Matroids motives, and a conjecture of Kontsevich

Abstract

We show that a certain class of varieties with origin in physics generates (additively) the Denef-Loeser ring of motives. In particular, this disproves a conjecture of M. Kontsevich on the number of points of these varieties over finite fields.

The Galois coaction on $\phi^4$ periods

“…Inspired by the occurrence of MZVs in P φ 4 , Kontsevich raised the question if [X G ] is a polynomial in q [43]. While for graphs with at most 13 edges this is true [56,58], it is known that [X G ] is of general type [6] and fails to be polynomial even for φ 4 graphs [27,34,56]. For every graph with at least three vertices, [X G ] q is divisible by q 2 [56].…”

“…The new letters f 6 2 , f 6 4 , f 6 6 , f 6 8 appear only at the left-most position. On the right-hand side of the tensor product in ∆ P 7,11 (which gives the non-trivial Galois conjugates of P 7,11 ) only odd letters and thus MZVs appear (this is the point of Theorem 3.7).…”

“…We hope that the reader will find this (purely notational) inconvience not too confusing when moving between our article and [23,24]. 6 Maple is a trademark of Waterloo Maple Inc.…”

“…Maple TM readable files with all known data are attached to this article, see Section 6. 6 The labeling of periods refers to these files and is consistent with [55]. 4 For example, π and the numbers ζ (2n + 1) are expected to be algebraically independent over the rationals, but ζ (3) is the only odd Riemann zeta value which is known to be irrational [5].…”

Traintrack Calabi-Yaus from twistor geometry

“…While it was initially hoped that the integrals which appear in computations in planar N = 4 SYM are expressible in terms of generalized polylogarithms, it has by now become clear that this is not the case. 1 Not only are the generalized polylogarithms insufficient but, by any reasonable measure, most of the integrals in N = 4 SYM seem to require more complicated classes of functions, which are as of yet very poorly understood.…”