Generalised Complex and Spinor Relations

TL;DR

This paper explores relations between generalized complex and Kähler structures, T-duality, and Courant algebroids, establishing new links and conditions for their compatibility in geometric and physical contexts.

Contribution

It introduces a framework for relations between generalized structures, characterizes their reductions, and connects T-duality with supergravity equations and supersymmetric sigma-models.

Findings

T-duality induces spinor relations linking Dirac operators of T-dual Courant algebroids.

Relations between generalized complex structures are characterized and their reductions are described.

Existence results for T-dual structures and their compatibility with supergravity are demonstrated.

Abstract

Courant algebroid relations are used to define notions of relations between Dirac structures and spinors. It is shown under which circumstances a spinor relation gives a Courant algebroid relation and how it descends to a relation between Dirac structures. A converse to this result is proved: a T-duality relation induces a spinor relation that links the Dirac generating operators defining T-dual Courant algebroids, generalising the isomorphism of twisted cohomology associated with topological T-duality. We introduce the notion of relation between generalised complex structures and characterise their reduction. We also define relations between generalised K\"ahler structures, and rephrase them in terms of bi-Hermitian structures which induce T-duality relations between $\mathcal{N}=(2,2)$ supersymmetric sigma-models. We prove existence results for T-dual structures, and demonstrate compatibility of T-duality relations with Type II supergravity equations.