Understanding a lemma in "Loop Spaces and Langlands Parameters" article

Question

First, some background. I was trying to read the article Loop Spaces and Langlands Parameters but I get immediately stuck at Theorem 2.1 in the introduction.

This was actually forward-referring to Chapter 5, and I am able to read until Chapter 4 inclusively, but Proposition 5.1 knocks me off, despite providing some examples. The short statement of it is

The category L_qcoh (Z/S^1) is equivalent to the category of comodule objects in L_qcoh(Z) for the coalgebra p_∗act^∗ in End(L_qcoh (Z)).

Now since there are some references included, I tried looking there, but I think I just don't "get" something about localization. In fact, I remember some physical examples (Chern-Simons theory, I believe) about localizing the path integral onto the fixed points of some torus -- I'm sure this is relevant here, but I don't know how.

So what I have in mind for this question is that perhaps somebody could provide some simpler examples of localization, or connect it to other places where it naturally arises, e.g. physics.

Answer 1

S^1 localization is a fascinating subject IMHO, but the statement you quote is actually about something else, namely descent (or in this case, equivariance). The question is how do you describe sheaves on a quotient X/G (aka G-equivariant sheaves on X) in terms of sheaves on X. The answer is sheaves on X for which the two pushforwards to G x X (under projection and action maps) are identified, plus associativity and higher coherences. Using adjunction, one can rewrite this isomorphism as a coaction of functions on G (with coproduct given by the group structure). That's all that's being asserted (of course in an oo-categorical context, but all the juice is the theory of adjunctions, namely Lurie's oo-categorical version of the Barr-Beck theorem).

For classical S^1 localization you should look at Atiyah and Bott's localization theorem.. for the physics version, Witten's papers on the Atiyah-Singer index theorem and on elliptic genus. My favorite reference in the subject (which gives many other references) is the paper by Goresky-Kottwitz-MacPherson on Koszul duality, localization and equivariant cohomology (which inspired the paper you quote).