Conformal Field Theory Of AdS Background With Ramond-Ramond Flux

We review a formalism of superstring quantization with manifest six-dimensional spacetime supersymmetry, and apply it to AdS 3 × S 3 backgrounds with Ramond-Ramond flux. The resulting description is a conformal field theory based on a sigma model whose target space is a certain supergroup SU In the RNS formulation of superstring theory, powerful conformal field theory methods of computation are available. A price one pays is that spacetime supersymmetry is not visible as a classical symmetry of the worldsheet action. It can be seen by incorporating spin fields and picture changing [1]. On the other hand, the Green-Schwarz description [2] makes spacetime supersymmetry manifest, but quantization becomes difficult, except in light cone gauge. Though a fully covariant quantization of the Green-Schwarz string is not known, one does know a partial substitute in that it is possible to reformulate the RNS description in terms of Green-Schwarz-like variables, in a way that makes manifest a portion of the spacetime symmetry group. For example [3], one can exhibit four-dimensional Poincaré invariance with manifest N = 1 supersymmetry (or, for Type II superstrings, N = 1 from left-movers and N = 1 from right-movers), giving a framework in which one can study superstring compactification on a Calabi-Yau manifold, with all symmetries more or less manifest. By a further extension of these ideas [4], one can exhibit six-dimensional N = 1 super-Poincaré invariance (or N = 1 for left-movers and N = 1 for right-movers), giving a framework in which all symmetries of superstring compactification on a K3 surface are visible. These constructions are hybrids in which the spacetime symmetries of four or six of the ten dimensions are described in Green-Schwarz-like variables, while the remaining dimensions are described using RNS variables. These constructions may thus loosely, but only loosely, be described as covariant quantization of the Green-Schwarz string in four or six dimensions. There also exists a quantization of the ten-dimensional superstring which, after Wick-rotating, preserves manifest U (5) super-Poincaré invariance [5]. One important property of the RNS approach to Type II superstrings is that in the RNS case, some of the bosonic fields – those from the Ramond-Ramond sector – are represented by spin fields. As a result, it is exceedingly difficult to describe and understand backgrounds in which Ramond-Ramond or RR fields are present in the vacuum. At best, one would have to describe such backgrounds by worldsheet Lagrangians with spin …