A spectral description of the spin Ruijsenaars-Schneider system

Fix a Weierstrass cubic curve \(E\), and an element σ in the Jacobian variety $\Jac E$ corresponding to the line bundle \mathcal L_\sigma . We introduce a space \mathsf{RS}_{σ, n}(E, V) of pure 1-dimensional sheaves living in S_\sigma = \PP(\OO \oplus \mathcal L_\sigma) together with framing data at the \(0\) and \(\infty\) sections E₀, E_\infty \subset S_\sigma . For a particular choice of \(V\), we show that the space \mathsf{RS}_{σ, n}(E, V) is isomorphic to a completed phase space for the spin Ruijsenaars-Schneider system, with the Hamiltonian vector fields given by tweaking flows on sheaves at their restrictions to E₀ and E_\infty. We compare this description of the RS system to the description of the Calogero-Moser system in \cite{MR2377220}, and show that the two systems can be assembled into a universal system by introducing a σ \rightarrow 0 limit to the CM phase space. We also shed some light on the effect of Ruijsenaars' duality between trigonometric CM and rational RS spectral curves coming from the two descriptions in terms of supports of spectral sheaves.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2019

Author dc:creator

• Penciak, Matej

Contributors dc:contributor

• Nevins, Thomas
• Pascaleff, James
• Dodd, Christopher
• Bergvelt, Maarten

• Ruijsenaars-Schneider, spectral curves, integrable systems, algebraic geometry

Statement dc:rights

• Copyright 2019 Matej Penciak

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/105808

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/105808