Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Probabilistic construction of the $\mathbb{H}^3$-Wess-Zumino-Witten conformal field theory and correspondence with Liouville theory

Colin Guillarmou, Antti Kupiainen, Rémi Rhodes

TL;DR

The paper constructs a rigorous probabilistic path integral for the $\mathbb{H}^3$-WZW conformal field theory by combining Gaussian free field techniques with Gaussian multiplicative chaos to renormalize the hyperbolic target. It establishes a conformal structure with central charge $c(k)=\tfrac{3k}{k-2}$ and develops a precise H^3-WZW -- Liouville correspondence on the sphere, including DOZZ structure constants and BPZ-like relations for degenerate insertions. The framework is extended to gauged WZW models via holomorphic vector bundles, and generalized to arbitrary rank-2 bundles on higher-genus surfaces, with a careful treatment of Witten Laplacians, determinants, and gauge covariance. The results provide a probabilistic foundation for the WZW path integral, its holographic-like Liouville dual, and a route toward a gauge-covariant, geometric construction of the associated CFT data and correlation functions.

Abstract

Wess-Zumino-Witten (WZW) models are among the most basic and most studied Conformal Field Theories (CFT). They have had a huge influence not only in physics but also in mathematics, in representation theory and geometry. However their rigorous probabilistic construction and analysis starting from the path integral is still missing and all their properties have been obtained algebraically from their postulated affine Lie algebra symmetry. Initially considered as taking values in a compact semisimple Lie Group G, the WZW model also has a "dual" formulation where the group $G$ is replaced by the homogenous space $G^{\mathbb{C}}/G$, where $G^{\mathbb{C}}$ is the complexification of $G$, and it has been argued that the former can be (re-)constructed from the latter. For $G={\rm SU}(2)$, the space ${\rm SL}(2,\mathbb{C})/{\rm SU}(2)$ can be identified with the three dimensional hyperbolic space $\mathbb{H}^3$ and, in physics, the corresponding CFT has been studied as the simplest example of the AdS/CFT correspondence. A surprising correspondence between the $\mathbb{H}^3$-WZW CFT and the Liouville CFT was found by Ribault and Teschner and later generalised by Hikida and Shomerus. This correspondence has been dubbed by Gaiotto-Teschner as the "quantum analytic Langlands correspondence" since the analytic Langlands correspondence of Etingof, Frenkel and Kazhdan seems to emerge in its formal semi classical limit. In this paper we give a rigorous construction of the path integral for the $\mathbb{H}^3$-WZW model on a closed Riemann surface $Σ$, twisted by an arbitrary smooth gauge field on $Σ$. Using the probabilistic path integral we prove a correspondence between the correlation functions of the primary fields of the $\mathbb{H}^3$ model and those of Liouville CFT extending the expressions proposed by Ribault-Teschner and by Hikida-Schomerus to this general setup.

Probabilistic construction of the $\mathbb{H}^3$-Wess-Zumino-Witten conformal field theory and correspondence with Liouville theory

TL;DR

The paper constructs a rigorous probabilistic path integral for the -WZW conformal field theory by combining Gaussian free field techniques with Gaussian multiplicative chaos to renormalize the hyperbolic target. It establishes a conformal structure with central charge and develops a precise H^3-WZW -- Liouville correspondence on the sphere, including DOZZ structure constants and BPZ-like relations for degenerate insertions. The framework is extended to gauged WZW models via holomorphic vector bundles, and generalized to arbitrary rank-2 bundles on higher-genus surfaces, with a careful treatment of Witten Laplacians, determinants, and gauge covariance. The results provide a probabilistic foundation for the WZW path integral, its holographic-like Liouville dual, and a route toward a gauge-covariant, geometric construction of the associated CFT data and correlation functions.

Abstract

Wess-Zumino-Witten (WZW) models are among the most basic and most studied Conformal Field Theories (CFT). They have had a huge influence not only in physics but also in mathematics, in representation theory and geometry. However their rigorous probabilistic construction and analysis starting from the path integral is still missing and all their properties have been obtained algebraically from their postulated affine Lie algebra symmetry. Initially considered as taking values in a compact semisimple Lie Group G, the WZW model also has a "dual" formulation where the group is replaced by the homogenous space , where is the complexification of , and it has been argued that the former can be (re-)constructed from the latter. For , the space can be identified with the three dimensional hyperbolic space and, in physics, the corresponding CFT has been studied as the simplest example of the AdS/CFT correspondence. A surprising correspondence between the -WZW CFT and the Liouville CFT was found by Ribault and Teschner and later generalised by Hikida and Shomerus. This correspondence has been dubbed by Gaiotto-Teschner as the "quantum analytic Langlands correspondence" since the analytic Langlands correspondence of Etingof, Frenkel and Kazhdan seems to emerge in its formal semi classical limit. In this paper we give a rigorous construction of the path integral for the -WZW model on a closed Riemann surface , twisted by an arbitrary smooth gauge field on . Using the probabilistic path integral we prove a correspondence between the correlation functions of the primary fields of the model and those of Liouville CFT extending the expressions proposed by Ribault-Teschner and by Hikida-Schomerus to this general setup.

Paper Structure

This paper contains 42 sections, 38 theorems, 486 equations.

Table of Contents

  1. Introduction
  2. Sigma models
  3. WZW model
  4. Hyperbolic space sigma models
  5. Probabilistic definition of the $\mathbb{H}^3$-WZW path integral
  6. The $\mathbb{H}^3$-WZW -- Liouville correspondence
  7. Gauged WZW model and holomorphic vector bundles
  8. Probabilistic definition, general case.
  9. General $\mathbb{H}^3$-WZW -- LCFT correspondence
  10. Preliminary background
  11. Geometric background
  12. Determinant of Laplacians
  13. Green's function and resolvent of the Laplacian
  14. Gaussian Free Field
  15. Metric regularisation of the GFF
  16. ...and 27 more sections

Key Result

Theorem 1.1

For $k>3$ with $b=(k-2)^{-1/2}\in (0,1)$, the path integral defPI_intro defines a CFT with central charge $\mathbf{c}(k):=\frac{3k}{k-2}$. For ${\bf z}=(z_1,\dots,z_m)$ distinct points in $\hat{\mathbb{C}}$ and $\boldsymbol{j}=(j_1,\dots,j_m)$ some associated weights in $(-\infty,-1/2)$ such that $s where $\delta_{V^0}$ is the uniform measure supported on the hyperplane $V^0=\{ \boldsymbol{\mu}\in

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Definition 3.1
  • Definition 3.2: GFF-Witten pair
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Definition 3.5
  • ...and 62 more