On Mirror Symmetry in Three Dimensional Abelian Gauge Theories

TL;DR

This paper establishes a mathematical identity linking the partition functions of N=4 supersymmetric QED and its mirror dual, revealing how mirror symmetry acts as a strong-weak coupling duality in three-dimensional abelian gauge theories.

Contribution

It introduces a generalized Fourier transform identity for partition functions, enabling derivation of mirror transforms and demonstrating exact mirror symmetry at all scales for N=4 SQED.

Findings

Partition function identity as a generalized Fourier transform

Mirror symmetry acts as a strong-weak coupling duality

Construction of a theory exactly mirror to N=4 SQED at all scales

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.