Periodicity and the Determinant Bundle |
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Authors: | Richard Melrose Frédéric Rochon |
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Institution: | (1) Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA;(2) Department of Mathematics, State University of New York, Stony Brook, New York 11794, USA |
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Abstract: | The infinite matrix ‘Schwartz’ group G
−∞ is a classifying group for odd K-theory and carries Chern classes in each odd dimension, generating the cohomology. These
classes are closely related to the Fredholm determinant on G
−∞. We show that while the higher (even, Schwartz) loop groups of G
−∞, again classifying for odd K-theory, do not carry multiplicative determinants generating the first Chern class, ‘dressed’ extensions, corresponding to a star product,
do carry such functions. We use these to discuss Bott periodicity for the determinant bundle and the eta invariant. In so
doing we relate two distinct extensions of the eta invariant to self-adjoint elliptic operators and to elliptic invertible
suspended families and show that the corresponding τ invariant is a determinant in this sense.
The first author acknowledges the support of the National Science Foundation under grant DMS0408993, the second author acknowledges
support of the Fonds québécois sur la nature et les technologies and NSERC while part of this work was conducted. |
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Keywords: | |
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