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

HTML articles powered by AMS MathViewer

- 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.## References

- M. F. Atiyah, N. J. Hitchin, and I. M. Singer,
*Self-duality in four-dimensional Riemannian geometry*, Proc. Roy. Soc. London Ser. A**362**(1978), no.Β 1711, 425β461. MR**506229**, DOI 10.1098/rspa.1978.0143 - M. F. Atiyah, V. K. Patodi, and I. M. Singer,
*Spectral asymmetry and Riemannian geometry. I*, Math. Proc. Cambridge Philos. Soc.**77**(1975), 43β69. MR**397797**, DOI 10.1017/S0305004100049410 - Howard D. Fegan and Peter Gilkey,
*Invariants of the heat equation*, Pacific J. Math.**117**(1985), no.Β 2, 233β254. MR**779919**, DOI 10.2140/pjm.1985.117.233 - Peter B. Gilkey,
*The residue of the local eta function at the origin*, Math. Ann.**240**(1979), no.Β 2, 183β189. MR**524666**, DOI 10.1007/BF01364633 - Peter B. Gilkey,
*The residue of the global $\eta$ function at the origin*, Adv. in Math.**40**(1981), no.Β 3, 290β307. MR**624667**, DOI 10.1016/S0001-8708(81)80007-2 - Peter B. Gilkey,
*Invariance theory, the heat equation, and the Atiyah-Singer index theorem*, Mathematics Lecture Series, vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984. MR**783634** - David Groisser and Thomas H. Parker,
*Semiclassical Yang-Mills theory. I. Instantons*, Comm. Math. Phys.**135**(1990), no.Β 1, 101β140. MR**1086754**, DOI 10.1007/BF02097659 - Thomas Parker and Steven Rosenberg,
*Invariants of conformal Laplacians*, J. Differential Geom.**25**(1987), no.Β 2, 199β222. MR**880183** - Steven Rosenberg,
*The determinant of a conformally covariant operator*, J. London Math. Soc. (2)**36**(1987), no.Β 3, 553β568. MR**918645**, DOI 10.1112/jlms/s2-36.3.553 - A. S. Schwarz,
*Instantons and fermions in the field of instanton*, Comm. Math. Phys.**64**(1979), no.Β 3, 233β268. MR**520092**, DOI 10.1007/BF01221733 - A. S. Schwarz,
*The partition function of a degenerate functional*, Comm. Math. Phys.**67**(1979), no.Β 1, 1β16. MR**535228**, DOI 10.1007/BF01223197 - R. T. Seeley,
*Complex powers of an elliptic operator*, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp.Β 288β307. MR**0237943** - Edward Witten,
*Supersymmetry and Morse theory*, J. Differential Geometry**17**(1982), no.Β 4, 661β692 (1983). MR**683171**

## 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