Homological perturbation theory for nonperturbative 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.

22 pages. Minor revisions from previous version

References in corpus (3)

• Introduction to Integral Discriminants
• Twisted de Rham cohomology, homological definition of the integral and "Physics over a ring"
• Good bases for tame polynomials

Cited by in corpus (7)

• Quantum $L_{\infty }$ Algebras and the Homological Perturbation Lemma
• Correlation functions on the lattice and twisted cocycles
• Non-perturbative computation of lattice correlation functions by differential equations
• Symmetry Factors of Feynman Diagrams and the Homological Perturbation Lemma
• Perturbative Quantum Field Theory and Homotopy Algebras
• Weyl n-algebras
• Perturbative path-integral of string field and the $A_{\infty }$ structure of the BV master equation