Can any Clifford module bundle be extended to a Dirac bundle?

I assume that the question in the title is clear, so let me talk about its relevance:

According to theorem 4.3 in Heat Kernels and Dirac Operators the index theorem $$$$\tag{1} \mathrm{ind}(D)=(4\pi)^{-n/2}\int_M\hat{A}(M)\wedge \mathrm{ch}(E/S)$$$$ holds for every Dirac operator on a given Clifford module bundle (the set of Dirac operators is always non-empty as explained on page $$117$$). The theorem is proven for a Dirac bundle - a Clifford module bundle equipped with a Clifford connection and a compatible metric - and it follows that the theorem actually holds for all Dirac operators on $$E$$, because both the left-hand side and the right-hand side of $$(1)$$ are independent of the Dirac operator. But of course this leaves us with the question whether any Clifford module bundle can be extended to a Dirac bundle.

• This is precisely the content of Corollary 3.41 and its proof: every Clifford module bundle admits a Clifford connection (and hence, in particular, a Clifford superconnection). Commented Nov 11, 2023 at 14:12
• @BranimirĆaćić Thanks for the comment! I only read the corollary and hence I didn't see that they even prove the existence of a Clifford connection. Now we still need a metric though (for the proof of McKean Singer) :) Commented Nov 11, 2023 at 15:12
• Locally, your Clifford module bundle is a twisted spinor bundle $W \otimes S$, where, by restricting further to a local trivialisation of $W$, you may assume that $W$ is trivial. On the one hand, the spinor bundle $S$ canonically defines a self-adjoint Clifford module; on the other hand, use the triviality of $W$ to give $W$ a Hermitian metric. Then the tensor product Hermitian metric on $W \otimes S$ makes it into a self-adjoint Clifford module. Commented Nov 11, 2023 at 15:20
• @BranimirĆaćić All these proofs (existence of Clifford connection and metric) only work for complex Clifford module bundles though (since we assume that we locally have $E=S\otimes W$), right? Commented Nov 11, 2023 at 15:35
• Off the top of my head, I suppose so, but this is the only setting considered by Berline--Getzler--Vergne anyway. I have a vague recollection that Lawson--Michelsohn (for comparison) only consider twisted spinor bundles amongst real Clifford module bundles, but I may be wrong. Commented Nov 11, 2023 at 17:13

You haven't quoted all the definitions, so I'm not sure what level of generality you are interested in. However, for usual spinor bundles over a Riemannian manifold $$M$$, every such bundle has a unique connection induced by the unerlying Levi-Civita connection (see also §II.4 in Lawson & Michelson). This is a special case of a more general construction: the Levi-Civita connection can be lifted to a connection on $$P_{\mathrm{SO}}$$ the orthogonal (or $$P_{\mathrm{Spin}}$$ the spin) principal bundle over the Riemannian manifold $$M$$, which in turn induces a connection on any bundle associated to the principal bundle.