**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.


For the projection to be consistent, the set $A$ of operators retained by the projection must satisfy:

- Closure — The operator product expansion (OPE) of any two operators in $A$ contains only operators which are in $A$.

- Mutual locality — There are no branch cuts in the OPE of any two operators in the set $A$.

- Modular invariance — The partition function on the two-torus of the theory containing only the operators in $A$ respects modular invariance.

My naive questions are that
> (1)  whether there is a **mathematical branch** highly relevant for formulating **GSO (Gliozzi-Scherk-Olive) projection** and determine the *consistency of projection*?

My guess is that the "Modular invariance," "Closure" and "Mutual locality" may have something to do with the **symplectic geometry and Lagrangian submanifolds (of certain space)**. But I am not sure what is the precise mathematics to put these ideas together?

> (2) These above are spin structure determined by Ramond (periodic) and Neveu-Schwarz (antiperiodic), say $H^1(M, \mathbb{Z}_2)$. I wonder, do we have higher dimensional analogous $H^d(M, \mathbb{Z}_2)$? If so, what is the analogous GSO projection for $H^d(M, \mathbb{Z}_2)$?