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On correlation numbers $V_{0,4}$ and $V_{1,1}$ in Virasoro Minimal String Theory

Dmitry Khromov, Alexey Litvinov

TL;DR

The paper uses conformal-field-theory techniques for Liouville theories, including spacelike and timelike formulations and higher equations of motion, to analytically verify the Virasoro minimal string volumes $V_{0,4}$ and $V_{1,1}$ proposed in prior work. By treating the volumes as analytic functions of external Liouville momenta and exploiting contour-analytic arguments, triality, and HEM, it shows that the sphere four-point volume is a polynomial in the momenta with specific coefficients that can be fixed in tractable cases. It also establishes a direct relation between the sphere and torus volumes, and provides numerical evidence supporting a general degeneracy-related ansatz for discrete contributions, namely $v_{0,4}^{m,n}=-mn$ for certain degenerate configurations. The results strengthen the field-theoretic foundation of the Virasoro minimal string, connecting Liouville CFT data to Mirzakhani-like volume recursions, with implications for non-critical string amplitudes and matrix-model dualities. The analysis relies on polynomial growth assumptions for volumes and paves the way for extending the approach to more general $(g,n)$ via further control of analytic structures.

Abstract

We study correlation numbers in Virasoro minimal string \cite{Collier:2023cyw}. Using analytic properties of correlation functions in spacelike and timelike Liouville theories, we verify exact expressions for correlation numbers for the four punctured sphere and the once punctured torus conjectured in \cite{Collier:2023cyw}.

On correlation numbers $V_{0,4}$ and $V_{1,1}$ in Virasoro Minimal String Theory

TL;DR

The paper uses conformal-field-theory techniques for Liouville theories, including spacelike and timelike formulations and higher equations of motion, to analytically verify the Virasoro minimal string volumes and proposed in prior work. By treating the volumes as analytic functions of external Liouville momenta and exploiting contour-analytic arguments, triality, and HEM, it shows that the sphere four-point volume is a polynomial in the momenta with specific coefficients that can be fixed in tractable cases. It also establishes a direct relation between the sphere and torus volumes, and provides numerical evidence supporting a general degeneracy-related ansatz for discrete contributions, namely for certain degenerate configurations. The results strengthen the field-theoretic foundation of the Virasoro minimal string, connecting Liouville CFT data to Mirzakhani-like volume recursions, with implications for non-critical string amplitudes and matrix-model dualities. The analysis relies on polynomial growth assumptions for volumes and paves the way for extending the approach to more general via further control of analytic structures.

Abstract

We study correlation numbers in Virasoro minimal string \cite{Collier:2023cyw}. Using analytic properties of correlation functions in spacelike and timelike Liouville theories, we verify exact expressions for correlation numbers for the four punctured sphere and the once punctured torus conjectured in \cite{Collier:2023cyw}.

Paper Structure

This paper contains 37 sections, 220 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Liouville theory
  3. Conformal bootstrap for Liouville field theory
  4. Degenerate fields in Liouville field theory
  5. Spacelike Liouville field theory
  6. Analytic continuation of OPE and degenerate fields
  7. Pinching poles are from series $\Circled{1}$ and $\Circled{6}$ and from $\Circled{3}$ and $\Circled{8}$.
  8. Pinching poles are from series $\Circled{1}$ and $\Circled{8}$ or from $\Circled{2}$ and $\Circled{7}$.
  9. Higher equations of motion
  10. Analytic structure of general N-point correlation function
  11. Triality transformation for 4-point functions
  12. Analytic structure of the 4-point function
  13. Two degenerate fields or two screening conditions.
  14. One degenerate field and a screening condition.
  15. Timelike Liouville field theory
  16. ...and 22 more sections

Figures (4)

  • Figure 1: Structure of poles of the spacelike Liouville thee-point function $C(-P,P_1,P_2)$ (shown by black bullets) for real momenta $P_2>P_1>0$ and of OPE coefficients $\beta_{\boldsymbol{\lambda}}(P)$ (shown by red bullets). The poles of OPE coefficients are exactly canceled by the zeroes of three-point function (provided that the correlation function is at least $4$-point). The contour of integration is the real axis.
  • Figure 2:
  • Figure 3: The "on-shell" poles are drawn for $P_0\in\mathbb{R}$. The two ways to analytically continue the discrete contributions in $P_0^2$ are shown by the two arrows. The region shaded in red is where the poles lie if $\mathop{\mathrm{Re}}\nolimits P_0^2<0$. The contour of integration is shown in blue.
  • Figure 4: Numerical results for the $v_{0,4}^{1,1}(P_2, P_2, P_2)$ and $v_{0,4}^{2,1}(P_2, P_2, P_2)$. The hypothesis is $v_{0,4}^{m,n}(P_2, P_2, P_2) = -mn$, shown as a dashed line. The values of $b$ are as follows: $b_1 = \frac{1}{e}$, $b_2 = \frac{1}{\pi}$, $b_3 = \frac{1}{2} \left(\frac{1}{e} + \frac{1}{\pi}\right)$, $b_4 = \frac{e}{\pi}$.