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Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories

Marcos Marino

Abstract

In these lectures I give a pedagogical presentation of some of the recent progress in supersymmetric Chern-Simons-matter theories, coming from the use of localization and matrix model techniques. The goal is to provide a simple derivation of the exact interpolating function for the free energy of ABJM theory on the three-sphere, which implies in particular the N^{3/2} behavior at strong coupling. I explain in detail part of the background needed to understand this derivation, like holographic renormalization, localization of path integrals, and large N techniques in matrix models

Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories

Abstract

In these lectures I give a pedagogical presentation of some of the recent progress in supersymmetric Chern-Simons-matter theories, coming from the use of localization and matrix model techniques. The goal is to provide a simple derivation of the exact interpolating function for the free energy of ABJM theory on the three-sphere, which implies in particular the N^{3/2} behavior at strong coupling. I explain in detail part of the background needed to understand this derivation, like holographic renormalization, localization of path integrals, and large N techniques in matrix models

Paper Structure

This paper contains 42 sections, 547 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Supersymmetric Chern--Simons--matter theories
  3. Conventions
  4. Vector multiplet and supersymmetric Chern--Simons theory
  5. Supersymmetric matter multiplets
  6. ABJM theory
  7. A brief review of Chern--Simons theory
  8. Perturbative approach
  9. The one--loop contribution
  10. The free energy at weak coupling
  11. Chern--Simons theory on ${\mathbb S}^3$
  12. Matter fields
  13. ABJM theory at weak coupling
  14. Strong coupling and AdS duals
  15. Holographic renormalization
  16. ...and 27 more sections

Figures (1)

  • Figure 1: The contour ${\cal C}$ encircling the support of the density of eigenvalues, which can be regarded as a contour in the spectral curve $y(p)$.

Theorems & Definitions (1)

  • Example 7.1