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Mirror Symmetry and Toric Geometry in Three-Dimensional Gauge Theories

Nick Dorey, David Tong

TL;DR

Three-dimensional N=2 gauge theories are studied in the context of mirror symmetry; Coulomb branches can be made compact by dynamical generation of Chern-Simons terms, and a new class of mirror pairs is found in which both Coulomb and Higgs branches have toric descriptions. The review clarifies the field content, the role of FI parameters and real masses, and the quantum corrections that generate effective CS terms and FI renormalisation, which are crucial for the toric interpretation of the moduli spaces. The authors also present a simple self-mirror abelian model in which the Coulomb branch becomes a compact two-sphere and the Higgs branch is its mirror, with the one-loop metric reproducing the standard two-sphere metric. They then develop a general toric-geometry framework for mirror symmetry, showing that the Coulomb branch of Theory A and the Higgs branch of Theory B are the same toric variety encoded by toric data, and illustrate this with a standard complex projective space example that is self-mirror.

Abstract

We study three dimensional gauge theories with N=2 supersymmetry. We show that the Coulomb branches of such theories may be rendered compact by the dynamical generation of Chern-Simons terms and present a new class of mirror symmetric theories in which both Coulomb and Higgs branches have a natural description in terms of toric geometry.

Mirror Symmetry and Toric Geometry in Three-Dimensional Gauge Theories

TL;DR

Three-dimensional N=2 gauge theories are studied in the context of mirror symmetry; Coulomb branches can be made compact by dynamical generation of Chern-Simons terms, and a new class of mirror pairs is found in which both Coulomb and Higgs branches have toric descriptions. The review clarifies the field content, the role of FI parameters and real masses, and the quantum corrections that generate effective CS terms and FI renormalisation, which are crucial for the toric interpretation of the moduli spaces. The authors also present a simple self-mirror abelian model in which the Coulomb branch becomes a compact two-sphere and the Higgs branch is its mirror, with the one-loop metric reproducing the standard two-sphere metric. They then develop a general toric-geometry framework for mirror symmetry, showing that the Coulomb branch of Theory A and the Higgs branch of Theory B are the same toric variety encoded by toric data, and illustrate this with a standard complex projective space example that is self-mirror.

Abstract

We study three dimensional gauge theories with N=2 supersymmetry. We show that the Coulomb branches of such theories may be rendered compact by the dynamical generation of Chern-Simons terms and present a new class of mirror symmetric theories in which both Coulomb and Higgs branches have a natural description in terms of toric geometry.

Paper Structure

This paper contains 5 sections, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review of N=2 Gauge Theories
  3. A Self-Mirror Example
  4. Mirror Symmetry and Toric Geometry
  5. Mirror Symmetry and Toric Geometry

Figures (2)

  • Figure 1: The 5-brane web. The cross denotes the origin of the $(X^5,X^6)$ plane. The dotted lines represent the zero mode of the web.
  • Figure 2: The 5-brane web for various parameters. The dot denotes the position of the D3-brane. The top two diagrams are, from left to right, 2i) and 2ii) and the bottom three 2iii), 2iv) and 2v). In diagram 2iii), the D3-brane is restricted to lie on the interval as shown by the arrow.