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.
DOI : 10.4171/aihpd/156
Mots-clés : tensor models, melonic graphs, graph hypersurfaces, Grothendieck ring of varieties, log concavity
@article{AIHPD_2023__10_3_503_0,
author = {Aluffi, Paolo and Marcolli, Matilde and Qaisar, Waleed},
title = {Motives of melonic graphs},
journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
pages = {503--554},
volume = {10},
number = {3},
year = {2023},
doi = {10.4171/aihpd/156},
mrnumber = {4620363},
zbl = {1530.14008},
language = {en},
url = {http://www.numdam.org/articles/10.4171/aihpd/156/}
}
TY - JOUR AU - Aluffi, Paolo AU - Marcolli, Matilde AU - Qaisar, Waleed TI - Motives of melonic graphs JO - Annales de l’Institut Henri Poincaré D PY - 2023 SP - 503 EP - 554 VL - 10 IS - 3 UR - http://www.numdam.org/articles/10.4171/aihpd/156/ DO - 10.4171/aihpd/156 LA - en ID - AIHPD_2023__10_3_503_0 ER -
Aluffi, Paolo; Marcolli, Matilde; Qaisar, Waleed. Motives of melonic graphs. Annales de l’Institut Henri Poincaré D, Tome 10 (2023) no. 3, pp. 503-554. doi : 10.4171/aihpd/156. http://www.numdam.org/articles/10.4171/aihpd/156/
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