**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 question is that

Whether there is a

mathematical branchhighly relevant for formulatingGSO (Gliozzi-Scherk-Olive) projectionand determine theconsistency 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?