Probabilistic construction of simply-laced Toda conformal field theories
aa r X i v : . [ m a t h - ph ] F e b PROBABILISTIC CONSTRUCTION OF SIMPLY-LACED TODACONFORMAL FIELD THEORIES
BAPTISTE CERCL´E, R´EMI RHODES, AND VINCENT VARGAS
Abstract.
Following the 1984 seminal work of Belavin, Polyakov and Zamolodchikov ontwo-dimensional conformal field theories, Toda conformal field theories were introduced inthe physics literature as a family of two-dimensional conformal field theories that enjoy,in addition to conformal symmetry, an extended level of symmetry usually referred toas
W-symmetry or higher-spin symmetry. More precisely Toda conformal field theoriesprovide a natural way to associate to a finite-dimensional simple and simply-laced Liealgebra a conformal field theory for which the algebra of symmetry contains the Virasoroalgebra. In this document we use the path integral formulation of these models to providea rigorous mathematical construction of simply-laced Toda conformal field theories basedon probability theory. By doing so we recover expected properties of the theory such as theWeyl anomaly formula with respect to the change of background metric by a conformalfactor and the existence of Seiberg bounds for the correlation functions. Introduction
Toda Conformal Field Theories in the physics literature.
In 1981, Polyakovpresented in a pioneer work [32] a canonical way of defining the notion of random surface,usually called Liouville conformal field theory (LCFT hereafter), which is now considered tobe an essential feature in the understanding of non-critical string theory and two-dimensionalquantum gravity. A few years later Belavin, Polyakov and Zamolodchikov (BPZ) presented intheir 1984 seminal work [2] a systematic procedure to solve models which like LCFT possesscertain conformal symmetries, now referred to as two-dimensional conformal field theories (CFTs in the sequel). The main input of their method was to exploit the constraints imposedby conformal symmetry through the study of the algebra of its generators, the
Virasoroalgebra , which in turn completely determines (up to the so-called structure constants) themain quantities of interest, namely the correlation functions of certain special operators,thanks to a recursive procedure dubbed the conformal bootstrap .A natural question which appeared shortly after these developments was: what happenswhen the algebra of symmetry stricly contains the Virasoro algebra? In other words, dothe same techniques apply when Virasoro symmetry is extended to feature an additionallevel of symmetry? Certain extensions of the Virasoro algebra, called W -algebras, have beenfirst studied by Zamolodchikov in his work [36] where was presented the notion of higher-spin symmetry, and following this work two-dimensional CFTs having this extended levelof symmetry appeared in the physics literature in [11, 13]. In addition to being an objectof interest in itself, the study of a CFT having what is called W -symmetry is crucial inthe understanding of W -strings, W -gravity theories or certain statistical physics systems(the articles [12] and [23] provide explicit instances of such systems) and can be applied to the understanding of some Wess-Zumino-Novikov-Witten models (such a link is explainedin [14]). From the representation theory viewpoint the study of W-algebras has proved tobe a seminal topic with numerous applications ranging from integrable hierarchies to thegeometric Langlands program (see [1] and the references therein).Toda Conformal Field Theories (TCFTs hereafter) may be thought of as realizations ofthese algebras of symmetry since they are assumed to provide highest-weight representationsof W -algebras. In this context, the primary fields, defined as random fields on a Riemannsurface Σ with special conformal covariance properties, are called Vertex Operators andtheir correlations are defined as an average of the product of the fields (taken at differentpoints of the surface Σ) with respect to the law of a random map called the Toda field . Thisconstruction can be made somehow explicit thanks to the fundamental fact that TCFTsadmit a path integral formulation based on the following Lagrangian(1.1) S T, g ( ϕ, g ) := 14 π Z Σ (cid:16) h ∂ g ϕ ( x ) , ∂ g ϕ ( x ) i g + R g h Q, ϕ ( x ) i + 4 π r X i =1 µ i e γ h e i ,ϕ ( x ) i (cid:17) v g ( dx ) , where the Toda field ϕ is a map Σ → h and: • g is a Riemannian metric on a Riemann surface Σ with associated scalar curvature R g , gradient ∂ g and volume form v g , • h is the Cartan subalgebra of some finite-dimensional simple and simply-laced com-plex Lie algebra g , equipped with its standard scalar product h· , ·i and norm | · | (seeSubsection 2.1.2), • the e i , 1 i r are the simple roots of the Lie algebra g relative to h , • h· , ·i g is the scalar product associated to the tangent space of h -valued functionsdefined on Σ, • the constants µ i (1 i r ) are positive and are dubbed the cosmological constants, • γ > • Q is the h -valued background charge.In order to ensure conformal symmetry, the background charge is related to the couplingconstant via the relation Q := ( γ + γ ) ρ where ρ is the Weyl vector associated to the Cartansubalgebra h . Let us emphasize that one recovers LCFT when g is the Lie Algebra sl (of2 × r = 1) and that the conventionfor γ in this paper differs by a scaling factor of √ W -geometry , introduced by Gervais and Matsuo in[17]. Indeed, in the semi-classical limit γ → i , µ i = Λ γ with fixed Λ > γ ) converges towards the solution u : Σ → h (provided that it exists and is unique) of the Toda equation :(1.2) 2∆ g u = R g ρ + 4 π Λ r X i =1 e i e h e i ,u i on Σ . This means that, in the quantum theory, the Toda field (rescaled by γ ) will have a tendencyto remain close to the (classical) solution of this h -valued equation. It is explained in [17] IMPLY-LACED TODA CONFORMAL FIELD THEORIES 3 that when Σ is the two-dimensional sphere S , sl n Toda equations (where one adds ap-propriate conical singularities) can in some sense be interpreted as compatibility equationsfor a meromorphic embedding of a two-sphere into a complex projective plane C P n , andtherefore establishes a correspondence between solutions of the Toda equation (1.2) andcertain meromorphic embeddings from C P to C P n , a problem which somehow provides ageneralization of the celebrated uniformisation of Riemann surfaces .In the context of the quantum theory, the Toda field ϕ should be understood as a randommap from a Riemannian surface (Σ , g ) to h whose law can be defined via the expression(here F is some real-valued functional on a space of maps Σ → h ):(1.3) E gγ, µ [ F ( ϕ )] := h F i T,g h i T,g where the quantities that appear in this definition are defined by the (formal) path integral(1.4) h F i T,g := Z F ( X ) e − S T, g ( X,g ) DX and DX refers to a “uniform measure” on the space of square integrable h -valued maps de-fined on Σ; the normalization factor h i T,g corresponds to the partition function R e − S T, g ( X,g ) DX of TCFTs. In the sequel, we will focus on the case where Σ is the Riemann sphere S (thoughthe present framework can be extended to other topologies). In this case, the partition func-tion is not defined ( i.e. infinite) and one can only define quantities with insertion of a certainnumber of Vertex Operators, whose parameters obey the Seiberg bounds [30] in order to en-sure existence of the corresponding correlation function (see next subsection).1.2.
A probabilistic construction.
Though an algebraic approach to CFTs was devel-opped shortly after the BPZ paper (see the notion of Vertex Operator Algebra [3, 15]), aprobabilistic approach to conformal invariance was only developed recently following the in-troduction by Schramm [34] of random curves, called Schramm-Loewner Evolutions (SLEs),which describe (conjecturally at least) the interfaces of critical models of statistical physics(such as percolation or the Ising model). More recently, there has been a huge effort inprobability theory to make sense of LCFT within the realm of random conformal geometryand the scaling limit of random planar maps (see [28, 29, 20, 9, 6, 8]). Another approach,based on the path integral formulation of LCFT in the physics literature, was developedin [4, 5, 21, 19] to give a rigorous probabilistic construction of the correlation functionsof LCFT. This construction initiated a program [25] to lay the mathematical foundationsof the conformal bootstrap procedure envisioned in physics by BPZ, namely that one canexpress the LCFT correlation functions in terms of representation theoretical special func-tions. The building blocks are an explicit formula for the three point correlation functions(or equivalently the structure constants) and a recursive procedure for the higher correla-tions. The explicit formula for the three-point structure constants discovered in the physicsliterature, the celebrated DOZZ formula, was recovered probabilistically in [26] and a prob-abilistic justification of the conformal bootstrap formalism for the higher order correlationswas provided recently in [18].Building on [4], our goal here is to provide a probabilistic definition of TCFTs in the casewhere we consider the underlying Lie algebra to be a finite-dimensional simple and simply-laced complex Lie algebra, and when the manifold on which the theory is constructed is
BAPTISTE CERCL´E, R´EMI RHODES, AND VINCENT VARGAS the (Riemann) sphere. To do so we follow the ideas developed in the case of LCFT (whichcorresponds to the sl case) in [4] and interpret the path integral formulation of the theory asa formal way of defining a measure on some functional space. More precisely we interpret themapping defined via the path integral (1.4) as a measure F
7→ h F i T,g on the Sobolev spacewith negative index H − ( S → h , g ) (which we define in (2.4)); in order to construct thismeasure we introduce a probabilistic framework which involves two objects: the GaussianFree Field (GFF) and the exponential of the GFF called Gaussian Multiplicative Chaos(GMC).The presence of the GFF is related to the presence of the square gradient term in theToda field action and has proven to be particularly relevant in the context of constructiveconformal field theory. But as opposed to LCFT where only one GFF is involved in theconstruction, in TCFTs we have to consider several GFFs that are coupled in a way thatis prescribed by the underlying structure of the Lie algebra. For this coupling to be math-ematically meaningful we need to consider simple Lie algebras that are simply-laced, thatis for which the underlying Cartan matrix (which is used as a covariance matrix in ourconstruction) is symmetric. Under this assumption, the fields of TCFTs are well-definedbut non-regular since they exist only in the sense of Schwartz distributions; therefore theexponential terms that appear in (1.1) are not well-defined objects. However, GMC theoryprovides a way of making sense of these terms as (random) Radon measures.This interpretation allows to construct a regularized partition function by taking F = 1in (1.4); however and similarly to the existence of Seiberg bounds in LCFT [30] this par-tition function will not converge in relation with an obstruction of geometrical nature: theGauss-Bonnet theorem entails that classical Toda equations ( i.e the equations of motionassociated to this action) cannot admit solutions on the Riemann sphere. This difficulty canbe overcome by looking at special functionals F that admit Vertex Operators as factors;by adding these extra terms to the measure the partition function becomes the correlationfunction of Vertex Operators and is predicted to exist as long as some conditions on theseoperators are satisfied. To define these Vertex Operators one relies on a regularization of theToda field and introduces the regularized Vertex Operator V α,ε ( z ), which is expressed for z on the Riemann sphere in terms of the Toda field ϕ and a weight α ∈ h : up to constantterms, V α,ε ( z ) is defined as ε | α | / e h α,ϕ ε i where ϕ ε is the field ϕ smoothed up at scale ε (itsdefinition will be made precise in Subsection 2.2.3). Our main result provides a necessaryand sufficient condition that ensures the existence of the correlation functions (defined aslimits of h Q Nk =1 V α k ,ε ( z k ) i T,g when the cut-off ε is sent to 0). Moreover, the correlations areindeed conformally covariant as predicted by CFT: Theorem 1.1.
Let g be a finite-dimensional simple and simply-laced complex Lie algebraand assume that γ ∈ (0 , √ . If g is any Riemannian metric in the conformal class of thestandard round metric on the sphere ˆ g , let h Q Nk =1 V α k ,ε ( z k ) i T,g be the regularized correlationfunction of the g -Toda theory. Then:
1. (Seiberg bounds):
The limit h N Y k =1 V α k ( z k ) i T,g := lim ε → h N Y k =1 V α k ,ε ( z k ) i T,g
IMPLY-LACED TODA CONFORMAL FIELD THEORIES 5 exists and is non trivial if and only if the two following conditions hold for all i = 1 , . . . , r : • h N X k =1 α k − Q, ω i i > where the ( ω i ) i r are the fundamental weights of h , • for all k N , h α k , e i i < h Q, e i i = γ + γ .
2. (Conformal covariance):
For any M¨obius transform of the plane ψ h N Y k =1 V α k ( ψ ( z k )) i T,g = N Y k =1 | ψ ′ ( z k ) | − αk h N Y k =1 V α k ( z k ) i T,g . where the conformal weights are given by ∆ α j := h α j , Q − α j i .
3. (Weyl anomaly):
For appropriate ϕ (more precisely ϕ ∈ ¯ C ( R ) : see notations inSection 2) then h N Y k =1 V α k ( z k ) i T,e ϕ ˆ g = e c T π S L ( ϕ, ˆ g ) h N Y k =1 V α k ( z k ) i T, ˆ g where S L is the Liouville functional (with vanishing cosmological constant) S L ( ϕ, ˆ g ) := Z S (cid:0) | ∂ ˆ g ϕ | g + 2 R ˆ g ϕ (cid:1) d v ˆ g , and the central charge c T is given by c T = r + 6 | Q | . The value of the central charge can be described explicitly in terms of the coupling con-stant γ and the underlying Lie algebra. Indeed, by the classification of finite-dimensionaland simple complex Lie algebra we know that g , which is simply-laced, is necessarily isomor-phic to sl n for n > so n for n > E , E or E for which the central charge is explicit:see (2.18) or (3.6). Our main statement can be understood as a rigorous definition of thecorrelation functions of TCFTs, but also of the law of the Toda field ϕ (when we have fixedmarked points ( z , α ) = ( z k , α k ) k N that satisfy the Seiberg bounds) by setting(1.5) E ( z , α ) [ F ( ϕ )] := h F Q Ni =1 V α i ( z i ) i T,g h Q Ni =1 V α i ( z i ) i T,g where F is any bounded and continuous functional on H − ( S → h , g ) and h F Q Ni =1 V α i ( z i ) i T,g is the limit of h F Q Nk =1 V α k ,ε ( z k ) i T,g when the cut-off ε goes to 0 (this more general case canbe handled similarly to the case F = 1). Remark 1.2.
The above construction can be generalized when one considers as underlyingLie algebra any simply-laced, semisimple and complex Lie algebra. Indeed, from the clas-sification of semisimple Lie algebras, such a Lie algebra can be written as a direct sum of simple
Lie algebras g = ⊕ pk =1 g k . Moreover a general property of TCFTs (which can be de-rived from the form of the Toda field action) is that for A, B two semisimple and simply-lacedLie algebras (1.6) h N Y k =1 V ( α k ,β k ) ( z k ) i A ⊕ BT,g = h N Y k =1 V α k ( z k ) i AT,g h N Y k =1 V β k ( z k ) i BT,g , BAPTISTE CERCL´E, R´EMI RHODES, AND VINCENT VARGAS where with the notation hi g T,g we have stressed the dependence on the Lie Algebra g . Thisprovides a way of constructing correlation functions for general finite-dimensional simply-laced and semisimple complex Lie algebras. The latter equation also implies that the centralcharges add up, in the sense that (1.7) c T,A ⊕ B = c T,A + c T,B where here again the notation c T, g stresses the dependence on the Lie Algebra g . Background and notations
Some reminders on conformal geometry and Lie algebras.
Conformal geometry on the Riemann sphere.
The sphere S can be mapped by stere-ographic projection to the (compactified) plane ( i.e. the Riemann sphere) which we viewboth as R ∪ {∞} and C ∪ {∞} . We will work under this more convenient framework in thesequel. Metrics on the Riemann sphere.
We will consider differentiable conformal metrics on the two-dimensional sphere S ; theycan can be identified via stereographic projection with metrics on the plane of the form g = e ϕ ˆ g with ˆ g is the standard round metric(2.1) ˆ g := 4(1 + | x | ) | dx | , and ϕ ∈ ¯ C ( R ) where, for k >
0, ¯ C k ( R ) stands for the space of functions ϕ : R → R that are k -times differentiable with continuous derivatives as well as x ϕ (1 /x ) in aneighbourhood of x = 0. The reader may check that the metric ˆ g is the pushforward (viastereographic projection) of the standard metric on the Riemann sphere S . We will thuswork with such metrics g on the plane, for which we will denote by ∂ g the gradient, △ g the Laplace-Beltrami operator, R g = −△ g ln √ det g the Ricci scalar curvature and v g thevolume form. If u, v ∈ R , we denote by ( u, v ) g the inner product with respect to the metric g ( | · | g stands for the associated norm). When no index is given, this means that the objecthas to be understood in terms of the usual Euclidean metric on the plane ( i.e. ∂ , △ , R , v and( · , · )). Since the stereographic projection is an isometry, we already know that the sphericalmetric ˆ g is such that R ˆ g = 2 (its Gaussian curvature is 1) with total mass v ˆ g ( R ) = 4 π .More generally, two metrics g and g ′ will be said to be conformally equivalent when g = e ϕ g ′ for ϕ ∈ ¯ C ( R ). It is readily seen that these conditions imply that R R (cid:0) | ∂ g ′ ϕ | g ′ + 2 R g ′ ϕ (cid:1) d v g ′ < ∞ as soon as g ′ is in the conformal class of the spherical metric—that is when g ′ = e ϕ ˆ g where ϕ is as above. Furthermore, for ϕ ∈ ¯ C ( R ), the curvatures of two such metrics are relatedby the relation(2.2) R g = e − ϕ (cid:0) R g ′ − ∆ g ′ ϕ (cid:1) . IMPLY-LACED TODA CONFORMAL FIELD THEORIES 7
In what follows and for given metrics g and h ∈ ¯ C ( R ), we will denote by m g ( h ) themean value of h in the metric g , that is the quantity(2.3) m g ( h ) := 1v g ( R ) Z R h ( x ) v g ( dx )and work in the Sobolev space H ( R , g ), which is the closure of C ∞ c ( R ) with respect tothe Hilbert-norm(2.4) Z R h ( x ) v g ( dx ) + Z R | ∂ g h ( x ) | g v g ( dx ) . The standard dual of H ( R , g ) will be denoted H − ( R , g ). It may be useful to note thatthe Dirichlet energy is a conformal invariant, that is to say is independent of the metricwithin a given conformal class:(2.5) Z R | ∂ g ′ h ( x ) | g ′ v g ′ ( dx ) = Z R | ∂ g h ( x ) | g v g ( dx ) . Green kernels.
Given a metric g on the Riemann sphere that is conformally equivalentto the spherical metric ˆ g , we denote by G g the Green function of the problem △ g u = − π ( f − m g ( f )) on R , Z R u ( x ) v g ( dx ) = 0where f belongs to the space L ( R , g ) and u is in H ( R , g ). This means that the solution u can be expressed as(2.6) u = Z R G g ( · , x ) f ( x )v g ( dx ) =: G g f with m g ( G g ( x, · )) = 0 for all x ∈ R . The kernel G g has an explicit expression given by (see[4, Equation (2.9)])(2.7) G g ( x, y ) = ln 1 | x − y | − m g (cid:18) ln 1 | x − ·| (cid:19) − m g (cid:18) ln 1 | y − ·| (cid:19) + θ g where θ g := 1v g ( R ) Z R Z R ln 1 | x − y | v g ( dx )v g ( dy ) . For instance for the spherical metric this becomes(2.8) G ˆ g ( x, y ) = ln 1 | x − y | −
14 (ln ˆ g ( x ) + ln ˆ g ( y )) + ln 2 − . Another well-known property of these Green functions (see [4, Proposition 2.2] for in-stance) is that they are conformally covariant in the sense that:
Lemma 2.1 (Conformal covariance) . Let ψ be a M¨obius transform of the Riemann sphereand g be a Riemannian metric conformally equivalent to the spherical one. Then (2.9) G g ψ ( x, y ) = G g ( ψ ( x ) , ψ ( y )) where g ψ ( z ) = | ψ ′ ( z ) | g ( ψ ( z )) is the pullback of the metric g by ψ . BAPTISTE CERCL´E, R´EMI RHODES, AND VINCENT VARGAS
Again let us register what happens for the spherical metric:(2.10) G ˆ g ( ψ ( x ) , ψ ( y )) = G ˆ g ( x, y ) −
14 ( φ ( x ) + φ ( y ))where φ is such that e φ = ˆ g ψ ˆ g .2.1.2. Lie algebras and the Toda field action. . Finite-dimensional simple Lie algebras.
The special linear group SL n ( C ) = { A ∈ M n ( C ); det( A ) = 1 } can be endowed with the structure of a smooth manifold of (complex)dimension n −
1. When equipped with the group operation of matrix multiplication, itbecomes a Lie group. Its tangent space at identity sl n consists of all n × n complex matriceswith trace 0 and becomes a Lie algebra, called the special linear Lie algebra, when equippedwith the usual Lie bracket(2.11) [ A, B ] := AB − BA for A, B ∈ sl n . The adjoint action ad is the map ad A ( B ) := [ A, B ] for
A, B ∈ sl n . The sl n Lie algebra isone of the simplest instances of complex simple Lie algebras . The interested reader may findmore details on the notion of Lie algebra for instance in the textbook [22]; for the purposeof the present document we shed light on the fact that in the study of finite-dimensionalcomplex simple Lie algebras there is a key subalgebra that naturally arises: the so-called
Cartan subalgebra of the Lie algebra g , usually denoted h . In our context it can be defined asa maximal commutative subalgebra of g for which the adjoint action ad H is diagonalizablefor each H ∈ h . Such Cartan algebras exist and are unique up to isomorphism: in the caseof sl n , it can be realized as the set of diagonal matrices whose diagonal entries sum up to 0,hence can be identified with h n := { x = ( x , . . . , x n ) ∈ C n | P ni =1 x i = 0 } .When working with complex simple Lie algebras we are naturally led to introduce the roots of the Lie algebra g with respect to the Cartan subalgebra h ; these are the linear functionals α acting on h such that the set { x ∈ g | ∀ H ∈ h , [ H, x ] = α ( H ) x } is not restricted to { } . Among these roots it is possible to find a basis ( e i ) i n of the space h ∗ (of linearfunctionals on h ) such that any root can be written as a linear combination of this basisusing only integers with same sign: these are the so-called simple roots of g relative to h .This Cartan subalgebra also comes equipped with a scalar product, the so-called Killingform B , inherited from the Lie algebra g and which is defined by(2.12) B ( x, y ) := tr (ad x ◦ ad y ) . A convenient way to classify finite-dimensional and simple complex Lie algebras is to usetheir Cartan matrix. This r × r matrix ( r is the rank of the simple Lie algebra, i.e. thedimension of h ) is defined by(2.13) A i,j := 2 B ( e i , e j ) B ( e i , e i ) · The entries of this matrix are integral, equal to 2 on the diagonal and non-positive elsewhere;the matrix is invertible. For instance the sl n Cartan matrix is tridiagonal with 2 on thediagonal and − i, j ) with | i − j | = 1. In the sequel and instead of working withthe Killing form B we will use the scalar product on h ∗ defined on its basis by h e i , e j i := A i,j .These two scalar products only differ by a multiplicative constant, usually referred to as the IMPLY-LACED TODA CONFORMAL FIELD THEORIES 9
Dynkin index of the adjoint representation , and which is equal to 2 g , where g is the so-called dual Coxeter number .It is very natural to introduce the basis of the fundamental weights ( ω i ) i r , that isthe basis of h ∗ dual to that of the simple roots:(2.14) ω i := r X l =1 ( A − ) i,l e l . They are defined so that ( δ ij is the Kronecker symbol)(2.15) h e ∨ i , ω j i = δ ij , h ω i , ω j i = r X l,l ′ =1 ( A − ) i,l A l,l ′ ( A − ) l ′ ,j = ( A − ) i,j where e ∨ i := 2 e i h e i ,e i i is the coroot associated to e i . The Weyl vector which is defined by(2.16) ρ := r X i =1 ω i naturally enjoys the property that h ρ, e ∨ i i = 1 for all 1 i r . The squared norm of thisvector can be expressed explicitly depending on the Lie algebra under consideration via the Freudenthal-de Vries strange formula for simple Lie algebras [16, Equation (47.11)] (2.17) | ρ | = g dim g . For finite-dimensional simple complex Lie algebras this quantity is given by(2.18) ( n − n ( n + 1)12 for sl n , ( n − n (2 n − so n and 78, , 620 for the exceptional Lie algebras E , E and E . To finish this quick intro-ductory part on simple Lie algebras let us mention the following duality relation betweentwo vectors α, β ∈ h ∗ (2.19) h α, β i = r X i =1 h α, ω i ih β, e ∨ i i . In the sequel we will work with simple Lie algebras whose Cartan matrices are symmetric(this is actually the definition of simply-laced simple Lie algebras); under this frameworkcoroots and roots are equal, i.e. e ∨ i = e i . By the classification of finite-dimensional Liealgebras, the Lie algebra is then isomorphic to the special linear Lie algebra sl n (for n > so n (for n >
4) or one of the exceptionalLie algebras E , E and E . This equation differs from the one in [16] by a multiplicative factor 2 g . This is due to our normalizationconvention for the scalar product h· , ·i on h ∗ . Toda field action.
Given ϕ and φ two differentiable maps from R to h ∗ , ϕ = P ri =1 ϕ i ω i and φ = P ri =1 φ i ω i , let us set(2.20) h ∂ g ϕ, ∂ g φ i g := r X i,j =1 h ω i , ω j i ( ∂ g ϕ i , ∂ g φ j ) g . Recall that we have defined the Toda field action S T, g in the metric g for the Lie algebra g by the expression(2.21) S T, g ( φ, g ) := 14 π Z R (cid:16) h ∂ g φ ( x ) , ∂ g φ ( x ) i g + R g h Q, φ ( x ) i + 4 π r X i =1 µ i e γ h e i ,ϕ ( x ) i (cid:17) v g ( dx )where(2.22) Q := ( γ + 2 γ ) ρ is the background charge, µ := ( µ > , · · · , µ r >
0) are the cosmological constants and γ > γ satisfies the condition(2.23) γ ∈ (0 , √ . The condition on γ is the optimal condition ensuring that the probabilistic constructionmakes sense : this will become clear later when connecting to GMC theory. From the defi-nition of Q we already know that for all 1 i r , h Q, e i i = γ + 2 γ and | Q | := h Q, Q i = ( γ + 2 γ ) | ρ | . The path integral we aim to construct corresponds to a measure on a suitable space of maps φ : R → h ∗ formally corresponding to(2.24) e − S T, g ( φ,g ) Dφ.
As we will explain below, this can be achieved thanks to the introduction of the GFF.2.2.
Probabilistic interpretation of the path integral.
Gaussian measure interpretation of the squared gradient.
The Toda field action canbe decomposed as a sum of different terms. The first one is the quadratic term12 π Z R h ∂ g ϕ ( x ) , ∂ g ϕ ( x ) i g v g ( dx ) = 12 π Z R h ϕ ( x ) , −△ g ϕ ( x ) i g v g ( dx ) =: h ϕ, −△ g ϕ i g which is reminiscent of a Gaussian measure. Indeed, the measure formally written as(2.25) e − h ϕ, −△ g ϕ i g Dϕ, The scalar field ϕ being studied in TCFTs usually has values in h . To keep notations simple we adoptthe convention that ϕ actually takes values in the space of roots h ∗ . This identification is possible thanksto the Riesz representation theorem. Recall that here our convention on γ is different from the standard convention for LCFT in the proba-bilistic literature by a factor of √ IMPLY-LACED TODA CONFORMAL FIELD THEORIES 11 when restricted to the spaceΣ := { ϕ ∈ H − ( R → h ∗ , g ); Z R ϕ ( x ) v g ( dx ) = 0 } where H − ( R → h ∗ , g ) is the set of h ∗ -valued (generalized) functions with each component(with respect to a basis of h ∗ ) in H − ( R , g ), can be understood as the measure on a Gaussianspace ( h ϕ, −△ g f i g ) f ∈ H with covariance kernel given by h h, −△ g f i g . In other words we arelooking for a Gaussian field enjoying the property that E [ h ϕ, −△ g f i g h ϕ, −△ g h i g ] = h h, −△ g f i g for f, g ∈ H ( R → h ∗ , g ). When r = 1, this is achieved by introducing the GFF X g with vanishing v g -mean on the sphere, that is a centered Gaussian random distributionwith covariance kernel given by the Green function G g (see [7, 35] for more details on thisobject). An important feature of the GFF is that it is not defined pointwise but ratherbelongs to the distributional space H − ( R , g ). For generic rank this is done by consideringadditional fields and setting(2.26) X g := r X i =1 X gi ω i , where X g , . . . , X gr are r such GFFs with covariance structure given by(2.27) E [ X gi ( x ) X gj ( y )] = A i,j G g ( x, y ) . Remark 2.2.
The reason why we must assume the underlying Lie algebra to be simply-lacedis due to the fact that the Cartan matrix is symmetric only in that case, so that it is indeedpossible to construct GFFs with covariance kernel given by (2.27) . To summarize we may wish to interpret the formal Gaussian measure (2.25) restricted toΣ as Z ( g ) − Z F ( ϕ ) e − π R R h ∂ g ϕ ( x ) ,∂ g ϕ ( x ) i g v g ( dx ) Dϕ = E h F ( X g ) i for each continuous and bounded functional F on H − ( R → h ∗ , g ), where Z ( g ) stands forthe total mass of the Gaussian integral (2.25)“ Z ( g ) := det( Σ ) − ” , where Σ is the covariance matrix(2.28) Σ := −△ g Vol g ( R ) A with A the Cartan matrix of g and det( Σ ) is given by a regularized determinant. We knowhow this factor varies when we consider a conformal change of metric: it is proved in [31,Equation (1.13)] that(2.29) log Z ( e ϕ g ) = log Z ( g ) + r π Z R (cid:16) | d ϕ | g + 2 R g ϕ (cid:17) v g ( dx ) . As a consequence, up to a global factor, one has(2.30) Z ( e ϕ ˆ g ) = det( A ) − e r π R R ( | d ϕ | g +2 R ˆ g ϕ ) v g ( dx ) within the conformal class of the spherical metric ˆ g .However in the above construction we do not take into account the fact that the GFF X g has zero mean in the metric g ; to overcome this issue we will introduce so-called zeromodes in the interpretation of the squared gradient term as a Gaussian measure. To do so weintroduce the Lebesgue measure d c on h ∗ by setting for each positive measurable function F : h ∗ → R (2.31) Z h ∗ F ( c ) d c := det( A ) Z R r F (cid:16) r X i =1 c i ω i (cid:17) dc . . . dc r , where dc , . . . , dc r stands for the Lebesgue measure with respect to each variable c i . Bydoing so we are led to the following probabilistic interpretation of the formal (full) Gaussianmeasure (2.25)(2.32) Z F ( ϕ ) e − h ϕ, −△ g ϕ i g Dϕ = Z ( g ) Z h ∗ E h F (cid:16) X g + c ) (cid:17)i d c for each continuous and bounded functional F on H − ( R → h ∗ , g ) and g in the conformalclass of the spherical metric.2.2.2. Gaussian Multiplicative Chaos interpretation of the exponential potential.
It remainsto treat the other terms that appear in the Toda field action (1.1):14 π Z R (cid:16) R g ( x ) h Q, ϕ ( x ) i + 4 π r X i =1 µ i e γ h e i ,ϕ ( x ) i (cid:17) v g ( dx ) . The first term perfectly makes sense if we remember that the GFF has a meaning in thedistributional sense. However the second term does not make sense because of the lackof regularity of the field. For it to be meaningful we need to make use of the notion of
Gaussian Multiplicative Chaos (see [24, 33]) which relies on a proper renormalization ofsome regularization of the GFF.
Definition 2.3.
Let η ε := ε η ( · ε ) be a smooth mollifier. We define the regularized field X ε by considering the convolution approximation of X : (2.33) X ε := X ∗ η ε . Usually one considers the regularization given by averaging the field on circles with radius ε . The reasoning is exactly the same. Definition 2.4.
Assume that γ < √ . From (2.38) and basics of GMC theory [24, 33] ,we know that the following convergence holds in probability in the space of Radon measures(equipped with the weak topology): (2.34) M γe i ,g ( dx ) := lim ε → e h γe i ,X gε ( x ) i− E [ h γe i ,X gε ( x ) i ]v g ( dx ) . The random measure M γe i ,g ( dx ) is non trivial (i.e. different from ) and is called the Gauss-ian Multiplicative Chaos (GMC) measure associated to the field h γe i , X g i . The prefactor det( A ) in the equation comes from the fact that the basis ( ω i ) i r is not orthonormal. IMPLY-LACED TODA CONFORMAL FIELD THEORIES 13
More generally the GMC measure M α,g ( dx ) := lim ε → ε | α | e h α,X gε ( x ) i v g ( dx )exists and is non trivial if and only if | α | := h α, α i < Remark 2.5.
The statement of [4, Proposition (2.5)] can be easily adapted to show that theregularized GFF thus defined has a variance which evolves as E h X ˆ gi,ε ( x ) X ˆ gj,ε ( x ) i = A i,j (cid:18) − ln ε −
12 ln ˆ g ( x ) + θ η + o (1) (cid:19) when ε goes to , and where we have set θ η := R C R C η ( x ) η ( y ) ln | x − y | v ( dx ) v ( dy ) + ln 2 − (recall that v denotes the standard Lebesgue measure). As a consequence the GMC measuredefined above and the limiting measure defined by lim ε → ε γ e h γe i ,X ˆ gε ( x )+ Q ln ˆ g − γeiθη i v ( dx ) actually define the same random measure. In the end we interpret the path integral of TCFTs in a probabilistic way by making theidentification for F ∈ H − ( R → h ∗ , g ):(2.35) Z ( g ) − Z Σ F ( ϕ ) e − S T, g ( ϕ,g ) Dϕ := lim ε → Z h ∗ E h F (cid:16) X gε + Q g + c ) (cid:17) e − π R R R g ( x ) h Q,X gε ( x )+ c i v g ( dx ) − P ri =1 µ i e γ h ei, c i M γei,gε ( C ) i d c , when the limit exists and where g = e ϕ ˆ g is in the conformal class of the spherical metric.2.2.3. Vertex Operators.
There is a class of functionals F which play a key role in the studyof TCFTs; usually referred to as Vertex Operators , computing their correlation functions isoften one of the main issue in the study of two-dimensional CFTs. As we will see below, theyenjoy a certain conformal covariance identity which is the starting point in the understandingof the theory.In TCFTs these Vertex Operators formally correspond to taking F = e h α,ϕ ( z ) i for z ∈ R and α ∈ h ∗ but since such F are not defined on H − ( R → h ∗ , g ), one must use aregularization procedure in order to define their correlations. This motivates the followingdefinition: Definition 2.6.
For z ∈ C and α ∈ h ∗ the regularized Vertex Operator V gα,ε ( z ) is defined by (2.36) V gα,ε ( z ) := ε | α | e h α,X gε ( z )+ Q ln g + c − αθη i where X gε ( z ) is the field regularized as above. Similarly to the GMC measure, this regularized Vertex Operator has same limit when ε → e h α,X ˆ gε ( x )+ c i− E [ h α,X ˆ gε ( x ) i ]ˆ g ( x ) h α ,Q − α i . Indeed for α ∈ h ∗ the variable h α, X gε i is given by the following formula h α, X gε i = r X i =1 h α, ω i i X gi,ε and thus has the following covariance structure, for α, β ∈ h ∗ (2.38) E [ h α, X gε ( x ) ih β, X gε ( y ) i ] = h α, β i G g,ε ( x, y )where G g,ε is the covariance kernel of the field X gε .3. General definitions and existence theorems
Having introduced the probabilistic tools allowing to translate in mathematical terms thepath integral formulation of TCFTs, we are now ready to investigate the existence of thepartition function and of the correlation functions.3.1.
TCFT measure and correlation functions.
To start with, recall that we have givenin (2.35) a meaning to the measure on the space H − ( R → h ∗ , g ) formally defined by theexpression (2.24) by setting for F a positive measurable function on H − ( R → h ∗ , g ): h F i T,g := Z ( g ) lim ε → Z h ∗ E h F (cid:16) X gε + Q g + c ) (cid:17) e − π R R R g ( x ) h Q,X gε ( x )+ c i v g ( dx ) − P ri =1 µ i e γ h ei, c i M γei,gε ( C ) i d c . The mapping F
7→ h F i T,g generates a measure on H − ( R → h ∗ , g ). It is worth noticingthat this measure has infinite mass because of the c → −∞ behaviour of the integrand, butis non trivial as we will see below.Indeed, there is a special family of functions F that deserve special attention: these arethe Vertex Operators which allow to define the correlations function of TCFTs. Fix aninteger N > N distinct points z , . . . , z N ∈ C with respective associated weights α , . . . , α N ∈ h ∗ . The regularized correlation function is defined by the expression :(3.1) h V α ,ε ( z ) · · · V α N ,ε ( z N ) i T,g := Z ( g ) Z h ∗ E "(cid:16) N Y k =1 V gε,α k ( z k ) (cid:17) e − π R R R g ( x ) h Q,X gε ( x )+ c i v g ( dx ) − P ri =1 µ i e γ h c ,ei i M γei,gε ( C ) d c . The correlation function is then set to be the limit when ε → h V α ( z ) · · · V α N ( z N ) i T,g := lim ε → h V α ,ε ( z ) · · · V α N ,ε ( z N ) i T,g . The next subsection is devoted to the study of the convergence of this correlation function.3.2.
Existence of the correlation function.
The form of the correlation function (3.1)is not really convenient when it comes to investigating its convergence as ε →
0. To obtaina reformulation of the correlation functions, we introduce the random measures(3.3) Z γe i ( z , α ) ( dx ) := e γ P Nj =1 h α j ,e i i G ˆ g ( x,z j ) M γe i , ˆ g ( dx )and we define(3.4) s := P Nj =1 α j − Qγ as well as, for all i , s i := h P Nj =1 α j − Q, ω i i γ · IMPLY-LACED TODA CONFORMAL FIELD THEORIES 15
The expression of the Vertex Operator as a Wick exponential (2.37) allows us to interpretthe product in the regularized correlation function as a Girsanov transform (see Theorem6.3), and has the effect of shifting the law of the GFF X g . This reformulation is essential toprove the following result: Theorem 3.1.
Existence and non triviality of the correlation function h V α ( z ) · · · V α N ( z N ) i T,g do not depend on the background metric g in the conformal class of the spherical metric.Furthermore:
1. (Seiberg bounds):
The correlation function h V α ( z ) · · · V α N ( z N ) i T,g exists and isnon trivial if and only if the two following conditions hold for all i = 1 , . . . , r : • s i > , • for all k N , h α k , e i i < h Q, e i i = γ + γ .
2. (Conformal covariance):
Let ψ be a M¨obius transform of the plane. Then h V α ( ψ ( z )) · · · V α N ( ψ ( z N )) i T,g = N Y k =1 | ψ ′ ( z k ) | − αk h V α ( z ) · · · V α N ( z N ) i T,g . where the conformal weights are given by ∆ α j := h α j , Q − α j i .
3. (Weyl anomaly): If ϕ ∈ ¯ C ( R ) then we have the following relation h V α ( z ) · · · V α N ( z N ) i T,e ϕ ˆ g = e c T π S L ( ϕ, ˆ g ) h V α ( z ) · · · V α N ( z N ) i T, ˆ g where S L is the Liouville functional S L ( ϕ, ˆ g ) := Z R (cid:0) | ∂ ˆ g ϕ | g + 2 R ˆ g ϕ (cid:1) d v ˆ g , and the central charge c T is given by c T = r + 6 | Q | .
4. (GMC representation):
In the particular case where g = ˆ g is the round metric,one gets the following explicit expression for the correlation function (3.5) h V α ( z ) · · · V α N ( z N ) i T, ˆ g = r Y i =1 Γ( s i ) µ − s i i γ ! N Y k =1 ˆ g ( z k ) ∆ αk e P k To start with, and anticipating on the conformal anomaly formula, we can assumethat we work with the spherical metric ˆ g , which is such that, since X ˆ g has zero mean valuein the metric ˆ g , π R C h Q + X ˆ g , c i R ˆ g ( x ) v ˆ g ( dx ) = 2 h Q, c i . As a consequence the expressionof the Vertex operators V α k ,ε ( z k ) as Wick exponentials (2.37) allow to interpret them asGirsanov weights that have the effect of shifitng the law of the GFF by a additive term G ˆ g,ε ( · , z k ) where G ˆ g,ε is a mollified version of G ˆ g (see Theorem 6.3). This allows to rewritethe regularized correlation function as N Y k =1 ˆ g ( z k ) ∆ αk e P k 1. Therefore we can proceed byinduction on r so that the only point to check is lim ε → E h(cid:12)(cid:12)(cid:12) Z γe ( z , α ) ,ε ( C ) − s − Z γe ( z , α ) ( C ) − s (cid:12)(cid:12)(cid:12) p i p = 0.This fact has already been proved by the authors in [4, Lemma 3.3]. For the second bulletpoint, let us introduce the set P := { i = 1 , . . . , r |∃ k N, h α i − Q, e i i > } and assumethat it is non-empty. Then we can write that, for positive p i and q summing to one, E " r Y i =1 Z γe i ( z , α ) ,ε ( C ) − s i Y i ∈P E h Z γe i ( z , α ) ,ε ( C ) − p i s i i pi E "Y i Z γe i ( z , α ) ,ε ( C ) − qs i q . Then we have already seen that the second expectation in the right-hand-side had a fi-nite limit as ε → i ∈ P ,lim ε → E h Z γe i ( z , α ) ,ε ( C ) − p i s i i pi = 0. This proves the first item in the statement of Theorem 3.1.We now turn to the proofs of the second and third items. Let us start with the thirditem. By making a standard change of variable in the zero mode c , and using the fact that X g − m ˆ g ( X g ) and X ˆ g have same law, we can assume that X is a GFF with vanishing meanwith respect to the round metric. Since g = e ϕ ˆ g , we have that h V α ,ε ( z ) · · · V α N ,ε ( z N ) i g = Z ( g ) N Y k =1 e h αk,Q i ϕ ( x k ) × lim ε → Z h ∗ e γ h s , c i E " k Y j =1 e V ˆ gε,α j ( z j ) e − π R C R g h Q,X i ( x ) v g ( dx ) − P ri =1 µ i e γci R C e γ ϕ ( x )2 V ˆ gε,ei ( x ) v g ( dx ) d c where this time regularization is done with respect to the round metric, and e V is the vertexoperator without constant mode. Recall that for all i , h Q, e i i = γ + γ ; therefore: E (cid:20)(cid:18)Z C R g ( y ) h Q, X i ( y ) v g ( dy ) (cid:19) h e i , X i ( x ) (cid:21) = h Q, e i i Z C R g ( y ) G ˆ g ( x, y ) v g ( dy ) = ( γ + 2 γ )2 π ( ϕ ( x ) − m ˆ g ( ϕ ))where we have used that R g ( y ) v g ( dy ) = ( − ∆ ˆ g ϕ ( y ) + 2) v ˆ g ( dy ) (here in the weak sense since ϕ ∈ ¯ C ( R )) and the definition of the Green function G ˆ g . More generally when α ∈ h ∗ : E (cid:20)(cid:18) π Z C R g ( y ) h Q, X i ( y ) v g ( dy ) (cid:19) h α, X i ( x ) (cid:21) = h Q, α i ( ϕ ( x ) − m ˆ g ( ϕ )) . Next we want to consider the exponential of this term R C R g ( y ) h Q, X i ( y ) v g ( dy ) as a Girsanovtransform. Its variance is given by E "(cid:18) π Z C R g ( x ) h Q, X i ( x ) v g ( dx ) (cid:19) = 116 π Z C × C R g ( x ) R g ( y ) E [ h Q, X i ( x ) h Q, X i ( y )] v g ( dx ) v g ( dy )= 116 π | Q | Z C × C R g ( x ) R g ( y ) G ˆ g ( x, y ) v g ( dx ) v g ( dy )= 116 π | Q | Z C R g ( x ) (cid:18)Z C R g ( y ) G ˆ g ( x, y ) v g ( dy ) (cid:19) v g ( dx )= 18 π | Q | Z C R g ( x ) ( ϕ ( x ) − m ˆ g ( ϕ )) v g ( dx )= 18 π | Q | Z C ( − ∆ ˆ g ϕ ( x ) + 2) ( ϕ ( x ) − m ˆ g ( ϕ )) v ˆ g ( dx )= 18 π | Q | Z C | ∂ ˆ g ϕ | v ˆ g ( dx )and as seen above it has the effect of shifting the law of X by an additional factor Q ( ϕ ( x ) − m ˆ g ( ϕ )).Therefore h V α ( z ) · · · V α N ( z N ) i g = Z ( g ) e | Q | π R C | ∂ ˆ g ϕ | v ˆ g × lim ε → Z h ∗ e γ h s , c i E " N Y k =1 e V ε,α k ( z k ) e − P ri =1 µ i e γci R C e ϕ ( x )( γ | ei | − γ h Q,ei i e V ε,ei ( x ) v g ( dx ) d c . Since for all i we have | γe i | − γ h Q,e i i = − 1, by the change of variable c i → c i − h Q, e i i m ˆ g ( ϕ )we get that h V α ( z ) · · · V α N ( z N ) i g = Z ( g ) e | Q | π R C | ∂ ˆ g ϕ | v ˆ g + | Q | π R C ϕv ˆ g × lim ε → Z h ∗ E " N Y k =1 V ˆ gε,α k ( z k ) e − P ri =1 µ i R C V ˆ gε,ei ( x ) v ˆ g ( dx ) d c , whence the result, by using the expression (2.30) for the regularized determinant Z ( g ). IMPLY-LACED TODA CONFORMAL FIELD THEORIES 19 For the second item, we see that according to our proof of the first item, h F ( ϕ ˆ g ) Q Nk =1 V α k ( z k ) i ,where F is continuous and bounded on H − ( S → h ∗ , ˆ g ), is actually given by N Y k =1 ˆ g ( z k ) ∆ αk e P k The conditional volume measures We wish to extend here the validity of the probabilistic representation (3.5). The point isthat the explicit expression (3.5) allows us to isolate the constraints s i > Lemma 4.1 (Extended Seiberg bounds) . The bound (4.1) E " r Y i =1 ( Z γe i ( z , α ) ( R )) − s i < ∞ holds if and only if for all i = 1 , · · · , r one has (4.2) − s i < γ ∧ min k =1 ,...,N γ h Q − α k , e i i Proof. We suppose that condition (4.2) holds. Let us consider the families of indices P := { i = 1 , . . . , r | s i > } and N := { i = 1 , . . . , r | s i < } . Choose p > i ∈ N , − ps i < γ ∧ min k =1 ,...,N γ h Q − α k , e i i and fix the conjugateexponent q > p + q = 1. By H¨older inequality we can write that E " r Y i =1 ( Z γe i ( z , α ) ( R )) − s i E "Y i ∈P ( Z γe i ( z , α ) ( R )) − ps i /p E "Y i ∈N ( Z γe i ( z , α ) ( R )) − qs i /q . The product running over i ∈ P is finite because GMC admits negative moments of allorder (see [33, Theorem 2.12]). For the product running over i ∈ N , we use Corollary 6.2 inappendix as well as the relation (2.38), which shows that the GFFs h γe i , X g i and h γe j , X g i ,for i = j , are negatively correlated since h γe i , γe j i = γ A i,j (recall that A is the Cartanmatrix) and all off-diagonal elements of A are nonpositive. Hence(4.3) E "Y i ∈N ( Z γe i ( z , α ) ( R )) − qs i Y i ∈N E h ( Z γe i ( z , α ) ( R )) − qs i i . We conclude that each expectation in the product above is finite because [4, Lemma A.1]ensures that it is indeed the case provided that − ps i < γ ∧ min k =1 ,...,N γ h Q − α k , e i i for all i , which we assumed to hold.Conversely, assume that the expectation (4.1) is finite. By Corollary 6.2 in the appendixapplied to the function G ( x , . . . , x d ) = Y i ∈N x − s i i Y i ∈P x − s i i , to r = |N | , and to the GFFs ( h γe i , X g i ) i =1 ,...,r we deduce E " r Y i =1 ( Z γe i ( z , α ) ( R )) − s i > E "Y i ∈P ( Z γe i ( z , α ) ( R )) − s i E "Y i ∈N ( Z γe i ( z , α ) ( R )) − s i . Since GMC admits negative moments of all order [33, Theorem 2.12], the first expectation inthe right-hand side above is a finite constant C > 0. This implies that the second expectationis finite too. From now on, we fix i ∈ N and j ∈ { , . . . , N } . Without loss of generality andfor the sake of simplicity, we may assume that z j = 0. Then we can choose δ > j ′ = j | z ′ j | > × δ and we can choose non empty balls ( B i ) i = i ,i ∈N all of them at distanceat least 10 × δ > × δ from all the z j ’s. Set B i := B (0 , δ ). Obviously we have E "Y i ∈N ( Z γe i ( z , α ) ( R )) − s i > E "Y i ∈N ( Z γe i ( z , α ) ( B i )) − s i . IMPLY-LACED TODA CONFORMAL FIELD THEORIES 21 Consider the mean value of the field Y := πi H | x | = δ X g ( x ) dxx . A simple check of covariancesshows that the law of the field X g − Y is the independent sum of the field X hg —whichcoincides with X g − Y outside of B (0 , δ ) and corresponds inside B (0 , δ ) to the har-monic extension (component by component) of the field X g − Y restricted to the boundary ∂B (0 , δ )—plus the Dirichlet field X d defined by(4.4) X d = r X i =1 ω i X d,i , where ( X gd, , . . . , X gd,r ) is a family of centered correlated Dirichlet GFFs inside B (0 , δ ) withcovariance structure given by E [ X gd,i ( z ) X gd,j ( z ′ )] = A i,j G d ( z, z ′ )and G d ( z, z ′ ) stands for the Dirichlet Green function inside B (0 , δ ). From now on we willwrite Z h γe i ,X g i ( z , α ) ( d x ) instead of Z γe i ( z , α ) ( d x ) to indicate in the notations the dependence onthe underlying Gaussian field. This means that, generally speaking, we will write Z h γe i ,X i ( z , α ) for Z h γe i ,X i ( z , α ) ( d x ) := lim ε → e P Nj =1 h α j ,γe i i G ˆ g ( x,z j ) ε | γei | e h γe i ,X ε ( x ) i v g ( dx )where X ε stands for the ε -regularization of the field X in the metric g . So we can write E "Y i ∈N ( Z h γe i ,X g i ( z , α ) ( B i )) − s i = E " e − P i ∈N s i h γe i ,Y i Y i ∈N ( Z h γe i ,X g − Y i ( z , α ) ( B i )) − s i . Now we use the Girsanov transform to remover the factor e − P i ∈N s i h γe i ,Y i , which we renor-malize by its variance. We do not need to compute explicitly the variance, nor the covarianceof Y with X g − Y . Indeed, the variance is bounded as well as the covariance of Y with X g − Y .This entails the existence of some constant C > E " e − P i ∈N s i h γe i ,Y i Y i ∈N ( Z h γe i ,X g − Y i ( z , α ) ( B i )) − s i > C E "Y i ∈N ( Z h γe i ,X g − Y i ( z , α ) ( B i )) − s i . Using the decomposition of the law of X g − Y = X d + X gh and independence of X d and X gh ,we get E "Y i ∈N ( Z h γe i ,X g − Y i ( z , α ) ( B i )) − s i > E h ( Z h γe i ,X d i ( z , α ) ( B i )) − s i i E " e min x ∈ Bi X gh ( x ) Y i = i ,i ∈N ( Z h γe i ,X g − Y i ( z , α ) ( B i )) − s i . This implies that both expectations in the right-hand side are finite (they are obviouslynonzero). Finiteness of the first expectation above entails that − s i < γ ∧ γ h Q − α j , e i i (see [4, Lemma A.1]). Since the argument is valid for all i ∈ N and all j , this yields theresult. (cid:3) Perspectives TCFTs provide natural extensions of LCFT with a higher level of symmetry in additionto the Weyl anomaly which encodes the local conformal structure. In this document wehave constructed TCFTs but we have not really shed light on where this W-symmetry doesappear in the model and how useful it can be. We review below some interesting questionsrelated to this observation.5.1. W-symmetry and local conformal structure of Toda theories. Characterizing the sl Lie algbera using W-symmetry. Let us consider the first non-trivial extension of the Liouville theory, i.e. when one works with two negatively correlatedGFFs—like in the construction of the sl Toda theory. Then we can create a one-parameterfamily of QFTs by changing the covariance between these two GFFs, Lie algebras arisingonly for some special values of this parameter, that is when the covariance matrix coincideswith the Cartan matrix of some Lie algebra. More explicitly one can assume to be workingwith a pair of GFFs whose covariance matrix is given by E [ X gi ( x ) X gj ( y )] = A i,j G g ( x, y ) , but where A = (cid:18) cc (cid:19) with c ∈ ( − , 2) would no longer be the Cartan matrix of somesemisimple Lie algebra (except for the values c = 0 , − sl ⊕ sl and sl Lie algebras). Interestingly these theories also enjoy conformal invariance thus define CFTs,but are not supposed to enjoy higher-spin symmetry. In this context it seems natural towonder what is so specific about the theories defined via the sl Lie algebra, and how canone see where W-symmetry does appear ? These questions are being investigated in a workin progress. More precisely it is expected that the existence of higher-spin currents that are holomorphic is granted if and only if the covariance matrix of the GFFs is given by theGreen kernel times the Cartan matrix of the sl Lie algebra.5.1.2. Local conformal structure of Toda Field Theories. In LCFT, the Weyl anomaly (com-bined with diffeomorphism invariance of the theory) is in some sense equivalent to the exis-tence of a holomorphic current: the stress-energy tensor T ( z ). The expression of this tensorcan be obtained by formally derivating the correlation function with respect to the metric h T µ,ν ( z ) V α ( z ) · · · V α k ( z N ) i g := 4 π ∂∂g µ,ν h V α ( z ) · · · V α k ( z N ) i g , and then setting T := T z,z + c t where t is explicit and depends on the background metric g . The first Ward identity(5.1) h T ( z ) N Y k =1 V α k ( z k ) i = N X k =1 ∆ α k ( z − z k ) + N X k =1 ∂ z k z − z k ! h N Y k =1 V α k ( z k ) i and the asymptotic behaviour of the stress-energy tensor T ( z ) ∼ z near infinity (whichcomes from the fact that we have conformally mapped the sphere to the plane) usuallyensure conformal covariance of the model. In a similar way TCFTs feature higher-spincurrents W ( k ) ( z ) which should encode higher-spin symmetry via equations that take thesame form as the Ward identity. As an application of our formalism it should be possible to IMPLY-LACED TODA CONFORMAL FIELD THEORIES 23 check that these identities hold (at least in the simplest sl case) and study the propertiesof integrability provided by the W-symmetry. However the structure of W -algebras is muchmore complicated that the Virasoro one (for instance higher-spin tensors feature higherderivatives and the commutation relations of their modes are no longer linear) and the toolsused to prove integrability of Liouville theory come with additional complications. Howeverthere is still some hope of actually solving these models: for instance it is predicted in thephysics literature that one can find a differential equation for some four-point correlationfunctions, which would allow to derive the value of certain three-point correlation functions.See [10, Equation (14)] for a precise statement. These questions will be addressed in anupcoming series of work by Y. Huang and one of the author. The probabilistic frameworkshould in particular allow to write down an explicit expression for the descendants of theprimary fields V α ( x ), expression which was left unidentified in the physics literature.5.2. The semi-classical limit and Toda equations in W-geometry. Let us commenthere on the geometrical signification of TCFTs. Their path integral formulation rely onthe action functional (1.1) that corresponds to the quantization ( i.e. where we have intro-duced a coupling constant and considered an appropriate renormalization of the field andcosmological constants) of the action S cT ( ϕ, g ) := 14 π Z Σ (cid:16) h ∂ g ϕ ( x ) , ∂ g ϕ ( x ) i g + 2 R g h ρ, ϕ ( x ) i + 2Λ n − X i =1 e h e i ,ϕ ( x ) i (cid:17) v g ( dx ) , whose critical point is given by the solution of the Toda equation (1.2). Such a critical pointexists and is unique as soon as Σ admits a metric for which R g is negative and constant;when the surface has the topology of the sphere or the torus one may need the field to havecertain logarithmic singularities in order for such a problem to admit a unique solution.In the simplest case where n − − Λ within the conformal class of (Σ , g ). In general, such an interpretationremains possible but instead of working in the setup of conformal geometry the good frame-work to consider is the one of W-geometry . The interested reader may find more details on W -geometries in the work of Gervais and Matsuo [17]. One of its important features is that,in a way similar to the fact that W -algebras admit the Virasoro algebra as a subalgebra, W -geometries in some sense contain two-dimensional Riemannian geometry.From the quantum theory viewpoint, since we have constructed TCFTs via a quanti-zation of this classical model, it is natural to expect that when the coupling constant γ (which characterizes the level of randomness) is taken to zero—and under appropriaterenormalizations—we recover the solution of the classical Toda equation (1.2) (when it ex-ists and is unique). Such a result has been recently proved by H. Lacoin and the last twoauthors in [27] for LCFT; it should be reasonable to expect that the result extends for thegeneral sl n TCFTs on the sphere. 6. Appendix In the appendix, we gather rather general and classical results on Gaussian vectors: firstcomparison lemmas and then a statement of the Girsanov theorem. Lemma 6.1. Let F be some smooth function defined on ( R n ) d with at most polynomialgrowth at infinity for F as well as for its derivatives up to order . Assume that for ( x , · · · , x d ) ∈ ( R n ) d (where x i = ( x i , · · · , x in ) ) the following inequalities hold:for all i = j and k, k ′ ∂ F∂x ik ∂x jk ′ > . Let X := ( X , · · · , X d ) and e X := ( e X , · · · , e X d ) be two centered Gaussian vectors in ( R n ) d such that1) for all i = j and k, k ′ E [ X ik X jk ′ ] E [ e X ik e X jk ′ ] . 2) for all i , X i as the same law as e X i .Then the following inequality holds: E [ F ( X , · · · , X d )] E [ F ( e X , · · · , e X d )] . Proof. For t ∈ [0 , X t = √ tX + √ − t e X , where X and e X are independent, and G ( t ) = E [ F ( X t )] . By using Gaussian integration by parts, we get the following relation G ′ ( t ) = 12 d X i =1 n X k =1 E (cid:20) ∂F∂x ik ( X t )( 1 √ t X ik − √ − t e X ik ) (cid:21) = 12 d X i =1 n X k =1 d X i ′ =1 n X k ′ =1 E (cid:20) ∂ F∂x ik ∂x i ′ k ′ ( X t ) (cid:21) E (cid:20) ( √ tX i ′ k ′ + √ − t e X i ′ k ′ )( 1 √ t X ik − √ − t e X ik ) (cid:21) = 12 d X i =1 n X k =1 d X i ′ = i n X k ′ =1 E (cid:20) ∂ F∂x ik ∂x i ′ k ′ ( X t ) (cid:21) (cid:0) E [ X i ′ k ′ X ik ] − E [ e X i ′ k ′ e X ik ] (cid:1) . Therefore G (1) G (0). (cid:3) Corollary 6.2. Let G be some smooth function defined on ( R + ) d with at most polynomialgrowth at infinity for G as well as for its derivatives up to order and consider a partition P , . . . , P m of the set { , . . . , d } . Assume that for ( x , · · · , x d ) ∈ R d + , the following inequalityholds for all r, r ′ ∈ { , . . . , m } with r = r ′ , all i ∈ P r and all j ∈ P r ′ ∂ G∂x i ∂x j > . Further assume that X , · · · , X d is a family of continuous centered Gaussian fields respec-tively defined over domains D i ⊂ R n (for i = 1 , . . . , d ) such that for all r, r ′ ∈ { , . . . , m } with r = r ′ , all i ∈ P r and all j ∈ P r ′ ∀ x ∈ D i , ∀ x ′ ∈ D j , E [ X i ( x ) X j ( x ′ )] . IMPLY-LACED TODA CONFORMAL FIELD THEORIES 25 Let e X = ( e X , · · · , e X d ) be another family of continuous centered Gaussian fields such that:1) for all r = 1 , . . . , m , ( e X i ) i ∈ P r has same distribution as ( X i ) i ∈ P r .2) the families ( e X i ) i ∈ P , . . . , ( e X i ) i ∈ P m are independent.Eventually, let f , . . . , f d be a family of positive functions each of which respectively definedon D i . For i = 1 , . . . , d , we set M i := Z D i e X i ( x ) − E [ X i ( x ) ] f i ( x ) dx and f M i := Z D i e e X i ( x ) − E [ X i ( x ) ] f i ( x ) dx. Then the following inequality holds E [ G ( M , · · · , M d )] E [ G ( f M , · · · , f M d )] . Proof. Up to a discretization of the fields, it suffices to apply Lemma 6.1 with F ( x , . . . , x d ) := G X k p k e γX k − γ E [( X k ) ] , · · · , X k d p dk d e γX dkd − γ E [( X dkd ) ] ! for some nonnegative numbers p ik i obtained by discretizing f i over D i . (cid:3) After these two comparison lemmas we recall the statement of the Girsanov theorem. Itcan be adapted in a way similar to the above proof via a regularization procedure in orderto fit to the GFF we have considered throughout the present document: Theorem 6.3 (Girsanov theorem) . Let D be a subdomain of C and ( X ( x )) x ∈ D := ( X ( x ) , · · · , X n − ( x )) x ∈ D be a family of smooth centered Gaussian field; also consider Z any Gaussian variable be-longing to the L closure of the subspace spanned by ( X ( x )) x ∈ D . Then, for any boundedfunctional F over the space of continuous functions one has that E " e Z − E [ Z ] F ( X ( x )) x ∈ D = E (cid:2) F ( X ( x ) + E [ Z X ( x )]) x ∈ D (cid:3) . References [1] T. 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