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The disk 1-point function in timelike Liouville theory

Gaston Giribet, Bruno Sivilotti

Abstract

We compute the disk 1-point function in timelike Liouville theory. Using the Coulomb gas formalism and analytically continuing in the number of screening operators, we derive an explicit formula, which is shown to satisfy the correct reflection symmetry, to have the expected self-dual properties, to fulfill the bootstrap shift-equations, and to reduce to previous known results in the appropriate limits. In the limit of zero cosmological constant, our result reproduces the one recently obtained in arXiv:2505.09390.

The disk 1-point function in timelike Liouville theory

Abstract

We compute the disk 1-point function in timelike Liouville theory. Using the Coulomb gas formalism and analytically continuing in the number of screening operators, we derive an explicit formula, which is shown to satisfy the correct reflection symmetry, to have the expected self-dual properties, to fulfill the bootstrap shift-equations, and to reduce to previous known results in the appropriate limits. In the limit of zero cosmological constant, our result reproduces the one recently obtained in arXiv:2505.09390.
Paper Structure (13 sections, 94 equations, 3 figures)

This paper contains 13 sections, 94 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Spacelike disk 1-point function
  3. Timelike disk 1-point function
  4. Consistency checks
  5. Timelike disk 1-point function at $\Lambda =0$
  6. Comparison with our result
  7. Resonant correlators at $\Lambda =0$
  8. The zero-mode: a cautionary note
  9. Reflection symmetry
  10. Fixed-length 1-point function
  11. Self-duality
  12. Bootstrap shift-equations
  13. Summary

Figures (3)

  • Figure 1: Scheme of the Liouville 1-point function.
  • Figure 2: Integration contours $\mathcal{C}$ and $\mathcal{C}_R$ in the $c\in \mathbb{C}$ plane.
  • Figure 3: Scheme of how the integration contour for the integrals in $\Lambda_B\in \mathbb{C}$ involved changes when going from the spacelike (left) to the timelike (right) fixed-length 1-point function. Here, $\kappa=\sqrt{{\Lambda }/{\sin(\pi \beta^2)}}$