A spectral description of the spin Ruijsenaars-Schneider system

Penciak, Matej

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https://hdl.handle.net/2142/105808 Copy

Description

Title

A spectral description of the spin Ruijsenaars-Schneider system

Author(s)

Penciak, Matej

Issue Date

2019-07-10

Director of Research (if dissertation) or Advisor (if thesis)

Nevins, Thomas

Doctoral Committee Chair(s)

Pascaleff, James

Committee Member(s)

• Dodd, Christopher
• Bergvelt, Maarten

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

Fix a Weierstrass cubic curve \(E\), and an element $\sigma$ in the Jacobian variety $\Jac E$ corresponding to the line bundle \(\mathcal L_\sigma \). We introduce a space \(\mathsf{RS}_{\sigma, n}(E, V)\) 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 \(E_0, E_\infty \subset S_\sigma \). For a particular choice of \(V\), we show that the space \(\mathsf{RS}_{\sigma, n}(E, V)\) 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 \(E_0\) 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 \(\sigma \rightarrow 0\) 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.

Graduation Semester

2019-08

Type of Resource

text

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

Copyright and License Information

Copyright 2019 Matej Penciak

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