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

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 $SL(2,\mathbf{Z})$. The generator $S$ of $SL(2,\mathbf{Z})$ 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 $SL(2,\mathbf{Z})$ action in three dimensions is related by the AdS/CFT correspondence to $SL(2,\mathbf{Z})$ duality of low energy U(1) gauge fields in four dimensions.

28 pp, significant added references and reworking of arguments in section 2; v. 3: additional references and clarification

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