Abstract:
We study the dependence of the dimension h 0(g,A) of the kernel of the Atyiah-Singer Dirac operator \({\cal D}_{g,A}\) on a spinc-manifold M on the metric g and the connection A. The main result is that in the case of spin-structures the value of h 0(g) for the generic metric is given by the absolute value of the index provided \(\dim M\in\{3,4\}\). In dimension 2 the mod-2 index theorems have to be taken into a account and we obtain an extension of a classical result in the theory of Riemann surfaces. In the spinc-case we also discuss upper bounds on h 0(g,A) for generic metrics, and we obtain a complete result in dimension 2. The much simpler dependence on the connection A and applications to Seiberg–Witten theory are also discussed.
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Received: 3 July 1996 / Accepted: 27 February 1997
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Maier, S. Generic Metrics and Connections on Spin- and Spinc-Manifolds . Comm Math Phys 188, 407–437 (1997). https://doi.org/10.1007/s002200050171
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DOI: https://doi.org/10.1007/s002200050171