Conformal Courant Algebroids and Orientifold T-duality

TL;DR

This paper introduces conformal Courant algebroids, generalizing Courant algebroids with a conformal structure, and develops a T-duality framework for orientifolds with implications for twisted cohomology and $KR$-theory.

Contribution

It defines conformal Courant algebroids, classifies exact cases via pairs $(L,H)$, and constructs a T-duality for orientifolds with potential extensions to twisted $KR$-theory.

Findings

Exact conformal Courant algebroids classified by pairs $(L,H)$.

Construction of T-duality for orientifolds with free involution.

Isomorphism of 4-periodic twisted cohomology under T-duality.

Abstract

We introduce conformal Courant algebroids, a mild generalization of Courant algebroids in which only a conformal structure rather than a bilinear form is assumed. We introduce exact conformal Courant algebroids and show they are classified by pairs $(L,H)$ with $L$ a flat line bundle and $H \in H^3(M,L)$ a degree 3 class with coefficients in $L$. As a special case gerbes for the crossed module $({\rm U}(1) \to \mathbb{Z}_2)$ can be used to twist $TM \oplus T^*M$ into a conformal Courant algebroid. In the exact case there is a twisted cohomology which is 4-periodic if $L^2 = 1$. The structure of Conformal Courant algebroids on circle bundles leads us to construct a T-duality for orientifolds with free involution. This incarnation of T-duality yields an isomorphism of 4-periodic twisted cohomology. We conjecture that the isomorphism extends to an isomorphism in twisted $KR$-theory and give some calculations to support this claim.