Dirac generating operators of split Courant algebroids

Abstract

Given a vector bundle A over a smooth manifold M such that the square root $L$ of the line bundle ${\wedge }^{\mathrm{top}}{A}^{⁎}\otimes {\wedge }^{\mathrm{top}}{T}^{⁎}M$ exists, the Clifford bundle associated to the standard split pseudo-Euclidean vector bundle $(E=A\oplus A^{⁎},〈\cdot ,\cdot 〉)$ admits a spinor bundle ${\wedge }^{•}A\otimes L$, whose section space consists of Berezinian half-densities of the graded manifold $A^{⁎}\left[1\right]$. Inspired by Kosmann-Schwarzbach's formula of deriving operator of split Courant algebroid (or proto-bialgebroid) structures on $A\oplus A^{⁎}$, we give an explicit construction of the associate Dirac generating operator introduced by Alekseev and Xu. We prove that the square of the Dirac generating operator is an invariant of the corresponding split Courant algebroid, and also give an explicit expression of this invariant in terms of modular elements.