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Development of combinatorial, homological and geometric methods in the theory of moduli spaces of algebraic curves and their mappings, with applications to problems of mathematical physics

Priority areas of development: mathematics

Goal of research

Development of a common approach to a variety of issues at the interface between the theory of  integrable systems and representation theory of quantum and infinite-dimensional groups and algebras.


Algebraic and representation-theoretic analysis of the classical and quantum field theory, statistical physics and stochastic processes, investigation of integrable structure behind gauge quiver theories and analysis of their correspondence with the two-dimensional conformal field theories, the development of combinatorial, homological and geometric methods in the theory of moduli spaces of various geometric and analytical structures with applications to problems of mathematical physics

Empirical base of research

Not applicable to this research.

Results of research

Organizational results:

The laboratory has published 26 articles in journals indexed by systems "World of Science" and Scopus (all in Q1/Q2 quartiles, all of them with an affiliation of NRU HSE and with an acknowledgement of a support of the Programme “5-100”), as well as 4 papers accepted for publication  and 40 talks at the international conferences.

The laboratory has organized two international conferences: "Classical and quantum integrable systems" (St.Petersburg, July 22-26, 2019) and "Vertex Algebras and Geometry of Moduli Spaces” (Moscow, April 20-24, 2019); two  international schools: the 4-th international school-conference on “String Theory, Integrable Models and Representation Theory” (Moscow, January 20-26, 2019) and an international school-conference «Advanced methods of modern theoretical physics: Integrable and Stochastic Systems» (Dubna, July 28 - August 2, 2019). Laboratory is organising a weekly  mathematical physics seminar and invited leading experts who gave 4 minicourses on advanced research topics in various areas of mathematics and mathematical physics. Among them are Profs. V. Roubtsov, V. Dotsenko (France), Prof. P. Tamaroff (Ireland), Prof. N.Masatoshi (Japan).

Most valuable scientific results obtained by the members of the Laboratory in 2019 are listed below.

We consider elliptic solutions to integrable nonlinear partial differential and difference equations (Kadomtsev-Petviashvili equations, their B-versions, Toda equations) and derive equations of motion for poles of the solutions. The pole dynamics is given by an integrable many-body system (Calogero-Moser, Ruijsenaars-Schneider).  The basic tool is the auxiliary linear problems for the wave function, which yields equations of motion together with their Lax representation. We also discuss integrals of motion and properties of the spectral curves.

We consider a special class of quantum nondynamical R-matrices in the fundamental representation of GL_N with spectral parameter given by trigonometric solutions of the associative Yang-Baxter equation. In the simplest case N=2  these are the well-known 6-vertex R-matrix and its 7-vertex deformation. The R-matrices are used for construction of the classical relativistic integrable tops of the Euler-Arnold type.  Namely, we describe the Lax pairs with spectral parameter, the inertia tensors  and the Poisson structures.  The latter are given by the linear Poisson-Lie brackets for the non-relativistic models and by the classical Sklyanin-type algebras in the relativistic cases. In some particular cases, the tops are gauge equivalent to the Calogero-Moser-Sutherland or trigonometric Ruijsenaars-Schneider models.

For quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl(N)-invariant R-matrix we study two types of Bethe vectors. The first type corresponds to the original monodromy matrix. The second type is associated to a monodromy matrix closely related to the inverse of the monodromy matrix. We show that these two types of the Bethe vectors are identical up to normalization and reshuffling of the Bethe parameters. To prove this correspondence we use the current approach. This identity gives new combinatorial relations for the scalar products of the Bethe vectors.

We study certain representations of quantum toroidal gl_1 algebra for q=t. We construct explicit bosonization of the twisted Fock modules with nontrivial slope n'/n. As a vector space, it is naturally identified with the basic level 1 representation of affingl_n. We also study twisted W-algebras  of sl_n acting on these Fock modules. As an application, we prove the relation on q-deformed conformal blocks which was conjectured in the study of q-deformation of isomonodromy/CFT correspondence.

We study the extension of Painleve - gauge theory correspondence to circular quivers by focusing on the special case of SU(2), N=2* theory. We show that the Nekrasov-Okounkov partition function of this gauge theory provides an explicit combinatorial expression and a Fredholm determinant formula for the tau-function describing isomonodromic deformations of SL_2 flat connections on the one-punctured torus. We also generalize these results to the case of SL(N) flat connections on torus with arbitrary number of punctures.

We explained relation between the cluster integrable systems and spin chains in the context of their correspondence with 5d supersymmetric gauge theories. It was shown that gl_N XXZ-type spin chain on M sites is isomorphic to a cluster integrable system with NxM rectangular Newton polygon and NxM fundamental domain of a 'fence net' bipartite graph.

We study the family of commutative Bethe subalgebras B(C) in the Yangian Y(g). We investigate different initial data sets and find cases where Bethe subalgebra free, and cases where Bethe subalgebra is the maximal commutative subalgebra of the Yangian. We study two ways to compactify the parameter space of Bethe subalgebras: construction of limit subalgebras, which gives us the flat family of subalgebras and the wonderful compactification of group G, which gives us non-flat family of subalgebras. We also describe subalgebras corresponding to any stratum of the wonderful compactification.

For a wide class of infinite dimensional algebras (braided Yangians) we construct elementary symmetric functions depending on a spectral parameter and prove their commutativity. Thus, the coefficients of the symmetric functions expansion in inverse power of the spectral parameter generate the commutative Bethe subalgebra in the braided Yangian.

The exact laws of large numbers for two time additive quantities in the raise and peel model were established, the number of tiles removed by avalanches and the number of global avalanches happened by given time were found.

We systematize and develop a graphical approach to the investigations of quantum integrable vertex statistical models and the corresponding quantum spin chains. The graphical forms of the unitarity and various crossing relations are introduced. The commutativity conditions for the transfer operators of lattices with a boundary are derivedby the graphical method.

For the inhomogeneous reduced density matrix in case of an arbitrary simple Lie algebra we find functional equations of the form of the reduced quantum Knizhnik-Zamolodchikov equation. This equation is the starting point for the investigation of correlation functions at arbitrary temperature and notably for the ground state.

We present a construction of an integrable model as a projective type limit of Calogero-Sutherland models of N fermionic particles, when N tends to infinity. Explicit formulas    for limits of Dunkl operators and of commuting Hamiltonians by means of vertex operators are given.


Ogievetsky O., Shlosman S. Critical con gurations of solid bodies and the Morse theory of MIN functions // Успехи математических наук. 2019. Vol. 74. No. 4. P. 631-657. doi
Bershtein M., Gavrylenko P., Marshakov A. Cluster Toda Chains and Nekrasov Functions // Теоретическая и математическая физика. 2019. Vol. 198. No. 2. P. 157-188. doi
Bershtein M., Tsymbaliuk A. Homomorphisms between different quantum toroidal and affine Yangian algebras // Journal of Pure and Applied Algebra. 2019. Vol. 223. No. 2. P. 867-899. doi
Bershtein M., Shchechkin A. Painlevé equations from Nakajima–Yoshioka blowup relations // Letters in Mathematical Physics. 2019. Vol. 109. No. 11. P. 2359-2402. doi
Берштейн М. А., Гавриленко П. Г., Маршаков А. В. Кластерные цепочки Тоды и функции Некрасова // Теоретическая и математическая физика. 2019. Т. 198. № 2. C. 179-214. doi
Shapiro B., Yurii B. On Hurwitz--Severi numbers // Annali della Scuola normale superiore di Pisa - Classe di scienze. 2019. Vol. XIX. No. 1. P. 155-167. doi
Бычков Б. С., Михайлов А. В. Полиномиальные инварианты графов и иерархии линейных уравнений // Успехи математических наук. 2019. Т. 74. № 2. C. 189-190. doi
Bonelli G., Gavrylenko P., Tanzini A., Del M. F. Circular quiver gauge theories, isomonodromic deformations and $W_N$ fermions on the torus // Working papers by Cornell University. Series math "arxiv.org". 2019. 
Cafasso M., Gavrylenko P., Lisovyy O. Tau Functions as Widom Constants // Communications in Mathematical Physics. 2019. Vol. 365. No. 2. P. 741-772. doi
Bai Yuzhe, Gorsky E., Kivinen O. Quadratic ideals and Rogers–Ramanujan recursions // The Ramanujan Journal. 2019. Vol. 13. P. 1-30. doi
Gorsky E., Carlsson E., Mellit A. The A_{q,t} algebra and parabolic flag Hilbert schemes. // Mathematische Annalen. 2019. 
Горский Е. А. Гомологии Хегора-Флоера // Успехи математических наук. 2019. Т. 74. № 1. C. 3-40. doi
Grekov A., Sechin I., Zotov A. Generalized model of interacting integrable tops // Journal of High Energy Physics. 2019. Vol. 10. No. 081. P. 1-32. doi
Zabrodin A. Matrix modified Kadomtsev-Petviashvili hierarchy // Theoretical and Mathematical Physics. 2019. Vol. 199. No. 3. P. 771-783. doi
Grekov A., A. Zabrodin, A. Zotov Supersymmetric extension of qKZ-Ruijsenaars correspondence // Nuclear Physics B. 2019. Vol. 939. P. 174-190. doi
Zabrodin A. Time discretization of the spin Calogero-Moser model and the semi-discrete matrix KP hierarchy // Journal of Mathematical Physics. 2019. Vol. 60. P. 033502-1-033502-14. doi
Зубов Д. И. Конечно-аддитивные меры на неустойчивых слоях диффеоморфизмов Аносова // Функциональный анализ и его приложения. 2019. Т. 53. № 3. C. 92-97. doi
Ильин А. И. О максимальности некоторых коммутативных подалгебр Янгианов // Функциональный анализ и его приложения. 2019. Т. 53. № 4. C. 85-88. doi
Krasnov T., Zotov A. Trigonometric Integrable Tops from Solutions of Associative Yang–Baxter Equation // Annales Henri Poincare. 2019. Vol. 20. No. 8. P. 2671-2697. doi
Черняков Ю. Б., Харчев С. М., Левин А. М., Ольшанецкий М. А., Зотов А. В. Обобщенные модели Калоджеро и Тоды // Письма в Журнал экспериментальной и теоретической физики. 2019. Т. 109. № 2. C. 131-138. doi
Ivan L. Derived Equivalences for Symplectic Ref lection Algebras // International Mathematics Research Notices. 2019. Vol. 0. No. 00. P. 1-31. doi
Lyashik A., Pakuliak S., Ragoucy E., Slavnov N. New symmetries of gl(N)-invariant Bethe vectors // Journal of Statistical Mechanics: Theory and Experiment. 2019. Vol. 2019. No. 4. P. 1-23. doi
Marshakov A., Семенякин Н. С. Cluster integrable systems and spin chains // Journal of High Energy Physics. 2019. Vol. 100. No. 10. P. 1-52. doi
Makhlin I. FFLV-type monomial bases for type B // Algebraic Combinatorics. 2019. P. 305-322. doi
Nirov K. S., Razumov A. Vertex Models and Spin Chains in Formulas and Pictures // Symmetry, Integrability and Geometry - Methods and Applications. 2019. Vol. 15. No. 68. P. 1-67. doi
Хельминк Г. Ф., Poberezhny V. A., Поленкова С. В. Strict versions of integrable hierarchies in pseudodifference operators and related Cauchy problems // Теоретическая и математическая физика. 2019. Vol. 198. No. 2. P. 197-214. doi
Povolotsky A. M. Laws of large numbers in the raise and peel model // Journal of Statistical Mechanics: Theory and Experiment. 2019. No. 074003. P. 1-22. doi
Leonid R., Mikhail Z. Gelfand–Tsetlin Degeneration of Shift of Argument Subalgebras in Types B, C and D // Arnold Mathematical Journal. 2019. Vol. 5. No. 2-3. P. 285-313. doi
Saponov P. A., Gurevich D. Braided Yangians // Journal of Geometry and Physics. 2019. Vol. 138. P. 124-143. doi
Gurevich D., Pavel S., Talalaev D. KZ equations and Bethe subalgebras in generalized Yangians related to compatible R-matrices // Journal of Integrable Systems. 2019. Vol. 4. No. 1. P. xyz005-. doi
Sergeev A. On rings of supersymmetric polynomials // Journal of Algebra. 2019. No. 517. P. 336-364. doi
Takebe T. Q-operators for higher spin eight vertex models with a rational anisotropy parameter // Letters in Mathematical Physics. 2019. Vol. 109. No. 8. P. 1867-1890. doi
S.M. Khoroshkin, M. G. Matushko Fermionic limit of the Calogero-Sutherland system // Journal of Mathematical Physics. 2019. Vol. 60. No. 7. P. 071706-1-071706-22. doi
Gorsky E., Wedrich P. Evaluations of annular Khovanov-Rozansky homology / Cornell University. Series arxive "math". 2019. 
Gorsky E., Schilling A., Hawkes G., Rainbolt J. Generalized q,t-Catalan numbers / Cornell University. Series arxive "math". 2019. 
Ogievetsky O., Pyatov P. N. Quantum Matrix Algebras of BMW type: Structure of the Characteristic Subalgebra / Cornell University. Series math "arxiv.org". 2019. No. arXiv:1910.08551 . 
Gorsky E., Mazin M., Vazirani M. Recursions for rational q,t-Catalan numbers / Cornell University. Series arxive "math". 2019. 
Gorsky E., Hogancamp M., Mellit A., Nakagane K. Serre duality for Khovanov-Rozansky homology / Cornell University. Series arxive "math". 2019. 
Gorsky E., Liu Beibei, Moore A. Surgery on links of linking number zero and the Heegaard Floer d-invariant. / Cornell University. Series arxive "math". 2018.