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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.
Buryak A., Shadrin S., Epijournal de Geometrie Algebrique 2024 Vol. 8 Article 12
We present a family of conjectural relations in the tautological cohomology of the moduli spaces of stable algebraic curves of genus g with n marked points. A large part of these relations has a surprisingly simple form: the tautological classes involved in the relations are given by stable graphs that are trees and that are decorated only by powers ...
Added: June 23, 2026
Buryak A., Rossi P., Zvonkine D., Geometry and Topology 2024 Vol. 28 P. 2793–2824
We prove that the cohomology classes of the moduli spaces of residueless meromorphic differentials, ie the closures, in the moduli space of stable curves, of the loci of smooth curves whose marked points are the zeros and poles of prescribed orders of a meromorphic differential with vanishing residues, form a partial cohomological field theory (CohFT) of ...
Added: June 23, 2026
Buryak A., Труды Математического института им. В.А. Стеклова РАН 2024 Т. 325 С. 26–66
The main goal of the paper is to show that the DR hierarchies, introduced by the author in an earlier paper, allow one to establish, in the most clear way, a relation between the topology of the Deligne–Mumford compactification of the moduli space of smooth algebraic curves of genus g with n marked points and integrable systems ...
Added: June 23, 2026
Buryak A., Tessler R., Troshkin M., Journal of Geometry and Physics 2026 Vol. 223 Article 105783
We give a natural definition of open Hurwitz numbers, where the weight of each ramified covering includes an integer parameter N taken to the power that is equal to the number of boundary components of a Riemann surface with boundary mapping to . We prove that the resulting sequence of partition functions, depending on , is a tau-sequence of ...
Added: June 19, 2026
Buryak A., Rossi P., Communications in Mathematical Physics 2025 Vol. 406 Article 205
Of the two approaches to integrable systems associated to semisimple cohomological field theories (CohFTs), the one suggested by Dubrovin and Zhang and the more recent one using the geometry of the double ramification (DR) cycle, the second has the advantage of being very explicit. The Poisson operator of the DR hierarchy is , where is the metric ...
Added: June 19, 2026
Buryak A., Mikhail Troshkin, Moscow Mathematical Journal 2024 Vol. 24 No. 4 P. 513–527
We prove that the DR hierarchy corresponding to the family of F-cohomological field theories without unit considered in a previous work of the first author together with D. Gubarevich can be “trivialized”, i.e., reduced to two copies of the KdV hierarchy, using a simple nonlinear reciprocal transformation. This gives the first manifestation of a role ...
Added: December 4, 2024
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
Buryak A., Rossi P., Journal of the Institute of Mathematics of Jussieu 2023 Vol. 22 No. 6 P. 2987–3009
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
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
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
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
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
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
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