Abstract
Using an approach inspired by the theory of the anomalous divergence of the axial vector current, we derive trace formulas for the resolvents of Dirac operators on open spaces of odd dimension. These formulas readily yield index theorems for these operators. As applications we determine the index of the Dirac operator for a particle of arbitrary isospin in the background field of a static system of SU(2) monopoles; and we find formulas in essentially closed form for certain determinants involving these operators.
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This index formula in one dimension can be derived as a corollary to the index theorem on compact manifolds with boundary proved in: Atiyah, M., Patodi, V., Singer, I.: Math. Proc. Camb. Phil. Soc.77, 43 (1975). Here, however, we have given a direct open space proof that can be adapted to higher dimensions
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Communicated by A. Jaffe
This work is supported in part through funds provided by the U.S. Department of Energy (DOE) under contract EY-76-C-02-3069
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Callias, C. Axial anomalies and index theorems on open spaces. Commun.Math. Phys. 62, 213–234 (1978). https://doi.org/10.1007/BF01202525
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DOI: https://doi.org/10.1007/BF01202525