Higher category theory, renormalization, and non-perturbative QFTs

Question

I'm (vaguely) aware of certain uses of higher category theory in attempts to mathematically understand quantum field theories -- for example, Lurie's work on eTQFTs, the recent-ish book by Paugam, and a bunch of work by people like Urs Schreiber.

What I'm wondering is: what work has been done at understanding the role of renormalization in quantum field theory in these terms (specifically in terms of homotoptic geoemtry or something along those lines)? And since much of the work I've seen along these lines tends to focus on perturbative QFT (with good reason, of course), are there are good references which try to capture the non-perturbative aspects of QFT from this perspective?

This post imported from StackExchange Physics at 2016-10-18 13:58 (UTC), posted by SE-user OperaticDreamland

Comments

Well, the starting point for that would be the nLab page on renormalization, wouldn't it? There's a load of references to various formalizations of renormalization there.

This post imported from StackExchange Physics at 2016-10-18 13:58 (UTC), posted by SE-user ACuriousMind

---

I updated the question to be slightly more specific -- I'm wondering if there are any references using something along the lines of homotopical geometry to understand either of these aspects of QFT. I didn't see anything on the nLab page seemed to do so (granted, that probably means that there aren't any references, otherwise I would definitely expect them to be on there...).

This post imported from StackExchange Physics at 2016-10-18 13:58 (UTC), posted by SE-user OperaticDreamland

---

The recent book by Costello on factorization algebras is nonperturbative in the sense that it delivers asymptotic series in $\hbar$ rather than in a coupling constant.

Without a small parameter, a coupling or $\hbar$ or the inverse of the number of sites of a discretization, only exactly solvable problems (which, in dimension 4 means free theories) can be treated nonperturbatively in a very strict sense, in the absence of a rigorous mathematical foundation.