Conformal Field Theory for the Superstring in a Ramond-Ramond Plane Wave Background

A quantizable worldsheet action is constructed for the superstring in a supersymmetric plane wave background with Ramond-Ramond flux. The action is manifestly invariant under all isometries of the background and is an exact worldsheet conformal field theory. It has recently been recognized that the Penrose limit of the AdS 5 × S 5 background with Ramond-Ramond (R-R) flux is a supersymmetric plane wave [1], and the superstring in this background is described in the light-cone Green-Schwarz (GS) formalism [2] by a quadratic worldsheet action [3] [4]. The spectrum of physical states with nonzero P + momentum can easily be computed using this light-cone GS action, which has been useful for checking aspects of the Maldacena conjecture [5]. However, to compute tree amplitudes or to describe physical states with vanishing P + momentum, the light-cone GS formalism is problematic even in a flat background. As will be discussed below, the problems associated with light-cone gauge become even more troublesome in the plane wave background since there is no J +− isometry. Furthermore, it would be convenient to have a quantizable worldsheet action in which all isometries of the plane wave background are manifest, and not just those isometries which commute with the light-cone gauge choice. Although the covariant GS action [6][3] can classically describe the plane wave background in a manner which preserves all isometries, it is not known how to covariantly quantize the GS action. Over the last eight years, an alternative formalism for the su-perstring has been developed which can be covariantly quantized [7][8] [9]. In a flat background , this formalism has a quadratic worldsheet action and tree amplitudes can be easily computed in a manifestly super-Poincaré covariant manner. The formalism generalizes to curved backgrounds [10] and was used to construct quantizable actions for the superstring In this paper, this alternative formalism will be used to construct a quantizable action for the superstring in the plane wave background obtained by taking the Penrose limit of these AdS D 2 × S D 2 backgrounds. By using the " pure spinor " or " hybrid " versions of the formalism, all isometries of the plane wave background can be made manifest. The action is not quadratic, which is not surprising since the action for the bosonic plane wave background is not quadratic when written in conformal gauge. However, the action is simpler than its AdS D 2 × S …