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Rigorous results for timelike Liouville field theory

Sourav Chatterjee

TL;DR

This work provides a rigorous probabilistic foundation for timelike Liouville field theory, where the action carries a negative sign in the kinetic term. By developing a theory of wrong-sign Gaussian variables and a right way to perform analytic continuation, the authors construct and evaluate timelike correlation functions, proving a timelike DOZZ formula in a charge-neutral regime and giving explicit k-point expressions. They extend these results to the sphere and plane via Coulomb gas and complex Selberg integral techniques, study SL(2, C) invariance and poles of the structure constants, and establish a detailed semiclassical limit with heavy operators that concentrates on a complex critical point, connecting to JT gravity-like dynamics. Collectively, these results advance the mathematical understanding of timelike Liouville theory and its connections to quantum gravity and probabilistic structures, while providing rigorous tools for exploring general parameter regions and pole structures.

Abstract

Liouville field theory has long been a cornerstone of two-dimensional quantum field theory and quantum gravity, which has attracted much recent attention in the mathematics literature. Timelike Liouville field theory is a version of Liouville field theory where the kinetic term in the action appears with a negative sign, which makes it closer to a theory of quantum gravity than ordinary (spacelike) Liouville field theory. Making sense of this `wrong sign' requires a theory of Gaussian random variables with negative variance. Such a theory is developed in this paper, and is used to prove the timelike DOZZ formula for the $3$-point correlation function when the parameters satisfy the so-called `charge neutrality condition'. Expressions are derived also for the $k$-point correlation functions for all $k\ge 3$, and it is shown that these functions approach the correct semiclassical limits as the coupling constant is sent to zero

Rigorous results for timelike Liouville field theory

TL;DR

This work provides a rigorous probabilistic foundation for timelike Liouville field theory, where the action carries a negative sign in the kinetic term. By developing a theory of wrong-sign Gaussian variables and a right way to perform analytic continuation, the authors construct and evaluate timelike correlation functions, proving a timelike DOZZ formula in a charge-neutral regime and giving explicit k-point expressions. They extend these results to the sphere and plane via Coulomb gas and complex Selberg integral techniques, study SL(2, C) invariance and poles of the structure constants, and establish a detailed semiclassical limit with heavy operators that concentrates on a complex critical point, connecting to JT gravity-like dynamics. Collectively, these results advance the mathematical understanding of timelike Liouville theory and its connections to quantum gravity and probabilistic structures, while providing rigorous tools for exploring general parameter regions and pole structures.

Abstract

Liouville field theory has long been a cornerstone of two-dimensional quantum field theory and quantum gravity, which has attracted much recent attention in the mathematics literature. Timelike Liouville field theory is a version of Liouville field theory where the kinetic term in the action appears with a negative sign, which makes it closer to a theory of quantum gravity than ordinary (spacelike) Liouville field theory. Making sense of this `wrong sign' requires a theory of Gaussian random variables with negative variance. Such a theory is developed in this paper, and is used to prove the timelike DOZZ formula for the -point correlation function when the parameters satisfy the so-called `charge neutrality condition'. Expressions are derived also for the -point correlation functions for all , and it is shown that these functions approach the correct semiclassical limits as the coupling constant is sent to zero

Paper Structure

This paper contains 29 sections, 47 theorems, 432 equations.

Table of Contents

  1. Introduction
  2. The action for timelike Liouville theory
  3. Correlation functions of timelike Liouville theory
  4. Main results
  5. Correlation functions as wrong sign expectations
  6. A theory of wrong sign Gaussian random variables
  7. Problem formulation
  8. Ideas that do not work
  9. The wrong way to do analytic continuation
  10. Analytic continuation done right
  11. Examples
  12. Integration by parts
  13. Application to the backward heat equation
  14. Semiclassical approximation
  15. Correlation functions in a special region
  16. ...and 14 more sections

Key Result

Theorem 1.3.1

Suppose that $k\ge 3$, $\operatorname{Re}(\alpha_j) > -1/2b$ for each $j$, and the parameter $w := (Q-\sum_{j=1}^k \alpha_j)/b$ is a positive integer. Let $x_1,\ldots,x_k$ be distinct points on $S^2$, none of which are the north pole $e_3 = (0,0,1)$. Let $z_1,\ldots,z_k$ be their stereographic proje

Theorems & Definitions (88)

  • Theorem 1.3.1: Formula for $k$-point correlations
  • Theorem 1.3.2: Timelike DOZZ formula
  • Theorem 1.3.3: Semiclassical limit with heavy operators
  • Lemma 1: Nonexistence of real critical points
  • Theorem 1.3.4: Validity of the semiclassical limit
  • Lemma 2
  • proof
  • Theorem 2.4.1
  • proof
  • Theorem 2.6.1
  • ...and 78 more