Graded Hermitian Clifford modules and geometric Dirac operators

Let $(M,g)$ be a smooth Riemannian manifold of even dimension $2m$, and let $V=V^{+}\oplus V^{-}$ be a finite-rank smooth complex vector bundle with a Hermitian metric linear in its first variable. Write ${\mathrm{C}\mathrm{l}}_{\mathbb{C}}(T^{\ast }M,-g)$ for the complex Clifford bundle generated by cotangent vectors with relation $\xi \eta +\eta \xi =-2g^{-1}(\xi ,\eta )$. A graded Hermitian Clifford action is a smooth real-linear map $c:T^{\ast }M\to {\mathrm{End}}_{\mathbb{C}}(V)$ such that $c(\xi {)}^{\ast }=-c(\xi )$, $c(\xi )V^{\pm }\subseteq V^{\mp }$, and $c(\xi )c(\eta )+c(\eta )c(\xi )=-2g^{-1}(\xi ,\eta )I$. It extends complex-linearly to the complex Clifford bundle.

A graded Hermitian Clifford connection is a unitary connection ${\mathrm{\nabla }}^V$ preserving $V^{\pm }$ and satisfying ${\mathrm{\nabla }}_X^V(c(\xi )v)=c({\mathrm{\nabla }}_X^{LC}\xi )v+c(\xi ){\mathrm{\nabla }}_X^{V}v$ for smooth $X,\xi ,v$, where ${\mathrm{\nabla }}^{LC}$ is the Levi–Civita connection. In a local orthonormal coframe $(e^i)$ define $D_{V}v={\sum }_{i}c(e^i){\mathrm{\nabla }}_{e_i}^{V}v\mathrm{.}$ The operator is odd, so its chiral part is $D_V^{+}:C^{\mathrm{\infty }}(V^{+})\to C^{\mathrm{\infty }}(V^{-})$. Under Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the formula's frame independence, formal symmetry on closed $M$, and global bundle well-definedness are proved in The geometric Dirac operator is globally well defined and formally symmetric ↗. A domain and index are assigned only under the compact hypotheses of later items.