PreprintCC BY 4.0arXiv
Bethe Cycles, Frobenius Identities, and Quantum/Classical Duality
Abstract
We study Frobenius action on the trigonometric Ruijsenaars--Schneider system associated with cotangent bundles to complete flag varieties. The construction continues the program initiated by Etingof and Varchenko and developed in quantum K-theory by the author and Smirnov. We explain how the monodromy eigenvectors recently constructed by Tarasov and Varchenko fit the quantum/classical duality between Bethe equations and tRS level sets. At a root of unity, all partial shifts cancel in the Newton--Adams combinations of the tRS Hamiltonians, leaving precisely the operators obtained by Frobenius action. This proves compatibility of the joint eigenvalue problem with Frobenius action in every rank and at every root order.
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BibTeX
@misc{koroteev2026bethe,
title = {{Bethe Cycles, Frobenius Identities, and Quantum/Classical Duality}},
author = {Peter Koroteev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36975},
eprint = {2610.36975},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36975},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bethe Cycles, Frobenius Identities, and Quantum/Classical Duality
AU - Peter Koroteev
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36975
UR - https://arxiv.org/abs/2610.36975
AB - We study Frobenius action on the trigonometric Ruijsenaars--Schneider system associated with cotangent bundles to complete flag varieties. The construction continues the program initiated by Etingof and Varchenko and developed in quantum K-theory by the author and Smirnov. We explain how the monodromy eigenvectors recently constructed by Tarasov and Varchenko fit the quantum/classical duality between Bethe equations and tRS level sets. At a root of unity, all partial shifts cancel in the Newton--Adams combinations of the tRS Hamiltonians, leaving precisely the operators obtained by Frobenius action. This proves compatibility of the joint eigenvalue problem with Frobenius action in every rank and at every root order.
ER -