SL(2,Z) Action On Three-Dimensional Conformal Field Theories With Abelian Symmetry

TL;DR

This paper explores the action of the SL(2,Z) group on 3D conformal field theories with U(1) symmetry, linking background gauge field transformations to dualities and mirror symmetry.

Contribution

It elucidates how SL(2,Z) acts on 3D CFTs with U(1) symmetry, connecting background gauge field operations to dualities and mirror symmetry via AdS/CFT.

Findings

SL(2,Z) acts on 3D CFTs with U(1) symmetry.

The S operation makes the background gauge field dynamical.

The T operation shifts the Chern-Simons coupling.

Abstract

On the space of three-dimensional conformal field theories with U(1) symmetry and a chosen coupling to a background gauge field, there is a natural action of the group $SL(2,{\bf Z})$. The generator $S$ of $SL(2,{\bf Z})$ acts by letting the background gauge field become dynamical, an operation considered recently by Kapustin and Strassler in explaining three-dimensional mirror symmetry. The other generator $T$ acts by shifting the Chern-Simons coupling of the background field. This $SL(2,{\bf Z})$ action in three dimensions is related by the AdS/CFT correspondence to $SL(2,{\bf Z})$ duality of low energy U(1) gauge fields in four dimensions.