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Homological Perturbation Theory for Nonperturbative Integrals

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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.

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  1. Mathematics Department, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208, USA

    Theo Johnson-Freyd

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Johnson-Freyd, T. Homological Perturbation Theory for Nonperturbative Integrals. Lett Math Phys 105, 1605–1632 (2015). https://doi.org/10.1007/s11005-015-0791-9

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