Quantum mirror curve of periodic chain geometry

TL;DR

This paper computes mirror curves and their quantizations for chain geometries in topological string theory, revealing connections to hypergeometric functions and potential links between 5d and 6d gauge theories.

Contribution

It introduces the calculation of mirror curves for periodic chain geometries and explores their quantization, extending the understanding of topological strings on toric Calabi-Yau threefolds.

Findings

Derived mirror curves for N-chain and periodic N-chain geometries.

Identified difference equations involving elliptic hypergeometric functions.

Suggested a connection between large N limits and higher-dimensional gauge theories.

Abstract

The mirror curves enable us to study B-model topological strings on non-compact toric Calabi--Yau threefolds. One of the method to obtain the mirror curves is to calculate the partition function of the topological string with single brane. In this paper, we discuss two types of geometries; one is the chain of $N$ $\mathbb{P}^1$'s which we call `$N$-chain geometry,' the other is the chain of $N$ $\mathbb{P}^1$'s with a compactification which we call `periodic $N$-chain geometry.' We calculate the partition functions of the open topological strings on these geometries, and obtain the mirror curves and their quantization. Through the computation, we find some difference equations of (elliptic) hypergeometric functions. We also find a relation between the periodic chain and $\infty$-chain geometries, which implies a possible connection between 5d and 6d gauge theories in the large $N$ limit.