Self-Duality and Self-Similarity of Little String Orbifolds

TL;DR

This paper explores the self-duality and self-similarity of certain little string theories derived from M-brane orbifolds, revealing that their BPS degeneracies can be reconstructed from a fundamental configuration, indicating deep symmetries.

Contribution

It demonstrates that in a specific limit, the free energies of these theories exhibit self-similarity, allowing reconstruction of complex configurations from a basic one, and uncovers their T-duality symmetry.

Findings

BPS degeneracies are self-similar across different (N,M) configurations.

In the Nekrasov-Shatashvili-limit, free energies scale with NM times the fundamental case.

The theories exhibit self-duality under T-duality in a special moduli space region.

Abstract

We study a class of ${\cal N}=(1,0)$ little string theories obtained from orbifolds of M-brane configurations. These are realised in two different ways that are dual to each other: either as $M$ parallel M5-branes probing a transverse $A_{N-1}$ singularity or $N$ M5-branes probing an $A_{M-1}$ singularity. These backgrounds can further be dualised into toric, non-compact Calabi-Yau threefolds $X_{N,M}$ which have double elliptic fibrations and thus give a natural geometric description of T-duality of the little string theories. The little string partition functions are captured by the topological string partition function of $X_{N,M}$. We analyse in detail the free energies $\Sigma_{N,M}$ associated with the latter in a special region in the K\"ahler moduli space of $X_{N,M}$ and discover a remarkable property: in the Nekrasov-Shatashvili-limit, $\Sigma_{N,M}$ is identical to $NM$ times $\Sigma_{1,1}$. This entails that the BPS degeneracies for any $(N,M)$ can uniquely be reconstructed from the $(N,M)=(1,1)$ configuration, a property we refer to as self-similarity. Moreover, as $\Sigma_{1,1}$ is known to display a number of recursive structures, BPS degeneracies of little string configurations for arbitrary $(N,M)$ as well acquire additional symmetries. These symmetries suggest that in this special region the two little string theories described above are self-dual under T-duality.