2005/06/08 by Lucian M. Ionescu, Ionescu, Lucian M.
Mathematics · Physics and Astronomy · #18G55 #81Q30 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #math-ph #math.MP #math.QA #msc:18G55 #msc:81Q30
paper · pdf · doi:10.48550/arxiv.math/0506142
Dec. 2003, LaTeX, 15 pages; appeared in Focus on Quantum Field Theory, Nova Publishers Inc. 2005
arxiv created 2005/06/08 · arxiv updated 2009/12/01
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.