[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](https://mathoverflow.net/q/315201/27004)

**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](https://mathoverflow.net/q/315201/27004).

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?