BRST Cohomology and Nonlocal Conserved Charges

A relation is found between nonlocal conserved charges in string theory and certain ghost-number two states in the BRST cohomology. This provides a simple proof that the nonlocal conserved charges for the superstring in an AdS 5 × S 5 background are BRST-invariant in the pure spinor formalism and are κ-symmetric in the Green-Schwarz formalism. 1. Introduction Maldacena's conjecture that d = 4 N = 4 super-Yang-Mills theory is dual to super-string theory in an AdS 5 ×S 5 background has been difficult to prove since the perturbative descriptions of these two theories do not overlap. To obtain non-perturbative information about the two theories, one possible tool could be integrability and there have been various papers discussing this possibility both on the super-Yang-Mills side and on the superstring side. On the superstring side, Bena, Polchinski and Roiban [1] constructed an infinite set of nonlocal conserved charges for the Green-Schwarz (GS) superstring in an AdS 5 × S 5 background 2 , suggesting an integrable structure. Vallilo [4] then constructed an analogous set of nonlocal conserved charges using the pure spinor formalism for the superstring in an AdS 5 × S 5 background [5]. These nonlocal charges for the superstring were related in [6] to a corresponding set of nonlocal charges on the super-Yang-Mills side. In the GS formalism for the superstring, invariance under κ-transformations is crucial for determining the physical spectrum. For example, classical κ-symmetry is preserved in a curved background when the background satisfies the low-energy supergravity equations of motion, so onshell massless vertex operators must be κ-symmetric at least at the classical level. Although quantum κ-transformations and massive GS vertex operators are not yet understood, it is reasonable to expect that all physical GS states should be κ-symmetric. This would imply that the conserved charges for physical symmetries should also be κ-symmetric. In the pure spinor formalism for the superstring, the role of κ-symmetry is replaced by BRST invariance. In this case, quantization and massive vertex operators are well-understood, and the physical spectrum is described by states in the BRST cohomology. So physical symmetries in the pure spinor formalism must be BRST-invariant. Surprisingly, it has not been previously verified if the nonlocal conserved charges of [1] are κ-symmetric, or if the nonlocal conserved charges of [4] are BRST-invariant. This led some people (including this author) to conclude that the charges were not κ-symmetric and to question their physical …