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On Mirror Symmetry in Three Dimensional Abelian Gauge Theories

Anton Kapustin, Matthew J. Strassler

Abstract

We present an identity relating the partition function of N=4 supersymmetric QED to that of its dual under mirror symmetry. The identity is a generalized Fourier transform. Many known properties of abelian theories can be derived from this formula, including the mirror transforms for more general gauge and matter content. We show that N=3 Chern-Simons QED and N=4 QED with BF-type couplings are conformal field theories with exactly marginal couplings. Mirror symmetry acts on these theories as strong-weak coupling duality. After identifying the mirror of the gauge coupling (sometimes called the ``magnetic coupling'') we construct a theory which is exactly mirror -- at all scales -- to N=4 SQED. We also study vortex-creation operators in the large $N_f$ limit.

On Mirror Symmetry in Three Dimensional Abelian Gauge Theories

Abstract

We present an identity relating the partition function of N=4 supersymmetric QED to that of its dual under mirror symmetry. The identity is a generalized Fourier transform. Many known properties of abelian theories can be derived from this formula, including the mirror transforms for more general gauge and matter content. We show that N=3 Chern-Simons QED and N=4 QED with BF-type couplings are conformal field theories with exactly marginal couplings. Mirror symmetry acts on these theories as strong-weak coupling duality. After identifying the mirror of the gauge coupling (sometimes called the ``magnetic coupling'') we construct a theory which is exactly mirror -- at all scales -- to N=4 SQED. We also study vortex-creation operators in the large limit.

Paper Structure

This paper contains 13 sections, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Mirror Transform is a Fourier Transform
  4. Inverse and repeated Fourier transforms
  5. Mapping of operators
  6. The convolution theorem
  7. ${\cal N}=2$ mirror symmetry
  8. ${\cal N}=4$ theories with BF couplings
  9. ${\cal N}=3$ theories with Chern-Simons couplings
  10. Piecewise Mirror Transformations
  11. Mirror symmetry away from the infrared limit
  12. Vortex-creation operators
  13. Outlook