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Don’t we need a geometric Spin^c structure to define the Spin^c Dirac operator? Otherwise, how is the connection on the complex spinor bundle defined?
Does the nLab have an article about geometric Spin^c structures?
I’d say that, just as for the plain Dirac operator, the obstruction is the existence of the topological structure. Given that, a connection may be chosen.
We don’t have an entry on geometric/differential Spin^c-structures, I believe.
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