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Nonperturbative integrals, imaginary critical points, and homological perturbation theory

Theo Johnson-Freyd
(UC Berkeley)
Fredag, 24 august, 2012, at 12:45-13:45, (1530-326) QGM Lounge
The method of Feynman diagrams is a well-known example of algebraization of integration. Specifically, Feynman diagrams algebraize the asymptotics of integrals of the form $\int f \exp(s/\hbar)$ in the limit as $\hbar\to 0$ along the pure imaginary axis, supposing that $s$ has only nondegenerate critical points. (In quantum field theory, $s$ is the "action",' and $f$ is an "observable".') In this talk, I will describe an analogous algebraization when $\hbar = 1$ - no formal power series will appear - and $s$ is allowed degenerate critical points. Nevertheless, some features from Feynman diagrams remain: I will explain how to algebraically "integrate out the higher modes" and reduce any such integral to the critical locus of $s$; the primary tool will be a homological form of perturbation theory (itself almost as old as Feynman's diagrams). One of the main new features in nonperturbative integration is that the critical locus of $s$ must be interpreted in the scheme-theoretic sense, and in particular imaginary critical points do contribute. Perhaps this will shed light on questions like the Volume Conjecture, in which an integral over $\mathrm{SU}(2)$ connections is dominated by a critical point in $\mathrm{SL}(2,\mathbb{R})$.
Organiseret af: QGM
Kontaktperson: Jørgen Ellegaard Andersen