On one example and one counterexample in counting rational points on graph hypersurfaces

In this paper we present a concrete counterexample to the conjecture of Kontsevich about the polynomial countability of graph hypersurfaces. In contrast to this, we show that the "wheel with spokes" graphs $WS_n$ are polynomially countable

Dual graph polynomials and a 4-face formula

We study the dual graph polynomials and the case when a Feynman graph has no triangles but has a 4-face. This leads to the proof of the duality-admissibility of all graphs up to 18 loops. As a consequence, the $c_2$ invariant is the same for all 4 Feynman period representations (position, momentum, parametric and dual parametric) for any physically relevant graph

On B-type Open–Closed Landau–Ginzburg Theories Defined on Calabi–Yau Stein Manifolds

We consider the bulk algebra and topological D-brane category arising from the differential model of the open–closed B-type topological Landau–Ginzburg theory defined by a pair (X,W), where X is a non-compact Calabi–Yau manifold and W is a complex-valued holomorphic function. When X is a Stein manifold (but not restricted to be a domain of holomorphy), we extract equivalent descriptions of the bulk algebra and of the category of topological D-branes which are constructed using only the analytic space associated to X. In particular, we show that the D-brane category is described by projective factorizations defined over the ring of holomorphic functions of X. We also discuss simplifications of the analytic models which arise when X is holomorphically parallelizable and illustrate these in a few classes of examples. © 2018 Springer-Verlag GmbH Germany, part of Springer Natur © The Royal Society of Chemistry 2018111sciescopu