The semiclassical limit of Liouville conformal field theory

A rigorous probabilistic construction of Liouville conformal field theory (LCFT) on the Riemann sphere was recently given by David-Kupiainen and the last two authors. In this paper, we focus on the connection between LCFT and the classical Liouville field theory via the semiclassical approach. LCFT depends on a parameter $\gamma \in (0,2)$ and the limit $\gamma \to 0$ corresponds to the semiclassical limit of the theory. Within this asymptotic and under a negative curvature condition (on the limiting metric of the theory), we determine the limit of the correlation functions and of the associated Liouville field. We also establish a large deviation result for the Liouville field: as expected, the large deviation functional is the classical Liouville action. As a corollary, we give a new (probabilistic) proof of the Takhtajan-Zograf theorem which relates the classical Liouville action (taken at its minimum) to Poincar\'e's accessory parameters. Finally, we gather conjectures in the positive curvature case (including the study of the so-called quantum spheres introduced by Duplantier-Miller-Sheffield).


Introduction
The purpose of this paper is to relate classical Liouville theory to Liouville conformal field theory (also called quantum Liouville theory) through a semiclassical analysis. Before exposing the framework and the results of the paper, we provide a short historical introduction to classical Liouville theory.
We start by recalling a classical result of uniformization by Picard about the existence of hyperbolic metrics on the Riemann sphere, seen as the extended complex plane C = C ∪ {∞}, with prescribed conical singularities. In this context, given n 3 we let z 1 , . . . , z n ∈ C distinct and χ 1 , . . . , χ n ∈ R denote respectively the prescribed locations and orders of our singularities. We assume that our coefficients satisfy the following two conditions: ∀ k, χ k < 2 and n k=1 χ k > 4. (1.1) Let ∆ z denote the standard Laplacian on C, ∇ z the standard gradient and for z = x + iy we adopt the complex notation ∂ z = 1 2 (∂ x − i∂ y ). We use the notation d dz for the complex derivative of a holomorphic function. The classical result of Picard [19,20] (see also [33]) asserts that for any Λ > 0 the Liouville equation ∆ z φ = 2πΛe φ (1.2) possesses a unique smooth solution φ on C \ {z 1 , . . . , z n } with the following asymptotics near the singular points and at infinity We denote this solution by φ * . The first condition in (1.1) simply ensures that e φ * is integrable in a neighborhood of each z k so that e φ * (z) |dz| 2 defines indeed a compact metric on C. The second condition n k=1 is a negative curvature type condition. In the language of Riemannian geometry, the metric e φ * (z) |dz| 2 has constant negative Ricci scalar curvature (as this is the only notion of curvature used in this paper, we simply refer to it -1032 -as curvature in the remainder of the text) equal to −2πΛ on C \ {z 1 , . . . , z n } (recall that the curvature of the metric at z is given by −e −φ * (z) ∆ z φ * (z)) and a conical singularity of order χ k at z k for each k = 1, . . . , n. Using standard integration by parts (on the Riemann sphere equipped with the round metric), it is possible to show that the solution φ * to the system (1.2)-(1.3) must satisfy n k=1 χ k − 4 = Λ C e φ * (z) d 2 z . (1.5) where d 2 z denotes the standard Lebesgue measure. Therefore n k=1 χ k − 4 and Λ have same sign hence justifying the terminology negative curvature type condition for the condition (1.4). In a celebrated work [22], Poincaré showed how to relate this metric to the problem of the uniformization of C \ {z 1 , . . . , z n } (1) . More specifically, he introduced the (2, 0)-component of the "classical" stress-energy tensor Direct computations show that (1.2) implies that T φ * is a meromorphic function on C and then (1.3) implies that it displays second order poles at z 1 , . . . , z n . More precisely we must have with the asymptotics T φ * (z) = O(z −4 ) as z → ∞ where the real numbers c k are the so-called accessory parameters. Then, considering the second order Fuchsian equation for holomorphic functions on the universal cover (here the unit disk) d 2 u dz 2 + 1 2 T φ * (z)u(z) = 0, (1.8) Poincaré showed (2) that the ratio f = u 1 /u 2 of two independent solutions u 1 , u 2 solves the uniformization problem in the sense that the metric e φ * (z) |dz| 2 is the pull-back of the hyperbolic metric on the unit disk by f , i.e. the following holds (1.9) In particular, if one normalizes u 1 , u 2 to have Wronskian w = u 1 u 2 − u 1 u 2 equal to 1 then e −φ * (z)/2 = Λ/8(|u 2 | 2 − |u 2 1 |). The factor e −φ * (z)/2 thus (1) In fact, Poincaré considered the case χ k = 2 for all k. (2) Recall that in fact Poincaré considered the case χ k = 2 for all k. More than eighty years later, Polyakov and Zamolodchikov suggested (4) the following identity for the parameters c k (5) , (1.11) where the function φ ∈ Θ (χ k ,z k ) → S (χ k ,z k ) (φ) is a functional, called the (classical) Liouville action (with conical singularities), defined on some functional space Θ (χ k ,z k ) (a space of functions with logarithmic singularities of the form (1.3)) and that must be formally understood as (1.12) Let us mention here that relation (1.11) has already been proved in a rather simple way by Takhtajan-Zograf [30] based on geometrical considerations. Yet, expression (1.12) is ill-defined since the functions in Θ (χ k ,z k ) have logarithmic singularities at z k (just like the function φ * ). In order to give the precise definitions of Θ (χ k ,z k ) and S (χ k ,z k ) , we introduce the round metric g(z)|dz| 2 on the Riemann sphere where g(z) is given by g(z) = 4 (1 + zz) 2 with associated Green kernel G with vanishing mean on the sphere, i.e. C G(x, · )g(x)d 2 x = 0 (where d 2 x denotes the standard Lebesgue measure). If we consider the standard Sobolev space (3) The fact that e −φ * (z)/2 solves the PDE version of the Fuchsian equation can also be seen by definition of T φ * .
(4) See Takhtajan's lecture notes [32]. (5) Let us stress here that the function φ * also depends on the (χ k , z k ) though for notational simplicity we keep this dependence implicit.
-1034 -where the norm is defined by |h| H 1 (Ĉ) = C (|∇ z h(z)| 2 + |h(z)| 2 g(z))d 2 z 1/2 , then Θ (χ k ,z k ) is given by χ k G(z k , · ); h ∈ H 1 ( C) . (1.13) We endow Θ (χ k ,z k ) with the metric space structure induced by the H 1 -norm Now, for φ ∈ Θ (χ k ,z k ) , the action S (χ k ,z k ) (φ) is defined by a limiting procedure where one applies a regularization procedure around the points z k with logarithmic singularity and then adds diverging counter terms. More precisely, and where the integration domain in the first integral is defined by C := C \ n k=1 B(z k , ) ∪ {|z| > 1 } and the contour integrals are oriented counterclockwise.
The purpose of this paper is to relate via a semiclassical analysis the classical Liouville action S (χ k ,z k ) to the recent rigorous probabilistic construction of Liouville Conformal Field Theory (LCFT hereafter) given by David-Kupiainen and the last two authors in [3]. Recall that the construction of [3] is based on the Gaussian Free Field (GFF). Moreover, the local conformal structure of LCFT was studied in [9] (Ward and BPZ identities), paving the way to a proof of the DOZZ formula for the three point correlation function [10]. A byproduct of the semiclassical analysis is a new proof of relation (1.11) in the spirit of the way Polyakov and Zamolodchikov discovered it via non rigorous asymptotic expansions on path integrals. We believe that the connection between LCFT (in a probabilistic setting) and the classical Liouville action (1.14) is interesting per se as it is not straightforward (see Takhtajan's lecture notes and discussion [32,31]).
The construction of LCFT is based on the quantum Liouville action, the quantum analog of (1.12) (recall that (1.12) is a formal definition and the exact definition requires a regularization procedure). In order to introduce the quantum Liouville action, let us set the change of variable φ = ϕ + ln g in the formal definition of the classical action (1.12); we get up to a global constant that where in the last line we disregard a global constant (independent of the χ k , z k ) and we introduced the Ricci curvature of the round metric R g (z) := − 1 g(z) ∆ z ln g(z) (which in the particular case of the Riemann sphere is constant and equal to 2). Let us perform for γ > 0 the change of variables α k = χ k γ , µ = Λ γ 2 and work with γϕ instead of ϕ in which case we get The construction of LCFT is based on the "quantization" of the above action. Following general principles in quantum field theory, the theory should correspond to constructing the measure e −S (χ k ,z k ),γ (ϕ) Dϕ where Dϕ is the ill-defined "Lebesgue measure" on the space of functions (6) . Following a standard procedure in constructive field theory (see Barry Simon's book [29]), the probabilistic construction of LCFT is based on interpreting the mea- Dϕ as the Gaussian Free Field (GFF) (7) measure (6) It is a well known fact that the Lebesgue measure does not exist on infinite dimensional spaces. (7) There is an important subtelty here in the interpretation of this quadratic term; indeed, one must not forget to incorporate the spatial average C ϕ(z)g(z)d 2 z of the field ϕ (with respect to g) in the definition of the GFF measure. The correct measure on this -1036 - The semiclassical limit of Liouville conformal field theory and then expressing the other terms of the action as functionals of the GFF along the following factorization: However, the GFF is not a function but rather a Schwartz distribution hence the exponential term C e γϕ(z) g(z)d 2 z is ill-defined. In order to make sense of this term, a renormalization procedure is required and, in order to preserve the conformal invariance properties at the quantum level, one must modify the above action by replacing the classical value 2 γ in front of R g (z)ϕ(z)g(z) by the quantum value Q := 2 γ + γ 2 (see [3]). Of course, the extra correction term γ 2 vanishes when one considers the semiclassical regime γ → 0. This leads to the following formal definition of the quantum Liouville action (where one removes the logarithmic singularities) where γ is a positive parameter belonging to (0, 2), Q = γ 2 + 2 γ and µ > 0 is a positive parameter called the cosmological constant. As we will see shortly, the quantum Liouville measure e −S(ϕ,g) Dϕ is in fact defined on the dual space H −1 ( C) of H 1 ( C), which is defined as the completion of the set of smooth functions on C with respect to the following norm for f smooth The quantum Liouville theory is then defined by its functional expectation (called path integal in the physics literature) for every continuous function F on H −1 ( C) and where φ = ϕ + Q 2 ln g is the Liouville field. The main observables in LCFT are the correlations of the fields V α (z) = e αφ(z) under the measure (1.18). For α 1 , . . . , α n and F any bounded measurable functional, we set average is the standard Lebesgue measure on R: in physics, this average is called the zero mode.
We will recall the rigorous probabilistic definition in the next subsection; the case F = 1 corresponds to the correlations. Let us just mention that inserting quantities like V α k (z k ) correspond at the quantum level to adding a logarithmic singularity to φ at the point z k and with weight α k (this a consequence of the classical Girsanov theorem of probability theory); this property is of course to be expected from the previous discussion at the classical level. Now, gathering the above considerations, the semiclassical regime is the limit of LCFT when γ goes to 0 with α k = χ k γ and µ = Λ γ 2 for fixed Λ and χ k ; it is natural to expect that the following semi-classical limit holds when F is a continuous function on H −1 ( C) where φ * solves the Liouville Equation (1.2) with logarithmic singularities (1.3) and C > 0 is some constant. One of the main results of this paper is to show that this is indeed the case (see Proposition 2.2 below). Once statement (1.19) has been established rigorously, following the idea of Polyakov and Zamolodchikov, it is natural to exploit the above semiclassical limit to recover the accessory parameters by using the BPZ equations (8) .
The BPZ differential equations are the quantum analogues of (1.10); it was shown in [9] that the following BPZ differential equation holds for the field is called the conformal weight of V α (z) (while ∆ is also used for the Laplacians, our use of this notation should not yield any ambiguity). Since the BPZ differential equations are the quantum analogues of the classical equation (1.10), one should recover (1.10) by taking the semiclassical limit γ → 0 (recall that α k = χ k γ and µ = Λ γ 2 for fixed Λ and χ k ). Indeed, exploiting (1.19) and the convergence γ 2 ∆ α k → γ→0 χ k − χ 2 k /4 one gets (8) The argument of Polyakov and Zamolodchikov was in fact based on the stressenergy tensor of LCFT but this is a minor point. asymptotically for small γ that for some constant C > 0 This leads to the desired relation (1.11): this heuristic derivation can be made rigorous and is the content of Corollary 2.5. The main reason why the above derivation is not an immediate consequence of (1.19) is for regularity reasons; more specifically, one must justify that one can differentiate equivalence (1.19). One could also along the same lines take the semiclassical limit of the dual BPZ equation established for the field V − 2 γ (z) in [9] and recover rigorously a Hamilton-Jacobi type equation for S (χ l ,z l ) (φ * ) discovered recently in the physics literature in [12]. However, to keep the paper concise, we will not expand on this here.
Finally, we conclude this introduction by reminding that similar semiclassical results have also been obtained recently in the context of SLE. More precisely, in a series of works [1,18,35] (see the review [36]), the authors have derived large deviation results for SLE κ curves when κ goes to 0 or in the dual case where κ goes to infinity; on the CFT side, this corresponds to taking the limit where the central charge of the system tends to −∞. The main conceptual difference with the present work is that their results are expressed in terms of curves whereas in LCFT one works with fields. It would be interesting to relate both approaches (perhaps in the spirit of [34]). Now, we proceed with the statement of the main results of this paper as well as related open problems. In order to do so, we first recall the probabilistic definition of LCFT.

Background and notations
In this section, we recall the precise definition of the Liouville action and LCFT as given in [3].

Convention and notations
In what follows, in addition to the complex variable z, we will also consider variables x, y in C and for integer n 3 variables z 1 , . . . , z n which also belong to C.
The variables x, y (and sometimes z) will typically be variables of integration: we will denote by d 2 x and d 2 y (and d 2 z) the corresponding Lebesgue measure on C (seen as R 2 ). We will also denote | · | the norm in C of the standard Euclidean (flat) metric and for all r > 0 we will denote by B(x, r) the Euclidean ball of center x and radius r.

LCFT on C
To define the measure (1.18) it is natural to start with the quadratic part of the action functional (1.17) which naturally gives rise to a Gaussian measure, the Gaussian Free Field (GFF) (we refer to [5,Section 4] or [28] for an introduction to the topic). As is well known the GFF on the plane is defined modulo a constant but in LCFT this constant has to be included as an integration variable in the measure (1.18). The way to proceed is to replace ϕ in (1.17) by c + X where c ∈ R is integrated w.r.t. to the Lebesgue measure and X is the Gaussian Free Field on C centered with respect to the round metric, i.e. which satisfies C X(x)g(x)d 2 x = 0 for g(x) = 4 (1+|x| 2 ) 2 . The covariance of X (9) is given explicitly for x, y ∈ C by where κ := ln 2 − 1 2 . (9) The field X was denoted Xg in the article [3] or the lecture notes [26].

Gaussian multiplicative chaos
The field X is distribution valued and to define its exponential a renormalization procedure is needed. We will work with a mollified regularization of the GFF, namely uniformly on C where the constant a depends on the regularization function ρ. Define the measure where we have used the Wick notation for a centered Gaussian random variable :e Z : = e Z− 1 2 E[Z 2 ] (see Section 3.1). While the factor e γ 2 2 κ plays no role, this normalization of M γ, (d 2 x) has been chosen to match the standards of the physics literature. For γ ∈ [0, 2), this sequence of measures converges in probability in the sense of weak convergence of measures This limiting measure is non trivial and is (up to the multiplicative constant e γ 2 2 κ ) Gaussian multiplicative chaos (GMC) of the field X with respect to the measure g(x)d 2 x (see Berestycki's paper [2] for an elementary approach and references).
The theory of GMC also enables to make sense of measures on C defined as the limit of :e αX (x) : d 2 x. (2.5) when |α| < 2. We will denote these measures :e αX(x) : d 2 x (these GMC measures will appear when we integrate the fields V α ; in particular, we will be interested in the case α = − γ 2 ).

Liouville measure
The Liouville measure e −S(ϕ,g) Dϕ with S given by (1.17) is now defined as follows. Since R g = 2 and C X(x)g(x)d 2 x = 0 the linear term becomes C (c + X(x))g(x)d 2 x = 4πc. This leads to the definition (recall that µ > 0 is a fixed parameter) ν(dX, dc) := e −2Qc e −µe γc Mγ (C) P(dX) dc. (2.6) Here P(dX) denotes the Gaussian Free Field probability measure on H −1 ( C). Note that the random variable M γ (C) is almost surely finite and that This implies that the total mass of the measure ν is infinite since for M γ (C) < ∞ the c-integral diverges at −∞. While we have formally defined ν as a measure on H −1 ( C) × R, we are solely interested in the behavior of the Liouville field, defined by Note that by (2.2) (recalling the definition (2.3)), the term appearing in the exponential in (2.6) can be expressed as for A = ae κ and φ := ρ * φ being the smoothened version of φ. We denote averages with respect to ν by · γ,µ .

Liouville correlation functions
The vertex operators V α (z) = e αφ(z) need to be regularized as well. Equation (2.3) and (2.8) provide several ways of regularizing which end up to be equivalent when considering limits. For vertex operators, we introduce for α ∈ R and z ∈ C V α, (z) := e αc e α 2 κ 2 :e αX (z) : g(z) ∆α . (2.9) where recall that ∆ α = α 2 (Q − α 2 ). Let us fix z 1 , . . . , z n , n distincts points in C and associated weights α 1 , . . . , α n in R. Here and below we use the notation · γ,µ, for the regularized Liouville measure where in (2.6) we (2.10) exists and is finite if and only if n k=1 α k > 2Q. Moreover, under this condition, the limit is non zero if and only if α k < Q for all k. The conditions n k=1 α k > 2Q, ∀ k, α k < Q (2.11) (10) The global constant 4e −2κQ 2 which depends on γ plays no role but it is included to match with the standard physics literature which is based on the celebrated DOZZ formula. This constant was not included in the definitions in [3] or [26].
are the Seiberg bounds originally introduced in [27]. These two conditions are the quantum equivalent of those presented in (1.1). This is really transparent in the semiclassical regime where we set α k = χ k γ for fixed χ k 's and γ goes to 0; taking the limit in (2.11), one gets χ k 2 and n k=1 χ k 4. With the exception of Section 2.3 on open problems, we assume throughout this paper that the Seiberg bounds (2.11) are satisfied and (without entailing any restriction since V 0 (z) = 1) that α k = 0 for all k. Note that this also implies that n 3.

Reduction to Gaussian Multiplicative Chaos.
In order to keep this work as self contained as possible, we remind the basics of the construction of the Liouville correlations. The main idea is that one can express these correlations as functions of GMC measures with log singularities. As a first step using the explicit expression (2.9), and the change of variable a = γc + log M γ,ε (C) (11) we can factorize the respective role of X and the constant. Setting this yields (see [3] for more details): . Now using Girsanov's theorem (see [3]) we may trade :e α k X (z k ) : in the expectation for a shift of X. Then setting Here a is a variable and should not be confused with the constant a in (2.2).
we end up with the expression Thus, the Liouville correlations can be expressed in terms of the negative moments of the random variable Z 0 . In particular, the Seiberg bounds α k < Q for all k are the condition of integrability of e γ n k=1 α k G(z k ,x) against the chaos measure M γ (d 2 x) (see [3]). We recall the following result on the BPZ equations proved in [9]: Similarly to the correlation n k=1 V α k (z k ) γ,µ , we can also define a limit for the sequence of probability measures induced by the multiplication by vertex operators. Setting we have (in the topology of weak convergence of measures on Under the probability measure P µ,(α k ,z k ) (with expectation denoted by E µ,(α k ,z k ) ), the distribution of the Liouville field (2.7) is given by (for any continuous bounded function F : µ s e as e −µe a Γ(s) da.
(2. 19) where recall that in this expression s = n k=1 α k −2Q γ . This formula is obtained using the same change of variables performed in (2.13).

The semiclassical limit: statement of the main results
In this section, we state the main results of the paper which are derived in the semiclassical regime. In this regime we fix the values of (χ k ) n k=1 satisfying (1.1) (recall that this implies n 3) and Λ > 0 and set and we let γ tend to 0.

The semiclassical limit of the correlation functions
We introduce the following constant (depending on the χ k ) Our first achievement is to obtain sharp asymptotics for the correlation function (we direct the reader to Section 3.1 for details about the Wick notation :X 2 :. Let us just mention that :X 2 (x): is the limit of X 2 (x) − E[X 2 (x)] in an appropriate space as goes to 0). Recalling that φ * is the solution of (1.2)-(1.3) we set Notice that µ * is a probability measure on C thanks to (1.5). Recall also that .

Convergence of the Liouville field
Recall the distribution P µ,(α k ,z k ) of the Liouville field defined in Equation (2.19).
We have the following result: where the distribution of X m is absolutely continuous with respect to that of the original GFF X with density given by

Large deviations of the Liouville field
Now, we can state the following large deviation principal for the field γφ: -In the regime given by (2.20), the field γφ satisfies a large deviation principle on Θ (χ k ,z k ) with rate γ −2 and good rate function (2.23) -1046 -

A probabilistic proof of the Takhtajan-Zograf theorem on the accessory parameters
As a corollary of our techniques, we obtain a new (probabilistic) proof of relation (1.11) (previously proved in [30]):

Open problems
The semiclassical limit in the nonnegative curvature case The negative curvature condition n k=1 χ k > 4 is necessary to properly define our Liouville measure. As can be seen in Equation (2.19) the integral in a diverges if the quantum analogue of this condition (namely is not satisfied. However, we can bypass this obstruction by considering the measure conditioned on a fixed value of a (without loss of generality we can consider a = 0) Considering this expression is motivated by considering the law of the random measure on C defined by e γφ(x) d 2 x under the probability law (2.19). It is then easy to check that the total mass of this random measure has law Γ(s, µ) (i.e. with density proportional to µ −s A s−1 e −µA ), and that the law of the Liouville field, conditionally on the total mass of e γφ(x) d 2 x being 1, is described by (2.24). In order to state a conjecture in the unit volume setting, we need to find a natural variational problem associated to it. In this section, we assume the following conditions which were introduced by Troyanov [33] hold It is rather easy to check that condition (2.25) implies n 3. The reason for the above condition will become clear shortly; let us just notice that condition (2.25) extends condition (1.1). Consider φ ∈ Θ (χ k ,z k ) and its canonical decomposition where h ∈ H 1 ( C). Normalizing e φ to have unit volume amounts to replacing φ by The h satisfying such a relation is unique if we require h to be of vanish- where h satisfies the following equation is the Laplacian in the round metric g. Now, considering the decomposition (2.26) of the shifted version of φ corresponding to unit volume, we obtain that φ − ln( Note that such solutions to (2.28) in H 1 (C) can be obtained as critical points of the following action (2.29) Therefore, on the quantum level, it is natural to consider the image under transformation T (2.27) of the measure formally defined on H 1 (C) by This is precisely what is achieved by conditioning the Liouville field defined by (2.19) to have volume 1 and which leads to formula (2.24). For (2.24) to be well defined we only need to require E Z − k α k −2Q/γ 0 < ∞ and this is equivalent to the following bounds which are the quantum analogues of Troyanov's condition (2.25) In conclusion, when our parameters χ k satisfy n k=1 χ k 4 and provided that (2.25) is satisfied, the unit volume framework enables to investigate the semiclassical asymptotic with α k = χ k γ with χ k fixed and γ going to 0. Unfortunately, there are some technical obstructions for our proof to cover also the positive curvature case n k=1 χ k < 4. Indeed we would need to extend Lemma 4.3 below with α < 0 and this is currently out of reach with our method. Moreover, it could be the case that (2.29) admits several critical points in H 1 (C) and therefore equation (2.28) has several solutions; nonetheless, the work of Luo-Tian [13] ensures unicity in the case where χ k > 0 for all k (notice that this condition is automatically satisfied for n = 3) and therefore (2.29) admits a unique minimum in that case. It seems reasonable to conjecture that the unit volume Liouville field will converge to the unique minimum of (2.29) when it exists (12) and that a large deviation principle will hold with the (non convex) functional (2.29). Finally let us mention the works of Eremenko [7] and Mondello-Panov [15,16] where the authors construct solutions to (2.28) when condition (2.25) is not necessarily satisfied. However, there exists presently no quantum analogue of these constructions.

The semiclassical limit with two conical singularities
Another case where classical and quantum Liouville theory can be constructed without the negative curvature assumption is the case of metrics with two conical singularities of same weight. Indeed, one can also construct metrics with positive curvature and two conical singularities at 0 and ∞ with weight χ ∈ [0, 2). If the metric has unit volume than all the solutions are given by 2 − χ 2π λ 2 |λz| −χ (1 + |λz| 2−χ ) 2 |dz| 2 (2.31) (12) To the best of our knowledge, this point is not known and could be false in full generality assuming only (2.25).
where λ > 0. In a recent paper, Duplantier-Miller-Sheffield [6] introduced the quantum analogue of these measures (more specifically, they introduced the quantum analogue of the round metric which corresponds to χ = 0 and to α = γ in the following discussion). More precisely, they introduced an equivalence classe of random measures (defined up to dilations and rotations) with two marked points 0 and ∞. The random measures are defined on the cylinder R×[0, 2π] and we will identify the cylinder with the Riemann sphere via the conformal mapping z → e −z . If α ∈ ( γ 2 , Q) then we introduce where B α s , B α s are two independent Brownian motions with negative drift α − Q and conditioned to stay negative. Let Y be a log-correlated Gaussian field with covariance The (unit area or volume) α-quantum sphere is the unit volume random measure µ(dsdθ) defined on the cylinder R × [0, 2π] by and R(α) is the Liouville reflection coefficient defined by ]. For α = χ γ with χ ∈ (0, 2) fixed, we conjecture that the α-quantum sphere (mapped back to the Riemann sphere) converges (as γ → 0) to the positive curvature metric given by (2.31) for some λ > 0.

Organization of the paper
The rest of the paper is organized as follows. In the next section, we introduce general tools and notations on Gaussian variables; we will also give an alternative formula for the Liouville action S (χ k ,z k ) which is more adapted to our framework. In Section 4, we will state and prove Proposition 4.1 which is the key result of the paper; from Proposition 4.1, we will deduce in the remainder of Section 4 all the main results of the paper (which were stated in Section 2.2). In Section 5, we prove technical results which are used in the proof of Proposition 4.1. Finally, in the Appendix, we gather convexity considerations and general large deviation type results as well as the proof of the alternative formula for S (χ k ,z k ) .

Technical preliminaries
Let us introduce in this section a few technical tools which we use to prove our main results. This includes some basic notions concerning the Wick notation which we have used in a couple of equations in the previous Section, as well as some classical results concerning Gaussian processes.

Wick Notation
If Z is a Gaussian variable with mean zero and variance σ 2 , its Wick n-th power (n ∈ N) is defined by where H n is the n-th Hermite Polynomial. This definition is designed to makes the Wick monomials orthogonal to one another. More precisely if (Z, Y ) is a Gaussian vector we have The Wick exponential is defined formally as the result of the following expansion in Wick powers In the present paper, we mostly use the notation C :Y (x) 2 :µ(d 2 x) and C :e γY (x) :µ(d 2 x) for Gaussian fields defined on C and µ( While these integrals makes sense when (Y (x)) x∈C is a field with uniformly bounded covariance, some additional care is needed when we use the notation for distributional fields such as the GFF.

Wick Notation for Gaussian Fields
Let us consider Y a Gaussian field on C (or a subset) whose covariance satisfies Consider a sequence (Y n ) n 1 of Gaussian fields defined on the same space as Y and such that the full process [(Y n ), Y ] is Gaussian. Assume this sequence has bounded covariance and converges to Y in the following sense where the first inequality has to be satisfied for an arbitrary constant C > 0 that does not depend on n, and the convergence in the second line is in probability. An example of sequence satisfying these conditions is a convolution sequence as the one described in Section 2.1. It can be checked via elementary computations that for any fixed k the sequence is Cauchy in L 2 . We can thus define C :Y k (x):µ(d 2 x) as the limiting random variable which does not depend on the sequence (Y n ) n 1 .
The distribution :e γY : can be defined using the procedure described in Equations (2.2)-(2.4) (and detailed e.g. in [2]) as soon as γ < 2. Let us however provide a concise and self-contained argument which asserts the existence of C :e γY (x) : µ(d 2 x), as soon as γ 2 (1 + η −1 ) < 2. It can be checked that if Y n satisfies (3.6) the sequence C :e γYn(x) : µ(d 2 x) is Cauchy in L 2 provided that Hölder's inequality and (3.4) guarantees that the above holds as soon as

Girsanov/Cameron Martin Formula
The formula states how the distribution of elements of a Gaussian Hilbert space are modified by an exponential tilt of a random variable. In our context -1052 -it says that if Y is a centered Gaussian field with covariance funtion K( · , · ) with displays a logarithmic divergence (similar to (3.5)) then we have for any bounded continuous fonction on H −1 (C) and any signed measure µ such that The formula is easily checked for finite dimensional marginals and then extended by continuity. We are going to apply this formula also to :e γY : which is not a continuous fonction of Y . However :e γYε : is, and using the limiting procedure (2.8) we can deduce from the above that for every F continuous on H −1 (C) × R, γ < 2 and ν with density w.r.t. Lebesgue and finite total mass.

Positive association for positively correlated fields
A classical result of Gaussian analysis [21] states that if (X i ) i∈I (I finite) is a Gaussian vector such that E[X i X j ] 0 for all i, j 1 then for any pair of square integrable functions f, g : R I → R which are non-decreasing in all |I| variables we have (3.10) In order to apply this inequality to our fields which are indexed by C and defined in a space of distribution we simply apply a limiting procedure.

White Noise Decomposition
While it is a priori possible to write a proof of our results by working directly on the Riemann Sphere C, it turns out to be more convenient for notation to work with a field defined in the ball B(0, 1) := {z : |z| 1}. Instead of working directly with the restriction of X on B(0, 1) we are going to look at a randomly shifted version of it which we denote by X that possesses the convenient feature of having an explicit white noise decomposition for which correlations are positive on all scales, which is helpful in view of using positive association.
As an intermediate step we introduce X the GFF on the plane with average zero on the circle of radius 1. It can be obtained by considering the limit X − lim →0 (2π) −1 2π 0 X (θ)dθ where X is the mollified version of X considered in Section 2.1. As C X(x)g(x)d 2 x = 0 we also have The covariance of X in the ball B(0, 1) is given by E[X(x)X(y)] = ln 1 |y−x| as can be checked by a simple computation of covariances. Now to obtain a positively correlated field, we set X := X + √ ln 2 Y where Y is an independent standard Gaussian variable. Equation (3.11) is also satisfied with X replaced by X and the covariance of this last field satisfies [24, Example 2.3] Instead of using a convolution to approximate X by a smoothened field, we construct it as a limit (when t tends to infinity) of functional fields indexed by t 0. We let ( X t (x)) x∈B(0,1),t 0 be a bivariate field of covariance where Q u (x, y) := (1 − e u |x − y|/2) + .
(3.13) Note that Q u (x, y) is a positive definite function [17]. We have in particular that, letting K t (x, y) := t 0 Q u (x, y) + 2Q 0 (x, y) denote the covariance function of the field X t , there exists a constant C such that for all x, y ∈ B(0, 1), t 0 we have

The centered Liouville action S L,(χ k ,z k )
In what follows, it will be convenient to introduce the notation w(z) = e n k=1 χ k G(z k ,z) . (3.14) Note that in our setup, the field γφ displays logarithmic singularities at (z k ), cf. (2.19), and these singularities persist in the semiclassical limit. As it is easier to work with a field with no such singularities, we replace φ by a -1054 -more regular function h as done in Section 2.3 above. We introduce thus the centered Liouville action S L,(χ k ,z k ) on H 1 (C) which roughly corresponds to rewriting S (χ k ,z k ) (φ) as a function of h = φ − ln(gw). It admits the following expression One can extend S L,(χ k ,z k ) to H −1 ( C) by setting it to be equal to infinity outside H 1 ( C): the extension is convex and a good rate function in the terminology of large deviation theory. The following claim (proved in Appendix A.1) motivates our definition: where C ((χ k )) is defined by (2.21), and

Introducing the statement
The goal of this Section is to reduce the proof of our three main results Proposition 2.2, Theorem 2.3 and Proposition 2.4 to a general statement. Let us recall that we always assume that α k = χ k γ −1 , µ = Λγ −2 and that A statement concerning large deviations can be obtained by studying the asymptotic behavior of the Laplace transform of the field which is given by for appropriate ψ. On the other hand, to obtain results concerning the limiting law of φ−γ −1 φ * , we need to compute the following limit for any bounded continuous function on Hence we can prove both statements if we obtain sharp asymptotics for where φ * ,ψ is a function to be determined but which coincides with φ * when ψ ≡ 0. Using our factorisation of the measure into X and a given by (2.19) we can in fact compute a separate asymptotic for a and X.
Before discussing things in more details let us introduce further notations used in this section. Considering ψ a smooth function on C, we set (4.4) and set Note that µ ψ (d 2 x) has volume 1. We set κ ψ := ln C e h ψ (y) w(y)g(y)d 2 y and of course from above we have h ψ = h ψ − κ ψ .
The asymptotic for the integral in a is a standard computation. Using the following variant of Stirling's formula -1056 - The semiclassical limit of Liouville conformal field theory (1)). (4.6) Moreover (and this is only of interest when ψ = 0), we have for any bounded This is sufficient to show that, after recentering, the variable γ −1 a converges to a Gaussian of variance c −1 ψ . The part concerning X is the main probabilistic estimate of the paper. Given F a continuous bounded function in H −1 (C)×R we want to determine the precise asymptotic of the following Laplace functional Proposition 4.1.
-If ψ is such that c ψ > 0, then we have in the small γ asymptotics Let us also mention separately a consequence of the proof of the above proposition. Here H(0) and c 0 denote the respective value of H and c ψ for ψ ≡ 0. We get the following lemma (we will derive a sketch of the proof) Now we can combine Proposition 4.1 with Equation (4.7) to prove our main results.

Proof of Proposition 2.2 and Theorem 2.3
In what follows and in order to have more concise formulae, we will sometimes write dg in place of g(x)d 2 x or dµ * in place of µ * (d 2 x), etc. Also we will simply write without indicating the set on which we integrate: this should be clear from the context.
To obtain the limit of correlations, we use Proposition 4.1 and Equation (4.7) for ψ ≡ 0 and F ≡ 1. Notice that µ 0 := µ ψ=0 defined by (4.5) coincides with µ * defined by (2.22) since by definition h 0 + log wg is the solution of (1.2)+(1.3) up to constant, and c 0 := n k=1 χ k − 4. We obtain, recalling (2.15) and (3.17) In the appendix A.2, we prove that the quantity (see (A.4)) can be related to the quantity min h S L,(χ k ,z k ) (h) by the relation This is the content of Proposition A.4. Consequently Lemma (3.1) allows us express J 0 (h 0 ) as By plugging these relations into (4.9), we get the statement of Proposition 2.2.

Proof of Proposition 2.4
According to relation (3.16), the claim is equivalent to proving that the satisfies a large deviation principle on H −1 ( C) with good rate function the centered Liouville action (shifted by its minimum) S L,(χ k ,z k ) ( · )−S L,(χ k ,z k ) (h * ). Here we adopt the notations of Section A.3 in the appendix. With these notations, Proposition 4.1 implies straightforwardly for all ψ that In the language of large deviation theory, f(ψ) − f(0) is the limit of the logarithmic moment generating function of γX (under P Λ γ 2 ,( χ k γ ,z k ) ). In Proposition 4.3, we prove that the distribution of γX is exponentially tight under P Λ . Thanks to Lemma A.5 on the exposed points of S L,(χ k ,z k ) ( · ), we can conclude by using Baldi's theorem in [4, Theorem 4.5.20 p. 157].

Proof of relation (1.11)
Here we prove relation (1.11) by using the BPZ differential equations established in [9]. We have the following BPZ differential equation where formally V α (z) = e α(X(z)+ Q 2 ln g(z)+c) (and rigorously it is the limit of the corresponding regularized quantity) and ∆ α = α 2 (Q − α 2 ). We write α k = χ k γ and set η > 0 small. We consider smooth functions u k with compact support in B(z k , η) and u a smooth function with compact support in C \ -1060 -B(z k , η). Using (2.15) and (2.19), we have for any smooth function f with compact support in C \ B(z k , η) and z k ∈ B(z k , η) the following identity (the definition of K is given by (2.16 Now using Lemma 4.2 there exists a constant C > 0 such that Applying (4.11) with f (z) = ∂ 2 zz u(z) and using the fact that (4.11) is uniform for z k ∈ B(z k , η) (and f has compact support in C \ B(z k , η)), we get by integration by parts We also get for all j that (up to the same constant as in the former equivalents) One can then conclude by taking the equivalent γ → 0 of the BPZ equation integrated with respect to u(z) n k=1 u k (z k )d 2 z n k=1 d 2 z k and then taking the limit η to 0.

Proof of Proposition 4.1
For simplicity we write the proof in the case F ≡ 1. We explain how to adapt the proof for general F in the end. Note that using our notation we have Z 0 = :e γX :wdg = ∫ e h ψ wdg ∫ :e γX :e −h ψ dµ ψ (4.12) and hence we can rewrite the quantity we wish to estimate in the following manner The first part of our proof consists in checking that the exponential tilt produced by the second exponential factor exactly cancels the e −h ψ present in the first exponent.
Then we need to check that after taking into account this exponential tilt, the integral converges. This can be achieved by showing convergence in probability of the integrand and uniform integrability. This is the content of the following proposition, whose proof is detailed in the next section. Let Y ψ denote the Gaussian variable present in the second exponential in (4.13) We have (4.16) Using the integral version of (4.4) we have Hence we have E[Y ψ X(x)] = h ψ (x), and using integration by part and where in the last line we used the Cameron Martin formula (3.9) and By Proposition 4.3, the quantity in the integral is bounded in L 2 (as the product of two quantities which are bounded in L 4 ) and moreover it converges in probability when γ tends to zero to (∫ e h ψ wdg)e −c ψ ln : This is enough to conclude our proof. Let us now discuss the case where a functional F is added. The Cameron Martin shift in (4.18) has the effect of cancelling γ −1 h ψ in the first argument of F , it also transforms log Z 0 − κ ψ into log :e γX :dµ ψ . Using the boundedness in L 2 it is then sufficient to show that log :e γX :dµ ψ converges to Xdµ ψ in probability. This is a consequence of (4.15) applied to the measure µ ψ .

Proof of Lemma 4.2. -To obtain an asymptotic equivalent for
We repeat the computation above until (4.18) (the case ψ ≡ 0) and with an added prefactor. We obtain that the quantity we wish to estimate is equal to Now the inside of the expectation above converges in probability and is bounded in L 2 so the expectation converges to which yields the result since

Uniform integrability
In this section we always consider µ to be probability measure on C and denote by ρ its density with respect to our reference measure g(x)d 2 x. We assume that (3.4) holds for some η > 0.

Proof of Proposition 4.3
The proof of Proposition 4.3 requires a few technical estimates which we present now and prove at the end of the section. The first one allows us to assert that the second term in our chaos expansion is uniformly integrable. For practical reason, in the proof of (4.14) we wish to reduce our domain of integration to B(0, 1). This can be achieved by splitting the sphere in two and considering each half separately. Set q := |x| 1 ρ(x)g(x)d 2 x and let . (5.4) Now observe that the distribution X is invariant by the transformation x → (1/x), so the second term remains unchanged if we replace µ 2 by µ 3 , the image measure of µ 2 by the transformation x → (1/x). We have We can observe that the function (1 − q) −1 ρ(1/x)g(1/x) d 2 x |x| 4 satisfies (3.4), and hence both terms in the r.h.s. of (5.4) can be treated in the same manner. Also note that as X = X + Z (see the construction of Section 3.3) where Z is a Gaussian random variable (which is not independent of X), replacing X by X does alter the value of the function inside the expectation by a lot.
We introduce the event Our idea is that on A we can use Taylor expansion to get rid of :exp: and ln while the complement has such a small probability that a rough estimate will be sufficient. A simple application of Markov inequality implies (recall that Var( X t ) = t + 2) that We are going to prove that if γ is sufficiently small on the event A, we have ln :e γXt :dµ 1 − γ X t dµ 1 γ 2 2 : Using the formula e u 1 + u + u 2 2 + u 3 6 for u = γ X t − γ 2 2 (t + 2), we obtain that for some constant C > 0, for all γ 1 and t 1, we have as soon as Note that with our choice of t the last term is smaller than γ 2 /8 for small values of γ. Now, on the event A, using that : X 2 t : −(t + 2) almost surely, as a consequence of the event's definition, the missing parts in the integral are negligible and we have thus for γ sufficiently small :e γXt :dµ 1 Now the event A guarantees that the integral terms on the right hand side are at most of respective order t 2 γ and t 4 γ 2 . Using this information together with the inequality ln(1 + u) u − u 2 2 − |u| 3 which is valid when |u| is sufficiently small, we obtain (5.9). Now combining (5.7) and (5.9) we obtain that The first term can be controlled using (5.8)  and observing that the last term is smaller than γ 2 on B we have (in the second line we just use ab a 2 /2 The first term is bounded uniformly in γ > 0, cf. (5.14), while for the second one, it is sufficient to observe that U tγ = ( X − X tγ )dµ 1 is a Gaussian whose variance is small, the following being valid for some c > 0, as a consequence of (3.4) and Hölder inequality For the other part we have using Hölder's inequality The first term can be controlled by Lemma 5.2. As for the second one, its smallness is a consequence of the following result proved in [11] under slightly different assumptions for µ. The proof adapts however to this context, we replicate it in below for the sake of completeness Of course, as e −tγ /4 γ 3 , this completes the proof.

Proof of auxiliary Lemmas
Proof of Lemma 5.1. -Setting N t := |x| 1 X t dµ we have As E[N 2 t ] is uniformly bounded in t it is sufficient to control the first term in the r.h.s. . Let us set Fixing t 0 (its exact value which depends on α and ρ is to be chosen later), we assume that t > t 0 . Using orthogonality of the increments and the identity The first term is easily controlled since we have for some constant C(ρ) for every x ∈ B(0, 1) As for the two other terms, we rely on [8, Theorem 6.7] which states in particular that for some universal constant c 2 any square integrable variable Z [t0,t] which can be expressed as the L 2 limit of second degree polynomials in (X t (x)) t 0,x∈B(0,1) , we have Tedious but standard calculation allows to show that for some positive constant C (depending on the function ρ) These estimates are sufficient to show that Z 1 2 and Z 2 2 can be made arbitrarily large by choosing t 0 large.  [23]) and the use of Stirling's asymptotics for the Γ-function yields (in our range of parameters) The following holds The random function φ is almost surely differentiable and if X t := X − X t and K t (x, y) :=  Recall that the centered Liouville action is defined on H 1 ( C) by the following expression: where l((χ k , z k )) = − n k=1 χ k 1 − χ k 4 ln g(z k ) − 1 2 k =j χ k χ j G(z j , z k ) Now, we prove Lemma 3.1 on the link between S L,(χ k ,z k ) and S (χ k ,z k ) .

A.2. Existence of solutions to the Liouville equation
Here, we give a short proof of the existence and uniqueness to the equation (A.6).
Let ψ be some function defined on the Riemann sphere. We introduce the functional J ψ on functions h ∈ H 1 ( C) with vanishing mean on the sphere Recall that we have the following Moser-Trudinger inequality for all functions h ∈ H 1 ( C) with vanishing mean on the sphere (see [14] for example): Therefore, the functional J ψ is bounded from below if and only if c ψ > −4(1 ∧ inf i (1 − χ k /2)). In that case, the minimum solves the following equation: -1078 -Proposition A.1. -Assume c ψ > −4(1 ∧ inf i (1 − χ k /2)). Then the equation admits a unique solution h ψ with vanishing mean on the sphere.

A.3.1. General considerations
Recall that when f is a function taking values in ]−∞, ∞] on some Banach space B then we can define its lower semicontinuous enveloppe f sc by the following limit This lower semicontinuous enveloppe satisfies the following properties: (1) If f is convex then so is f sc .
(2) For all λ there exists some sequence (λ n ) n 1 such that λ n converges to λ and f(λ n ) converges to f sc (λ).
By item 2 above one can easily see that f * = (f sc ) * . Finally we recall the Fenchel-Moreau theorem f * * = f sc .
Hence we deduce the following lemma which we will need in the following: Proof. -We have I * = f sc and therefore by the Moreau-Legendre theorem we get I = I * * = (f sc ) * = f * . and therefore also S L,(χ k ,z k ) = (f sc ) * = f * .
Proof. -The supremum of the above proposition is the same (up to the −l (χ k ,z k ) term) as the supremum of the following function T (c, h) defined for c ∈ R and h ∈ H 1 ( C) by If c ψ < 0, one can fix h and take the limit c → −∞ which shows that the supremum of T (c, h) is infinity. Therefore, we suppose that c ψ 0. For fixed h, c → T (c, h) is maximal for Λe c C w(z)e h(z) g(z)d 2 z = c ψ . This yields Now, one can conclude by optimizing this expression on h.
Now, we introduce the following set called exposed points in the language of large deviation theory (see [4,  In words, F is the set of h ∈ H −1 ( C) with the property: there exists ψ ∈ H 1 ( C) such that u → ψ(z)u(z)g(z)d 2 z − S L,(χ k ,z k ) (u) admits a unique maximum at u = h. The corresponding ψ is called an exposed hyperplane. The following lemma shows that there are many exposed points: Lemma A.5. -All smooth h ∈ H −1 ( C) are exposed points for S L,(χ k ,z k ) , i.e. if h is smooth then there exists some ψ in H 1 ( C) such that the function u → ψ(z)u(z)g(z)d 2 z − S L,(χ k ,z k ) (u) admits a unique supremum at u = h. The corresponding exposed hyperplane ψ satisfies the following property: there exists some t > 1 such that f(tψ) < ∞. Moreover, one has for all open set G ⊂ H −1 ( C) inf h∈G∩F S L,(χ k ,z k ) (h) = inf h∈G S L,(χ k ,z k ) (h).
Proof. -Fix some smooth h. We can write this element as h 0 + β where h 0 ∈ H 1 ( C) has vanishing mean on the sphere. We set α + k χ k − 4 = Λe β C w(z)e h0(z) g(z)d 2 z and then .