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Cohomology of Feynman graphs and perturbative quantum field theory

Lucian M. Ionescu
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Abstract

An analog of Kreimer's coproduct from renormalization of Feynman integrals in quantum field theory, endows an analog of Kontsevich's graph complex with a dg-coalgebra structure. The graph complex is generated by orientation classes of labeled directed graphs. A graded commutative product is also defined, compatible with the coproduct. Moreover, a dg-Hopf algebra is identified. Graph cohomology is defined applying the cobar construction to the dg-coalgebra structure. As an application, L-infinity morphisms represented as series over Feynman graphs correspond to graph cocycles. Notably the total differential of the cobar construction corresponds to the L-infinity morphism condition. The main example considered is Kontsevich's formality morphism. The relation with perturbative quantum field theory is considered by interpreting L-infinity morphisms as partition functions, and the coefficients of the graph expansions as Feynman integrals.

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Lucian M. Ionescu. 2005-06-08. Cohomology of Feynman graphs and perturbative quantum field theory. https://arxiv.org/abs/math/0506142

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