String Theory on AdS 3 Revisited ∗

We discuss string theory on AdS 3 × S 3 × M 4 with particular emphasis on unitarity and state-operator correspondence. The AdS-CFT correspondence, in the Minkowski signature, is reexamined by taking into account the only allowed unitary representation: the principal series module of the affine current algebra SL(2,R) supplemented with zero modes. Zero modes play an important role in the description of on-shell states as well as of windings in space-time at the AdS 3 boundary. The theory is presented as part of the supersymmetric WZW model that includes the supergroup SU(2/1, 1) (or OSp(4/2) or D(2, 1; α)) with central extension k. A free field representation is given and the vertex operators are constructed in terms of free fields in SL(2,R) principal series representation bases that are labeled by position space or momentum space at the boundary of AdS 3. The vertex operators have the correct operator products with the currents and stress tensor, all of which are constructed from free fields, including the subtle zero modes. It is shown that as k → ∞, AdS 3 tends to flat 3D-Minkowski space and the AdS 3 vertex operators in momentum space tend to the vertex operators of flat 3D-string theory (furthermore the theory readjusts smoothly in the rest of the dimensions in this limit).