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Probabilistic conformal blocks for Liouville CFT on the torus

Promit Ghosal, Guillaume Remy, Xin Sun, Yi Sun

Abstract

Virasoro conformal blocks are a family of important functions defined as power series via the Virasoro algebra. They are a fundamental input to the conformal bootstrap program for 2D conformal field theory (CFT) and are closely related to four dimensional supersymmetric gauge theory through the Alday-Gaiotto-Tachikawa correspondence. The present work provides a probabilistic construction of the 1-point toric Virasoro conformal block for central change greater than 25. More precisely, we construct an analytic function using a probabilistic tool called Gaussian multiplicative chaos (GMC) and prove that its power series expansion coincides with the 1-point toric Virasoro conformal block. The range $(25,\infty)$ of central charges corresponds to Liouville CFT, an important CFT originating from 2D quantum gravity and bosonic string theory. Our work reveals a new integrable structure underlying GMC and opens the door to the study of non-perturbative properties of Virasoro conformal blocks such as their analytic continuation and modular symmetry. Our proof combines an analysis of GMC with tools from CFT such as Belavin-Polyakov-Zamolodchikov differential equations, operator product expansions, and Dotsenko-Fateev type integrals.

Probabilistic conformal blocks for Liouville CFT on the torus

Abstract

Virasoro conformal blocks are a family of important functions defined as power series via the Virasoro algebra. They are a fundamental input to the conformal bootstrap program for 2D conformal field theory (CFT) and are closely related to four dimensional supersymmetric gauge theory through the Alday-Gaiotto-Tachikawa correspondence. The present work provides a probabilistic construction of the 1-point toric Virasoro conformal block for central change greater than 25. More precisely, we construct an analytic function using a probabilistic tool called Gaussian multiplicative chaos (GMC) and prove that its power series expansion coincides with the 1-point toric Virasoro conformal block. The range of central charges corresponds to Liouville CFT, an important CFT originating from 2D quantum gravity and bosonic string theory. Our work reveals a new integrable structure underlying GMC and opens the door to the study of non-perturbative properties of Virasoro conformal blocks such as their analytic continuation and modular symmetry. Our proof combines an analysis of GMC with tools from CFT such as Belavin-Polyakov-Zamolodchikov differential equations, operator product expansions, and Dotsenko-Fateev type integrals.

Paper Structure

This paper contains 37 sections, 64 theorems, 310 equations.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Summary of method
  4. Outlook
  5. Probabilistic construction of the conformal block
  6. Definition of Gaussian multiplicative chaos
  7. Definition and analyticity of $\mathcal{G}^\alpha_{\gamma, P}(q)$
  8. Nekrasov partition function and Zamolodchikov's recursion
  9. A one-step reduction
  10. Analyticity: proof of Lemma \ref{['lem:a-props']}
  11. BPZ equation for deformed conformal blocks
  12. Proof of Proposition \ref{['prop:u-block']}
  13. Proof of Theorem \ref{['thm:bpz']}
  14. From the BPZ equation to hypergeometric differential equations
  15. Proof of Proposition \ref{['prop:bpz-to-hgf']}
  16. ...and 22 more sections

Key Result

Theorem 1.1

For $\gamma \in (0, 2)$, $\alpha \in (-\frac{4}{\gamma}, Q)$, and $P \in \mathbb{R}$, the probabilistic conformal block $\mathcal{G}^\alpha_{\gamma, P}(q)$ admits an analytic extension on a complex neighborhood of $q=0$, whose $q$-series expansion around $q=0$ agrees with $\mathcal{F}^\alpha_{\gamma

Theorems & Definitions (128)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 2.3: GMC
  • Definition 2.4
  • Proposition 2.5
  • Lemma 2.6
  • proof : Proof of Proposition \ref{['prop:analytic-G']} given Lemma \ref{['lem:a-props']}
  • Remark 2.7
  • ...and 118 more