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
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.