Dirac generating operators of split Courant algebroids

TL;DR

This paper constructs explicit Dirac operators for split Courant algebroids on vector bundles, showing their squares produce invariants that characterize these structures.

Contribution

It provides an explicit construction of Dirac generating operators for split Courant algebroids and proves their squares yield invariants of the structures.

Findings

Explicit Dirac operators are constructed for split Courant algebroids.

The square of these operators produces invariants of the algebroid.

The approach links Dirac operators with the algebraic structure of split Courant algebroids.

Abstract

Given a vector bundle $A$ over a smooth manifold $M$ such that the square root $\mathcal{L}$ of the line bundle $\wedge^{\mathrm{top}}A^\ast \otimes \wedge^{\mathrm{top}}T^\ast M$ exists, the Clifford bundle associated to the split pseudo-Euclidean vector bundle $(E = A \oplus A^\ast, \langle \cdot, \cdot \rangle)$, admits a spinor bundle $\wedge^\bullet A \otimes \mathcal{L}$, whose section space can be thought of as that of Berezinian half-densities of the graded manifold $A^\ast[1]$. We give an explicit construction of Dirac generating operators of split Courant algebroid (or proto-bialgebroid) structures on $A \oplus A^\ast$ introduced by Alekseev and Xu. We also prove that the square of the Dirac generating operator gives rise to an invariant of the split Courant algebroid.