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 =
S3. We further identify a class of homology 3–spheres
N for which Witten's conjecture takes a particular simple form.