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Supersymmetric localization in two dimensions

Francesco Benini, Bruno Le Floch

TL;DR

This article surveys localization techniques in two-dimensional supersymmetric gauge theories, detailing how to construct Lagrangians on curved spaces and compute exact partition functions on S^2, hemispheres, and related backgrounds as well as the elliptic genus on T^2. It presents two complementary localization channels for N=(2,2) theories on S^2: a Coulomb-branch form and a Higgs-branch form, both yielding exact, RG-invariant results that depend on FI/theta parameters, twisted masses, and background fluxes. The review also covers local operator insertions, Ω-deformed twists, and general backgrounds, and explains how the Jeffery–Kirwan residue organizes the elliptic genus computations for N=(2,2) and N=(0,2) theories. It then discusses dualities, including mirror symmetry and Seiberg-like dualities for unitary groups, plus variants and N=(0,2) trialities, with results cross-validated through sphere partition functions and elliptic genera. Overall, the work establishes a rich framework connecting exact QFT results to geometric, algebraic, and duality structures, with wide implications for Calabi–Yau physics, string worldsheet theories, and beyond.

Abstract

This is an introductory review to localization techniques in supersymmetric two-dimensional gauge theories. In particular we describe how to construct Lagrangians of N=(2,2) theories on curved spaces, and how to compute their partition functions and certain correlators on the sphere, the hemisphere and other curved backgrounds. We also describe how to evaluate the partition function of N=(0,2) theories on the torus, known as the elliptic genus. Finally we summarize some of the applications, in particular to probe mirror symmetry and other non-perturbative dualities.

Supersymmetric localization in two dimensions

TL;DR

This article surveys localization techniques in two-dimensional supersymmetric gauge theories, detailing how to construct Lagrangians on curved spaces and compute exact partition functions on S^2, hemispheres, and related backgrounds as well as the elliptic genus on T^2. It presents two complementary localization channels for N=(2,2) theories on S^2: a Coulomb-branch form and a Higgs-branch form, both yielding exact, RG-invariant results that depend on FI/theta parameters, twisted masses, and background fluxes. The review also covers local operator insertions, Ω-deformed twists, and general backgrounds, and explains how the Jeffery–Kirwan residue organizes the elliptic genus computations for N=(2,2) and N=(0,2) theories. It then discusses dualities, including mirror symmetry and Seiberg-like dualities for unitary groups, plus variants and N=(0,2) trialities, with results cross-validated through sphere partition functions and elliptic genera. Overall, the work establishes a rich framework connecting exact QFT results to geometric, algebraic, and duality structures, with wide implications for Calabi–Yau physics, string worldsheet theories, and beyond.

Abstract

This is an introductory review to localization techniques in supersymmetric two-dimensional gauge theories. In particular we describe how to construct Lagrangians of N=(2,2) theories on curved spaces, and how to compute their partition functions and certain correlators on the sphere, the hemisphere and other curved backgrounds. We also describe how to evaluate the partition function of N=(0,2) theories on the torus, known as the elliptic genus. Finally we summarize some of the applications, in particular to probe mirror symmetry and other non-perturbative dualities.

Paper Structure

This paper contains 27 sections, 182 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. N=(2,2) gauge theories on spheres
  3. Multiplets, Lagrangians and supersymmetry
  4. BPS equations
  5. Coulomb branch localization
  6. The A-system
  7. Higgs branch localization
  8. One-loop determinants
  9. Other N=(2,2) curved-space results
  10. Local operator insertions
  11. Twisted multiplets on the sphere
  12. Localization on the hemisphere
  13. $\boldsymbol{\Omega}$-deformed A-twist on the sphere
  14. General supersymmetric backgrounds
  15. Elliptic genera of N=(2,2) and N=(0,2) theories
  16. ...and 12 more sections

Figures (1)

  • Figure 1: Quivers for three $\mathcal{N}{=}(0,2)$ theories related by triality: $K_i=(N_1+N_2+N_3)/2-N_i$ and $(N_P,N_\Phi,N_\Psi)$ is one of $(N_2,N_3,N_1)$, $(N_3,N_1,N_2)$, $(N_1,N_2,N_3)$. The table lists how the chirals $P,\Phi$ and Fermis $\Psi,\Gamma,\Omega$ transform under the gauge group $U(K)=U\bigl(\frac{1}{2}(N_P+N_\Phi-N_\Psi)\bigr)$ and the classical flavor symmetry $S[U(N_P)\times U(N_\Phi)\times U(N_\Psi)\times U(2)]$ which loses one Abelian factor due to a flavor-gauge anomaly.