GSO projections via SPT phases

TL;DR

The paper shows that GSO projections in superstring theory can be understood as choices of invertible 1+1D fermionic SPT phases on the worldsheet, classified by bordism groups. It demonstrates that the Arf invariant accounts for the two Type II theories and that ABK-type phases yield a mod $8$ family of unoriented Type 0 theories, with an essentially unique Type I worldsheet theory. The approach also connects GSO choices to D-brane K-theory, explaining why Type II branes map to $K^0(X)$ or $K^1(X)$ and how unoriented theories are classified by KO-theory $KO^{+n}(X)\oplus KO^{-n}(X)$. The work provides a unifying topological framework for GSO projections and enumerates possible GSO variants, with acknowledgments to related work by Witten.

Abstract

We point out that the choice of phases in GSO projections can be accounted for by a choice of fermionic SPT phases on the worldsheet of the string. This point of view not only easily explains why there are essentially two type II theories, but also predicts that there are unoriented type 0 theories labeled by n mod 8, and that there is an essentially unique choice of the type I worldsheet theory. We also discuss the relationship between this point of view and the K-theoretic classification of D-branes.