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Localization on three-dimensional manifolds

Brian Willett

TL;DR

This article develops and applies localization for 3d $N=2$ theories on compact manifolds, enabling exact partition functions on $S^3_b$, $S^3_b/\mathbb{Z}_p$, and $S^2\times S^1$ via finite-dimensional matrix models. By embedding theories into rigid 3d supergravity backgrounds, it constructs SUSY-preserving actions on curved spaces and derives explicit $S^3$ and $S^3_b$ partition functions, lens-space generalizations, and the $S^2\times S^1$ index, including operator insertions and real mass/FI deformations. The framework yields powerful checks of dualities, reveals factorization into holomorphic blocks, and connects to the $3d$-3d correspondence and Higgs-branch localization, with broad implications for RG flows ($F$-theorem/maximization), holography (ABJM), and lower-dimensional reductions. Overall, the localization program provides a unifying, exact toolset for understanding strongly coupled 3d $N=2$ dynamics on diverse geometries and their interrelations with topology and higher-dimensional theories.

Abstract

In this review article we describe the localization of three dimensional N=2 supersymmetric theories on compact manifolds, including the squashed sphere, S^3_b, the lens space, S^3_b/Z_p, and S^2 x S^1. We describe how to write supersymmetric actions on these spaces, and then compute the partition functions and other supersymmetric observables by employing the localization argument. We briefly survey some applications of these computations.

Localization on three-dimensional manifolds

TL;DR

This article develops and applies localization for 3d theories on compact manifolds, enabling exact partition functions on , , and via finite-dimensional matrix models. By embedding theories into rigid 3d supergravity backgrounds, it constructs SUSY-preserving actions on curved spaces and derives explicit and partition functions, lens-space generalizations, and the index, including operator insertions and real mass/FI deformations. The framework yields powerful checks of dualities, reveals factorization into holomorphic blocks, and connects to the -3d correspondence and Higgs-branch localization, with broad implications for RG flows (-theorem/maximization), holography (ABJM), and lower-dimensional reductions. Overall, the localization program provides a unifying, exact toolset for understanding strongly coupled 3d dynamics on diverse geometries and their interrelations with topology and higher-dimensional theories.

Abstract

In this review article we describe the localization of three dimensional N=2 supersymmetric theories on compact manifolds, including the squashed sphere, S^3_b, the lens space, S^3_b/Z_p, and S^2 x S^1. We describe how to write supersymmetric actions on these spaces, and then compute the partition functions and other supersymmetric observables by employing the localization argument. We briefly survey some applications of these computations.

Paper Structure

This paper contains 22 sections, 319 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of $3d$ ${\cal N}=2$ Field Theories
  3. Supersymmetry on the $3$-sphere
  4. Round sphere
  5. Supersymmetry on general $3$-manifolds
  6. Squashed $3$-sphere
  7. Localization of the partition function on the $3-$sphere
  8. Round $S^3$
  9. Squashed $S^3$
  10. Operator insertions
  11. Lens spaces
  12. Localization on $L(p,1)$
  13. $S^2 \times S^1$ partition function
  14. Supersymmetric backgrounds
  15. Localization on $S^2 \times S^1$
  16. ...and 7 more sections

Figures (3)

  • Figure 1: The (squashed) sphere in toroidal coordinates, cut open along the torus at $\chi=\frac{\pi}{4}$.
  • Figure 2: Two dual brane configurations. In the top, the low energy gauge theory is $U(3) \times U(3)$ with bifundamental hypermultiplets and a fundamental hypermultiplet in one of the gauge factors. In the bottom, the low energy gauge theory is $U(3)$ with an adjoint hypermultiplet and two fundamental hypermultiplets. These gauge theories are mirror dual.
  • Figure 3: In the top, the Coulomb branch of the undeformed theory, with a single interacting fixed point at $\sigma_j=0$. In the bottom, the Coulomb branch of the theory with a real mass $m$, which deforms the moduli space, and there are now two points with non-trivial SCFTs.