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# Advances in Theoretical and Mathematical Physics

## Volume 8 (2004)

### Number 5

### Four Dimensional Abelian Duality and DL (2,**Z**) Action in Three Dimensional Conformal Field Theory

Pages: 895 – 936

DOI: http://dx.doi.org/10.4310/ATMP.2004.v8.n5.a5

#### Author

#### Abstract

Recently, Witten showed that there is a natural action of the group *SL*(2, **Z**) on the space of 3 dimensional conformal field theories with U (1) global symmetry and a chosen coupling of the symmetry current to a background gauge field on a 3-fold *N*. He further argued that, for a class of conformal field theories, in the nearly Gaussian limit, this *SL*(2, **Z**) action may be viewed as a holographic image of the well-known *SL*(2, **Z**) Abelian duality of a pure *U* (1) gauge theory on AdS-like 4-folds *M* bounded by *N*, as dictated by the AdS/CFT correspondence. However, he showed that explicitly only for the generator *T*; for the generator *S*, instead, his analysis remained conjectural. In this paper, we propose a solution of this problem. We derive a general holographic formula for the nearly Gaussian generating functional of the correlators of the symmetry current and, using this, we show that Witten's conjecture is indeed correct when *N* = *S*^{3}. We further identify a class of homology 3-spheres *N* for which Witten's conjecture takes a particular simple form.