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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 \begin{equation}\tag{1} \mathrm{ind}(D)=(4\pi)^{-n/2}\int_M\hat{A}(M)\wedge \mathrm{ch}(E/S) \end{equation} 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.

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  • $\begingroup$ 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). $\endgroup$
    – Branimir Ćaćić
    Commented Nov 11, 2023 at 14:12
  • $\begingroup$ @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) :) $\endgroup$
    – Filippo
    Commented Nov 11, 2023 at 15:12
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    $\begingroup$ 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. $\endgroup$
    – Branimir Ćaćić
    Commented Nov 11, 2023 at 15:20
  • $\begingroup$ @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? $\endgroup$
    – Filippo
    Commented Nov 11, 2023 at 15:35
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    $\begingroup$ 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. $\endgroup$
    – Branimir Ćaćić
    Commented Nov 11, 2023 at 17:13

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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.

Anyway, this connection preserves the Clifford multiplication, the Clifford module action, and any fiber metric that with respect to which the Clifford module action is orthogonal.

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