Seminar: Nonperturbative integrals, imaginary critical points, and homological perturbation theory

Speaker: Theo Johnson-Freyd (UC Berkeley)

2012.09.20 | Christine Dilling

Date Fri 24 Aug
Time 12:45 13:45
Location QGM Lounge (1530-326)

Abstract

The method of Feynman diagrams is a well-known example of \emph{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 \emph{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 \emph{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 $\mathrm{SL}(2,\mathbb{R})$.

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