Quantum Geometry of Refined Topological Strings

TLDR: In this paper, the authors consider branes in refined topological strings and show that their wave-functions satisfy a Schrodinger equation depending on multiple times and prove this in the case where the topological string has a dual matrix model description.

Abstract: We consider branes in refined topological strings We argue that their wave-functions satisfy a Schrodinger 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 Schrodinger 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

Figures

Figure 1: The skeleton of the B-model geometry of local P1 × P1. A-period contours are drawn in red while B-period contours in blue.

Figure 2: The skeleton of the B-model geometry of local P2. A-period contours are drawn in red while B-period contours in blue. Case (i) shows the effective geometry of the parametrization (7.18) under the limit z ≪ 1, while (ii) of (7.21).