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 Title: A spectral description of the spin Ruijsenaars-Schneider system Author(s): Penciak, Matej Director of Research: Nevins, Thomas Doctoral Committee Chair(s): Pascaleff, James Doctoral Committee Member(s): Dodd, Christopher; Bergvelt, Maarten Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(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. Issue Date: 2019-07-10 Type: Text URI: http://hdl.handle.net/2142/105808 Rights Information: Copyright 2019 Matej Penciak Date Available in IDEALS: 2019-11-26 Date Deposited: 2019-08
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