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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Anomalies associated to the polar decomposition of $\textrm {GL}(n,\textbf {C})$
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by Steven Rosenberg PDF
Trans. Amer. Math. Soc. 334 (1992), 749-760 Request permission

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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Additional Information
  • © Copyright 1992 American Mathematical Society
  • 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