Little String Instanton Partition Functions and Scalar Propagators

TL;DR

This paper explores the instanton partition functions of certain Little String Theories, expressing them through Kronecker-Eisenstein series and revealing recursive structures and geometric interpretations involving Calabi-Yau threefolds.

Contribution

It introduces a novel representation of LST instanton partition functions using Kronecker-Eisenstein series and uncovers recursive relations and geometric links to Calabi-Yau manifolds.

Findings

Partition functions expressed via Kronecker-Eisenstein series

Identification of propagators with Nekrasov subfunctions

Recursive structures relating instanton contributions

Abstract

We discuss a class of Little String Theories (LSTs) whose low energy descriptions are supersymmetric gauge theories on the $\Omega$-background with gauge group $U(N)$ and matter in the adjoint representation. We show that the instanton partition function of these theories can be written in terms of Kronecker-Eisenstein series, which in a particular limit of the deformation parameters of the $\Omega$-background organise themselves into Greens functions of free scalar fields on a torus. We provide a concrete identification between (differences of) such propagators and Nekrasov subfunctions. The latter are also characterised by counting specific holomorphic curves in a Calabi-Yau threefold $X_{N,1}$ which engineers the LST. Furthermore, using the formulation of the partition function in terms of the Kronecker-Eisenstein series, we argue for new recursive structures which relate higher instanton contributions to products of lower ones.