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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 $\exp(i \pi \, \eta)$ 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 $n$Lab entry.