Annotation of the results obtained in 2021
Using the intertwining matrix of the IRF-Vertex correspondence we proposed a determinant representation for the generating function of the commuting Hamiltonians of the double elliptic integrable system. More precisely, it is a ratio of the normally ordered determinants, which turns into a single determinant in the classical case. With its help we reproduced the recently suggested expression for the eigenvalues of the Hamiltonians for the dual to elliptic Ruijsenaars model. The classical counterpart of our construction gives expression for the spectral curve and the corresponding L-matrix. This matrix is obtained explicitly as a weighted average of the Ruijsenaars and/or Sklyanin type Lax matrices with the weights as in the theta function series definition. By construction the L-matrix satisfies the Manakov triple representation instead of the Lax equation. We began to study form factors and correlation functions in the XYZ Heisenberg chain using the generalized algebraic Bethe ansatz. At the first stage, the XY chain was considered, since for this model there is an alternative approach based on representing of the Hamiltonian eigenvectors in the form of coherent states. We have established a connection between coherent states and on-shell Bethe vectors. We managed to construct generalized coherent states that depend on two additional parameters and coincide with the off-shell Bethe vectors for chains of small length. A new geometric description of the quantum difference equation for the Hilbert scheme of points in the complex plane has been developed. The monodromy of this equation is characterized in terms of the K-theoretic and elliptic stable envelope classes. In particular, complete description of the monodromy for the quantum connection discovered by Okounkov and Panharipande is obtained. It is shown that the monodromy operators for the Hilbert scheme equal to the K-theoretic R-matrices associated with the symplectic dual Hilbert scheme. Explicit formulas for the generating functions of the index numbers of quasimap moduli spaces were obtained. It has been proven that these generating functions are equal to the equivariant Euler characteristics of the K-theoretic stable envelopes for the symplectic-dual varieties. In particular, it is shown that these generating functions are Taylor series expansions of rational functions. A new hierarchy of integrable equations in dimension (2+1) was constructed. A characteristic property of this hierarchy is the inclusion of times with negative numbers, i.e. integrable equations generated by negative powers of the operator A in commutators. The construction is carried out on the example of the Davy-Stewartson hierarchy, for which the existence of a hierarchy for positive times is well known. We assume here B_{t_n} = [A^n ,B], B_{x_n} = [σA^n ,B], where n can take negative values. Thus we arrive at the following linear equation: B_{x_1 x_1 t_{-1}}−B_{t_1 t_1 t_{-1}}+4B_{t_1}=0. Similarly, higher versions of these equations are obtained. To construct the corresponding hierarchies, we introduce commutator identities containing commutators with A^{-n} and σA^{-n}. The dressing procedure applied to these linear equations has the form of the "d-bar" problem, the core of which is the operator B with a symbol that depends, in addition to the specified times, on the additional complex variable z, by conjugate to which the inverse problem is considered. In this case, the symbol of the operator B is normalized to 1 with z tending to infinity. This definition of the inverse problem guarantees the commutativity of all derivatives of the dressing operator with respect to t_n, x_n for all n. This means that evolutions over all times, both with positive and negative numbers, are in involution. This is where the analogy of the inverse problem for negative time and positive time number ends. It turns out that evolution in negative time leads to the inverse problem, the core of which requires a similarity transformation: B→ A^{-1}BA^{1}. To solve this problem, we introduce an additional discrete variable n, so that the shift n→ n+1 is given by the similarity transformation. These two equations make it possible to exclude the shift of the dressing operator, so that we come to a second-order equation specifying the evolution in time with a negative number. The condition of its compatibility with evolution with respect to a positive time gives a system of two equations that is nonlinear and integrable by construction. In this case, one of the equations is a connection. In this way, new integrable equations have been obtained that have not previously been found in the literature. These equations admit a natural generalization to the non-Abelian case. In addition, they lead to dimensional integrable reductions to equations in dimension 1+1. Similarly, we construct infinite hierarchies of such equations. The embedding of dynamics of poles of elliptic solutions to the KP hierarchy of type B into the dynamics of N-particle Calogero-Moser system is found and investigated. It is shown that in the phase space of the Calogero-Moser system there is a half-dimensional subspace invariant with respect to odd flows. It corresponds to the configuration in which the particles pairwise stick together. It is also shown that the earlier derived equations of motion for poles of elliptic solutions of the KP equation of type B coincide with the restriction of the third Hamiltonian Calogero-Moser flow onto the invariant subspace corresponding to such pairs of particles. In the third year of the project, the results obtained earlier for rational quantum integrable models associated with Yangian doubles were extended to trigonometric models. These results include formulas for the action of the elements of the monodromy matrix on the Bethe vectors, which can be interpreted as recurrence relations for state vectors and allow one to obtain explicit formulas for these vectors in quantum integrable models in terms of the elements of the corresponding monodromy matrix. Explicit formulas for state vectors are not necessary when one considers scalar products of Bethe vectors and form factors of local operators. The combination of the recurrence relation for the left Bethe vector in the scalar product of Bethe vectors with the formulas for the action of the lower triangular elements of the monodromy matrix on the right Bethe vector makes it possible to obtain recurrence relations for the higher coefficients of the scalar products. In some cases, such recurrence relations yield determinant representation for the scalar products. This determinant representation can be used in the thermodynamic limit to obtain physically significant results in various quantum integrable systems. The research initiated in this project develops a systematic approach to the study of quantum integrable systems associated with g- and Uq(g)-invariant R-matrices for all simple Lie algebras g. A new generalized gl(NM) Gaudin model is described. A corresponding NM-NM Lax pair with n poles on an elliptic curve is obtained. For this system, the equations of motion are obtained. In the case M = 1, the family of elliptic Gaudin models is reproduced, in the case N = 1, the multispin Calogero-Moser system is reproduced, and the case n=1 corresponds to the dynamics of interacting integrable tops. We also introduce generalization of this system constructed by means of R-matrices satisfying the associative Yang-Baxter equation. A natural extension of the obtained models to the Schlesinger systems is given as well.

 

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Annotation of the results obtained in 2019
One of the most important tasks of quantum mechanics is to calculate the correlation functions of quantum systems, since it is the correlation functions that are measured experimentally. In the framework of the quantum inverse scattering method, they are constructed from simpler blocks, which, in turn, are scalar products of a special type. To successfully compute correlation functions, it is necessary to have simple, compact expressions for these scalar products. Traditionally, such convenient expressions are representations in the form of determinants of matrices of finite order. However, such representations are not known in all quantum models. In our work, we considered a spin chain in which the original spatial symmetry is broken due to nontrivial boundary conditions. This violation leads to the fact that it is necessary to modify the approaches and methods traditionally applied to the study of spin chains. We have calculated scalar products that contain the physical states of the quantum system under consideration. The answers are obtained in a compact form and are determinants of matrices whose size is equal to the length of the spin chain. These scalar products can be used to compute the correlation functions of spin chains with broken spatial symmetry. The work carried out within the framework of this project is available at the link https://www.emis.de/journals/SIGMA/2019/066/sigma19-066.pdf We proposed a new class of quantum dynamical GL(NM)-valued R-matrices of mixed type. In particular case of N=1 they reproduce the widely known Felder’s dynamical quantum R-matrices, while in the M=1 case these are the non-dynamical vertex type R-matrices. The answer is presented in the block-matrix form constructed by means of GL(N) vertex R-matrices satisfying not only the ordinary quantum Yang-Baxter equation but also a quadratic relation known as the associative Yang-Baxter equation. The obtained results allow to construct R-matrix quantization of the model of interacting tops. They were published in the Russian Mathematical Surveys and are available at the links http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=9897&option_lang=rus https://arxiv.org/pdf/1905.08724.pdf We described a new elliptic integrable model of interacting tops of the Euler-Arnold type. The Lax representation of size NM by NM with the spectral parameter on elliptic curve was suggested. The dynamics describes a motion of M integrable relativistic tops related to GL(N) Lie group. Each top is endowed also with position and velocity. In the M=1 case we deal with a single top – a someone from a wide family of relativistic integrable tops. In the case N=1 the spin elliptic Ruijsenaars-Schneider model is reproduced. For the obtained models the equations of motion and the analogues of the inertia tensors describing interaction are described explicitly. The work carried out within the framework of this project is available at the links http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=9767&option_lang=rus https://arxiv.org/pdf/1910.08246.pdf We have analysed elliptic solutions to integrable nonlinear partial differential and difference equations (Kadomtsev-Petviashvili, B-version of the Kadomtsev-Petviashvili equation, two-dimensional Toda lattice) and derived equations of motion for poles of the solutions. The pole dynamics is given by an integrable many-body system of Calogero-Moser or Ruijsenaars-Schneider type. The basic tool is the auxiliary linear problems for the wave function which yield equations of motion together with their Lax or Manakov's triple representation. We have also studied integrals of motion and properties of the spectral curves. In our work we have investigated solutions of the B-version of the Kadomtsev-Petviashvili equation (BKP), which are elliptic functions of the variable x (the first time of the BKP hierarchy). It was shown that poles of the elliptic solutions as functions of the third time of the hierarchy move according to equations of motion of some new, previously unknown many-body system which are obtained in the explicit form. A characteristic feature of these equations of motion is the presence of three-body interaction and dependence on first time derivatives. It was also shown that instead of the Lax representation the equations of motion admit a matrix representation of the type of the Manakov's triple. The self-dual form of the equations of motion for poles of elliptic solutions to the BKP equation is also found. We have studied properties of the corresponding spectral curve (defined by the characteristic polynomial of the Lax matrix). It is shown that the equation of the spectral curve is the generating function of the integrals of motion. Several integrals of motion are found in the explicit form for arbitrary number of particles N. We have put forward the hypothesis that the dynamical system in question is integrable. Analytical properties of the wave function (common solution to the auxiliary linear problems) on the spectral curve are investigated, and it is shown that it has properties of the Baker-Akhiezer function. The obtained results are available at the links https://www.sciencedirect.com/science/article/pii/S0393044019301883?via%3Dihub https://arxiv.org/pdf/1905.11383.pdf During the reported year, A.Smirnov, in the collaboration with Z. Zhou (Stanford University,USA), investigated quantum geometry of hypertoric varieties. In particular, explicit formulas for the elliptic stable envelopes, vertex functions and the corresponding q-difference equations in the hypertoric case were obtained. They used these results to prove that in the equivariant elliptic cohomology of the product of a hypertoric variety with its dual there is a distinguished class whose restrictions coincide with the elliptic stable envelopes. As a corollary, the 3d-mirror symmetry conjecture for hypertoric varieties was proven. In the joint work of A.Smirnov and H.Dinkins (University of North Carlina, USA) the vertex functions (solutions of qKZ equations) for Nakajima varieties of type A were investigated. The representation of the vertex functions by power series in dynamical parameters and by contour integrals were obtained. The asymptotic behavior of the vertex functions is also investigated. It was shown that in the limit of infinite equivariant parameters the vertex functions converge to a product of q-gamma functions evaluated in some special points. The corresponding special points are determined from the characters of the tangent spaces at the torus fixed points on the dual varieties. These results are available at the arXiv: https://arxiv.org/pdf/1908.01199.pdf In the first year of the project, a program of studies of soN-invariant quantum integrable models using the current formalism of rational deformations of Katz–Moody algebras and the method of projections to the intersection of various Borel subalgebras was started. Whithin this method, the off-shell Bethe vectors are defined in terms of generators of the corresponding an infinite-dimensional algebra. The RTT formulation of this algebra uses the same R-matrix as the intertwining relation for the monodromy matrix of the soN -invariant quantum integrable model. The main goal of this work was to find formulas for the action of elements of the monodromy matrix on Bethe vectors using the current approach and the projection method. It was shown that this approach makes it possible to obtain such formulas and obtain the results of the action of the elements of the monodromy matrix as finite linear combinations of off-shell Bethe vectors. Note that explicit representations for Bethe vectors in the form of polynomials in elements of the monodromy matrix acting on a cyclic vector were not used to obtain these formulas. Such explicit representations have not yet been obtained in general, however, they are not necessary for the problem of calculating scalar products of Bethe vectors. It has been shown that the projection method completely avoids such representations. Although so3-invariant quantum integrable models are actually equivalent to the models based on the gl2 algebra, the calculations that lead to Bethe vectors and formulas for the action of matrix elements are quite different. For the models with gl2 symmetry, these formulas reflect the general glN scheme, while for the models with so3 symmetry, they are closer to the soN symmetry-based approach. These results show that the projection method for determining Bethe vectors works in the case of soN- and sp2n-invariant models. The corresponding study will be performed in the next two years of the project. The work carried out within the framework of this project is available at the link http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=9762&option_lang=rus https://arxiv.org/pdf/1906.03202.pdf

 

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Annotation of the results obtained in 2020
As part of the study of non-equilibrium processes in quantum integrable systems, the scalar product of wave functions of Hamiltonians of an isotropic spin chain that enjoy different boundary conditions was calculated. It is shown that for certain relations between the twist parameters that determine the boundary conditions, the resulting formula reduces to a single determinant of the finite-dimensional matrix. The size of the matrix is equal to the number of Bethe parameters that define one of the wave functions. A system of linear equations is obtained whose solutions are scalar products of on-shell and off-shell Bethe vectors in the 8-vertex model. Its solvability is proved. The solution of the system is obtained in the form of a determinant of a matrix whose size is equal to the number of roots of the Bethe equations that define the on-shell vector. Thus, a determinant representation is found for the scalar product of on-shell and off-shell Bethe vectors in the 8-vertex Baxter model. The normalized scalar product is free of ambiguities. https://arxiv.org/abs/2005.11224 A relativistic integrable GL(NM) model constructed via any skew-symmetric unitary quantum GL(N) R-matrix in the fundamental representation of GL(N), satisfying the quadratic associative Yang-Baxter equation, is described. On the one hand it generalizes the classical spin Ruijsenaars–Schneider systems (the case N= 1), and on the other hand it generalizes the relativistic integrable tops on GL(N) Lie group (the case M= 1). In particular, choosing the elliptic Baxter-Belavin quantum R-matrix, one gets for N=1 the elliptic spin Ruijenaars-Schneider system, described by Krichever and Zabrodin, and the relativistic GL(N) integrable top for M=1. The described models are obtained by means of the Lax pair with spectral parameter. Equations of motion are derived. The results are published in the paper I.Sechin, A.Zotov “Integrable system of generalized relativistic interacting tops”, Theoret. and Math. Phys., 205:1 (2020), 1291–1302. https://arxiv.org/abs/2011.09599 We investigated a special class of singular solutions to the matrix Kadomtsev-Petviashvili (KP) hierarchy and established their connection with spin generalization of integrable Calogero-Moser system. Namely, we considered solutions of the matrix KP hierarchy which are trigonometric functions of the first hierarchical time. It is shown that the evolution of poles and matrix residues at the poles of trigonometric solutions in k-th hierarchical time of the matrix KP hierarchy is given by Hamiltonian flow which is a linear combination of the first k Hamiltonian flows corresponding to higher Hamiltonians of the spin Calogero-Moser system. This linear combination is found in explicit form. Another result, closely connected with the previous one, is the introduction of discrete time version of the trigonometric spin Calogero-Moser system. Its equations of motion are obtained in the explicit form. The method by which this result was obtained is based on pole dynamics of trigonometric solutions of the matrix KP hierarchy with discrete time (more precisely, so-called semi-discrete matrix KP hierarchy in which the space variable remains continuous while the time variable is discrete). In the joint work of Ya. Kononov and A.Smirnov the conjecture of A. Okounkov, describing the factorization of K-theoretic limits of the elliptic stable envelopes, is proven. The Fourier-Mukai kernel describing tree-dimensional mirror symmetry in K-theory is constructed. It is shown that this kernel transforms the K-theoretic stable bases of a symplectic variety to the K-theoretic stable basis of the dual symplectic variety. As an application of this construction, the conjectures of E. Gosky and A. Negut, which relate the toroidal gl(1) R-matrices with trigonometric gl(N) R-matrices, are also proven. The results of this work are published in arXiv and are currently under review for publication in scientific journals. https://arxiv.org/abs/2004.07862 https://arxiv.org/abs/2008.06309 In the joint paper by A. Smirnov and Z. Zhou new combinatorial formulas for the elliptic stable envelope classes of hypertoric varieties are found. These formulas were used to construct an elliptic version of the Fourier-Mukai transform mapping the K-theoretic vertex functions of a symplectic variety to the K-theoretic vertex functions of the dual symplectic variety. The paper is available on arXiv and is submitted for publication in Advances in Mathematics. https://arxiv.org/abs/2006.00118 In the joint papers by A. Smirnov and H. Dinkins new combinatorial formulas for the descendent vertex functions of zero-dimensional Nakajima quiver varieties are obtained. The monodromy of these functions is computed explicitly. It is shown that the monodromy of the vertex functions is described as the ratio of the elliptic stable envelope classes for the dual symplectic varieties. As an application, new combinatorial formulas for the characters of the tautological bundles are obtained for arbitrary stability conditions of Nakajima varieties. This investigation is resulted in two papers, first of which is already published in International Mathematics Research Notices and the second is currently under review in Advances in Mathematics. https://arxiv.org/abs/1912.04834 https://arxiv.org/abs/2005.12980 In the second year of the project, one of the main objectives of this study was completed. Convincing evidence was obtained that the projection method for studying the space of states in the quantum integrable models associated with deformations of different Kac-Moody algebras works in cases of algebras corresponding to all classical series of simple Lie algebras. The spaces of states in quantum integrable models associated with B-series have been investigated in most details. In particular, it was demonstrated the possibility of representing the off-shell Bethe vectors in such models as a projection of an ordered product of currents corresponding to simple roots of the underlying simple Lie algebra. Further, explicit formulas were obtained for the action of the entries of the monodromy matrix on these vectors, and these calculations were based on relatively simple commutation relations in the corresponding deformed current algebra. Further, the action formulas for the upper-triangular elements of the monodromy matrix were interpreted as recurrence relations on the Bethe vectors, which can be further used to study the scalar products of Bethe vectors and form factors in the models under consideration. Similar studies for the C and D series will be continued next year and will represent the solution of various technical problems specific to each of the remaining series. The work carried out within the framework of this project is available at the link https://www.emis.de/journals/SIGMA/2020/120/ Commutator identities that give commutator of [A^n,B], n>2, of elements A and B of an associative algebra are considered. Expressions of these commutators in terms of multiple commutators of B with elements A and A^2 is presented. Derived hierarchy of commutator identities transforms in hierarchy of linear differential equations under introduction of dependenсe of independent times t_1, t_2,..., t_n by mean s of relations B_{t_i}=[A^i,B]. We performed a dressing procedure based on the d-bar problem, where element B plays role of the scattering data, and proved that higher members of this hierarchy lead to nonlinear integrable equations of the Kadomtsev-Petviashvili hierarchy. Next, we introduced element s of associative algebra such, that [A,s]=0, {s,B}=0, where {,}denotes anticommutator, and s^2=1. It is easy to see that identically [A^2,B]=[A,[As,B]]. Introducing a new set x_1,...,x_n of independent variables by means of B_{x_i}=[A^is,B], we get a new set of commutator identities, where commutators A^i and A^is with element B are decomposed by multiple commutators with A and As. As a result, we get two hierarchies of linear differential equations, where the lowest elements are linearized versions of the Devi-Stewartson equations, more exactly DS-1 (or DS-2) and DS-3. We performed dressing procedure and proved that every of these linear equations can be developed till nonlinear integrable equation of these hierarchies. Thus, new hierarchies of integrable equations are constructed by means of the dressing method in associative algebras.

 

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