The closest relation that I am aware of is that when treating the Chern-Simons path integral in perturbation theory then it becomes a product of powers/exponentials of analytic torsion and the eta invariant.

I had had dug out fairly good reviews of this late last night, but I lost the collected links this morning.

The story is however already told by Witten in his original article (here): in (2.8) the determinant of the Laplace operator appears, then in (2.14) the eta invariant. The latter term is also discussed for instance in Lott 90.

I had some references that produced the full CS path integral expression including both the analytic torsion and the eta invariant more neatly. Will have to search for them again, though.

]]>Lott’s clearer about the relationship here where eta invariant and analytic torsion appear as analogues in Table 1.

]]>You often see analytic torsion and eta invariant mentioned together. E.g., John Lott says

They have the flavor of being “secondary” invariants. In recent years much progress has been made in making this precise, in showing how [they] arise via transgression from “primary” invariants.

Presumably that’s in the same sense of the secondary invariants of sec 3.1 of the Regensberg proposal.

]]>Thanks. I made a brief note on that at *analytic torsion*, need to look into it later.

By the way, there is a vague similarity between Reznikov’s variant of arithmetic topology and the 2-framings via cobounding 4-manifolds which appear in the definition of Chern-Simons theory.

Given the general thrust of the analogy, this might be relevant, but I’d need to think more about it.

]]>In arithmetic topology it is claimed by Morishita that

]]>the Iwasawa main conjecture, which relates Iwasawa polynomials with p-adic analytic zeta functions, may be seen as an analogue of the relation between Reidemeister torsions and analytic torsions. (p. 28)

added the actual definition at *analytic torsion*

Another radical expansion of determinant line bundle.

]]>Significant improvements and expansion in determinant line bundle and new related stub analytic torsion.

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