Motives of melonic graphs
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arXiv
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| Format: | Preprint |
| Published: |
2020
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| _version_ | 1866916668067807232 |
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| author | Aluffi, Paolo Marcolli, Matilde Qaisar, Waleed |
| author_facet | Aluffi, Paolo Marcolli, Matilde Qaisar, Waleed |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_08565 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Motives of melonic graphs Aluffi, Paolo Marcolli, Matilde Qaisar, Waleed Algebraic Geometry Mathematical Physics 14C15, 18F30, 81T15 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. |
| title | Motives of melonic graphs |
| topic | Algebraic Geometry Mathematical Physics 14C15, 18F30, 81T15 |
| url | https://arxiv.org/abs/2007.08565 |