Courant Algebroid Relations, T-Dualities and Generalised Ricci Flow
Thomas C. De Fraja, Vincenzo Emilio Marotta, Richard J. Szabo
TL;DR
The paper develops a rigorous relational framework for Courant algebroids to relate divergence operators and generalized Ricci tensors, enabling a geometric understanding of T-duality beyond diffeomorphisms. It defines related divergences via Courant algebroid relations and proves existence/uniqueness results for dual divergences, along with Ricci-type transformation rules (Ricci-Buscher) that explain the dilaton shift under duality. A central achievement is showing that generalized Ricci flow is compatible with generalized T-duality: a unique dual flow exists for dual backgrounds, and fixed points of the flow are preserved under duality, thereby aligning flow dynamics with string background equations. The framework is illustrated through diverse explicit examples, including Hamilton’s cigar soliton, hyperbolic spaces, and Klein bottle fibrations, highlighting both local and global dualities and the role of correspondence spaces.
Abstract
The notion of Courant algebroid relation is used to introduce a definition of relation between divergence operators on Courant algebroids. By introducing invariant divergence operators, a notion of generalised T-duality between divergences is presented through an existence and uniqueness result for related divergence operators on T-dual pairs of exact Courant algebroids, which naturally incorporates the dilaton shift. When combined with the notion of generalised isometry, this establishes circumstances under which generalised Ricci tensors are related, proving that T-duality is compatible with generalised string background equations. This enables an analysis of the compatibility between T-duality and generalised Ricci flow, showing that the T-dual of a solution of generalised Ricci flow is also a solution of generalised Ricci flow. Our constructions are illustrated through many explicit examples.