Perturbative Quantum Field Theory and L∞-algebras

L∞−morphisms are investigated from the point of view of perturbative quantum field theory, as generalizations of Feynman expansions. Ideas from TQFT and Hopf algebra approach to renormalization are exploited. It is proved that the algebra of graphs with Kontsevich graph homology differential and Kreimer’s coproduct is a DG-coalgebra. The weights of the corresponding expansions are proved to be cycles of the DGcoalgebra of Feynman graphs, leading to graph cohomology via the cobar construction. Moreover, the moduli space of L-infinity morphims (partition functions/QFTs) is isomorphic to the cohomology of Feynman graphs. The weights constructed via integrals over configuration spaces represent a prototypical example of “Feynman integrals”. The present cohomological point of view aims to construct the coefficients of formality morphisms using an algebraic machinery, as an alternative to the analytical approach using integrals over configuration spaces. It is also expected to yield a categorical formulation for the Feynman path integral quantization, which is presently sketched in the context of L∞-algebras.

Keywords: L∞-algebra, configuration spaces, renormalization, QFT Primary: 18G55: Secondary: 81Q30, 81T18