AdS from CFT for scalar QED

We construct an explicit bulk dual in anti-de Sitter space, with couplings of order 1/N, for the SU(N)-singlet sector of QED in d space-time dimensions (2<d<4) coupled to N scalar fields. We begin from the bulk dual for the theory of N complex free scalar fields that we constructed in our previous work, and couple this to U(1) gauge fields living on the boundary in order to get the bulk dual of scalar QED (in which the U(1) gauge fields become the boundary value of the bulk vector fields). As in our previous work, the bulk dual is non-local but we can write down an explicit action for it. We focus on the CFTs arising at low energies (or, equivalently, when the U(1) gauge coupling goes to infinity). For d=3 we discuss also the addition of a Chern-Simons term for U(1), modifying the boundary conditions for the bulk gauge field. We also discuss the generalization to QCD, with U(Nc) gauge fields coupled to N scalar fields in the fundamental representation (in the large N limit with fixed Nc).<br />