Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Curves and D-Modules

Robbert Dijkgraaf, Lotte Hollands, Piotr Sułkowski

TL;DR

The paper develops a unified framework where chiral fermions on quantum curves, realized physically via I-branes with a B-field, are governed by holonomic ${\cal D}$-modules and noncommuting coordinates with $[x,y]=\lambda$. This approach yields a prescription to extract all-genus partition functions and invariants across diverse systems, including matrix models, $c=1$ string theory, and Seiberg-Witten geometries, by quantizing spectral curves and constructing associated fermionic states whose determinants reproduce KP/Toda tau-functions. It demonstrates concrete D-module realizations for double-scaled and Hermitian matrix models (recovering KdV and topological gravity), the $c=1$ string via two asymptotic patches and Fourier-type gluing, and dual Nekrasov-Okounkov partition functions from SW curves, all tied to topological string theory and its five-dimensional uplift. The framework reveals deep connections among matrix models, integrable hierarchies, topological strings, and gauge theories, and highlights both perturbative and nonperturbative facets, including Stokes phenomena and wall-crossing, with potential links to the geometric Langlands program and quantum field theory dualities.

Abstract

In this article we continue our study of chiral fermions on a quantum curve. This system is embedded in string theory as an I-brane configuration, which consists of D4 and D6-branes intersecting along a holomorphic curve in a complex surface, together with a B-field. Mathematically, it is described by a holonomic D-module. Here we focus on spectral curves, which play a prominent role in the theory of (quantum) integrable hierarchies. We show how to associate a quantum state to the I-brane system, and subsequently how to compute quantum invariants. As a first example, this yields an insightful formulation of (double scaled as well as general Hermitian) matrix models. Secondly, we formulate c=1 string theory in this language. Finally, our formalism elegantly reconstructs the complete dual Nekrasov-Okounkov partition function from a quantum Seiberg-Witten curve.

Quantum Curves and D-Modules

TL;DR

The paper develops a unified framework where chiral fermions on quantum curves, realized physically via I-branes with a B-field, are governed by holonomic -modules and noncommuting coordinates with . This approach yields a prescription to extract all-genus partition functions and invariants across diverse systems, including matrix models, string theory, and Seiberg-Witten geometries, by quantizing spectral curves and constructing associated fermionic states whose determinants reproduce KP/Toda tau-functions. It demonstrates concrete D-module realizations for double-scaled and Hermitian matrix models (recovering KdV and topological gravity), the string via two asymptotic patches and Fourier-type gluing, and dual Nekrasov-Okounkov partition functions from SW curves, all tied to topological string theory and its five-dimensional uplift. The framework reveals deep connections among matrix models, integrable hierarchies, topological strings, and gauge theories, and highlights both perturbative and nonperturbative facets, including Stokes phenomena and wall-crossing, with potential links to the geometric Langlands program and quantum field theory dualities.

Abstract

In this article we continue our study of chiral fermions on a quantum curve. This system is embedded in string theory as an I-brane configuration, which consists of D4 and D6-branes intersecting along a holomorphic curve in a complex surface, together with a B-field. Mathematically, it is described by a holonomic D-module. Here we focus on spectral curves, which play a prominent role in the theory of (quantum) integrable hierarchies. We show how to associate a quantum state to the I-brane system, and subsequently how to compute quantum invariants. As a first example, this yields an insightful formulation of (double scaled as well as general Hermitian) matrix models. Secondly, we formulate c=1 string theory in this language. Finally, our formalism elegantly reconstructs the complete dual Nekrasov-Okounkov partition function from a quantum Seiberg-Witten curve.

Paper Structure

This paper contains 20 sections, 272 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Quantum curves and invariants
  3. Krichever correspondence
  4. B-field, ${\cal D }$-modules and quantum curve
  5. Prescription
  6. Matrix model geometries
  7. Double scaled matrix models and the KdV hierarchy
  8. ${\cal D }$-module for topological gravity
  9. ${\cal D }$-module for Hermitean matrix models
  10. Gaussian example
  11. Conifold and $c=1$ string
  12. ${\cal D }$-module description of the $c=1$ string
  13. Seiberg-Witten geometries
  14. Dual partition functions and fermionic correlators
  15. Fermionic correlators as ${\cal D }$-modules
  16. ...and 5 more sections

Figures (9)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 4 more figures