Motives of melonic graphs (Q6083998)

This article concerns motives of melonic graphs. A recursive formula for the Grothendieck class of melonic graphs with arbitrary valences is provided. 1. A graph with two vertices and n parallel edges connecting them is referred to as a melon graph (or sometimes a banana graph). A melonic graph is the result of iterating the operation of replacing edges of a graph by strings of bananas. A melonic graph G is a graph determined by a melonic construction (see definition 2.1.) 2. In this article one studies the recursive properties of Grothendieck classes associated to melonic /tadpole graphs. A recursive formula for the Grothendieck class of melonic graphs with arbitrary valences is given. Note that classically the melonic/melon-tadpole graphs occurring in the massless CTKT models are such that all vertices have valence 4. See section 2 for a convenient formalism for the recursive construction of melonic graphs with arbitrary valences. 3. Recall that for the category of varieties over a field $K$ the Grothendieck group of varieties is the abelian group generated by isomorphism classes $[X]$ of varieties with an inclusion exclusion relation: $[X]=[Y]+[X \setminus Y]$, for closed subvarieties $Y$ in $X$. Varieties whose motive is in the category of mixed Tate motives will have a Tate Grothendieck class. The Grothendieck class of a graph $G$ is the class $U(G)$ of the complement of the affine cone in its ambient affine $n$-dimensional space. By construction, $U(G)$ is the class of a variety of dimension equal to the number of edges of $G$. In section 3 melonic constructions are used to obtain a recursive computation of the Grothendieck class of a melonic graph. This given recursive formula, for the Grothendieck classes associated to melon-tadpole graphs, provides a recursively constructed family of mixed-Tate motives. This includes the motives associated to the leading melonic terms of certain bosonic tensor models. A conjecture is outlined: The polynomial in $S$ expressing the Grothendieck class of a melonic graph is log-concave, where the class $S$ is generated by the projective line minus three points.