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**Four dimensional Abelian duality and \(\text{SL}(2,\mathbb Z)\) action in three dimensional conformal field theory.**
*(English)*
Zbl 1097.81048

Summary: Recently, Witten showed that there is a natural action of the group \(\text{SL}(2,\mathbb Z)\) 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 \(\text{SL}(2,\mathbb Z)\) action may be viewed as a holographic image of the well-known \(\text{SL}(2,\mathbb Z)\) 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.

### MSC:

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

81T13 | Yang-Mills and other gauge theories in quantum field theory |

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

58J90 | Applications of PDEs on manifolds |

11Z05 | Miscellaneous applications of number theory |