T-duality For Orientifolds and Twisted KR-theory

TL;DR

This paper introduces a new version of KR-theory accounting for sign variations of O-planes in orientifold string theories, computes it for specific compactifications, and verifies its consistency with T-duality.

Contribution

It defines and computes a novel 'KR-theory with a sign choice' for orientifolds with mixed O-plane signs, extending mathematical understanding of D-brane charge classification.

Findings

Defined 'KR-theory with a sign choice' for orientifolds.

Computed the theory for circle and 2-torus compactifications.

Confirmed T-duality consistency for elliptic curve orientifolds.

Abstract

D-brane charges in orientifold string theories are classified by the KR-theory of Atiyah. However, this is assuming that all O-planes have the same sign. When there are O-planes of different signs, physics demands a "KR-theory with a sign choice" which up until now has not been studied by mathematicians (with the unique exception of Moutuou, who didn't have a specific application in mind). We give a definition of this theory and compute it for orientifold theories compactified on a circle and 2-torus. We also explain how and why additional "twisting" is implemented. We show that our results satisfy all possible T-duality relationships for orientifold string theories on elliptic curves, which will be studied further in subsequent work.