# GSO projection and $H^d(M, \mathbb{Z}_2)$

This follows up the comment which suggests that asking the later 2nd part of subquestion in "GSO (Gliozzi-Scherk-Olive) projection and its Mathematics" as a new different question

GSO (Gliozzi-Scherk-Olive) projection is an ingredient used in constructing a consistent model in superstring theory. The projection is a selection of a subset of possible vertex operators in the worldsheet conformal field theory (CFT)—usually those with specific worldsheet fermion number and periodicity conditions. Such a projection is necessary to obtain a consistent worldsheet CFT.

For terminology, for a compact 1-manifold as a $$S^1$$ circle, there are two spin structures, let one be periodic or antiperiodic in going around the circle. In string theory, these are called

• Ramond (periodic)

• Neveu-Schwarz (antiperiodic)

of spin structures.

There are conditions for the projection to be consistent: Closure, Mutual locality, Modular invariance. See the earlier post.

My following question is that

The usual GSO projection above have spin structures determined by Ramond (periodic) and Neveu-Schwarz (antiperiodic), say $$H^1(M, \mathbb{Z}_2)$$, where $$M$$ is the spacetime manifold.

I wonder, do we have higher dimensional analogous $$H^d(M, \mathbb{Z}_2)$$, for higher integer $$d$$? If so, what is the analogous GSO projection for $$H^d(M, \mathbb{Z}_2)$$? How do we consistently define it say on a 2-manifold (say a 2-torus or genus-$$g$$ Riemann surface)? Or is there further higher $$d$$ generalization?