The disk 1-point function in timelike Liouville theory

TL;DR

This paper calculates the disk 1-point function in timelike Liouville theory using Coulomb gas formalism, deriving an explicit formula that satisfies key symmetries and properties, and connects to previous results in specific limits.

Contribution

It introduces a new explicit formula for the disk 1-point function in timelike Liouville theory via analytic continuation, satisfying key theoretical consistency conditions.

Findings

Formula satisfies reflection symmetry and self-duality.

Reduces to known results in the zero cosmological constant limit.

Validates the bootstrap shift-equations for the computed function.

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.