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## On the bi-Hamiltonian Structure of the Trigonometric Spin Ruijsenaars–Sutherland Hierarchy

P. 75-87.

Fehér L., Marshall I.

### In book

Cham : Birkhäuser, 2020

Fairon M., Fehér L., Marshall I., Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics 2021 Vol. 22 P. 615-675

We investigate the trigonometric real form of the spin Ruijsenaars–Schneider system introduced, at the level of equations of motion, by Krichever and Zabrodin in 1995. This pioneering work and all earlier studies of the Hamiltonian interpretation of the system were performed in complex holomorphic settings; understanding the real forms is a non-trivial problem. We explain ...

Added: October 6, 2021

Finkelberg M. V., Rybnikov L. G., Journal of the European Mathematical Society 2014 Vol. 16 No. 2 P. 235-271

Drinfeld Zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of the affine Lie algebra of the special linear group. We introduce an affine, reduced, irreducible, normal quiver variety Z isomorphic to the zastava space. The natural Poisson structure on the zastava space can ...

Added: January 16, 2014

Брауэр О., Buryak A., Функциональный анализ и его приложения 2021 Т. 55 № 4 С. 22-39

In a recent paper, given an arbitrary homogeneous cohomological field theory (CohFT), Rossi, Shadrin, and the first author proposed a simple formula for a bracket on the space of local functionals, which conjecturally gives a second Hamiltonian structure for the double ramification hierarchy associated to the CohFT. In this paper we prove this conjecture in ...

Added: September 14, 2022

Fehér L., Marshall I., Journal of Physics A: Mathematical and Theoretical 2017 Vol. 50 No. 31 P. 1-20

Integrable deformations of the hyperbolic and trigonometric BCn Sutherland
models were recently derived via Hamiltonian reduction of certain free
systems on the Heisenberg doubles of SU(n, n) and SU(2n) respectively.
As a step towards constructing action–angle variables for these models,
we here apply the same reduction to a different free system on the double
of SU(2n), and thereby obtain a ...

Added: February 5, 2018

Buryak A., Gubarevich D., Mathematical Physics Analysis and Geometry 2023 Vol. 26 No. 3 Article 23

One of many manifestations of a deep relation between the topology of the moduli spaces of algebraic curves and the theory of integrable systems is a recent construction of Arsie, Lorenzoni, Rossi, and the first author associating an integrable system of evolutionary PDEs to an F-cohomological field theory (F-CohFT), which is a collection of cohomology ...

Added: November 20, 2023

Arsie A., Buryak A., Lorenzoni P. et al., Communications in Mathematical Physics 2023 Vol. 397 P. 141-197

In this paper, we generalize the Givental theory for Frobenius manifolds and cohomological field theories to flat F-manifolds and F-cohomological field theories. In particular, we define a notion of Givental cone for flat F-manifolds, and we provide a generalization of the Givental group as a matrix loop group acting on them. We show that this action is transitive on semisimple flat F-manifolds. We then extend this ...

Added: December 8, 2022

Finkelberg M. V., Rybnikov L. G., Algebraic Geometry 2014 Vol. 1 No. 2 P. 166-180

Drinfeld zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of an affine Lie algebra g^. In case g is the symplectic Lie algebra spN, we introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the zastava space isomorphically in characteristic 0. The natural Poisson structure on ...

Added: October 25, 2013

Buryak A., Rossi P., Journal of the Institute of Mathematics of Jussieu 2022

In this paper, we formulate and present ample evidence towards the conjecture that the partition function (i.e. the exponential of the generating series of intersection numbers with monomials in psi classes) of the Pixton class on the moduli space of stable curves is the topological tau function of the noncommutative Korteweg-de Vries hierarchy, which we ...

Added: September 14, 2022

Marshall I., Letters in Mathematical Physics 2017 Vol. 107 No. 4 P. 619-642

Presentation of a method for generating Lax pairs for systems obtaibed by means of Hamilton reduction ...

Added: December 8, 2016

Marshall I., Communications in Mathematical Physics 2015 Vol. 338 No. 2 P. 563-587

Hamiltonian reduction is used to project a trivially integrable system on the Heisenberg double of SU(n; n), to obtain a system of Ruijsenaars type on a suitable quotient space. This system possesses BCn symmetry and is shown to be equivalent to the standard three-parameter BCn hyperbolic Sutherland model in the cotangent bundle limit. ...

Added: March 3, 2015