Quantum Geometry of Refined Topological Strings

TL;DR

This paper explores the quantum geometry of refined topological strings, showing that brane wave-functions obey Schrödinger equations and linking these findings to integrable systems and N=2 gauge theories.

Contribution

It demonstrates that brane wave-functions in refined topological strings satisfy Schrödinger equations and connects this to integrable systems and gauge theories, especially in specific limits.

Findings

Brane wave-functions satisfy Schrödinger equations depending on multiple times.

In a certain limit, the partition function obeys a time-independent Schrödinger equation.

Provides an explanation for the link between integrable systems and N=2 gauge theories.

Abstract

We consider branes in refined topological strings. We argue that their wave-functions satisfy a Schr\"odinger equation depending on multiple times and prove this in the case where the topological string has a dual matrix model description. Furthermore, in the limit where one of the equivariant rotations approaches zero, the brane partition function satisfies a time-independent Schroedinger equation. We use this observation, as well as the back reaction of the brane on the closed string geometry, to offer an explanation of the connection between integrable systems and N=2 gauge systems in four dimensions observed by Nekrasov and Shatashvili.