Zeta forms and the local family index theorem
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Abstract:
For a family $F$ of elliptic pseudodifferential operators we show there is a natural zeta-form $\zeta (F,S)$ and zeta-determinant form $\operatorname {det}_\zeta (F)$ in the ring of smooth differential forms on the parameterizing manifold, generalizing the classical single operator zeta-function and zeta-determinant. We show that the zeta forms extend the Atiyah-Bott-Seeley formula for the index of an elliptic operator to a family of elliptic operators, while the zeta-determinant form leads to a graded Chern class form for the index bundle. Globally, the zeta-form and zeta-determinant form exist only at the level of $K$-theory as maps to cohomology.References
- Jean-Michel Bismut and Daniel S. Freed, The analysis of elliptic families. I. Metrics and connections on determinant bundles, Comm. Math. Phys. 106 (1986), no. 1, 159–176. MR 853982
- Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat kernels and Dirac operators, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 298, Springer-Verlag, Berlin, 1992. MR 1215720, DOI 10.1007/978-3-642-58088-8
- Jean-Michel Bismut, The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs, Invent. Math. 83 (1986), no. 1, 91–151. MR 813584, DOI 10.1007/BF01388755
- Gerd Grubb, A resolvent approach to traces and zeta Laurent expansions, Spectral geometry of manifolds with boundary and decomposition of manifolds, Contemp. Math., vol. 366, Amer. Math. Soc., Providence, RI, 2005, pp. 67–93. MR 2114484, DOI 10.1090/conm/366/06725
- Gerd Grubb and Lars Hansen, Complex powers of resolvents of pseudodifferential operators, Comm. Partial Differential Equations 27 (2002), no. 11-12, 2333–2361. MR 1944032, DOI 10.1081/PDE-120016160
- Gerd Grubb and Robert T. Seeley, Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems, Invent. Math. 121 (1995), no. 3, 481–529. MR 1353307, DOI 10.1007/BF01884310
- Gerd Grubb and Robert T. Seeley, Zeta and eta functions for Atiyah-Patodi-Singer operators, J. Geom. Anal. 6 (1996), no. 1, 31–77. MR 1402386, DOI 10.1007/BF02921566
- Daniel Quillen, Superconnections and the Chern character, Topology 24 (1985), no. 1, 89–95. MR 790678, DOI 10.1016/0040-9383(85)90047-3
- 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. 288–307. MR 0237943
- Scott, S., Zagier. D: ‘A symbol proof of the local Atiyah-Singer index theorem’, in preparation.
- M. A. Shubin, Pseudodifferential operators and spectral theory, 2nd ed., Springer-Verlag, Berlin, 2001. Translated from the 1978 Russian original by Stig I. Andersson. MR 1852334, DOI 10.1007/978-3-642-56579-3
Additional Information
- Simon Scott
- Affiliation: Department of Mathematics, King’s College London, London, WC2R 2LS England
- Email: simon.scott@kcl.ac.uk
- Received by editor(s): May 4, 2004
- Published electronically: December 19, 2006
- © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 1925-1957
- MSC (2000): Primary 58J40, 58J52
- DOI: https://doi.org/10.1090/S0002-9947-06-04321-2
- MathSciNet review: 2276607