Determinant representations for Garvan formulas

TL;DR

This paper derives explicit determinant formulas for powers of the classical eta function using correlation functions in conformal field theory, extending Garvan's formulas to genus two Riemann surfaces.

Contribution

It introduces a novel approach using determinant representations from conformal field theory to obtain formulas for eta function powers on higher genus surfaces.

Findings

Explicit determinant formulas for eta function powers on genus two surfaces

Extension of Garvan's formulas to higher genus cases

Connection between conformal field theory and elliptic function identities

Abstract

In this note, we demonstrate how determinant representations for correlation functions in conformal field theory can be used to derive explicit determinant formulas for powers of the classical $\eta$-function, expressed via deformed elliptic functions with parameters. In particular, we obtain counterparts of Garvan's formulas for the modular discriminant corresponding to the genus two Riemann surface case.