By Matilde Marcolli

This e-book provides fresh and ongoing study paintings geared toward figuring out the mysterious relation among the computations of Feynman integrals in perturbative quantum box idea and the idea of reasons of algebraic forms and their sessions. one of many major questions within the box is knowing while the residues of Feynman integrals in perturbative quantum box concept overview to classes of combined Tate factors. The query originates from the prevalence of a number of zeta values in Feynman integrals calculations saw by means of Broadhurst and Kreimer. varied ways to the topic are defined. the 1st, a “bottom-up” technique, constructs particular algebraic kinds and classes from Feynman graphs and parametric Feynman integrals. This process, which grew out of labor of Bloch–Esnault–Kreimer and used to be extra lately built in joint paintings of Paolo Aluffi and the writer, results in algebro-geometric and motivic types of the Feynman principles of quantum box idea and concentrates on specific buildings of factors and sessions within the Grothendieck ring of types linked to Feynman integrals. whereas the kinds got during this manner should be arbitrarily complex as causes, the a part of the cohomology that's concerned about the Feynman quintessential computation could nonetheless be of the targeted combined Tate type. A moment, “top-down” method of the matter, constructed within the paintings of Alain Connes and the writer, includes evaluating a Tannakian class built out of the knowledge of renormalization of perturbative scalar box theories, acquired within the type of a Riemann–Hilbert correspondence, with Tannakian different types of combined Tate reasons. The ebook attracts connections among those techniques and offers an outline of different ongoing instructions of analysis within the box, outlining the numerous connections of perturbative quantum box concept and renormalization to factors, singularity thought, Hodge constructions, mathematics geometry, supermanifolds, algebraic and n

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**Example text**

7) i where Corr (Y, X) is the abelian group of correspondences from Y to X given by algebraic cycles in Zi (X × Y ). Thus, the generalization of morphisms f : X → Y is given by correspondences of the form Z = j nj Zj , that are finite linear combinations with nj ∈ Z of irreducible smooth algebraic subvarieties Zj ⊂ X × Y . 8) for Z ⊂ X1 × X2 and Z ⊂ X2 × X3 . This means that, to obtain the composition Z ◦ Z one does the following operations: Motives and periods 31 −1 −1 • Take the preimages π32 (Z ) and π21 (Z) in X1 × X2 × X3 , with πij : X1 × X2 × X3 → Xi × Xj the projections.

54) one then gets ∞ 1 0 2 e−(λ(x−x )p2 +λ m2 ) 2 e−λ q dD q λ dλ dx 0 ∞ 1 = π D/2 0 2 e−(λ(x−x )p2 +λ m2 ) λ−D/2 λ dλ dx 0 1 = π D/2 Γ(2 − D/2) ((x − x2 )p2 + m2 )D/2−2 dx, 0 which makes sense for D ∈ C∗ and shows the presence of a pole at D = 6. It seems then that one could simply cure these divergences by removing the polar part of the Laurent series obtained by dimensional regularization. Slightly more complicated examples with nested divergent graphs 24 Feynman Motives (see [Connes and Marcolli (2008)] [Collins (1986)]) show why a renormalization procedure is indeed needed at this point.

Let V ⊂ V (Γ) be the set of vertices other than v that are endpoints of the edges adjacent to v. This set is a union V = V1 ∪ V2 where the vertices in the two subsets Vi are contained in at least two different connected components of Γ v. Then the splitting Γ of Γ at v obtained by inserting an edge e such that the endpoints v1 and v2 are connected by edges, respectively, to the vertices in V1 and V2 is not 1PI. Conversely, assume that there exists a splitting Γ of Γ at a vertex v that is not 1PI.