Algebraic loops, braids, and tropicalization
Abstract
Let Y be a smooth complex algebraic variety. We develop a tropical framework for studying algebraic loops, i.e., free homotopy classes in Yan associated with formal loops Spec ℂ( (t) )→ Y. When Y is very affine, every formal loop determines an integral point of Trop(Y). For schön varieties, we prove that the associated algebraic loop is determined by this tropical point together with a connected component of the corresponding initial variety. Consequently, when all initial varieties are connected, integral tropical points classify algebraic loops. We apply this theory to the case when Y is a hyperplane arrangement complement. In this case, we identify the initial varieties with complements of graded arrangements and obtain explicit representatives for algebraic loops as products of relative full twists. In type A, these representatives recover pure algebraic braids. For root arrangements, our construction gives a tropical interpretation of the split root valuation data of Goresky, Kottwitz, and MacPherson. Finally, we prove that the cohomology of braid varieties associated with positive algebraic braids depends only on their root valuation data.
Related
- Generalizations of the groups Gnk: graphs, moduli spaces, algebraic geometry, spherical braids
- Tropicalization and Polyhedral Completions of Weyl Algebras
- Tropicalization of Fock's inverse spectral transform
- Tropicalization and degeneration of short flag varieties and their fibers
- Analytification and tropicalization via hyperfields and definable types
BibTeX
@misc{jakob2026algebraic,
title = {{Algebraic loops, braids, and tropicalization}},
author = {Konstantin Jakob and Masoud Kamgarpour and Ian Le and Minh-Tâm Quang Trinh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.12074},
eprint = {2609.12074},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12074},
}
RIS
TY - UNPB
TI - Algebraic loops, braids, and tropicalization
AU - Konstantin Jakob
AU - Masoud Kamgarpour
AU - Ian Le
AU - Minh-Tâm Quang Trinh
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.12074
UR - https://arxiv.org/abs/2609.12074
AB - Let Y be a smooth complex algebraic variety. We develop a tropical framework for studying algebraic loops, i.e., free homotopy classes in Y^an associated with formal loops Spec ℂ( (t) )→ Y. When Y is very affine, every formal loop determines an integral point of Trop(Y). For schön varieties, we prove that the associated algebraic loop is determined by this tropical point together with a connected component of the corresponding initial variety. Consequently, when all initial varieties are connected, integral tropical points classify algebraic loops. We apply this theory to the case when Y is a hyperplane arrangement complement. In this case, we identify the initial varieties with complements of graded arrangements and obtain explicit representatives for algebraic loops as products of relative full twists. In type A, these representatives recover pure algebraic braids. For root arrangements, our construction gives a tropical interpretation of the split root valuation data of Goresky, Kottwitz, and MacPherson. Finally, we prove that the cohomology of braid varieties associated with positive algebraic braids depends only on their root valuation data.
ER -