Relative cohomology

Let us consider the BRST cohomology:

(40)

where  .    The cohomology of this complex will be called:

(41)

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On the other hand, consider the relative cohomology group:

(42)

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We claim that (42) coincides with (41):

(43)

In fact (41) is the usual (slightly generalized) BRST cohomology of the Berkovits formalism.
And (42) is a new description.

Let me explain why (41) is the usual BRST cohomology. Let us take as the space dual to the universal enveloping :

  (44)

On the RHS, we have acting on the space:

(45)

This is the space of sections (or rather, Taylor series of sections) of the pure spinor bundle over (was explained here). In other words, this is the “usual” BRST complex on the pure spinor worldsheet.

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And now let us see what we have on the left hand side of (43).