Related papers: Motivic geometry of two-loop Feynman integrals

We study type III contractions of Calabi-Yau threefolds containing a ruled surface over a smooth curve. We discuss the conditions necessary for the image threefold to by smoothable. We describe the change in Hodge numbers caused by this…

Following the work of Gangl, Goncharov and Levin in [GGL], we will give a combinatorial framework for motivic study of iterated integrals on the affine line. We will show that under a certain genericity condition these combinatorial objects…

In this paper we construct extensions of mixed Hodge structures coming from the mixed Hodge structure on the graded quotients of the group ring of the fundamental group of a smooth, projective, pointed curve. These extensions correspond to…

Given two semistable, non potentially isotrivial elliptic surfaces over a curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the…

A connection between one-loop $N$-point Feynman diagrams and certain geometrical quantities in non-Euclidean geometry is discussed. A geometrical way to calculate the corresponding Feynman integrals is considered. (This paper contains a…

Let $\Sigma$ be a compact manifold without boundary whose first homology is nontrivial. Hodge decomposition of the incompressible Euler's equation in terms of 1-forms yields a coupled PDE-ODE system. The $L^2$-orthogonal components are a…

In this paper, we develop a new and efficient approach to the computation of envelope surfaces. We interpret one-parameter systems of surfaces as curves in the homogeneous spaces of suitable Lie groups. Using the formalism of Lie groups and…

Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this thesis we explore this correspondence to classify smooth lattice…

We propose a strategy to study the analytic structure of Feynman parameter integrals where singularities of the integrand consist of rational irreducible components. At the core of this strategy is the identification of a selected stratum…

We study complex algebraic K3 surfaces of Picard ranks 11,12, and 13 of finite automorphism group that admit a Jacobian elliptic fibration with a section of order two. We prove that the K3 surfaces admit a birational model isomorphic to a…

Let $\Gamma_g$ be the fundamental group of a closed connected orientable surface of genus $g\geq2$. We introduce a combinatorial structure of "core surfaces", that represent subgroups of $\Gamma_g$. These structures are (usually)…

We study curvature-adapted submanifolds of general symmetric spaces. We generalize Cartan's theorem for isoparametric hypersurfaces of spheres and Wang's classification of isoparametric Hopf hypersurfaces in complex projective spaces to any…

A comprehensive study is performed of two-loop Feynman diagrams with three external legs which, due to the exchange of massless gauge-bosons, give raise to infrared and collinear divergencies. Their relevance in assembling realistic…

The subring of the Grothendieck ring of varieties generated by the graph hypersurfaces of quantum field theory maps to the monoid ring of stable birational equivalence classes of varieties. We show that the image of this map is the copy of…

We revisit certain one-parameter families of affine covers arising naturally from Euler's integral representation of hypergeometric functions. We introduce a partial compactification of this family. We show that the zeta function of the…

Let X be a complex algebraic manifold of dimension n+1 embedded in a sufficiently higher dimensional complex projective space, and Y a generic hyperplane section of X. We describe the mixed Hodge structure on H^p(X-Y,C) and the Hodge…

Under natural hypotheses we give an upper bound on the dimension of families of singular curves with hyperelliptic normalizations on a surface S with p_g(S) >0 via the study of the associated families of rational curves in Hilb^2(S). We use…

In this note, we introduce and study position space Feynman quadrics that are the loci of divergences of the position space Feynman integrals for Euclidean massless scalar quantum field theories. We prove that the Feynman quadrics define…

We use Hodge theory to prove a new upper bound on the ranks of Mordell-Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and…

Work is reported on finite integral representations for 2-loop massive 2-, 3- and 4-point functions, using orthogonal and parallel space variables. It is shown that this can be utilized to cover particles with arbitrary spin (tensor…