Motives of melonic graphs

TL;DR

This paper studies the algebraic and geometric properties of melonic graphs, revealing recursive structures and positivity of their Grothendieck classes, with conjectures on their log-concavity based on extensive computations.

Contribution

It introduces recursive relations for Grothendieck classes of melonic graphs and proves their positivity, providing new insights into their algebraic structure and conjecturing log-concavity.

Findings

Grothendieck classes are positive polynomials in the class of the moduli space

Explicit computations suggest polynomials are log-concave

Recursive relations reveal complex divisibility and structural relations

Abstract

We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence-$4$ internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure, in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space $\mathcal M_{0,4}$. We also conjecture that the corresponding polynomials are log-concave, on the basis of hundreds of explicit computations.