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Letters in Mathematical Physics

, Volume 105, Issue 11, pp 1605–1632 | Cite as

Homological Perturbation Theory for Nonperturbative Integrals

  • Theo Johnson-FreydEmail author
Article

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.

Mathematics Subject Classification

Primary 81S40 Secondary 18G40 

Keywords

homological perturbation theory path integrals Feynman diagrams Jacobian varieties 

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Mathematics DepartmentNorthwestern UniversityEvanstonUSA

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