\(T\)-duality for transitive Courant algebroids (Q6183448)

\(T\)-duality has its origin in theoretical physics (\(T\)-duality for principal torus bundles, by \textit{P. Bouwknegt} et al. [JHEP 03, Paper No. 18, 18 p. (2004; \url{doi:10.1088/1126-6708/2004/03/018})], but can be understood in the framework of generalized geometry as an isomorphism of Courant algebroids. These are bundles \(E\to M\) endowed with a non-degenerate pairing, a bracket and an anchor map \(E\to TM\) satisfying natural compatibility conditions. They were first introduced in [\textit{Z. Liu} et al., J. Differ. Geom. 45, No. 3, 547--574 (1997; Zbl 0885.58030)]. In this framework, \(T\)-duality was established for exact [\textit{G. R. Cavalcanti} and \textit{M. Gualtieri}, in: A celebration of the mathematical legacy of Raoul Bott. Based on the conference, CRM, Montreal, Canada, June 9--13, 2008. Providence, RI: American Mathematical Society (AMS). 341--365 (2010; Zbl 1200.53062)] and heterotic [\textit{D. Baraglia} and \textit{P. Hekmati}, Adv. Theor. Math. Phys. 19, No. 3, 613--672 (2015; Zbl 1342.81395)] Courant algebroids, both particular cases of transitive Courant algebroids (those whose anchor map is surjective). The present paper undertakes the technical endeavour to find sufficient conditions for a transitive Courant algebroid to have a \(T\)-dual (Theorem 85) and shows that, in the generality of transitive Courant algebroids, \(T\)-duality intertwines the canonical Dirac generating operators and maps invariant spinors to invariant spinors (Theorem 79). Previously known results for exact and heterotic Courant algebroids are then recovered as particular cases of this general setup.