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Published August 25, 2021 | Submitted
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Motives of melonic graphs


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 M_(0,4). We also conjecture that the corresponding polynomials are log-concave, on the basis of hundreds of explicit computations.

Additional Information

The first author acknowledges support from a Simons Foundation Collaboration Grant, award number 625561, and thanks the University of Toronto for hospitality. The second author is partially supported by NSF grant DMS-1707882, and by NSERC Discovery Grant RGPIN-2018-04937 and Accelerator Supplement grant RGPAS-2018-522593, and by the Perimeter Institute for Theoretical Physics. The third author worked on parts of this project as summer undergraduate research at the University of Toronto.

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August 19, 2023
August 19, 2023