Timeline for Can any Clifford module bundle be extended to a Dirac bundle?
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7 events
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| Nov 11, 2023 at 17:13 | comment | added | Branimir Ćaćić | 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. | |
| Nov 11, 2023 at 15:35 | comment | added | Filippo | @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? | |
| Nov 11, 2023 at 15:20 | comment | added | Branimir Ćaćić | 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. | |
| Nov 11, 2023 at 15:12 | comment | added | Filippo | @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) :) | |
| Nov 11, 2023 at 14:12 | comment | added | Branimir Ćaćić | 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). | |
| Nov 11, 2023 at 11:55 | answer | added | Igor Khavkine | timeline score: 4 | |
| Nov 11, 2023 at 4:11 | history | asked | Filippo | CC BY-SA 4.0 |