Homological perturbation theory for nonperturbative integrals

TL;DR

This paper develops an algebraic method using homological perturbation theory to compute nonperturbative integrals, revealing that quantum field theory integrals can be exactly reduced to algebraic integrals over critical loci.

Contribution

It introduces an explicit algebraic formula for nonperturbative integrals using homological perturbation, extending the scope of Feynman diagram methods beyond asymptotic analysis.

Findings

Exact algebraic reduction of quantum field integrals to critical loci

Inclusion of imaginary and degenerate critical points in the analysis

Non-asymptotic formulas for polynomial integrals

Abstract

We use the homological perturbation lemma to produce explicit formulas computing the class in the twisted de Rham complex represented by an arbitrary polynomial. This is a non-asymptotic version of the method of Feynman diagrams. In particular, we explain that phenomena usually thought of as particular to asymptotic integrals in fact also occur exactly: integrals of the type appearing in quantum field theory can be reduced in a totally algebraic fashion to integrals over an Euler--Lagrange locus, provided this locus is understood in the scheme-theoretic sense, so that imaginary critical points and multiplicities of degenerate critical points contribute.