Abstract
We construct the stress-energy tensor correlation functions in probabilistic Liouville conformal field theory (LCFT) on the two-dimensional sphere \({\mathbb {S}}^2\) by studying the variation of the LCFT correlation functions with respect to a smooth Riemannian metric on \({\mathbb {S}}^2\). In particular we derive conformal Ward identities for these correlation functions. This forms the basis for the construction of a representation of the Virasoro algebra on the canonical Hilbert space of the LCFT. In Kupiainen et al. (Commun Math Phys 371:1005–1069, 2019) the conformal Ward identities were derived for one and two stress-energy tensor insertions using a different definition of the stress-energy tensor and Gaussian integration by parts. By defining the stress-energy correlation functions as functional derivatives of the LCFT correlation functions and using the smoothness of the LCFT correlation functions proven in Oikarinen (Ann Henri Poincaré 20(7):2377–2406, 2019) allows us to control an arbitrary number of stress-energy tensor insertions needed for representation theory.
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1 Introduction and Main Result
1.1 Local Conformal Symmetry
Two-dimensional conformal field theory (CFT) is characterized by local conformal symmetry, an infinite dimensional symmetry that strongly constrains the theory. A formulation of this symmetry can be summarised as follows [9]. The basic data of CFT are correlation functions
of primary fields\(V_{\alpha }(z)\) defined on a compact two-dimensional surface \(\Sigma \) equipped with a smooth Riemannian metric g. In a probabilistic formulation of CFT the angular bracket \(\langle \cdot \rangle _{\Sigma ,g}\) is an expectation in a suitable positive (not necessarily probability) measure and the primary fields often are (distribution valued) random fields.
The local conformal symmetry arises from the transformation properties of the correlation functions under the action of the groups of smooth diffeomorphisms and local Weyl transformations of the metric. The former acts by pullback on the metric \(g \mapsto \psi ^*g\) and the latter acts by a local scale transformation \(g \mapsto e^\varphi g\) with \(\varphi \in C^\infty (\Sigma ,{\mathbb {R}})\). In axiomatic CFT one postulatesFootnote 1
where the conformal anomaly is given by
and the constant c is the central charge of the CFT, which in our case will belong to the interval \((25,\infty )\). The number \(\Delta _{\alpha }\) is called the conformal weight of the field \(V_{\alpha }\). We denoted by \(v_g\) the Riemannian volume measure and by \(R_{g}\) the curvature scalar (see “Appendix”).
The stress-energy tensor field \(T_{\mu \nu }(z)\) is a symmetric 2 by 2 complex matrix valued field defined indirectly through the formal variation of the (inverse of the) metric at a point \(z \in \Sigma \) in the correlation function (1.1) (the precise definition can be found in Sect. 3.2):
where \(g^{\mu _i \nu _i}\) denotes a component of the inverse of the metric g. The functions on the right-hand side of (1.5) turn out to be analytic or anti-analytic in the variables \(z_i\) in the complex coordinates (4.14) as long as \(z_i \ne z_j\), \(x_i \ne x_j\) for \(i\ne j\) and \(z_i \ne x_j\) for all i and j. These functions diverge when two variables merge but for certain choices of the indices \(\mu _i\) and \(\nu _i\) the functions turn out to be meromorphic with poles described by the conformal Ward identities. Let us specialize to the case of the sphere, \(\Sigma ={{\mathbb {S}}}^2\). Then every smooth metric g can be obtained from a given one \({\hat{g}}\) by the action of diffeomorphisms and Weyl transformations:
This fact together with the symmetries (1.2) and (1.3) yields the tools for defining and computing the functional derivatives on the right-hand side of (1.5). The result is a recursive formula, the conformal Ward identity, that allows to express (1.5) in terms of derivatives (in the \(x_i\)’s) of (1.1).
Ward identities take an especially simple form in complex coordinates. Recall that a Riemannian metric determines a complex structure on \(\Sigma \): a system of local coordinates where the metric takes the form
In such coordinates consider the zz-component of the stress-energy,
where \(\{T_{ij}\}_{i,j=1}^2\) are the components of T in the Euclidean coordinates of the plane. We define
where
Then for distinct points \(\{z_i,x_j\}\) the Ward identity in the case \(\Sigma = {\mathbb {S}}^2\) reads
The definition (1.6) was already introduced in [8], and it is natural in the sense that it makes T(z) meromorphic also in the regions where there are curvature, which is essential for the Virasoro algebra discussed in Sect. 4. Iterating this identity, the left-hand side of (1.5) in the case \((\mu _i,\nu _i)=(z,z)\) will be expressed in terms of the functions (1.1) and their derivatives in the \(x_i\)’s. A corollary of this identity is that the functions (1.5) are holomorphic in the variables \(\{z_i\}\) in the region \(\{z_i \ne z_j : i \ne j\} \cap \{z_i \ne x_j : \forall \, i, j\}\).
The conformal Ward identities have been studied before with theoretical physics level of rigour. For a flat background metric the conformal Ward identities were initially derived in [3]. For a general metric and surface the identities were derived in [8], where also a term dealing with variation of the moduli of the surface appears. This term originates from the fact that compact Riemann surfaces with positive genus have non-trivial moduli spaces, so variation of the metric can also vary the conformal class of the surface.
1.2 Path Integrals and Liouville Conformal Field Theory
In constructive quantum field theory one attempts to construct the expectation as a path integral
over some space of fields \(\phi :\Sigma \rightarrow {\mathbb {R}}\) (in the scalar case). The symmetries (1.2) and (1.3) should then arise from the corresponding symmetries of the action functional S with the anomaly (1.4) arising from the singular nature of the integral in (1.9). A case where this program can be carried out is the Liouville conformal field theory (LCFT hereafter) which was introduced in 1981 by Polyakov [18] in the context of developing a path integral theory for two-dimensional Riemannian metrics.
Liouville field theory is described by the Liouville action functional, which for \(\gamma \in (0,2)\) and \(\mu >0\) is given by
The term Q is given by
The primary fields for LCFT are the vertex operators
where \(\alpha \in {\mathbb {C}}\). Their conformal weights are given by
A rigorous construction of the path integral, and in particular the correlation functions of the vertex operators, was given in [5] and will be recalled in Sect. 2 in the present setup. In [13] the conformal Ward identities (1.8) were derived in the case of one or two T-insertions (\(n=1,2\)). Instead of deriving the conformal Ward identities by varying the background metric, the authors of [13] defined (the zz-component of) the stress-energy tensor directly as the field
and computed the correlation functions (1.5) for \(n=1,2\) for a specific metric by Gaussian integration by parts. Generalizing this approach to arbitrary n was obstructed by a lack of proof of smoothness of the correlation functions (1.1) (which was later proven [17]) and the difficulty of simplifying the expressions coming from the integration by parts. It is however necessary to have (1.8) for arbitrary n in order to construct the representation of the Virasoro algebra for LCFT. This is the motivation and the objective of the present paper. Its main technical input is the recent proof of smoothness of the LCFT correlation functions by the second author [17].
1.3 Main Result
Our main result is a proof of the conformal Ward identities for arbitrarily many T-insertions for arbitrary metrics on the sphere by varying the background metric. The precise result is formulated in Propositions 2.3, 3.3 and 3.6.
Theorem 1.1
Let \((x_1,\ldots ,x_N) \in ({\mathbb {S}}^2)^N\), with \(N \ge 3\), be a tuple of non-coinciding points on the two-dimensional sphere and assume that the real numbers \(\alpha _1,\dots ,\alpha _N\) satisfy the Seiberg bounds. The LCFT correlation functions (1.1) are smooth functions with respect to the Riemannian metric g, and they satisfy the diffeomorphism and Weyl symmetries (1.2), (1.3). The derivatives (1.5) exist and are smooth in \(z_i,x_j\) in the region of non-coinciding points. The correlations for the field T defined by (1.6) satisfy the Ward identities (1.8).
The content of the article is as follows. In Sect. 2 we recall the definition of the correlation functions (1.1) and formulate and prove the diffeomorphism and Weyl covariance of LCFT on a compact surface \(\Sigma \). In Sect. 3 we prove the Ward identities (1.8) for \(\Sigma = {\mathbb {S}}^2\) and in Sect. 4 we discuss future work on the construction of the Virasoro representation of LCFT. The appendix collects the elementary definitions and notations from Riemannian geometry used in the paper.
2 Covariant Formulation of LCFT
In this section we recall the construction of LCFT correlation functions given in [5] and extend it to include the diffeomorphism covariance (1.2). Similar discussion can be found in [10], Sects. 3 and 4, where the authors work on compact Riemann surfaces with genus 2 or higher, but the cases of the sphere and the torus work almost the same way. The main mathematical objects appearing in the construction are the Gaussian Free Field and Gaussian Multiplicative Chaos which we need to define in a covariant way. The appendix collects the elementary definitions and notations from Riemannian geometry used in this section.
2.1 Gaussian Free Field
Let \((\Sigma ,g)\) be a two-dimensional smooth compact Riemannian manifold and \(\Delta _g\) be the Laplace–Beltrami operator. It is well-known that \(\Delta _g\) is a positive self-adjoint operator on \(L^2(\Sigma ,dv_g)\). The set of of orthonormal eigenfunctions \(e_{g,n}\), \(n=0,1,\dots \),
is complete in the sense that the \(L^2\)-closure of \({\text {span}}(e_{g,n})_{n =0}^\infty \) is the whole space \(L^2(\Sigma ,dv_g)\). It holds that \(\lambda _{g,n} >0\) for \(n>0\) and \(\lambda _{g,0} =0\) with \(e_{g,0}\) the constant function.
The Gaussian Free Field (GFF) \(X_g\) on the Riemannian surface \((\Sigma ,g)\) is defined as the random generalised function
where \(a_n\) are independent and identically distributed standard Gaussians. The series converges in the negative order Sobolev space \(H^{-s}(\Sigma ,dv_g)\) for any \(s > 0\) (see e.g. Sect. 4.2 of [7]). The covariance of \(X_g\) has an integral kernel
where \(v_g\) is the volume measure of g,
and \((X_g,f)_g\) denotes the dual bracket, so formally \((X_g,f)_g=\int _\Sigma X_g f \, dv_g\). This justifies the notation
even though \(X_g\) is almost surely not a function. An application of the Plancherel theorem gives
that is, \(G_g(x,y)\) is the Green function of \(-\Delta _g\) having zero average on \(\Sigma \):
Define the average in the metric g as
Then the GFF satisfies the following covariance under diffeomorphisms and Weyl transformations:
Proposition 2.1
(a) Let \(\psi \in \mathrm{Diff}(\Sigma )\). Then
(b) Let g and \(g'\) be conformally equivalent, that is, \(g' = e^\varphi g\) with \(\varphi \in C^\infty (\Sigma ,{\mathbb {R}})\). Then
Proof
(a) Follows from covariance of the Laplacian:
where \(\psi ^*\) acts on functions by \(\psi ^*f=f\circ \psi \). Hence \(\psi ^*e_{n,g}=e_{n,\psi ^*g} \) from which (2.2) follows.
(b) We have \(g'=e^{\varphi }g\) for some \(\varphi \in C^\infty (\Sigma ,{\mathbb {R}})\). Since \(m_{g'}(X_g)=\frac{1}{v_{g'}(\Sigma )}(X_g,e^{\varphi })_{g}\) the field \(X=X_{g}-m_{g'}(X_g)\) has covariance
Since \(\Delta _{e^{\varphi }g}=e^{-\varphi }\Delta _g \) we get from (2.1)
and thus
This implies that \({\mathbb {E}}X(z)X(z') = G_{g'}(z,z')\), meaning that X has the same covariance as \(X_{g'}\). Since the fields are Gaussian, the equality in distribution follows. \(\square \)
Choose now a local conformal coordinate z on \(U\subset \Sigma \) so that the metric is
Then Eq. (2.1) becomes
where \(\Delta \) is the standard Laplacian. Hence, for \(z,z'\in U\) we have
where \(h_g\) is a smooth function ensuring that \(G_g\) has zero \(v_g\)-mean over \(\Sigma \) both in z and \(z'\). For later purpose we note that if \(\psi \) is conformal on U then
so that
In particular for \(\Sigma ={\mathbb {S}}^2\) we take \(U={\mathbb {C}}\) and have
2.2 Gaussian Multiplicative Chaos
Next, we want to define the measure \(e^{\gamma X_g} dv_g\). We regularise the GFF by setting
Then
as \(N\rightarrow \infty \). Hence it is natural to consider the measure
If \(\psi \in \mathrm{Diff(\Sigma )}\) then the properties \(\psi ^* e_{g,n} = e_{ \psi ^*g,n}\) and \(\lambda _{\psi ^* g,n} = \lambda _{g,n}\) imply
Hence
Let \(\Sigma _{N}\) be the sigma-algebra generated by \(a_{1},\dots ,a_{N}\). Then, for \(M<N\) and any continuous \(f:\Sigma \rightarrow {\mathbb {R}}\)
that is, the integrals against continuous functions are martingales. This leads to the almost sure existence of the weak limit
and the limiting measure is non-trivial for \(\gamma < 2\) (see e.g. [4]) which is the origin of the parameter range \(\gamma \in (0,2)\) that we mentioned in the introduction. The critical value \(\gamma =2\) also leads to a non-trivial measure (see e.g. [6]), but requires a different renormalisation so we choose not to include it. The limiting measure is an instance of the Gaussian Multiplicative Chaos and it inherits the property (2.10)
2.3 Weyl Invariance
To have the Weyl transformation law for LCFT we need to modify the chaos measure a bit. Fix conformal coordinates so that (2.4) holds and consider the circle average regularization of \(X_{g}\) given by
see e.g. Lemma 3.2. in [10] for the precise definition of the circle average. From (2.8) and
we deduce
where o(1) stands for terms that vanish as \(\epsilon \rightarrow 0\). It is known that the limit
exists in the sense of weak convergence in probability (see e.g. [4]). By uniqueness of the Gaussian Multiplicative Chaos measure (see [4]), we have the equality
where \(d^2z\) denotes the two-dimensional Lebesgue measure. Let
and define the measure
This definition ensures that we get the transformation law under \(g \mapsto e^\varphi g\) in Proposition 2.2, which will later turn out to be the correct one for the Liouville expectation in Sect. 2.4 in the sense that it leads to the property (1.3). We have
From this formula and the fact that \(X_{{{e^{\varphi } g}}}\overset{law}{=}X_{{{g}}}-m_{e^{\varphi }g }(X_{{{g}}})\) we infer the Weyl transformation law
Note that our definition of \(\rho _{\gamma ,g}\) so far depends on the choice of conformal coordinates. Let \(\psi \) be a diffeomorphism. We want to define \(\rho \) in different coordinates by
We have to check that this is well-defined, meaning that the above formula is consistent with (2.14) in the case when \(\psi ^* g\) is also a conformal metric. Hence suppose we have a metric g and two different conformal coordinates are given by the diffeomorphisms \(\psi _1\) and \(\psi _2\). Thus we have \(g = \psi _1^* g_1\) and \(g=\psi _2^* g_2\) with
From \(\psi _{1}^{*}g_{1}=\psi _{2}^{*}g_{2}\) we get \(g_1=(\psi _1^{-1})^* \psi _2^* g_1 = (\psi _2 \circ \psi _1^{-1})^* g_2\) which implies \(\psi := \psi _2 \circ \psi _1^{-1}\) is a conformal map and
where \(\psi '\) denotes the complex derivative. Using the above formula and (2.7) we conclude
and this implies that (2.14) does not depend on the choice of conformal coordinates and (2.16) is consistent with it.
Now from Proposition 2.1 we infer
We can summarize these considerations to
Proposition 2.2
Let \(\psi \in \mathrm{Diff}(\Sigma )\) and \(\varphi \in C^{\infty }(\Sigma )\). Then we have the following equality of joint probability distributions
where the random variable \(c_{g}(\varphi )\) is given by
2.4 Liouville Expectation
The GFF \(X_g\) is almost surely an element of \(H^{-s}(\Sigma , dv_g)\) for any \(s>0\). Let \(F:H^{-s}(\Sigma ,dv_g)\rightarrow {\mathbb {C}}\). The Liouville expectation of F, initially defined in Sect. 3 of [5], is given by
where we use the notation \( M_{g,\gamma }(f)=\int _\Sigma fd M_{g,\gamma }\) so that \(M_{g,\gamma }(1)\) denotes the total mass of the measure \(M_{\gamma ,g}\). The factor \(Z( \Sigma ,g)\) is the “partition function of the GFF”, explicitly
where the zeta function is defined as
for real part of s small enough and \(\zeta _{\Sigma ,g}'(0)\) is defined by analytic continuation, see Sect. 1 of [20] for details. We include \(Z(\Sigma ,g)\) in the definition (2.18) to match physics literature conventions. Especially this has the effect of shifting the central charge of the theory from \(6Q^2\) to \(1+6Q^2\), see Proposition 2.3. For a diffeomorphism \(\psi \in \mathrm {Diff}(\Sigma )\) the property \(\lambda _{\psi ^* g,n} = \lambda _{g,n}\) implies
and furthermore, Eq. (1.13) in [20] gives
where
Proposition 2.3
Suppose \(F: H^{-s}(\Sigma ,dv_g) \rightarrow {\mathbb {R}}\) is such that \(\langle |F| \rangle _{\Sigma ,g}<\infty \). Then we have the diffeomorphism covariance
where \((\psi _*F)(X) := F(X\circ \psi )\) and the Weyl covariance
where \(\varphi \in C^\infty (\Sigma ,{\mathbb {R}})\) and \(c=1+6Q^2\).
Proof
The first claim follows directly from the identities
which follow from Proposition 2.2, and the fact that \(R_g\) is a scalar function (a 0-form). The second claim follows along the same lines as in [5]. For completeness we give the main steps. Let \(g'=e^{\varphi }g\). By Proposition 2.2 and a shift \(c'=c-c_g(\varphi )\) in the c integral we have
Using \( R_{g'} v_{g'}=(R_{g} -\Delta _{g} \varphi )v_{g}\) and dropping the prime from c this becomes
Next we apply the Girsanov theorem to the factor \(e^{ \frac{Q}{4\pi } (X_g,\Delta _g\varphi )_g}\). Denoting the rest of the integrand by \(H(X_{g},M_{g,\gamma })\) we have
with
From (2.1) we obtain \(G_g\Delta _g\varphi =-2\pi \varphi +{{\tilde{c}}} \) with \({{\tilde{c}}}=\tfrac{Q}{2}\frac{\int \varphi dv_g}{\int dv_g}\) so that \(h=-\tfrac{Q}{2}\varphi +{{\tilde{c}}}\) and thus
After a shift in the c-integral we obtain
with
The claim follows since
\(\square \)
2.5 Liouville Correlation Functions
Choose a local conformal coordinate. We define
and we define the regularized vertex operators by
Again we set
which is well defined by the same argument as with \(\alpha =\gamma \) earlier. Hence
Proposition 2.4
The correlation functions
exist and are non-zero if and only if the \(\alpha _i\)’s satisfy the Seiberg bounds
where \(\chi (\Sigma ) = 2 - 2 \mathrm {genus}(\Sigma )\) is the Euler characteristic. Furthermore they satisfy the diffeomorphism and Weyl transformation laws (1.2) and (1.3) with \(\Delta _\alpha = \frac{\alpha }{2}(Q-\frac{\alpha }{2}).\)
Proof
The strategy for the proof of convergence is the following. We first switch the integration order and argue that the c-integral converges. Then we evaluate the c-integral which then yields a different representation of the correlation function in terms of an expectation of a moment of a GMC integral with no c-integral remaining. Consider (2.18) with \(F(X)=e^{(X,f)_g}\) with \(f \in C^\infty (\Sigma )\). For the scalar curvature the Gauss–Bonnet theorem takes the formFootnote 2\(\int R_gdv_g= 4 \pi \chi (\Sigma )\), so we get
where we used Fubini’s theorem and defined
Since \(M_{g,\gamma }(1)>0\) almost surely, the c-integral converges provided
and after evaluating the c-integral we get
where
Finally a shift in the Gaussian integral (Girsanov theorem) gives
For the correlation functions we take
with \(f=\sum _{i=1}^n \alpha _i\sum _{n=0}^Ne_{g,n}e_{g,n}(z_i)\), because then \((X_g,f)_g = \sum _{i=1}^n \alpha _i X_{g,N}(z_i)\). Then, the condition (2.25) becomes the first of the conditions (2.23). As \(N\rightarrow \infty \) in a neighborhood of \(z_i\)
and the condition \(\alpha _i<Q\) is needed for the \(M_{ \gamma ,g}\) integrability of this singularity, see Lemma 3.3 in [5].
Diffeomorphism covariance follows from (2.22) in the limit.
For the Weyl covariance let us again choose conformal coordinates around the insertion points \(z_i\) and use the circle average regularization
where
By (2.20) we have then
which implies the claim since \(\varphi _\epsilon (z_i)\rightarrow \varphi (z_i)\) as \(\epsilon \rightarrow 0\). \(\square \)
3 Conformal Ward Identities
We will now specialize to the case \(\Sigma ={\mathbb {S}}^2={\hat{{\mathbb {C}}}}\) and consider the metric dependence of the vertex correlation functions:
where from now on we drop the \(\Sigma \) from the notation. Our objective is to construct the derivatives (1.5) and prove the identities (1.8). The identities (1.2) and (1.3) make the dependence on the metric quite explicit since the sphere has only one conformal class (see below for the definition), a fact we will recall next.
3.1 Beltrami Equation
Let \({{\mathcal {M}}}\) be the set of smooth metrics in \({\hat{{\mathbb {C}}}}\). We may work on the coordinate chart \({\mathbb {C}}\) and identify \(g\in {{\mathcal {M}}}\) with a smooth function \(g(z)=\{g_{\alpha \beta }(z)\}\) taking values in positive matrices such that \(D\zeta ^T (g\circ \zeta ) D\zeta \) is smooth as well where \(\zeta (z)=z^{-1}\) (this means that g is smooth at infinity). Similarly \(\psi \in \mathrm{Diff}({\hat{{\mathbb {C}}}})\) can be identified with a diffeomorphism \(\psi \in \mathrm{Diff}( {\mathbb {C}})\) satisfying the additional condition \(\psi \circ \zeta \in \mathrm{Diff}({\mathbb {C}})\).
The sphere has only one conformal class of metrics, which means that for a fixed metric \({\hat{g}} \in {\mathcal {M}}\), any other metric \(g \in {\mathcal {M}}\) can be written as \(g=e^\varphi \psi ^* {\hat{g}}\), where \(\varphi \in C^\infty ({{\hat{{\mathbb {C}}}}})\) and \(\psi \in {\text {Diff}}({{\hat{{\mathbb {C}}}}})\). We aim to prove the conformal Ward identities by varying the metric \({\hat{g}}\) and seeing how this affects the correlation function \(\langle \prod _{i=1}^N V_{\alpha _i,{\hat{g}}}(z_i) \rangle _{{\hat{g}}}\). To this end, we want to compute how \(\varphi \) and \(\psi \) depend on the perturbed metric g, or more concretely, if g is roughly of the form \(g={\hat{g}} + \epsilon f\), what is the f dependency of \(\varphi \) and \(\psi \) in the first order in \(\epsilon \). The purpose of this section is to find the relation between the perturbation f and the functions \(\psi \) and \(\varphi \), and we will see that the equation \(g=e^{\varphi } \psi ^* {\hat{g}}\) will lead us to the Beltrami equation.
Let \(g\in {{\mathcal {M}}}\) and set
where \(\sigma \in C^\infty ({{\hat{{\mathbb {C}}}}})\) and \(\delta \) denotes the Euclidean metric. Let us look for a function \(\varphi \) and a diffeomorphism \(\psi \) such that
where the second equality comes from (4.9). By taking determinants we get
and by plugging this back into the equation we get
where
Hence \(D \psi ^T = \gamma (D \psi )^{-1} \det D \psi \) which in complex coordinates becomes the Beltrami equation (see Theorem 10.1.1. in [2])
where
It is readily checked that
Indeed, we have
Above we used (4.15) (note that \(\overline{\gamma _{{\bar{z}} \bar{z}}} = \gamma _{zz}\) always in the complex coordinates), \(\det \gamma =1\), and the fact that \(\gamma _{z {\bar{z}}}\) is always real (to simplify the denominator) and positive, which follows from the facts that \(\gamma \) is positive definite and \(4\gamma _{z {\bar{z}}} = {\text {tr}} \gamma \). Now \(\Vert \mu \Vert _\infty < 1\) follows.
The Beltrami equation is solved by writing
whereby (3.4) becomes
To solve this recall the Cauchy transform \({\mathcal {C}}:C_0^\infty ({\mathbb {C}})\rightarrow C^\infty ({\mathbb {C}})\)
and the Beltrami transform \({\mathcal {B}}:C_0^\infty ({\mathbb {C}})\rightarrow C^\infty ({\mathbb {C}})\) given by \({\mathcal {B}} := \partial _z {\mathcal {C}} = {\mathcal {C}} \partial _z\) (see Chap. 4 of [2] for basic properties of these integral transforms). We have \(\partial _{\bar{z}}{{\mathcal {C}}}f = {\mathcal {C}} \partial _{{\bar{z}}} f=f\) so that (3.7) can be written for \(u\in C^\infty ({\mathbb {C}})\) as
and then as a Neumann series
where
We will also denote
The convergence of this series is classical and for what follows we state it in a slightly more general setup for a smooth family of Beltrami coefficients \(\mu (z,\epsilon )\). Our proof is a slight variation of the proof [2], Sects. 5.1 and 5.2, so we will be brief. For a multi-index \(l=(l_1,l_2,l_3)\) let \(D^l=\partial _\epsilon ^{l_1}\partial _z^{l_2}\partial _{{\bar{z}}}^{l_3}\).
Proposition 3.1
Let \(\mu \in C_0^\infty ({\mathbb {C}}\times {\mathbb {R}})\) with \(k:=\Vert \mu \Vert _{L^\infty ({\mathbb {C}}\times {\mathbb {R}})} < 1\). Then the series (3.9) converges uniformly together with all its derivatives and setting \(u_n={{\mathcal {C}}}v_n\) we have
Furthermore, for each \(\epsilon \) the function \(\psi (z, \epsilon )=z+u(z, \epsilon )\) defines a smooth diffeomorphism of \({\hat{{\mathbb {C}}}}\).
Proof
First, Theorem 4.5.3 in [2] implies \(S_p:=\Vert {{\mathcal {B}}}\Vert _{L^p({\mathbb {C}})\rightarrow L^p({\mathbb {C}})}\rightarrow 1\) as \(p\rightarrow 2\) with \(S_2=1\). From this we obtain
for p close enough to 2. For the rest of the proof we fix any such p, say \(p=2\).Footnote 3 It follows that the series \(\sum _{n=0}^\infty (\mu {\mathcal {B}})^n\mu \) converges in \( L^2({\mathbb {C}})\) uniformly in \(\epsilon \). Next we show that the limit belongs to the Sobolev space \(W^{k,2}({\mathbb {C}})\) for all k and \(\epsilon \). By applying the product rule of the derivative and \(\partial _z {\mathcal {B}} \mu = {\mathcal {B}} \partial _z \mu \), \(\partial _{{\bar{z}}} {\mathcal {B}} \mu = {\mathcal {B}} \partial _{{\bar{z}}} \mu \), we get for \(v_n=(\mu {\mathcal {B}})^n \mu \)
By using \(\Vert D^{k_i} \mu {\mathcal {B}} D^{k_j} \mu \Vert _2 \le \Vert D^{k_i} \mu \Vert _\infty S_2 \Vert D^{k_j} \mu \Vert _2\), we get the upper bound
where the constant C(l, n) depends on \(\Vert D^k\mu \Vert _\infty \) for \(|k|\le |l|\). This shows \(z \mapsto v(z,\epsilon ):=\sum _{n=0}^\infty v_n(z,\epsilon )\) is in the Sobolev space \(W^{k,2}({\mathbb {C}})\) for all k and \(\epsilon \), and since \(\mu (\cdot ,\epsilon ) \in C_0^\infty ({\mathbb {C}})\) we conclude \(v_n(\cdot ,\epsilon ) \in C^\infty _0({\mathbb {C}})\) and \(v(\cdot ,\epsilon ) \in C_0^\infty ({\mathbb {C}})\). For the bounds (3.11) recall that \(u_n = {\mathcal {C}} v_n\) and \(u = {\mathcal {C}} v\). Since \({\mathcal {C}}\) maps \(C^\infty _0({\mathbb {C}})\) into \(C^\infty ({\mathbb {C}})\), the functions \(z \mapsto u_n(z,\epsilon )\) and \(z \mapsto u(z,\epsilon )\) belong to \(C^\infty ({\mathbb {C}})\) and are bounded on compact sets. Now we get the bound
where \({\text {supp}} v\) denotes the support of v. Same argument yields the same bound for \(u_n\). Now we have shown that \(\psi (z,\epsilon )=z+u(z,\epsilon )\) belongs to \(C^\infty ({\mathbb {C}})\) for each \(\epsilon \) and \(\psi (z,\epsilon ) = z + {\mathcal {O}}(1/z)\) as \(z \rightarrow \infty \). Theorems 5.2.3 and 5.2.4 in [2] then imply that \(\psi (z,\epsilon ) = z+u(z,\epsilon )\) is a smooth diffeomorphism of \({{\hat{{\mathbb {C}}}}}\) for each \(\epsilon \). \(\square \)
Next we want to argue that the function \(\varphi \) in (3.2) is smooth. Equation (3.3) implies that \(\varphi \) is given by
where we used \(\det D\psi =|\partial _{z}\psi |^{2}-|\partial _{\bar{z}}\psi |^{2}\) and \(\psi (z)=z+u(z)\). Since g is a metric on the Riemann sphere \({\hat{{\mathbb {C}}}}\), the volume density \(\sqrt{\det g}\) must be smooth at infinity. This means that \(\sqrt{\det \zeta ^* g}\) is a smooth (and positive) function at the origin, where \(\zeta (z) = \tfrac{1}{z}\). Thus we can write \(\sqrt{\det \zeta ^* g} = e^{\rho (z,\epsilon )}\) where \(\rho \in C^\infty ({\mathbb {C}}\times {\mathbb {R}})\). We have
Thus we get
On compact sets this function is bounded, and as \(z \rightarrow \infty \) the absolute value is dominated by \(C|z|^{-1}\). Thus the bound (3.11) holds when we replace u by \(\ln \sqrt{\det g}\) or \(\sigma \) (by the same argument). We conclude then
Thus \(\varphi (\cdot ,\epsilon ) \in W^{k,p}({\mathbb {C}})\) for all k and \(p>2\) and we conclude \(\varphi \in C^\infty ({\mathbb {C}}\times {\mathbb {R}})\).
3.2 The Stress-Energy Tensor
In this section we give the precise definition of the derivatives (1.5).
Definition 3.2
Let \(S: C^\infty ({\mathbb {C}}) \rightarrow {\mathbb {C}}\) be a functional (not necessarily linear) such that for all \(h \in C^\infty ({\mathbb {C}})\) the function \(\epsilon \mapsto S(h+\epsilon f)\) is differentiable at 0 for all \(f \in C^\infty _0({\mathbb {C}})\). If it also holds that the derivative is linear and continuous in f, we denote by \(\frac{\delta S}{\delta h} \in {\mathcal {D}}'({\mathbb {C}})\) the distribution
These derivatives are also known as the Gateaux differentials. If \(\tfrac{\delta S}{\delta h}\) can be realized as a continuous function, that is, if we have
for some \(s \in C({\mathbb {C}})\), then we define \(\frac{\delta S}{\delta h(z)} := s(z)\).
The Liouville correlation functions (1.1) are functions of the metric g. As was explained in the beginning of Sect. 3, we identify g with a positive matrix valued smooth function \(g(z)= \{g_{\mu \nu }(z)\}\) on \({\mathbb {C}}\) such that also \(D \zeta ^T (g \circ \zeta ) D \zeta \) is smooth, where \(\zeta (z) = z^{-1}\). We denote the inverse matrices by \(\{g^{\mu \nu }(z)\}\). In this picture the correlation functions are functions of the smooth functions \(g_{\mu \nu }\) and by perturbing these functions we may compute derivatives of the form
where this derivative is to be understood in the way that was explained above. Note that we have to show that the derivatives
exist, where \(g_\epsilon ^{\mu \nu } = g^{\mu \nu } + \epsilon f^{\mu \nu }\), \(f^{\mu \nu } \in C^\infty _0({\mathbb {C}})\). Once the existence of these derivatives is established, they are shown to be given by distributions evaluated at the perturbation functions \(f^{\mu \nu }\). Then we prove that these distributions can be represented by functions, which will be denoted by \(\langle T_{\mu \nu }(z) \prod _{i=1}^N V_{\alpha _i,g}(z_i) \rangle _g\).
Let
and let \(f\in C_0^\infty ({\mathbb {C}}_\mathbf{x }, M_2)\) be a smooth function with compact support in \({\mathbb {C}}_\mathbf{x }\) and taking values in symmetric \(2\times 2\) matrices. Consider the perturbed metric \(g(z,\epsilon )\) with the inverse given by
where \({\hat{g}}\) is given by (3.1) so that \( {\hat{g}}^{\mu \nu }=e^{-\sigma }\delta ^{\mu \nu }\). It follows that g defines a metric (i.e. is positive definite) if \(\epsilon \) is small enough. We will also use the notation
Then we have
Proposition 3.3
The function \(F({\hat{g}},\cdot ): \epsilon \mapsto \langle \prod _{i=1}^{N}V_{\alpha _{i},g}(x_{i})\rangle _{g}\)Footnote 4 is smooth in a neighbourhood of the origin. Furthermore, for any positive integer n we have
where the n-linear function \({{\mathcal {T}}}_{n}\) defines a distribution \({{\mathcal {T}}}_{n}\in (C_0^\infty ({\mathbb {C}}_\mathbf{x }, M_2)^{\otimes n})^*\)Footnote 5.
Furthermore, let \(f_i\in C_0^\infty ({\mathbb {C}}_\mathbf{x }, M_2)\), \(i=1,\dots , n\) have disjoint compact supports and set
Then for the function \(F({\hat{g}}, \cdot ): (\epsilon _1,\ldots ,\epsilon _n) \mapsto \langle \prod _{i=1}^{N}V_{\alpha _{i},g}(x_{i})\rangle _{g}\) we have
where we sum over repeated indices and \(F^{{\hat{g}}}_{\mu _{1} \nu _{1}\dots \mu _{n}\nu _{n}}(z_{1},\dots ,z_{n})\) are smooth functions in the region \(z_{i}\in {\mathbb {C}}_\mathbf{x }\) with \(z_{i}\ne z_{j}\) when \(i\ne j\).
Proof
By Proposition 3.1 for \(\epsilon \) small enough we have
Writing
the Beltrami coefficient (3.5) is
and the function \(\varphi \) is given by
Now by the diffeomorphism and Weyl transformation laws in Proposition 2.4
We will now argue that smoothness of \(F^{{\hat{g}}}\) in \(\epsilon \) will follow from smoothness of \(\varphi \) and u in \(\epsilon \) (recall that \(\psi (z)=z+u(z)\)). First, to prove smoothness of the expectation on the right-hand side of (3.20) we use the result of the second author [17] that the correlation function \((x_1,\ldots ,x_N) \mapsto \langle \prod _{i=1}^N V_{\alpha _i, {\hat{g}}}(x_i) \rangle _{{\hat{g}}}\) is smooth in the region of non-coinciding points. Since \(\psi \) is a diffeomorphism the points \(\psi (x_i)\) are non-coinciding as well and smoothness of the expectation in \(\epsilon \) follows. Smoothness of the anomaly term follows from the bounds (3.11) and (3.12) which guarantee convergence of the integrals over \({\mathbb {C}}\).
We will now deduce from Eqs. (3.6), (3.19) and (3.20) that to compute the derivative (3.15) it is fundamental to compute \(\partial _\epsilon ^mu|_{\epsilon =0}\) and \(\partial _\epsilon ^m \zeta _{\mu \nu }|_{\epsilon =0}\) for \(m\le n\). Before starting the computations it is useful to remark that \(\zeta \) can be written in terms of the functions \(f^{\alpha \beta }\) by combining (3.13) and (3.17). This yields
where \(f^{-1}\) is the matrix with entries \(\{f^{\alpha \beta }\}\) and \({\hat{g}} f^{-1}\) is a matrix product. It follows that \(\partial ^m \zeta _{\mu \nu }|_{\epsilon =0}\) is a homogeneous polynomial of degree m in the variables \(\{f^{\alpha \beta }(z)\}\) and
Hence \(\mu ={\mathcal {O}}(\epsilon )\) and from (3.10) we get \(u_{k}={\mathcal {C}}(\mu {\mathcal {B}})^k \mu ={{\mathcal {O}}}(\epsilon ^{k+1})\). Now \(u=\sum _k u_k\) implies
From (3.18) and (3.21) we have
where \(p_l(z)\) is a homogeneous polynomial of degree l in the variables \(\{f^{\alpha \beta }(z)\}\). In particular \(p_l\in C_0^\infty ({\mathbb {C}}_\mathbf{x })\). Hence
Now we know how the basic terms \(\partial _\epsilon ^m u|_{\epsilon =0}\) and \(\partial _\epsilon ^m \zeta _{\mu \nu }|_{\epsilon =0}\) look like.
Let us now look at the various contributions to the derivative (3.15). From (3.20) we see that we get derivatives of the form
with \(k \le n\). The anomaly term equals
By recalling that \((\psi ^* {\hat{g}})^{\alpha \beta } = (D \psi ({\hat{g}} \circ \psi ) D\psi ^T)^{\alpha \beta }\), \(dv_{\psi ^* {\hat{g}}} = |\det D \psi | dv_{{\hat{g}}} \) and \(R_{\psi ^* {\hat{g}}} = R_{{\hat{g}}} \circ \psi \) we see that to compute \(\partial _\epsilon A(\varphi , \psi ^* {\hat{g}})\) it suffices to compute \(\partial _\epsilon \varphi \) and \(\partial _\epsilon \psi = \partial _\epsilon u\). For the derivatives \(\partial _\epsilon ^k \varphi (x_i)\) it suffices to know \(\partial _\epsilon ^k u\) because of (3.19) and (3.21). Finally, for the last term in (3.23) we note that
Thus, we see that all the derivatives in (3.23) reduce to computing derivatives of u and derivatives of \(\zeta _{\mu \nu }\). The derivatives of u we already computed above and the derivatives of \(\zeta \) are easily read off of (3.21).
Next we argue that the left-hand side of (3.15) can really be expressed in terms of a distribution. Let l be a positive integer and fix a set of positive integers \((l_i)_{i=1}^k\) such that \(\sum _i l_i = l\). Let \(m^i_{l_i}(f)\) denote a monomial of degree \(l_i\) in the variables \(\{D^Kf^{\alpha \beta }(x)\}\) where \(D=\partial _x,\partial _{{\bar{x}}}\) and \(K\le 2\). We will now explain that the above and (3.19) imply that the derivative (3.15) consists of products and sums of terms that are l-linear functionals of \(\{f^{\alpha \beta }(x)\}\) of the form
with \(p=0,1\) and \(\mathbf{l }=(l_1,\ldots ,l_k)\). The fact that the above expression can be written in terms of an l-linear functional \({{\mathcal {T}}}_{p,\mathbf{l }}\) follows from the definition of \(m^i_{l_i}\). The derivatives containing the integral over z arise from taking derivatives of \(A(\varphi ,\psi ^* {\hat{g}})\) in (3.20) and the terms with the \(x_i\)’s come from the derivatives of \(\varphi (x_i)\) and \(\langle \prod _{i=1}^N V_{\alpha _i, {\hat{g}}} (x_i) \rangle _{{\hat{g}}}\). Then the expressions (3.24) come from the observation above that everything reduces to derivatives of u and \(\zeta \) and using (3.21) and (3.22). The maps \(f\mapsto m^i_{l_i}(f)\) and \((f,g)\mapsto f{{\mathcal {B}}}g\) are continuous maps \(C_0^\infty \rightarrow C_0^\infty \) and \(C_0^\infty \times C_0^\infty \rightarrow C_0^\infty \), respectively (in the Fréchet topology of \(C_0^\infty \)), and \(f\mapsto ({{\mathcal {C}}}f)(z)\) is continuous \(C_0^\infty \rightarrow {\mathbb {C}}\). We conclude that \({{\mathcal {T}}}_{p,\mathbf{l }}\) is continuous in its arguments and by the nuclear theorem defines a distribution in \({{\mathcal {D}}}'({\mathbb {C}}_\mathbf{x }^l)\).
The fact that the derivative (3.15) can be expressed in terms of an n-linear functional \({{\mathcal {T}}}_n\) then comes from the fact that we take n-derivatives, so that the result is a sum of terms of the form
with \(\sum _j |\mathbf{l }_j| + \sum _k |\mathbf{l }_k| =n\), where \(|\mathbf{l }|=\sum _i l_i\). From the definitions of \({{\mathcal {T}}}_{p,\mathbf{l }}\) and \(m^i_{l_i}\) it follows that such integrals can be written in terms of an n-linear functional on \(C^\infty _0({\mathbb {C}}_\mathbf{x },M_2)\) and in the end we define \({{\mathcal {T}}}_n\) to be the resulting sum of such functionals.
For the third claim let \(f_i\in C_0^\infty ({\mathbb {C}}_\mathbf{x },M_2)\), \(i=1,\dots ,n\) have disjoint supports and consider the perturbed metric
Next we explain that from the definition of \({\mathcal {T}}_n \) and the fact that the \(f_i\)’s have disjoint support we get
To see this note that in the previous computation with (3.13) for example the term \(\partial ^2_\epsilon \mu |_{\epsilon =0}(z)\) was a homogeneous polynomial of degree 2 in the variables \(\{f^{\alpha \beta } (z)\}\). If we were to do the same computation with the perturbation \(\epsilon _1 f_1 + \epsilon _2 f_2\) of the metric and then compute \(\partial _{\epsilon _1}|_0 \partial _{\epsilon _2}|_0 \mu (z)\) the result vanishes if \(f_1\) and \(f_2\) have disjoint support. On the other hand, the term \((\partial _\epsilon \mu |_0) (\partial _\epsilon \mu |_0)\) (meaning we hit different terms with the derivatives) is a 2-linear functional \({\mathcal {S}}\) on \(C^\infty _0({\mathbb {C}}_\mathbf{x },M_2)\) so can be written as \({\mathcal {S}}(f,f)\) and in this case we have \({\mathcal {S}}(f_1,f_2)=(\partial _{\epsilon _1}\mu |_0) (\partial _{\epsilon _2}\mu |_0)\). Applying the same logic for all the terms appearing in the computation of the derivatives we end up with (3.25).
From the assumption that the \(f_i\)’s have disjoint supports it follows (as in (3.21)) that
In particular, this implies \(\partial _{\epsilon _i}\partial _{\epsilon _j}\mu (z)|_{(\epsilon _1, \ldots ,\epsilon _n)=0}=0\) for \(i\ne j\) so that \(l_i=1\) in (3.24) for all i. Furthermore
Hence
where
From (3.19) we also get terms which contain the derivatives of \(\zeta \)
Thus the derivatives of \(\zeta \) in the anomaly term in (3.20) give contributions of the form.
The term \(e^{-\sum _i \Delta _{\alpha _i} \varphi (x_i)}\) in (3.20) contributes only derivatives of u since \(\zeta (x_i)=0\) by the assumption on the supports of the \(f_i\)’s.
We conclude that the functions \(F^{{\hat{g}}}_{\mu _1 \nu _1 \ldots \mu _n \nu _n}\) in (3.16) are polynomials in the variables
and their complex conjugates. The complex conjugates come from the derivatives of \(\ln (|1+\partial _{z}u|^{2}-|\partial _{{\bar{z}}}u|^{2})\) in (3.19) since \(|1+\partial _z u|^2 = (1+\partial _z u)(1+\partial _{{\bar{z}}} {\bar{u}})\) and the derivatives of \(A(\varphi ,\psi ^* {\hat{g}})\) since \((\psi ^* {\hat{g}})^{\alpha \beta }\) contains elements of the derivative matrix
Thus the functions \(F^{{\hat{g}}}_{\mu _1 \nu _1 \ldots \mu _n \nu _n}\) are smooth on the claimed region. \(\square \)
To avoid confusion with the notations in the following computations, we now carefully explain what the previous result says about the functional derivatives of the LCFT correlation function. Recall that we are denoting \(g^{\mu \nu } = {\hat{g}}^{\mu \nu } + \sum _{i=1}^n \epsilon _i f_i^{\mu \nu }\) and \(F({\hat{g}},\epsilon _1,\ldots ,\epsilon _n) = \langle \prod _{i=1}^{N}V_{\alpha _{i}, g}(x_{i})\rangle _{ g}\). First of all, we want to interpret (3.16) using the functional derivative from Definition 3.2. Note that we can think of \({\hat{g}} \mapsto F({\hat{g}},0)\) as being a function of the four smooth functions given by the components of the inverse metric \({\hat{g}}^{\mu \nu }\). Denote \(g_{\epsilon _1}^{\mu \nu } = {\hat{g}}^{\mu \nu } + \epsilon _1 f_k^{\mu \nu }\). Now, provided that the functional derivatives \(\frac{\delta F(\cdot ,0)}{\delta {\hat{g}}^{\mu \nu }}\) exist, we have
The right-hand side of the above equation has four terms, which arise from the fact that on the left-hand side \(F({\hat{g}},\epsilon _1)\) is a function of all the four components \(g_{\epsilon _1}^{\mu \nu }\), and the derivative \(\partial _{\epsilon _1}\) has to operate on all these four arguments. By taking more derivatives we get
Now comparing with (3.16) and noting that
we get
where the factor \(e^{\sum _k \sigma (z_k)}\) comes from the fact that in (3.16) the right-hand side contains the volume forms \(dv_{{\hat{g}}}(z_k) = e^{\sigma (z_k)} d^2 z_k\). This shows that Proposition 3.3 gives the existence of the functional derivatives of the LCFT correlation functions with respect to the metric and that these functional derivatives are given in terms of the functions \(F^{{\hat{g}}}_{\mu _1 \nu _1 \ldots \mu _n \nu _n}\). Thus, we introduce the notation
We add the factor of \((4\pi )^{n}\) to match conventions of physics literature. This means that the left-hand side on the above expression is not the Liouville expectation of some function, but rather it is just a function of the \(z_k\)’s and \(x_i\)’s.
At this point we remark that Proposition 3.3 applies to an arbitrary metric \({{\tilde{g}}}\) on the Riemann sphere \({{\hat{{\mathbb {C}}}}}\), although in the proof we considered a diagonal metric \({\hat{g}} = e^\sigma \delta \). Indeed, there exists a smooth diffemorphism \({\tilde{\psi }} \) of \({{\hat{{\mathbb {C}}}}}\) such that \({{\tilde{g}}} = {\tilde{\psi }}^*( e^{{\tilde{\sigma }}} \delta )\) where \({\tilde{\sigma }}: {{\hat{{\mathbb {C}}}}}\rightarrow {\mathbb {R}}\) is a smooth function. Now a perturbed metric \({{\tilde{g}}}_\epsilon ^{\mu \nu } = {{\tilde{g}}}^{\mu \nu } + \epsilon f^{\mu \nu }\) can be written in the form
After using the diffeomorphism transformation law from Proposition 2.4 the more general result follows. Then, using (3.16) for the metric \(g_{\epsilon _1}^{\mu \nu } = {\hat{g}}^{\mu \nu } + \epsilon _1 f_1^{\mu \nu }\) instead of \({\hat{g}}\) we get
Now Definition 3.2 applied to the derivative \(\partial _{\epsilon _1}|_{\epsilon _1=0}\) on the right-hand side implies
where we sum over repeated indices. Comparing this with (3.16) gives
We will use this equation later.
Definition 3.4
It is natural to define the notations
where \(a \in {\mathbb {C}}\) and \(f: {\mathbb {C}}\rightarrow {\mathbb {C}}\) is a smooth function. We will also denote
and
where \(\psi : {{\hat{{\mathbb {C}}}}}\rightarrow {{\hat{{\mathbb {C}}}}}\) is a smooth diffeomorphism. Denote
The notation (3.34) has an obvious generalisation for multiple \({{\tilde{T}}}_{\mu _i \nu _i} + f_i\)-insertions, where \(f_i: {\mathbb {C}}\rightarrow {\mathbb {C}}\) is a smooth function, given recursively by
For the time being let us consider the variation of these T-correlation functions under diffeomorphisms and Weyl transformations.
Let us define
In conformal coordinates \({\hat{g}}_{\alpha \beta }=e^\sigma \delta _{\alpha \beta }\) we have from a computation similar to Lemma 4.2
Furthermore we have the locality property for \(z\ne z'\):
Indeed by the definition of the functional derivative in Definition 3.2
for \(z' \ne z\).
Then
Proposition 3.5
Let \((x_1,\ldots ,x_N) \in {\mathbb {C}}^N\) and \((z_1,\ldots ,z_n) \in {\mathbb {C}}_\mathbf{x }^n\) be tuples of disjoint points. Then the stress-energy tensor correlation functions satisfy
where \({{\tilde{T}}}_{\mu \nu } = \sum _{\alpha ,\beta } (D\psi ^{T})_{\mu \alpha }(T_{\alpha \beta }\circ \psi ) (D\psi )_{\beta \nu }\) and the right-hand side above is defined in (3.35). Also,
where the right-hand side above is defined in (3.35) by setting \(f_k(z)=a_{\mu _k \nu _k}(\varphi ,g,z_k)\) and \(\psi (z)=z\).
Proof
For the first claim let
where the \(f_i\)’s have disjoint supports. Then
where
with \({{\tilde{f}}}=(D\psi ^{-1})^T (f\circ \psi ^{-1}) (D\psi ^{-1})\) and \(\gamma = {\mathcal {O}}(\epsilon _i \epsilon _j)\) meaning \(\partial _{\epsilon _i} \gamma |_{(\epsilon _1,\ldots ,\epsilon _n)=0} =0\). We denote \( F(g,x_1,\ldots ,x_N)=\langle \prod _{i=1}^N V_{\alpha _i,g}(x_i)\rangle _g\). Thus by (3.16), (3.28) and the diffeomorphism covariance in Proposition 2.4 we get (we sum over repeated indices)
In the last equality we did a change of variables \(z_i \rightarrow \psi (z_i)\) and used
where we used \(D \psi ^{-1}(\psi (z)) = (D \psi ^T)(z)\) and the symmetry of T and f. Now the first claim is proven.
The second claim follows from the Weyl transformation law in Proposition 2.4 in the following way. Let
and
Now \(h = e^{\varphi } {{\tilde{h}}}\). Denote \(F(g) = \langle \prod _{j=1}^N V_{\alpha _j, g}(x_j) \rangle _{g}\). Then
where the first equality comes from (3.16) and (3.28), and the second equality uses the Weyl transformation law from Proposition 2.4. To compute the derivative on the last line we apply the locality (3.38). Indeed, it implies
where we sum over repeated indices. Now differentiating the product and using (3.42) gives
Using (3.16) and (3.28) again gives
Plugging the result back into (3.41) yields
The claim now follows from \(e^\varphi v_g=v_{e^\varphi g}\). \(\square \)
3.3 Ward Identities
In this section we will demonstrate that Propositions 3.3 and 3.5 allow us to compute T-correlations inductively. By Proposition 3.5 it suffices to do this in the conformal coordinates. Furthermore, only \(T_{zz}\) and \(T_{{\bar{z}}{\bar{z}}}\) correlations are non-trivial and they may be computed separately (see Remark 3.7 below). We have then
Proposition 3.6
The Ward identity (1.8) holds whenever \(z_i\ne z_j\) for \(i \ne j\), and \(z_i \ne x_j\) for all i and j.
Proof
Let \({\hat{g}}=e^\sigma \delta \), where \(\delta \) denotes the Euclidean metric, and consider the perturbed metric g with the inverse metric having the components
where f is real and has compact support outside the points \(\{z_i\}_{i=2}^n\) and \(\{x_j\}_{j=1}^N\), and \(\epsilon \in {\mathbb {C}}\) (so that \(g^{{\bar{z}}{\bar{z}}}= {\bar{\epsilon }} f\)). We will denote \(\epsilon =\epsilon _1+i\epsilon _2\), \(\epsilon _1,\epsilon _2 \in {\mathbb {R}}\) and \(\partial _\epsilon = \tfrac{1}{2}(\partial _{\epsilon _1}-i \partial _{\epsilon _2})\) so that \(\partial _\epsilon {\bar{\epsilon }} = 0\). By (3.28) and (3.31)
Next we apply Definition 3.2 and (3.28)
To compute the \(\epsilon \)-derivative we want to utilize the Weyl and diffeomorphism transformation laws from Proposition 3.5. To this end we write (see Sect. 3.1)
Note that the dependence on \(\epsilon \) is in \(\varphi \) and \(\psi \). Using the Weyl transformation law from Proposition 3.5 we get
Note that the last line uses the notation introduced in Definition 3.4. We can further simplify this by using the diffeomorphism transformation law from Proposition 3.5 (and again using notations from Definition 3.4)
We arrive at
To proceed with the computation we now compute \(\psi \) and \(\varphi \) to first order in \(\epsilon \). First, recall Eqs. (3.17) and (3.18). To first order in \(\epsilon \) the metric is \(g_{\bar{z}{\bar{z}}}=-\frac{\epsilon }{4} e^{2\sigma }f\). Hence
and \(\zeta _{ z{\bar{z}}}=0\). Then, for the \(\gamma \) appearing in (3.5) we have \(\gamma _{{\bar{z}}\bar{z}}=-\frac{\epsilon }{4}e^{\sigma }f + {\mathcal {O}}(\epsilon ^2)\) so that
Now recall from Sect. 3.1
Thus
From (3.19) we infer
Based on these formulas we remark that
The latter equation together with (3.37) further imply \(a_{zz}(\varphi ,\psi ^* {\hat{g}},z)|_{\epsilon =0}=0\). Thus, continuing from (3.43) we get
Next we evaluate all the derivatives appearing above. Since \(\varphi =(-u \partial _z \sigma - \partial _z u) \epsilon + {\mathcal {O}}(\epsilon ^2)\) and \( e^{\sigma }R_{ e^{\sigma } \delta }=-4\partial _{z}\partial _{{\bar{z}}}\sigma \) we get
where t is defined in (1.7).
The second term on the right-hand side of (3.46) is readily evaluated by using (3.45)
For the third term on the right-hand side of (3.46) we have
where we again used \(a_{zz}(\varphi , \psi ^* {\hat{g}}, z_i)|_{\epsilon =0} = 0\). Denote \(b(z_{j},f)=\partial _{\epsilon }\big |_{0}a_{zz}(\varphi ,\psi ^{*} {\hat{g}},z_{j})\). Using \(\varphi ={{\mathcal {O}}}(\epsilon )\), (3.36) and (3.45) we get
The derivative is calculated in Lemma 4.2:
Some algebra then gives
Next we compute (recalling Definition 3.4)
where we used \(\psi (z)|_{\epsilon =0} = z\). We have
Thus
where in the second equality we used \(\partial _{{\bar{z}}} {\mathcal {C}}(e^\sigma f) = e^\sigma f\) and the assumption that the support of f does not intersect \(\{z_2,\ldots ,z_k\}\) and in the third and the fourth equalities we used \(\partial _\epsilon {\bar{\epsilon }} = 0\). We get
By the chain rule the last term on the right-hand side of (3.46) takes the form
Collecting all the derivatives we computed together we get
and stripping out the arbitrary test function f we conclude
This is equivalent with (1.8) once one uses the definition (1.6). \(\square \)
Remark 3.7
Using the computations from the proofs of Propositions 3.3 and 3.6 it is simple to check that
whenever \(z_i\ne z_j\) for \(i \ne j\) and \(z_i \ne x_j\) for all i and j, where t is the function from (1.7).
In similar fashion it is also possible to compute that
whenever \(z_i\ne z_j\) for \(i \ne j\) and \(z_i \ne x_j\) for all i and j, but we skip this computation since the identity is not relevant from the point of view of the Virasoro algebra discussed in the next section.
These formulae motivate the focus on the \(T_{zz}\) correlations: the \(T_{{\bar{z}} {\bar{z}}}\) correlations are obtained from \(T_{zz}\) correlations by complex conjugation, and the other correlations are rather trivial.
We finish this section with a formulation of the Ward identity for the case where the vertex operators are replaced by a smooth version as follows.
Definition 3.8
Let \(F: H^{-s}({{\hat{{\mathbb {C}}}}}) \rightarrow {\mathbb {C}}\) be such that \(\langle |F| \rangle _g < \infty \).
-
(1)
We call the support of F, denoted by \({\text {supp}} F\), the smallest closed set K satisfying \(F(X)=0\) whenever \({\text {supp}}X \subset K^c\). Here \({\text {supp}} X\) denotes the support of X, which is the usual notion of the support of a distribution.
-
(2)
We say that F is smooth if the derivatives
$$\begin{aligned} \tfrac{d}{d \epsilon } F(X+ \epsilon f)|_{\epsilon =0} \end{aligned}$$exist for all \(X \in H^{-s}({{\hat{{\mathbb {C}}}}})\) and all \(f \in C^\infty _0({{\hat{{\mathbb {C}}}}})\) and they can be written as
$$\begin{aligned} \tfrac{d}{d \epsilon } F(X+ \epsilon f)|_{\epsilon =0} = \int _{\mathbb {C}}h(z) f(z) \, d^2z, \end{aligned}$$for all f where \(h:{{\hat{{\mathbb {C}}}}}\rightarrow {\mathbb {C}}\) is continuous. If F is smooth, we denote \(\frac{\delta F}{\delta X(z)} := h(z)\).
Let \(F: H^{-s}({{\hat{{\mathbb {C}}}}}) \rightarrow {\mathbb {C}}\) be smooth and g and \({\hat{g}}\) be as in the beginning of proof of Proposition 3.6, except now we assume that the support of f does not intersect the support of F (in the previous setting this assumption corresponded to assuming that support of f does not overlap with the points \(z_i\) and \(x_i\)). By Proposition 2.3 we have
so that recalling (3.44) and (3.45) we get
where the last term was computed in (3.47) and we denoted
Integrating by parts and using (3.44) we get
We conclude using \(\partial _\epsilon |_0 \langle F\rangle _g=:\frac{1}{4\pi } \int f(z) \langle T_{zz}(z) F \rangle _{{\hat{g}}} dv_{{\hat{g}}}(z)\) that
where T is given by (1.6). Note that all the terms are well-defined since F is smooth and if x is outside the support of F, then \(F_x=0\), which especially means that in the above integrals \(F_x\) vanishes in a neighbourhood of z.
In what follows we are interested in F of the form
where \(N\ge 0\) and the functions \(h_j\) are complex valued and smooth with compact support and \(X(h_i)=\int h_i(z)X(z)d^2z\). Furthermore we require \(\mathfrak {R}\int _{\mathbb {C}}h_0>2Q\), where \(\mathfrak {R}z\) denotes the real part of z, so that \(\langle |F|\rangle _{{\hat{g}}}<\infty \), as was discussed in the proof of Proposition 2.4. Let \({{\mathcal {V}}}\) denote the linear span of such F. Then by the definition of the functional derivative Definition 3.8 we get \((e^{X(h_0)})_z = h_0(z) e^{X(h_0)}\) and \(X(h_0)_z=h_0(z)\). We conclude
Note that \({\text {supp}}F\) is the union of the supports of the \(h_j\)’s. Then, if \(z\in ({\text {supp}}F)^c\)
where \({{\mathcal {T}}}_z F\) is determined by plugging (3.53) into (3.51) and integrating by parts. Explicitly
It is simple to check that \({{\mathcal {T}}}_zF\in {{\mathcal {V}}}\) and \({\text {supp}}({{\mathcal {T}}}_zF)\subset {\text {supp}}F\).
Now to get the Ward identities for the observables (3.52) we proceed as in the proof of Proposition 3.6 where the diffeomorphism and Weyl transformation laws from Proposition 3.5 were used. A direct consequence of Proposition 2.3 and the computations in the proof of Proposition 3.5 is that the transformation laws (3.39) and (3.40) in this case take the form
Now we get the Ward identity almost identically as in the proof of Proposition 3.6 by using the above transformation laws and the computations leading to (3.51). The result is
whenever F is as in (3.52), \(z_i \ne z_j\) for \(i \ne j\) and \(z_i \notin {\text {supp}} F\) for all i.
4 Prospects: Representation Theory
The Ward identities have well known algebraic consequences. To formulate these note that by iterating (3.56) for \(F\in {{\mathcal {V}}}\) we get
with \(G\in {{\mathcal {V}}}\). This can be viewed as an action \(T(z):{{\mathcal {V}}}\rightarrow {{\mathcal {V}}}\). This action gives rise to a representation of the Virasoro algebra on the physical Hilbert space \({{\mathcal {H}}}\) which is canonically related to LCFT. To describe the latter (see [16] for details) it is convenient to choose the metric \(g=e^\sigma |dz|^2\) with
i.e. the metric is Euclidean \( |dz|^2\) on the unit disc \({\mathbb {D}}\) and \(|z|^{-4} |dz|^2\) on \({\mathbb {D}}^c\). The curvature of g is concentrated on the equator: \(R_g(z) = 4 \delta (|z|-1)\). We denote the GFF \(X_g\) simply by X. It has the covariance
The function (2.14) becomes in this metric
and thus the chaos measure is
where the volume is \(dv(z)=e^\sigma d^2z\).
The Liouville expectation is then given by
Let \({{\mathcal {F}}}_{\mathbb {D}}\) consist of functions (3.52) with \(\mathrm{supp}\,F\subset {\mathbb {D}}\) (see Definition 3.8 for the definition of the support) and \(\mathfrak {R}\int _{\mathbb {D}}f>Q\), where \(\mathfrak {R}\) denotes the real part. Let \(\theta : {{\hat{{\mathbb {C}}}}}\rightarrow {{\hat{{\mathbb {C}}}}}\) be the reflection \(\theta (z) = 1/{\bar{z}}\). Given a \(F\in {{\mathcal {F}}}_{\mathbb {D}}\) define
where \(X\circ \theta \) refers to the distribution \(f\rightarrow X(f_\theta )\) where \(f_\theta =|z^{-4}|(f\circ \theta )\). Obviously \({\text {supp}}(\Theta F)\subset {\mathbb {D}}^c\).
We define a sesquilinear form \((\cdot , \cdot ): {\mathcal {F}}_{\mathbb {D}}\times {\mathcal {F}}_{\mathbb {D}}\rightarrow {\mathbb {C}}\) by
Since \(\mathfrak {R}(\int _{\mathbb {C}}(f+{\bar{f}}_\theta ))=2\mathfrak {R}\int _{\mathbb {D}}f>2Q\) this is well defined. Reflection positivity is the following statement:
Proposition 4.1
The form (4.5) is positive semidefinite:
For proof see [16]. It is simple to check that \((F,G) = \overline{(G,F)}\) by using a decomposition of the GFF described in Proposition 2.2 of [16], but we skip the computation to keep this discussion short.
We define the Hilbert space \({\mathcal {H}}\) of LCFT as the completion of \({\mathcal {F}}_{\mathbb {D}}/{\mathcal {N}}\), where
Let F and G have supports in the disc \({\mathbb {D}}_r\) with \(r<1\) and let \(C_i\) be circles of radii \(1>r_1>r_2\dots>r_k>r\). Define the objects
Note that the left-hand side is just a notation and as such does not define the operators \(L_{n_1} \ldots L_{n_k}: {\mathcal {F}}_{\mathbb {D}}\rightarrow {\mathcal {F}}_{\mathbb {D}}\). Even if we can guess what the definition of the \(L_{n_i}\)’s is supposed to be (which is readable from the above notation), showing that \(L_{n_1} \ldots L_{n_k}\) maps \({\mathcal {F}}_{\mathbb {D}}\) into \({\mathcal {F}}_{\mathbb {D}}\) seems difficult.
As a consequence of iterating the Ward identities, the integrand on the right-hand side of (4.7) is analytic in \(z_i\in {\mathbb {D}}\setminus {\mathbb {D}}_r\), \(z_i\ne z_j\), the right-hand side depends on the contours \(C_i\) only through their order, meaning that the value of the integral possibly changes if one swaps \(C_i\) and \(C_j\) for \(i \ne j\). The Ward identities then imply (see [9]) that the commutator \([L_n, L_m] := L_n L_m - L_m L_n\) is given by
where \(\delta _{i,j}\) is the Kronecker delta, in the obvious sense as a relation among the objects (4.7). However we would like to realize the \(L_n\)’s as operators acting on a suitable dense domain in \({{\mathcal {H}}}\) with the adjoints satisfying \(L_n^*=L_{-n}\) and construct a representation of the Virasoro algebra (4.8). This will be the subject of the forthcoming publication [15].
Notes
For scalar fields \(V_{\alpha }\).
Note that we are using the scalar curvature \(R_g\), which is twice the Gaussian curvature\(K_g\).
Note the slight difference between our proof and the proofs in Sects. 5.1 and 5.1 of [2]. In [2] \(p>2\) is assumed to establish integrability properties of u (see e.g. Lemma 5.2.1). For us the estimate (3.11) is enough (and implies \(u(\cdot ,\epsilon ) \in W^{k,p}({\mathbb {C}})\) for \(p>2\)) so we can fix \(p=2\).
The function F also depends on the points \(x_i\), but we omit this from the notation.
That is, \({{\mathcal {T}}}_n\) is a complex valued continuous linear map, taking as arguments functions from the n-fold tensor product \(\bigotimes _{k=1}^n C^\infty _0({\mathbb {C}}_x,M_2)\). We use the notation \({\mathcal {T}}_n(f_1,\ldots ,f_n) := {\mathcal {T}}_n (f_1\otimes \ldots \otimes f_n)\).
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Acknowledgements
Open access funding provided by University of Helsinki including Helsinki University Central Hospital. We thank Remi Rhodes, Vincent Vargas and Yichao Huang for helpful discussions. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Scaling limits, rough paths, quantum field theory where work on this paper was undertaken. We also thank the anonymous referees for careful reading of this work and for providing numerous valuable comments on it. The work is supported by the Academy of Finland and ERC Advanced Grant 741487. The work of J.O. is also supported by DOMAST.
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Appendix
Appendix
We collect here some notations from Riemannian geometry, see for example [11]. Let \((\Sigma ,g)\) be a smooth compact two-dimensional Riemannian manifold. In this section we use the Einstein summation convention. Given a local coordinate \(x=(x^1,x^2)\) we denote \(\partial _\alpha =\frac{\partial }{\partial x^\alpha }\). Hence vectors are given as \(u=u^\alpha \partial _\alpha \) and covectors as \(\lambda =\lambda _\alpha dx^\alpha \). The space of all diffeomorphisms \(\psi : \Sigma \rightarrow \Sigma \) (smooth maps with smooth inverse) is denoted by \(\mathrm {Diff}(\Sigma )\).
The Riemannian metric g is given by
where \(g_{\alpha \beta }(x)\) is a smooth function taking values in positive matrices. The metric g determines a volume measure \(v_g\) on \(\Sigma \) given in local coordinates by
where \(d^2x\) is the Lebesgue measure on \({\mathbb {R}}^2\). We denote the scalar product by
and then \(L^2(\Sigma ,g) := \{ f: \Sigma \rightarrow {\mathbb {C}}\mid (f,f)_g < \infty \}\).
The group of smooth diffeomorphisms \(\mathrm{Diff}(\Sigma )\) acts on the space of smooth metrics by \(g\rightarrow \psi ^*g\) where the pullback metric is given in coordinates as
Then we have the change of variables formula
We say that two Riemannian metrics \(g'\) and g belong to the same conformal class if there exists \(\varphi \in C^\infty (\Sigma )\) and \(\psi \in {\text {Diff}}(\Sigma )\) such that \(g'=e^\varphi \psi ^* g\). We say that \(g'\) and g are conformally equivalent if there exists \(\varphi \in C^\infty (\Sigma )\) such that \(g' = e^\varphi g\).
Let us denote the inverse of the matrix g(x) by \(g^{\alpha \beta }(x)\). The reason for the upper indices is that the tensor field \( g^{\alpha \beta } \partial _\alpha \otimes \partial _\beta \) is invariantly defined. It allows us to define the Dirichlet form
and the Sobolev space \(H^1(\Sigma ,g)=\{f\in L^2(\Sigma ,g):{\mathcal {D}}_g(f,f)<\infty \}\). The Dirichlet form gives rise to a positive self-adjoint operator \(-\Delta _g\) by
On smooth functions f by integration by parts one gets the formula for the Laplace–Beltrami operator as
The Dirichlet form satisfies the diffeomorphism invariance
which implies
The Laplace–Beltrami operator has a discrete spectrum \((\lambda _{g,n})_{n=0}^\infty \), \(0=\lambda _0 < \lambda _1 \le \lambda _2 \dots \), and a complete (in \(L^2(\Sigma ,g)\)) set of smooth eigenfunctions \((e_{g,n})_{n=0}^\infty \) with \(e_{g,0}\) the constant function. The property (4.11) implies
The zero-mean Green’s function is defined by the formula
Then (4.12) implies
We can view the diffeomorphisms also passively as changes of coordinates. Then locally we can find a coordinate so that \(g_{\alpha \beta }=e^\sigma \delta _{\alpha \beta }\) with a smooth \(\sigma \) (the proof is a small variation of Proposition 3.1). An atlas of such coordinates defines a complex structure on \(\Sigma \) since the transition functions are easily seen to be analytic. Indeed, if on \({\mathbb {R}}^2\) we have \(g=\psi ^*h\) with g and h diagonal matrices then \(\psi (x^1,x^2)=(u(x^1,x^2),v(x^1,x^2))\) where u, v satisfy the Cauchy–Riemann equations. We can then introduce complex coordinates \(z = x^1 + i x^2, {\bar{z}} = x^1 - i x^2\,\) and write tensors using them. E.g. \(T=T_{\alpha \beta }dx^\alpha \otimes dx^\beta \) becomes
where
and \(T_{{\bar{z}} {\bar{z}}} = \overline{T_{zz}}\). Furthermore, for a \(2 \times 2\) symmetric matrix f the formulae for the determinant and the trace in complex coordinates are
See for example Sect. 2.9.1 in [2] for some other basic properties of the complex coordinates.
We denote the scalar curvature of g by \(R_g\). It is defined by contracting the Ricci tensor \(R_{\mu \nu }\)
where the Ricci tensor comes from contracting the Riemann tensor
Finally, the Riemann tensor is defined by the formula
where the \(\Gamma \)’s are the Christoffel symbols
As \(R_g\) is defined by contractions of the metric and its derivatives, under diffeomorphisms it transforms as
In coordinates where \(g_{\alpha \beta }= e^\sigma \delta _{\alpha \beta }\) we have the formula
From this together with (4.18) and the existence of conformal coordinates it is easy to infer that in general
As an application of these definitions we have the Lemma used in the text:
Lemma 4.2
Let
Then
in coordinates where \(g_{\alpha \beta }= e^\sigma \delta _{\alpha \beta }\).
Proof
By definition
where \(g_\epsilon ^{z{\bar{z}}}=2e^{-\sigma }\) and \(g_\epsilon ^{z z}=\epsilon \phi \). Let \({{\tilde{g}}}_\epsilon =e^{-\sigma }g_\epsilon \). Then
and \({{\tilde{g}}}_\epsilon ^{z{\bar{z}}}=2\) and \({{\tilde{g}}}_\epsilon ^{z z}=\epsilon e^\sigma \phi \). We have
\(\partial _{\epsilon }\big |_{0}v_{{{\tilde{g}}}_\epsilon }=0\) and
Hence
which yields the claim. \(\square \)
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Kupiainen, A., Oikarinen, J. Stress-Energy in Liouville Conformal Field Theory. J Stat Phys 180, 1128–1166 (2020). https://doi.org/10.1007/s10955-020-02601-4
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DOI: https://doi.org/10.1007/s10955-020-02601-4