Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Geometry of Refined Topological Strings

Mina Aganagic, Miranda C. N. Cheng, Robbert Dijkgraaf, Daniel Krefl, Cumrun Vafa

TL;DR

<3-5 sentence high-level summary> The work shows that branes in refined topological strings produce wave-functions obeying a multi-time Schrödinger equation, with the times given by refined moduli and a clear NS-limit where the problem becomes time-independent. This framework unifies open/closed string dynamics, matrix-model duals, and the Nekrasov–Shatashvili relation to quantum integrable systems by interpreting side-by-side monodromies, Virasoro/BPZ constraints, and quantum Riemann-surface data as a single quantum-mechanical system. The authors validate the approach through genus-zero and genus-one examples, matching refined topological-string amplitudes and Liouville/CFT structures, and provide a conceptual mechanism for how NS limits extract Bethe-ansatz data from gauge theories. The results offer a robust, broadly applicable bridge between refined topological strings, matrix models, and four-dimensional ${\cal N}=2$ gauge dynamics, with several natural extensions proposed (e.g., Toda-type generalizations and non-NS regimes).

Abstract

We consider branes in refined topological strings. We argue that their wave-functions satisfy a Schrödinger 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.

Quantum Geometry of Refined Topological Strings

TL;DR

<3-5 sentence high-level summary> The work shows that branes in refined topological strings produce wave-functions obeying a multi-time Schrödinger equation, with the times given by refined moduli and a clear NS-limit where the problem becomes time-independent. This framework unifies open/closed string dynamics, matrix-model duals, and the Nekrasov–Shatashvili relation to quantum integrable systems by interpreting side-by-side monodromies, Virasoro/BPZ constraints, and quantum Riemann-surface data as a single quantum-mechanical system. The authors validate the approach through genus-zero and genus-one examples, matching refined topological-string amplitudes and Liouville/CFT structures, and provide a conceptual mechanism for how NS limits extract Bethe-ansatz data from gauge theories. The results offer a robust, broadly applicable bridge between refined topological strings, matrix models, and four-dimensional gauge dynamics, with several natural extensions proposed (e.g., Toda-type generalizations and non-NS regimes).

Abstract

We consider branes in refined topological strings. We argue that their wave-functions satisfy a Schrödinger 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.

Paper Structure

This paper contains 28 sections, 269 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review of Quantum Geometry in Topological Strings
  3. Quantum Mechanics and Loop Corrections
  4. Branes and Period Shifts
  5. Refined Topological Strings
  6. The A-model, its M-Theory Lift and Deformation
  7. Branes in the Refined A-model Topological String
  8. $\Omega$-Deformation of ${\cal N}=2,$ $d=4$ Theories
  9. The $\beta$-Ensemble as a Refinement of the Topological B-model
  10. Refinement, Topological Strings and Quantum Mechanics
  11. Matrix Models and Schrödinger Equations
  12. Time-Dependent Schrödinger Equation
  13. Virasoro Constraints and Hidden Conformal Symmetry
  14. Quantum Riemann Surface and the NS Limit
  15. Brane Monodromies and the NS Partition Function
  16. ...and 13 more sections

Figures (2)

  • Figure 1: The skeleton of the B-model geometry of local $\mathbb P^1\times\mathbb P^1$. A-period contours are drawn in red while B-period contours in blue.
  • Figure 2: The skeleton of the B-model geometry of local $\mathbb P^2$. A-period contours are drawn in red while B-period contours in blue. Case (i) shows the effective geometry of the parametrization (\ref{['P2curve1']}) under the limit $z\ll 1$, while (ii) of (\ref{['P2curve2']}).