Sign choices for orientifolds

TL;DR

This paper investigates sign choices for orientifold planes in type II string theory, establishing topological constraints, calculating specific examples, and connecting these choices to twisted K-theory and coboundary maps.

Contribution

It introduces a sequence of invariant p-gerbes to classify sign choices, derives topological constraints, and links sign configurations to twisted K-theory, providing new insights into orientifold topologies.

Findings

Sign choices are classified by invariant p-gerbes with coboundary relations.

Topological constraints limit possible sign configurations in orientifolds.

A connection between sign choices and twisted K-theory is established.

Abstract

We analyse the problem of assigning sign choices to O-planes in orientifolds of type II string theory. We show that there exists a sequence of invariant $p$-gerbes with $p\geq-1$, which give rise to sign choices and are related by coboundary maps. We prove that the sign choice homomorphisms stabilise with the dimension of the orientifold and we derive topological constraints on the possible sign configurations. Concrete calculations for spherical and toroidal orientifolds are carried out, and in particular we exhibit a four-dimensional orientifold where not every sign choice is geometrically attainable. We elucidate how the $K$-theory groups associated with invariant $p$-gerbes for $p=-1,0,1$ interact with the coboundary maps. This allows us to interpret a notion of $K$-theory due to Gao and Hori as a special case of twisted $KR$-theory, which consequently implies the homotopy invariance and Fredholm module description of their construction.