Comments on the little string partition functions of $K3\times T^2$ via the refined topological vertex

TL;DR

This paper computes partition functions for deformed M5-branes on K3×T^2 using advanced mathematical tools, revealing modular properties and connections to gauge theories and string theory invariants.

Contribution

It introduces a novel computation of partition functions via the refined topological vertex and Borcherds lift, linking string theory, modular forms, and gauge theory deformations.

Findings

Explicit modular covariant expressions for genus two Siegel modular forms.

Connection between genus-one free energy and Ray-Singer Torsion.

Demonstration of automorphic properties of the partition functions.

Abstract

We compute partition functions of the deformed multiple M5-branes theory on $K3\times T^2$ using the refined topological vertex formalism and the Borcherds lift. The deformation is related to the mass deformation in the corresponding four dimensional $N=4$ $SU(N)$ gauge theory on $K3$. The seed of the Borcherd-lift is calculated by taking the universal part of the type IIb little string free energy of the CY3-fold $X_{N,1}$. We provide explicit modular covariant expressions, as expansions in the mass parameter $m$, of the genus two Siegel modular forms produced by the Borcherds lift of the first few seed functions. We also discuss the relation between genus-one free energy and Ray-Singer Torsion, and the automorphic properties of the latter.