26 research outputs found
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 are
polynomially countable
Graph hypersurfaces and a dichotomy in the Grothendieck ring
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 Z generated by the class of a point. Thus, the span of the graph
hypersurfaces in the Grothendieck ring is nearly killed by setting the
Lefschetz motive L to zero, while it is known that graph hypersurfaces generate
the Grothendieck ring over a localization of Z[L] in which L becomes
invertible. In particular, this shows that the graph hypersurfaces do not
generate the Grothendieck ring prior to localization. The same result yields
some information on the mixed Hodge structures of graph hypersurfaces, in the
form of a constraint on the terms in their Deligne-Hodge polynomials.Comment: 8 pages, LaTe
The massless higher-loop two-point function
We introduce a new method for computing massless Feynman integrals
analytically in parametric form. An analysis of the method yields a criterion
for a primitive Feynman graph to evaluate to multiple zeta values. The
criterion depends only on the topology of , and can be checked
algorithmically. As a corollary, we reprove the result, due to Bierenbaum and
Weinzierl, that the massless 2-loop 2-point function is expressible in terms of
multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We
find that the coefficients in the Taylor expansion of planar graphs in this
range evaluate to multiple zeta values, but the non-planar graphs with crossing
number 1 may evaluate to multiple sums with roots of unity. Our
method fails for the five loop graphs with crossing number 2 obtained by
breaking open the bipartite graph at one edge
On Motives Associated to Graph Polynomials
The appearance of multiple zeta values in anomalous dimensions and
-functions of renormalizable quantum field theories has given evidence
towards a motivic interpretation of these renormalization group functions. In
this paper we start to hunt the motive, restricting our attention to a subclass
of graphs in four dimensional scalar field theory which give scheme independent
contributions to the above functions.Comment: 54
Feynman graph polynomials
The integrand of any multi-loop integral is characterised after Feynman
parametrisation by two polynomials. In this review we summarise the properties
of these polynomials. Topics covered in this article include among others:
Spanning trees and spanning forests, the all-minors matrix-tree theorem,
recursion relations due to contraction and deletion of edges, Dodgson's
identity and matroids.Comment: 35 pages, references adde
From polygons and symbols to polylogarithmic functions
We present a review of the symbol map, a mathematical tool that can be useful
in simplifying expressions among multiple polylogarithms, and recall its main
properties. A recipe is given for how to obtain the symbol of a multiple
polylogarithm in terms of the combinatorial properties of an associated rooted
decorated polygon. We also outline a systematic approach to constructing a
function corresponding to a given symbol, and illustrate it in the particular
case of harmonic polylogarithms up to weight four. Furthermore, part of the
ambiguity of this process is highlighted by exhibiting a family of non-trivial
elements in the kernel of the symbol map for arbitrary weight.Comment: 75 pages. Mathematica files with the expression of all HPLs up to
weight 4 in terms of the spanning set are include