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nLab > Latest Changes: eta invariant

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 9th 2014
    • (edited Dec 9th 2014)

    added to eta invariant a lightning section On manifolds with boundaries: as sections of the determinant line, essentially just a glorified pointer to Freed 95 for the time being.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 10th 2014
    • (edited Dec 10th 2014)

    I have added some more pointers to this section, and renamed it to Boundaries, determinant line bundles and perturbative Chern-Simons.

    In summary and schematically we have:

    1. the exponentiated eta invariant exp(iπη)\exp(i \pi \, \eta) on a suitable 3-manifold is the Selberg zeta function of odd type;

    2. it satisfies the sewing constraints that make it an Atiyah-style TQFT

    3. and in fact it is one factor (the actual quantum factor on top of the classical contribution) in the perturbative path integral of Chern-Simons theory.

    So if or to the extent that number-theoretic zeta functions are “really” analogous to Selberg zeta functions of odd type, this would give a rather suggestive way of thinking about them geometrically.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2015
    • (edited Jun 24th 2015)

    To that section Boundaries, determinant line bundles and perturbative Chern-Simons I have added pointer to Witten’s talk this week at Strings2015, which amplifies the result by Dai-Freed reviewed in the nnLab entry.

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