Instantons on ALE spaces and orbifold partitions

TL;DR

This paper links instanton partition functions on ALE spaces to affine characters and introduces 'orbifold partitions', connecting them to Frobenius partitions and free fermion systems, revealing new combinatorial structures in gauge theories.

Contribution

It introduces orbifold partitions and demonstrates their relation to instanton partition functions on ALE spaces, providing a new combinatorial perspective.

Findings

Partition functions match affine characters.

Orbifold partitions relate to Frobenius partitions.

Explicit interpretation via free fermion system.

Abstract

We consider N=4 theories on ALE spaces of $A_{k-1}$ type. As is well known, their partition functions coincide with $A_{k-1}$ affine characters. We show that these partition functions are equal to the generating functions of some peculiar classes of partitions which we introduce under the name 'orbifold partitions'. These orbifold partitions turn out to be related to the generalized Frobenius partitions introduced by G. E. Andrews some years ago. We relate the orbifold partitions to the blended partitions and interpret explicitly in terms of a free fermion system.