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Les Houches lectures on matrix models and topological strings

Marcos Marino

TL;DR

The notes establish a cohesive framework linking zero-dimensional matrix models to topological string theories, emphasizing the B-model open/closed duality and the Dijkgraaf–Vafa correspondence. By deriving one-cut and multicut spectral curves, master-field data, and geometric transitions, they show how perturbative matrix-model expansions resum into closed-string geometries on Calabi–Yau manifolds, yielding exact results for nonperturbative superpotentials in certain gauge theories. The treatment also includes a concrete Chern–Simons realization: the S^3 CS theory is recast as a Stieltjes–Wigert matrix model, whose planar limit reproduces the closed-string resolved conifold data, linking GV-type invariants to matrix-model free energies. Overall, the work demonstrates how matrix-model techniques provide powerful, calculable handles on open/closed string dualities and large-$N$ transitions in both B- and A-model contexts, with broad implications for gauge/string dualities and CY geometry.

Abstract

In these lecture notes for the Les Houches School on Applications of Random Matrices in Physics we give an introduction to the connections between matrix models and topological strings. We first review some basic results of matrix model technology and then we focus on type B topological strings. We present the main results of Dijkgraaf and Vafa describing the spacetime string dynamics on certain Calabi-Yau backgrounds in terms of matrix models, and we emphasize the connection to geometric transitions and to large N gauge/string duality. We also use matrix model technology to analyze large N Chern-Simons theory and the Gopakumar-Vafa transition.

Les Houches lectures on matrix models and topological strings

TL;DR

The notes establish a cohesive framework linking zero-dimensional matrix models to topological string theories, emphasizing the B-model open/closed duality and the Dijkgraaf–Vafa correspondence. By deriving one-cut and multicut spectral curves, master-field data, and geometric transitions, they show how perturbative matrix-model expansions resum into closed-string geometries on Calabi–Yau manifolds, yielding exact results for nonperturbative superpotentials in certain gauge theories. The treatment also includes a concrete Chern–Simons realization: the S^3 CS theory is recast as a Stieltjes–Wigert matrix model, whose planar limit reproduces the closed-string resolved conifold data, linking GV-type invariants to matrix-model free energies. Overall, the work demonstrates how matrix-model techniques provide powerful, calculable handles on open/closed string dualities and large- transitions in both B- and A-model contexts, with broad implications for gauge/string dualities and CY geometry.

Abstract

In these lecture notes for the Les Houches School on Applications of Random Matrices in Physics we give an introduction to the connections between matrix models and topological strings. We first review some basic results of matrix model technology and then we focus on type B topological strings. We present the main results of Dijkgraaf and Vafa describing the spacetime string dynamics on certain Calabi-Yau backgrounds in terms of matrix models, and we emphasize the connection to geometric transitions and to large N gauge/string duality. We also use matrix model technology to analyze large N Chern-Simons theory and the Gopakumar-Vafa transition.

Paper Structure

This paper contains 15 sections, 261 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Matrix models
  3. Basics of matrix models
  4. Matrix model technology I: saddle-point analysis
  5. Matrix model technology II: orthogonal polynomials
  6. Type B topological strings and matrix models
  7. The topological B model
  8. The open type B model and its string field theory description
  9. Topological strings and matrix models
  10. Open string amplitudes and multicut solutions
  11. Master field and geometric transition
  12. Extensions and applications
  13. Type A topological strings, Chern-Simons theory and matrix models
  14. Solving the Chern-Simons matrix model
  15. Extensions

Figures (8)

  • Figure 1: This diagram summarizes the different relations between closed topological strings, open topological strings, and gauge theories.
  • Figure 2: The index structure of the field $M_{ij}$ in the adjoint representation of $U(N)$ is represented through a double line.
  • Figure 3: The propagator in the double line notation.
  • Figure 4: The cubic vertex in the double line notation.
  • Figure 5: Contracting two cubic vertices in the double line notation: the $N^3$ contribution.
  • ...and 3 more figures