Torus one-point correlation numbers in minimal Liouville gravity

TL;DR

This paper introduces a first-principles method for calculating one-point correlation numbers on a torus in minimal Liouville gravity, using higher equations of motion to simplify the integrals involved.

Contribution

It extends the application of higher equations of motion in Liouville CFT from spherical to toroidal topology for minimal Liouville gravity.

Findings

Method reduces moduli integrals to boundary contributions

Results agree with matrix model calculations

Explicit boundary contributions can be calculated

Abstract

We present a method for the first principles calculation of tachyon one-point amplitudes in $(2,2p+1)$ minimal Liouville gravity defined on a torus. The method is based on the higher equations of motion in the Liouville CFT. These equations were earlier successfully applied for analytic calculations of the amplitudes in the spherical topology. We show that this approach allows to reduce the moduli integrals entering the definition of the torus amplitudes to certain boundary contributions, which can be calculated explicitly. The results agree with the calculations performed in the matrix models approach.