Cohomology of Feynman graphs and L-infinity algebra

The goal of the proposed research is twofold: 1) it will define and investigate the cohomology of Feynman graphs, investigating the relation to the moduli space of L-infinity morphisms; 2) based on the results and experience gained, the existence of a combinatorial formula for the coefficients of Kontsevich formula for the star product in the deformation quantization of Poisson manifolds will be investigated. A preliminary study conjectured the coincidence between the cohomology and muduli space, proving that the coefficients are cocycles. The study also isolated the axiomatic properties the coefficients must satisfy. The study suggests that a combinatorial formula based on binary trees may exist. This would be a simpler alternative to the current formula involving integrals over configuration spaces. Among the anticipated applications, it will yield a formula for the coefficients of the Haussdorff series of a Lie algebra (no explicit reasonable formula is available in the literature at this time!) as well as a better understanding of Feynman integrals and the connection between deformation quantization and renormalization. The targeted result is an important component of the author’s program, topic intensely studied at the present time and needed for the understanding of the two important features which mark the mathematical and physics research of the 21st century: noncommutativity and quantization (Sir [Atiyah, 2001], [Cuntz, 2001]). 1. PROBLEM AND NEED STATEMENT 1.1 Background in deformation quantization In classical physics observables like position x and momentum p=mv commute (real valued functions). In the quantum world they don’t. They are usually modeled by linear operators on Hilbert spaces, satisfying the celebrated Heisenberg uncertainty relation: x p – p x=hI. The “Quantization Problem” consists in finding a correspondence between classical and quantum variables, in order to pass (and benefit) from a classical description to the quantum description. A broad and “simple” approach, the so called deformation quantization approach, is based on the idea of deforming the commutative product into a non-commutative one: a “star-product” satisfying: x * p – p * x=hI. Now the state space of a classical system is modeled by a Poisson manifold (X, α). Here α denotes the Poisson structure. Assume for simplicity that X=R. In the case of a constant coefficients Poisson bracket, there is a nice exponential formula for such a star-product: f * g = exp(hα)(f⊗g) (Here f,g are functions on X). In 1997, in a ground breaking paper [Kont, 97], Maxim Kontsevich, recipient of the prestigious Fields medal (equivalent in mathematics of the Nobel price in physics), devised a formula for the star-product in the general case: