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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.
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:
the exponentiated eta invariant $\exp(i \pi \, \eta)$ on a suitable 3-manifold is the Selberg zeta function of odd type;
it satisfies the sewing constraints that make it an Atiyah-style TQFT
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.
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.
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