Feynman integrals and motives of configuration spaces
arXiv:1012.5485 · doi:10.1007/s00220-012-1484-1
Abstract
We formulate the problem of renormalization of Feynman integrals and its relation to periods of motives in configuration space instead of momentum space. The algebro-geometric setting is provided by the wonderful compactifications of arrangements of subvarieties associated to the subgraphs of a Feynman graph and a (quasi)projective variety. The motive and the class in the Grothendieck ring are computed explicitly for these wonderful compactifications, in terms of the motive of the variety and the combinatorics of the Feynman graph, using recent results of Li Li. The pullback to the wonderful compactification of the form defined by the unrenormalized Feynman amplitude has singularities along a hypersurface, whose real locus is contained in the exceptional divisors of the iterated blowup that gives the wonderful compactification. A regularization of the Feynman integrals can be obtained by modifying the cycle of integration, by replacing the divergent locus with a Leray coboundary. The ambiguities are then defined by Poincare' residues. While these residues give mixed Tate periods associated to the cohomology of the exceptional divisors and their intersections, the regularized integrals give rise to periods of the hypersurface complement in the wonderful compactification, which can be motivically more complicated.
arXiv comment35 pages, LaTeX
References in corpus (21)
- Deformation quantization of Poisson manifolds, I
- Operads and Motives in Deformation Quantization
- Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem
- Renormalization in quantum field theory and the Riemann-Hilbert problem II: the -function, diffeomorphisms and the renormalization group
- Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops
- On Motives Associated to Graph Polynomials
- On the periods of some Feynman integrals
- A K3 in phi4
- Mixed Hodge structures of configuration spaces
- From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory
- Perturbative 3-manifold invariants by cut-and-paste topology
- Mixed Hodge Structures and Renormalization in Physics
- On one example and one counterexample in counting rational points on graph hypersurfaces
- Chern-Simons Perturbation Theory
- Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura
- Graph hypersurfaces and a dichotomy in the Grothendieck ring
- The Hopf algebra of rooted trees in Epstein-Glaser renormalization
- Motives associated to sums of graphs
- Renormalization and resolution of singularities
- Cohomological analysis of the Epstein-Glaser renormalization
- Deformations of QFTs and Connes-Marcolli's renormalization group in Epstein-Glaser scheme