On the BRST Cohomology of Superstrings with/without Pure Spinors

Abstract

We replace our earlier condition that physical states of the superstring have non-negative grading by the requirement that they are analytic in a new real commuting constant t which we associate with the central charge of the underlying Kac-Moody superalgebra.The analogy with the twisted N=2 SYM theory suggests that our covariant superstring is a twisted version of another formulation with an equivariant cohomology. We prove that our vertex operators correspond in one-to-one fashion to the vertex operators in Berkovits' approach based on pure spinors. Also the zero-momentum cohomology is equal in both cases. Finally, we apply the methods of equivariant cohomology to the superstring, and obtain the same BRST charge as obtained earlier by relaxing the pure spinor constraints.

Relating the Green-Schwarz and Pure Spinor Formalisms for the Superstring

“…The BRST operator in these extended pure spinor formalisms have been related to the RNS BRST operator [10], to the light-cone GS spectrum [11], and to the original pure spinor BRST operator [12].…”

Theta expansion of first massive vertex operator in pure spinor

“…In [1], it was shown explicitly that the pure spinor BRST cohomology includes the states which appear at first massive level of open string as described above. The unintegrated form of the vertex operator for these states was also constructed in [1] (see also [19][20][21][22]). …”

Covariant map between Ramond-Neveu-Schwarz and pure spinor formalisms for the superstring

“…The definition of (1.8) for the unconstrained Λ α might be useful for understanding the relation with "extended" versions of the pure spinor formalism such as [20][21][22][23] in which the spinor ghosts were unconstrained. After defining Γ m and Γ m as in (1.5), the RNS γ ghost only appears in even powers so it is convenient to define a new ghost variable γ ≡ (γ) 2 .…”