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Transactions of the American Mathematical Society

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Anomalies associated to the polar decomposition of $ {\rm GL}(n,{\bf C})$


Author: Steven Rosenberg
Journal: Trans. Amer. Math. Soc. 334 (1992), 749-760
MSC: Primary 58G26; Secondary 58G10, 81T50
DOI: https://doi.org/10.1090/S0002-9947-1992-1075383-2
MathSciNet review: 1075383
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Abstract: Let $ D$ be a selfadjoint elliptic differential operator on a hermitian bundle over a compact manifold. For positive $ D$, the variation of the functional determinant of $ D$ under positive definite hermitian gauge transformations is calculated. This corresponds to computing a gauge anomaly in the nonunitary directions of the polar decomposition of the frame bundle $ {\text{GL}}(E)$. The variation of the eta invariant for general $ D$ is also calculated. If $ D$ is not selfadjoint, the integrand in the heat equation proof of the Atiyah-Singer Index Theorem is interpreted as an anomaly for $ {D^{\ast} }D$ . In particular, the gauge anomaly for semiclassical Yang-Mills theory is computed.


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DOI: https://doi.org/10.1090/S0002-9947-1992-1075383-2
Article copyright: © Copyright 1992 American Mathematical Society

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