2. The metric structure consists in gluing flat equilateral triangles: the exact definition of the metric structure is given in Les Houches lecture notes [RhVa2] in the case of the sphere and the case we consider here does not present additional difficulties for the definition. The uniformization theorem tells us that there exists a unique τ g along with an orientation preserving diffeomorphism ψ : T M and a conformal T ∈ M T → factor ϕT (with logarithmic singularities at the images of the vertices of the triangles) such that

ϕT hT = ψT∗ (e gτT ). (5.33)

Recall that in the decomposition (5.33), the functions ϕT and ψT are unique except if the metric gτT possesses non trivial isometries. In that case, the isometry group is ﬁnite of the (i) form (ψ )1 6 i 6 n and starting with a decomposition (5.33) all the other decompositions of (i) h are ((ψ(i)) 1 ψ ) (eϕT ψ g ). Therefore, in the following discussion, we will suppose T − ◦ T ∗ ◦ τT that the functions ϕT and ψT are uniquely determined by the triangulation T and if this is ϕT not the case (i.e. there exists a non trivial isometry group), we replace e gτT in what follows (i) 1 n ϕT ψ by the average n i=1 e ◦ gτT : these special metrics should play no role anyway as their equivalence classes are of measure 0 with respect to the Weil-Petersson volume form. P

Pure gravity. It is proved in [BeCa] that the following asymptotic holds:

5 µcN (g 1) 1 N,g C e N 2 − − (5.34) |T |N ∼ T →∞ where C > 0 and µc > 0 are constants. The constants C ,µc are non universal in the T T sense that one can consider quadrangulations say in the place of triangulations: in this setting, the number of quadrangulations N,g of size N will satisfy the asymptotic N,g Q |Q |N ∼ 5 →∞ µcN (g 1) 1 C ee N 2 − − where C is different from C and µc > 0 is different from µc. Q Q T We set 2 e µ¯ = µc + a µ, (5.35) where µ > 0 is fixed, and we consider the following random volume form on the surface M, defined in terms of its functional expectation 58 COLIN GUILLARMOU, REMI´ RHODES, AND VINCENT VARGAS

1 Ea µN¯ ϕT [F (νa)] = e− F (e dvgτ ), (5.36) Za T N > 1 T g X ∈TXN, for positive bounded functions F where Z is a normalization constant ensuring that Ea[ ] is a · the expectation of a probability measure. We denote by Pa the probability law associated to Ea.

We can now state a precise mathematical conjecture: Conjecture 1. Under Pa, the random measure ν converges in law as a 0 with µ¯ given by a → (5.35) in the space of Radon measures equipped with the topology of weak convergence towards the Liouville measure under E [ ] with parameter γ = 8 . Lγ (gτ )τ ,µ · 3 q 8 The fact that γ = 3 can be read of the total volume of space; indeed, thanks to (5.34), it is easy to show that inq the above asymptotic the total volume νa(M) converges to the Gamma 5 (g−1) µ 2 µx 5 (g 1) 1 law with density e x 2 1 > . This law matches the law of the total volume Γ( 5 (g 1)) − − − x 0 2 − ξ of in Theorem 5.1 for 2Q = 5 , i.e. γ = 8 . γ Lγ γ 2 3 Finally, let us mention that conjectures similarq to 1 have appeared in other topologies: the sphere [DKRV], the disk [HRV] and the torus [DRV]. However, in these other topologies, the corresponding conjectures are still completely open. Let us nevertheless mention some partial progress by Curien in [Cu] where appealing convergence results are proven assuming a reasonable condition that has unfortunately not been proven yet.

Bosonic string. Given a triangulation T , let us denote by VT the vertex set of the dual lattice. We consider the partition function of the bosonic string on T by

1 2 P ′ (xv x ′ ) Z(T ) := e− 2 v∼v − v dxv Z vY∈V where denotes adjacent vertices of the dual lattice. It is expected that ∼

µ′ N 2(g 1) 1 Z(T ) C′ e c N − − (5.37) N ∼ T T g →∞ ∈TXN, where C′ > 0 and µc′ > 0 are (non universal) constants. We set T 2 µ¯ = µc′ + a µ, (5.38) where µ > 0 is ﬁxed, and we consider the following random volume form on the surface M, deﬁned in terms of its functional expectation

1 Ea µN¯ ϕT [F (νa)] = e− F (e dvgτ )Z(T ), (5.39) Za T N > 1 T g X ∈TXN, for positive bounded functions F where Z is a normalization constant ensuring that Ea[ ] is a · the expectation of a probability measure. We denote by Pa the probability law associated to Ea. LQG AND BOSONIC 2d STRING THEORY 59

Conjecture 2. Assume g = 2. Under Pa, the random measure ν converges in law as a 0 a → with µ¯ given by (5.38) in the space of Radon measures equipped with the topology of weak convergence towards the Liouville measure under E [ ] with parameter γ = 2. Lγ (gτ )τ ,µ · The reader can ﬁnd much more material on 2d-string theory in the review [Kleb] or the lecture notes [Polch].

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DMA, U.M.R. 8553 CNRS, Ecole´ Normale Superieure, 45 rue d’Ulm, 75230 Paris cedex 05, France

E-mail address: [email protected]

Universite´ Paris-Est Marne la Vallee,´ LAMA, 5 boulevard Descartes, Champs sur Marne, 77454 Marne La Vallee,´ France., Research supported in part by ANR grant Liouville (ANR-15- CE40-0013)

E-mail address: [email protected]

DMA, U.M.R. 8553 CNRS, Ecole´ Normale Superieure, 45 rue d’Ulm, 75230 Paris cedex 05, France., Partially supported by the ANR project Liouville (ANR-15-CE40-0013)