On the timelike Liouville three-point function

TL;DR

This paper explores the computation of the timelike Liouville three-point function, demonstrating that a Coulomb gas approach and Selberg integral extensions reproduce known results, highlighting differences from spacelike cases.

Contribution

It introduces a Coulomb gas method and analytic extension of Selberg integrals to compute the timelike Liouville three-point function, confirming previous proposals.

Findings

Coulomb gas computation matches the proposed three-point function

Analytic extension of Selberg integrals reproduces the known expression

Differences between timelike and spacelike cases are identified

Abstract

In a recent paper, D. Harlow, J. Maltz, and E. Witten showed that a particular proposal for the timelike Liouville three-point function, originally due to Al. Zamolodchikov and to I. Kostov and V. Petkova, can actually be computed by the original Liouville path integral evaluated on a new integration cycle. Here, we discuss a Coulomb gas computation of the timelike three-point function and show that an analytic extension of the Selberg type integral formulas involved reproduces the same expression, including the adequate normalization. A notable difference with the spacelike calculation is pointed out.