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 $H^1(M, \mathbb{Z}_2)$, 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 $H^d(M, \mathbb{Z}_2)$?