Vertex operators of ghost number three in Type IIB supergravity

We study the cohomology of the massless BRST complex of the Type IIB pure spinor superstring in flat space. In particular, we find that the cohomology at the ghost number three is nontrivial and transforms in the same representation of the supersymmetry algebra as the solutions of the linearized classical supergravity equations. Modulo some finite dimensional spaces, the ghost number three cohomology is the same as the ghost number two cohomology. We also comment on the difference between the naive and semi-relative cohomology, and the role of b-ghost.
Comment: LaTeX 43pp; v2: expanded Introduction, analysis of discrete states, comments on b-ghost and dilaton (Section 5); v3: corrected mistakes, conclusions unchanged