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

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.