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Exact calculations beyond charge neutrality in timelike Liouville field theory

Sourav Chatterjee

TL;DR

The paper establishes exact, nonperturbative results for timelike Liouville theory beyond charge neutrality at the solvable coupling $b=1/\sqrt{2}$. By exploiting a Vandermonde-determinantal structure, it derives Mellin–Barnes representations for zero-, one-, two-, and a resonant three-point function, and rigorously handles the delicate integration over the zero mode via Gaussian regularization and contour prescriptions. It proves explicit formulas for the fixed-zero-mode correlators, their zero-mode integrals, and the renormalized partition function, obtaining distributional limits in momentum-conserving configurations and revealing poles consistent with timelike DOZZ predictions. The Hankel-contour analysis further clarifies how contour choices influence the results, highlighting both consistencies and ambiguities in existing physics prescriptions. Overall, the work provides the first rigorously controlled family of exact calculations in timelike Liouville theory outside charge neutrality and offers a concrete framework for comparing different zero-mode integration schemes.

Abstract

Timelike Liouville field theory (also known as imaginary Liouville theory or imaginary Gaussian multiplicative chaos) is expected to describe two-dimensional quantum gravity in a positive-curvature regime, but its path integral is not a probability measure and rigorous exact computations are currently available only in the charge-neutral (integer screening) case. In this paper we show that at the special coupling $b=1/\sqrt{2}$, the Coulomb-gas expansion of the timelike path integral becomes explicitly computable beyond charge neutrality. The reason is that the $n$-fold integrals generated by the interaction acquire a Vandermonde/determinantal structure at $b=1/\sqrt{2}$, which allows exact evaluation in terms of classical special functions. We derive Mellin-Barnes type representations (involving the Barnes $G$-function and, in a three-point case, Gauss hypergeometric functions) for the zero- and one-point functions, for an antipodal two-point function, and for a three-point function with a resonant insertion $α_2=b$. We then address the subtle zero-mode integration: after a Gaussian regularization we obtain an explicit renormalized partition function $C(1/\sqrt{2},μ)=e(4π\sqrt2 μ)^{-1}$, identify distributional limits in the physically relevant regime $α_j=\frac{1}{2}Q+\mathrm{i} P_j$, and compare with the Hankel-contour prescription recently proposed in the physics literature. These results provide the first rigorously controlled family of exact calculations in timelike Liouville theory outside charge neutrality.

Exact calculations beyond charge neutrality in timelike Liouville field theory

TL;DR

The paper establishes exact, nonperturbative results for timelike Liouville theory beyond charge neutrality at the solvable coupling . By exploiting a Vandermonde-determinantal structure, it derives Mellin–Barnes representations for zero-, one-, two-, and a resonant three-point function, and rigorously handles the delicate integration over the zero mode via Gaussian regularization and contour prescriptions. It proves explicit formulas for the fixed-zero-mode correlators, their zero-mode integrals, and the renormalized partition function, obtaining distributional limits in momentum-conserving configurations and revealing poles consistent with timelike DOZZ predictions. The Hankel-contour analysis further clarifies how contour choices influence the results, highlighting both consistencies and ambiguities in existing physics prescriptions. Overall, the work provides the first rigorously controlled family of exact calculations in timelike Liouville theory outside charge neutrality and offers a concrete framework for comparing different zero-mode integration schemes.

Abstract

Timelike Liouville field theory (also known as imaginary Liouville theory or imaginary Gaussian multiplicative chaos) is expected to describe two-dimensional quantum gravity in a positive-curvature regime, but its path integral is not a probability measure and rigorous exact computations are currently available only in the charge-neutral (integer screening) case. In this paper we show that at the special coupling , the Coulomb-gas expansion of the timelike path integral becomes explicitly computable beyond charge neutrality. The reason is that the -fold integrals generated by the interaction acquire a Vandermonde/determinantal structure at , which allows exact evaluation in terms of classical special functions. We derive Mellin-Barnes type representations (involving the Barnes -function and, in a three-point case, Gauss hypergeometric functions) for the zero- and one-point functions, for an antipodal two-point function, and for a three-point function with a resonant insertion . We then address the subtle zero-mode integration: after a Gaussian regularization we obtain an explicit renormalized partition function , identify distributional limits in the physically relevant regime , and compare with the Hankel-contour prescription recently proposed in the physics literature. These results provide the first rigorously controlled family of exact calculations in timelike Liouville theory outside charge neutrality.
Paper Structure (53 sections, 58 theorems, 558 equations, 2 figures)

This paper contains 53 sections, 58 theorems, 558 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Correlation functions and the charge-neutral barrier
  4. Expanding the path integral and why $b=\frac{1}{\sqrt2}$ is special
  5. Results
  6. The zero-point function at $b=\frac{1}{\sqrt{2}}$
  7. The one-point function at $b=\frac{1}{\sqrt{2}}$
  8. The two-point function at $b=\frac{1}{\sqrt{2}}$
  9. The three-point function at $b=\alpha_2=\frac{1}{\sqrt{2}}$
  10. Proof sketch and main ideas
  11. Proofs
  12. Zero-point function
  13. Proof of Theorem \ref{['zerocthm']}
  14. Proof of Theorem \ref{['zerothm']}
  15. Proof of Theorem \ref{['zerohankel']}
  16. ...and 38 more sections

Key Result

Lemma 1.1

The integral in equation andef and the series in equation zcor are absolutely convergent if Moreover, if $b\in (0,1)$, then $C({\boldsymbol \alpha}, {\boldsymbol x}, b, \mu,c)$ and $a_n$ are continuous in $({\boldsymbol \alpha},{\boldsymbol x},c)$ and analytic in $({\boldsymbol \alpha},c)$ in the region where $\operatorname{Re}(\alpha_j)>-\frac{1}{2b}$ for each $j$ and $x_1,\ldots,x_k$ are where

Figures (2)

  • Figure 1: Hankel-type contour $\mathcal{C}$ running from $\frac{\pi \mathrm{i}}{b}+\infty$ to $\frac{\pi \mathrm{i}}{b}$, then to $0$, and finally to $\infty$. Faded segments indicate the parts extending to infinity.
  • Figure 2: Rectangular contour $C_R$ with vertices $x_0\pm \mathrm{i} R$ and $N+x_0\pm \mathrm{i} R$, traversed counter-clockwise. Poles of $g$ at $0,\dots,N{-}1$ lie inside $C_R$.

Theorems & Definitions (82)

  • Lemma 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • ...and 72 more