Topological Superconductors on Superstring Worldsheets

TL;DR

This paper explores how different GSO projections in superstring theory relate to topological superconductors on the worldsheet, leading to a classification of D-branes via KO-theory and clarifying the structure of unoriented Type 0 and Type I string theories.

Contribution

It introduces a topological perspective on GSO projections using invertible phases and classifies D-branes with KO-theory, providing new insights into superstring worldsheet structures.

Findings

GSO projections correspond to topological superconductors on the worldsheet.

D-branes are classified by KO-theory groups in these models.

No new GSO projections exist for Type I string theory beyond known classifications.

Abstract

We point out that different choices of Gliozzi-Scherk-Olive (GSO) projections in superstring theory can be conveniently understood by the inclusion of fermionic invertible phases, or equivalently topological superconductors, on the worldsheet. This allows us to find that the unoriented Type $\rm 0$ string theory with $\Omega^2=(-1)^{\sf f}$ admits different GSO projections parameterized by $n$ mod 8, depending on the number of Kitaev chains on the worldsheet. The presence of $n$ boundary Majorana fermions then leads to the classification of D-branes by $KO^n(X)\oplus KO^{-n}(X)$ in these theories, which we also confirm by the study of the D-brane boundary states. Finally, we show that there is no essentially new GSO projection for the Type $\rm I$ worldsheet theory by studying the relevant bordism group, which classifies corresponding invertible phases. This paper provides the details for the results announced in the letter \cite{Kaidi:2019pzj}.