Dirac generating operators of split Courant algebroids

Given a vector bundle $A$ over a smooth manifold $M$ such that the square root $\mathcal{L}$ of the line bundle ${\wedge }^{\mathrm{t}\mathrm{o}\mathrm{p}}{A}^{\ast }\otimes {\wedge }^{\mathrm{t}\mathrm{o}\mathrm{p}}{T}^{\ast }M$ exists, the Clifford bundle associated to the split pseudo-Euclidean vector bundle $(E=A\oplus A^{\ast },⟨\cdot ,\cdot ⟩)$, 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.