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Motives of melonic graphs

Paolo Aluffi, Matilde Marcolli, Waleed Qaisar

TL;DR

The paper develops a recursive framework to compute the Grothendieck classes of affine graph hypersurface complements for melon-tadpole graphs arising in CTKT tensor models, showing these motives are mixed-Tate and expressible as positive polynomials in the moduli space class ${ t S}=[{ t P}^1ackslashackslashackslash{0,1, olinebreak olinebreak Binf}]=[{ t P}^1- ext{points}]$. By exploiting edge-splitting and parallel-edge replacement recursions, the authors derive explicit recursions for melonic graphs with arbitrary valence, and provide closed-form generating functions for key families such as the Γ_n graphs. They establish positivity of Grothendieck classes in ${ t S}$ and conjecture log-concavity of the coefficient sequences, linking these algebraic invariants to deeper geometric structures and Hodge–de Rham phenomena. The work also clarifies precise relations between vacuum and non-vacuum melonic graphs, including divisibility relations and tree-structured recursion for vacuum bubbles, offering computational tools and generating-function techniques with potential implications for motivic interpretations of Feynman integrals in tensor models. Overall, the paper advances a coherent, computable picture of the motivic content of melonic sectors in CTKT-type theories and highlights promising geometric patterns in graph hypersurface complements.”

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.

Motives of melonic graphs

TL;DR

The paper develops a recursive framework to compute the Grothendieck classes of affine graph hypersurface complements for melon-tadpole graphs arising in CTKT tensor models, showing these motives are mixed-Tate and expressible as positive polynomials in the moduli space class . By exploiting edge-splitting and parallel-edge replacement recursions, the authors derive explicit recursions for melonic graphs with arbitrary valence, and provide closed-form generating functions for key families such as the Γ_n graphs. They establish positivity of Grothendieck classes in and conjecture log-concavity of the coefficient sequences, linking these algebraic invariants to deeper geometric structures and Hodge–de Rham phenomena. The work also clarifies precise relations between vacuum and non-vacuum melonic graphs, including divisibility relations and tree-structured recursion for vacuum bubbles, offering computational tools and generating-function techniques with potential implications for motivic interpretations of Feynman integrals in tensor models. Overall, the paper advances a coherent, computable picture of the motivic content of melonic sectors in CTKT-type theories and highlights promising geometric patterns in graph hypersurface complements.”

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- 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 . We also conjecture that the corresponding polynomials are log-concave, on the basis of hundreds of explicit computations.

Paper Structure

This paper contains 28 sections, 13 theorems, 111 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Graph polynomials and CTKT models
  3. CTKT models and melonic Feynman graphs
  4. Families of melonic graphs
  5. Summary of the paper
  6. Melonic graphs
  7. The construction of melonic graphs
  8. Reduced melonic constructions
  9. Melonic graphs and bipartite rooted trees
  10. Valence-four melonic graphs
  11. Melonic vacuum bubbles
  12. Grothendieck classes of melonic graphs
  13. The Grothendieck ring of varieties
  14. Kirchhoff--Symanzik polynomials
  15. Grothendieck classes of graph hypersurface complements
  16. ...and 13 more sections

Key Result

Lemma 2.1

Every melonic graph admits a reduced construction.

Figures (2)

  • Figure 1: Tetrahedron, pillow, and double-trace contractions in CTKT models, BeGuHa.
  • Figure 2: Melon-tadpole graphs in CTKT models, BeGuHa.

Theorems & Definitions (45)

  • Conjecture
  • Example 2.1
  • Definition 2.1
  • Definition 2.2
  • Example 2.2
  • Definition 2.3
  • Lemma 2.1
  • Example 2.3
  • Definition 2.4
  • Definition 3.1
  • ...and 35 more