Surface Defects in $A$-type Little String Theories

TL;DR

This paper investigates surface defects in $A$-type Little String Theories, providing a combinatorial formula for their non-perturbative BPS partition functions and exploring symmetry preservation and algebraic structures in the Nekrasov-Shatashvili limit.

Contribution

It introduces a combinatorial expression for the non-perturbative BPS partition function of LSTs with surface defects and analyzes symmetry and algebraic structures in the NS limit.

Findings

Non-perturbative symmetries are preserved with defects.

A combinatorial formula for the defect partition function is established.

Cancellation of singularities in the NS limit is explained and generalized.

Abstract

$A$-type Little String Theories (LSTs) are engineered from parallel M5-branes on a circle $\mathbb{S}_\perp^1$, probing a transverse $\mathbb{R}^4/\mathbb{Z}_M$ background. Below the scale of the radius of $\mathbb{S}_\perp^1$, these theories resemble a circular quiver gauge theory with $M$ nodes of gauge group $U(N)$ and matter in the bifundamental representation (or adjoint in the case of $M=1$). In this paper, we study these LSTs in the presence of a surface defect, which is introduced through the action of a $\mathbb{Z}_N$ orbifold that breaks the gauge groups into $[U(1)]^N$. We provide a combinatoric expression for the non-perturbative BPS partition function for this system. This form allows us to argue that a number of non-perturbative symmetries, that have previously been established for the LSTs, are preserved in the presence of the defect. Furthermore, we discuss the Nekrasov-Shatashvili (NS) limit of the defect partition function: focusing in detail on the case $(M,N)=(1,2)$, we analyse two distinct proposals made in the literature. We unravel an algebraic structure that is responsible for the cancellation of singular terms in the NS limit, which we generalise to generic $(M,N)$. In view of the dualities of higher dimensional gauge theories to quantum many-body systems, we provide indications that our combinatoric expression for the defect partition are useful in constructing and analysing quantum integrable systems in the future.