Strings in Nontrivial Gravitino and Ramond-ramond Backgrounds

In this paper we discuss deformations of the BRST operator of the fermionic string. These deformations preserve inlpotency of the BRST operator and correspond to turning on infinitesimal Gravitino and Ramond-Ramond spacetime fields. [email protected] One of the outstanding problems of string theory is to understand the equations of motion for the fields of the theory ( massless and massive ) and the higher symmetries that relate them [1],[2]. Progress towards this direction can be achieved by studying infinitesimal deformations of the SuperVirasoro algebras that preserve superconformal invariance [3]. The problem of finding superconformal deformations is an interesting problem in its own right, but it also provides us with insights into the symmetry structure of string theory since spacetime symmetry transformations are particular superconformal deformations [4]. In a recent paper [5] we constructed a class of superconformal deformations, termed canonical deformations, in terms of superfields (see also [6]). Although properly speaking we need to discuss deformations of two copies of the SuperVirasoro algebra in the remainder of this paper we shall concentrate only on one copy. More specifically we found that a deformation of the form δT (σ) = δTF (σ) + θδT (σ) = ΦF (σ) + θΦB(σ) (1) where ΦB(ΦF ) is the bosonic (fermionic) component of a superfield of dimension ( 1 2 , 1 2 ), preserves superconformal invariance. Canonical deformations have a number of interesting features: superprimary fields of dimension ( 2 , 1 2 ) are in natural correspondence with the physical states of string theory, being the vertex operators. As such they have a nice spacetime interpretation in terms of turning on spacetime fields. Appealing though they are canonical deformations have also significant drawbacks. They do not appear to describe spacetime fermions and R-R bosonic fields which are written in terms of spin fields. Spin fields cannot be written as superfields. These string backgrounds have attracted interest recently due to the conjectured AdS/CFT equivalence [7]. We might attempt to identify the bosonic component ΦB(σ) of the canonical deformation with the appropriate spacetime gravitino vertex operator δT (σ) = ΦB(σ) = Ψ α μ(X)Sαe − φ 2 ∂X + Ψ̃αμ(X)(X)S̃αe − φ̃ 2 ∂X + ∂λΨ α μ(X)Sαe − φ 2 ψ̃ψ̃ + ∂λΨ̃ α μ(X)ψ ψS̃αe − φ̃ 2 . (2) In order to calculate δTF we need to calculate the commutator of ΦB(σ) with the supercurrent TF (σ). The commutator of the vertex operator which is written in terms of spin fields with the supercurrent TF is not well-defined since the corresponding OPE in the complex plane involves branch cut singularities TF (z)ΦB(w) = γλαβ̇∂λΨ α μ(X)S e φ 2 ∂X (z − w) 3 2 + γλαβ̇Ψ α μ(X)S e φ 2 ∂Xλ∂X μ (z − w) 1 2 + γραβ̇∂ρ∂λΨ α μ(X)S e φ 2 ψ̃ψ̃ (z − w) 3 2 + γραβ̇∂λΨ α μ(X)S e φ 2 ∂Xλψ̃ ψ̃ (z − w) 1 2 (3) where we have omitted terms that are either regular or have poles as singularities. This suggests then that the canonical deformations we have constructed in terms of superfields are not the most general solution to the deformation equations.