arXiv subjects

M. Olshanetsky

Publications and source records attributed to M. Olshanetsky.

At least 19 recordsLinked to original sources

arXiv2026

Epstein vector zeta functions related to the ADE Lie algebras

We introduce a vector-valued generalization of the Epstein zeta functions associated with the root lattices of ADE-type Lie algebras. The quadratic forms defining these lattices correspond to the Gram matrices of the simple roots. Using the discriminant group D = P/Q, we construct vector-valued theta series that realize the Weil representation of the metaplectic group Mp(2,Z). The proposed Epstein vector zeta functions are obtained as the Mellin transform of these theta series. By exploiting the equivariance properties of the theta vectors, we derive a matrix functional equation of the Riemann type. We show that the existence of this functional equation is governed by a selection rule: it holds specifically for the subspace of C-invariant vectors, where C is the central element of Mp(2,Z). Finally, we provide a complete classification of the lattices and invariant subspaces for which this matrix functional equation is satisfied.

math-ph↗

arXiv2025

Integrable deformations of principal chiral model from solutions of associative Yang-Baxter equation

We describe deformations of the classical principle chiral model and 1+1 Gaudin model related to ${\rm GL}_N$ Lie group. The deformations are generated by $R$-matrices satisfying the associative Yang-Baxter equation. Using the coefficients of the expansion for these $R$-matrices we derive equations of motion based on a certain ansatz for $U$-$V$ pair satisfying the Zakharov-Shabat equation. Another deformation comes from the twist function, which we identify with the cocentral charge in the affine Higgs bundle underlying the Hitchin approach to 2d integrable models.

math-ph↗

arXiv2024

Families of Kuramoto models and bounded symmetric domains

We define the families of Kuramoto models (KM) related to bounded symmetric domains. The families include the Lohe unitary model and the spherical models as special cases. Our approach is based on the construction proposed by Watanabe and Strogats WS. We replace the Poincare disc and its $S^1$ boundary in the WS construction on the bounded symmetric domains and on the its Bergman-Shilov (BS) boundaries. In Cartan classifications there are four classical domains of types I-IV. Here we consider the domains of types I,II and III. For a fixed domain there is a decreasing chain of the BS boundaries components. This leads to the KM families we described here.

math-ph↗

arXiv2022

2d Integrable systems, 4d Chern-Simons theory and Affine Higgs bundles

In this short review we compare constructions of 2d integrable models by means of two gauge field theories. The first one is the 4d Chern-Simons (4d-CS) theory proposed by Costello and Yamazaki. The second one is the 2d generalization of the Hitchin integrable systems constructed by means the Affine Higgs bundles (AHB). We illustrate this approach by considering 1+1 field versions of elliptic integrable systems including the Calogero-Moser field theory, the Landau-Lifshitz model and the field theory generalization of the elliptic Gaudin model.

hep-th↗

arXiv2021

Integrable extensions of classical elliptic integrable systems

In this article we consider two particular examples of general construction proposed in arXiv:2012.15529. We consider the integrable extensions of the classical elliptic Calogero-Moser model of N particles with spin and the integrable Euler-Arnold top related to the group SL(N,C). The extended systems has additional N-1 degrees of freedom and can be described in terms of the Darboux variables.

nlin.SI↗

arXiv2020

Odd supersymmetric Kronecker elliptic function and Yang-Baxter equations

We introduce an odd supersymmetric version of the Kronecker elliptic function. It satisfies the genus one Fay identity and supersymmetric version of the heat equation. As an application we construct an odd supersymmetric extensions of the elliptic $R$-matrices, which satisfy the classical and the associative Yang-Baxter equations.

math-ph↗

arXiv2020

Odd supersymmetrization of elliptic R-matrices

We study a general ansatz for an odd supersymmetric version of the Kronecker elliptic function, which satisfies the genus one Fay identity. The obtained result is used for construction of the odd supersymmetric analogue for the classical and quantum elliptic $R$-matrices. They are shown to satisfy the classical Yang-Baxter equation and the associative Yang-Baxter equation. The quantum Yang-Baxter is discussed as well. It acquires additional term in the case of supersymmetric $R$-matrices.

math-ph↗

arXiv2018

Quasi-compact Higgs bundles and Calogero-Sutherland systems with two types spins

We define the quasi-compact Higgs $G^{\mathbb C}$-bundles over singular curves introduced in our previous paper for the Lie group SL($N$). The quasi-compact structure means that the automorphism groups of the bundles are reduced to the maximal compact subgroups of $G^{\mathbb C}$ at marked points of the curves. We demonstrate that in particular cases this construction leads to the classical integrable systems of Hitchin type. The examples of the systems are analogues of the classical Calogero-Sutherland systems related to a simple complex Lie group $G^{\mathbb C}$ with two types of interacting spin variables. These type models were introduced previously by Feher and Pusztai. We construct the Lax operators of the systems as the Higgs fields defined over a singular rational curve. We also construct hierarchy of independent integrals of motion. Then we pass to a fixed point set of real involution related to one of the complex structures on the moduli space of the Higgs bundles. We prove that the number of independent integrals of motion is equal to the half of dimension of the fixed point set. The latter is a phase space of a real completely integrable system. We construct the classical $r$-matrix depending on the spectral parameter on a real singular curve, and in this way prove the complete integrability of the system. We present three equivalent descriptions of the system and establish their equivalence.

math-ph↗

arXiv2017

Calogero-Sutherland system with two types interacting spins

We consider the classical Calogero-Sutherland system with two types of interacting spin variables. It can be reduced to the standard Calogero-Sutherland system, when one of the spin variables vanishes. We describe the model in the Hitchin approach and prove complete integrability of the system by constructing the Lax pair and the classical $r$-matrix with the spectral parameter on a singular curve.

math-ph↗

arXiv2016

Noncommutative extensions of elliptic integrable Euler-Arnold tops and Painleve VI equation

In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on $R$-matrix description which provides Lax pairs in terms of quantum and classical $R$-matrices. First, we prove that for relativistic (and non-relativistic) tops such Lax pairs with spectral parameter follow from the associative Yang-Baxter equation and its degenerations. Then we proceed to matrix extensions of the models and find out that some additional constraints are required for their construction. We describe a matrix version of ${\mathbb Z}_2$ reduced elliptic top and verify that the latter constraints are fulfilled in this case. The construction of matrix extensions is naturally generalized to the monodromy preserving equation. In this way we get matrix extensions of the Painlevé VI equation and its multidimensional analogues written in the form of non-autonomous elliptic tops. Finally, it is mentioned that the matrix valued variables can be replaced by elements of noncommutative associative algebra. In the end of the paper we also describe special elliptic Gaudin models which can be considered as matrix extensions of the (${\mathbb Z}_2$ reduced) elliptic top.

math-ph↗

arXiv2015

Yang-Baxter equations with two Planck constants

We consider Yang-Baxter equations arising from its associative analog and study corresponding exchange relations. They generate finite-dimensional quantum algebras which have form of coupled ${\rm GL}(N)$ Sklyanin elliptic algebras. Then we proceed to a natural generalization of the Baxter-Belavin quantum $R$-matrix to the case ${\rm Mat}(N,\mathbb C)^{\otimes 2}\otimes {\rm Mat}(M,\mathbb C)^{\otimes 2}$. It can be viewed as symmetric form of ${\rm GL}(NM)$ $R$-matrix in the sense that the Planck constant and the spectral parameter enter (almost) symmetrically. Such type (symmetric) $R$-matrices are also shown to satisfy the Yang-Baxter like quadratic and cubic equations.

math-ph↗

arXiv2015

Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras, Calogero-Moser systems and KZB equations

We construct twisted Calogero-Moser (CM) systems with spins as the Hitchin systems derived from the Higgs bundles over elliptic curves, where transitions operators are defined by an arbitrary finite order automorphisms of the underlying Lie algebras. In this way we obtain the spin generalization of the twisted D'Hoker- Phong and Bordner-Corrigan-Sasaki-Takasaki systems. As by product, we construct the corresponding twisted classical dynamical r-matrices and Knizhnik-Zamolodchikov-Bernard equations related to the automorphisms of the Lie algebras.

math-ph↗

arXiv2015

Painleve Field Theory

We propose multidimensional versions of the Painlevé VI equation and its degenerations. These field theories are related to the isomonodromy problems of flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role of "time". Reduction of the field equations to the zero modes leads to ${\rm SL}(N, {\mathbb C})$ monodromy preserving equations. The latter coincide with the Painlevé VI equation for $N\!=\!2$. We consider two types of the bundles. In the first one the group of automorphisms is the centrally and cocentrally extended loop group $L({\rm SL}(N, {\mathbb C}))$ or some multiloop group. In the case of the Painlevé VI field theory in D=1+1 four constants of the Painlevé VI equation become dynamical fields. The second type of bundles are defined by the group of automorphisms of the noncommutative torus. They lead to the equations in dimension 2+1. In both cases we consider trigonometric, rational and scaling limits of the theories. Generically (except some degenerate cases) the derived equations are nonlocal. We consider Whitham quasiclassical limit to integrable systems. In this way we derive two and three dimensional integrable nonlocal versions of the integrable Euler-Arnold tops.

math-ph↗

arXiv2015

Quantum Baxter-Belavin R-matrices and multidimensional Lax pairs for Painleve VI

The quantum elliptic $R$-matrices of Baxter-Belavin type satisfy the associative Yang-Baxter equation in ${\rm Mat}(N,\mathbb C)^{\otimes 3}$. The latter can be considered as noncommutative analogue of the Fay identity for the scalar Kronecker function. In this paper we extend the list of $R$-matrix valued analogues of elliptic function identities. In particular, we propose counterparts of the Fay identities in ${\rm Mat}(N,\mathbb C)^{\otimes 2}$. As an application we construct $R$-matrix valued $2N^2\times 2N^2$ Lax pairs for the Painlevé VI equation (in elliptic form) with four free constants using ${\mathbb Z}_N\times {\mathbb Z}_N$ elliptic $R$-matrix. More precisely, the four free constants case appears for an odd $N$ while even $N$'s correspond to a single constant.

math-ph↗

arXiv2014

Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations

We construct special rational ${\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard (KZB) equations with $\tilde N$ punctures by deformation of the corresponding quantum ${\rm gl}_N$ rational $R$-matrix. They have two parameters. The limit of the first one brings the model to the ordinary rational KZ equation. Another one is $τ$. At the level of classical mechanics the deformation parameter $τ$ allows to extend the previously obtained modified Gaudin models to the modified Schlesinger systems. Next, we notice that the identities underlying generic (elliptic) KZB equations follow from some additional relations for the properly normalized $R$-matrices. The relations are noncommutative analogues of identities for (scalar) elliptic functions. The simplest one is the unitarity condition. The quadratic (in $R$ matrices) relations are generated by noncommutative Fay identities. In particular, one can derive the quantum Yang-Baxter equations from the Fay identities. The cubic relations provide identities for the KZB equations as well as quadratic relations for the classical $r$-matrices which can be halves of the classical Yang-Baxter equation. At last we discuss the $R$-matrix valued linear problems which provide ${\rm gl}_{\tilde N}$ Calogero-Moser (CM) models and Painleve equations via the above mentioned identities. The role of the spectral parameter plays the Planck constant of the quantum $R$-matrix. When the quantum ${\rm gl}_N$ $R$-matrix is scalar ($N=1$) the linear problem reproduces the Krichever's ansatz for the Lax matrices with spectral parameter for the ${\rm gl}_{\tilde N}$ CM models. The linear problems for the quantum CM models generalize the KZ equations in the same way as the Lax pairs with spectral parameter generalize those without it.

hep-th↗

arXiv2014

Classical integrable systems and soliton equations related to eleven-vertex R-matrix

In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum $R$-matrices. Here we study the simplest case -- the 11-vertex $R$-matrix and related ${\rm gl}_2$ rational models. The corresponding top is equivalent to the 2-body Ruijsenaars-Schneider (RS) or the 2-body Calogero-Moser (CM) model depending on its description. We give different descriptions of the integrable tops and use them as building blocks for construction of more complicated integrable systems such as Gaudin models and classical spin chains (periodic and with boundaries). The known relation between the top and CM (or RS) models allows to re-write the Gaudin models (or the spin chains) in the canonical variables. Then they assume the form of $n$-particle integrable systems with $2n$ constants. We also describe the generalization of the top to 1+1 field theories. It allows us to get the Landau-Lifshitz type equation. The latter can be treated as non-trivial deformation of the classical continuous Heisenberg model. In a similar way the deformation of the principal chiral model is also described.

math-ph↗

arXiv2014

Relativistic Classical Integrable Tops and Quantum R-matrices

We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical $R$-matrix even at the classical level, where the Planck constant plays the role of the relativistic deformation parameter in the sense of Ruijsenaars and Schneider (RS). The integrable systems (relativistic tops) are described as multidimensional Euler tops, and the inertia tensors are written in terms of the quantum and classical $R$-matrices. A particular case of ${\rm gl}_N$ system is gauge equivalent to the $N$-particle RS model while a generic top is related to the spin generalization of the RS model. The simple relation between quantum $R$-matrices and classical Lax operators is exploited in two ways. In the elliptic case we use the Belavin's quantum $R$-matrix to describe the relativistic classical tops. Also by the passage to the noncommutative torus we study the large $N$ limit corresponding to the relativistic version of the nonlocal 2d elliptic hydrodynamics. Conversely, in the rational case we obtain a new ${\rm gl}_N$ quantum rational non-dynamical $R$-matrix via the relativistic top, which we get in a different way -- using the factorized form of the RS Lax operator and the classical Symplectic Hecke (gauge) transformation. In particular case of ${\rm gl}_2$ the quantum rational $R$-matrix is 11-vertex. It was previously found by Cherednik. At last, we describe the integrable spin chains and Gaudin models related to the obtained $R$-matrix.

hep-th↗

arXiv2014

Classification of Isomonodromy Problems on Elliptic Curves

We consider the isomonodromy problems for flat $G$-bundles over punctured elliptic curves $Σ_τ$ with regular singularities of connections at marked points. The bundles are classified by their characteristic classes. These classes are elements of the second cohomology group $H^2(Σ_τ,{\mathcal Z}(G))$, where ${\mathcal Z}(G)$ is the center of $G$. For any complex simple Lie group $G$ and arbitrary class we define the moduli space of flat bundles, and in this way construct the monodromy preserving equations in the Hamiltonian form and their Lax representations. In particular, they include the Painlevé VI equation, its multicomponent generalizations and elliptic Schlesinger equations. The general construction is described for punctured curves of arbitrary genus. We extend the Drinfeld-Simpson (double coset) description of the moduli space of Higgs bundles to the case of flat connections. This local description allows us to establish the Symplectic Hecke Correspondence for a wide class of the monodromy preserving equations classified by characteristic classes of underlying bundles. In particular, the Painlevé VI equation can be described in terms of ${\rm SL}(2, {\mathbb C})$-bundles. Since ${\mathcal Z}({\rm SL}(2, {\mathbb C}))= {\mathbb Z}_2$, the Painlevé VI has two representations related by the Hecke transformation: 1) as the well-known elliptic form of the Painlevé VI(for trivial bundles); 2) as the non-autonomous Zhukovsky-Volterra gyrostat (for non-trivial bundles).

math-ph↗