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## Topology, Geometry, Integrable Systems, and Mathematical Physics: Novikov's Seminar 2012–2014

Providence:
American Mathematical Society, 2014.

, in: Topology, Geometry, Integrable Systems, and Mathematical Physics: Novikov's Seminar 2012–2014. .: Providence: American Mathematical Society, 2014.. Ch. 4. P. 49-74.

The Direct and the Inverse Scattering Problems for the heat operator with a potential being a perturbation of an arbitrary N soliton potential are formulated. We introduce Jost solutions and spectral data and present their properties. Then, giving the time evolution of the spectral data, the initial value problem of the Kadomtsev-Petviashvili II equation for ...

Added: September 15, 2016

Journal of Geometry and Physics 2012 Vol. 003 P. 16-36

We discuss the Poisson structures on Lie groups and propose an explicit construction of the integrable models on their appropriate Poisson submanifolds. The integrals of motion for the SL(N)-series are computed in cluster variables via the Lax map. This construction, when generalised to the co-extended loop groups, gives rise not only to alternative descriptions of relativistic Toda systems, but allows ...

Added: February 11, 2013

Journal of Geometry and Physics 2012 Vol. 62 P. 2294-2319

We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n graded commutative associative algebra. The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, we recover the classical Dieudonn\'e determinant of quaternionic ...

Added: September 28, 2015

Baxter-Belavin R-matrices as non-abelian generalization of elliptic functions / Cornell University. Series math "arxiv.org". 2015.

It was shown in our previous paper that quantum ${\rm gl}_N$ $R$-matrices
satisfy noncommutative analogues of the Fay identities in ${\rm gl}_N^{\otimes
3}$. In this paper we extend the list of $R$-matrix valued elliptic function
identities. We propose counterparts of the Fay identities in ${\rm
gl}_N^{\otimes 2}$, the symmetry between the Planck constant and the spectral
parameter, quasi-periodicities with respect ...

Added: February 3, 2015

International Mathematics Research Notices 2014 Vol. 252

A Poisson structure is defined on the space W of twisted polygons in R^\nu. Poisson reductions with respect to two Poisson group actions on W are described. The \nu=2 and \nu=3 cases are discussed in detail. Amongst the Poisson structures arising in examples are to be found the lattice Virasoro structure, the second Toda lattice ...

Added: November 28, 2014

Theoretical and Mathematical Physics 2021 Vol. 208 No. 2 P. 1093-1115

We consider solutions of the 2D Toda lattice hierarchy that are elliptic functions of the "zeroth" time t(0) = x. It is known that their poles as functions of t1 move as particles of the elliptic RuijsenaarsSchneider model. The goal of this paper is to extend this correspondence to the level of hierarchies. We show that the Hamiltonians that govern the dynamics of poles with respect to the mth hierarchical times t(m) and (t) over bar (m) of the 2D Toda lattice hierarchy are obtained from the expansion of the spectral curve for the Lax matrix of the Ruijsenaars-Schneider model at the marked points. ...

Added: September 7, 2021

Journal of High Energy Physics 2014 Vol. 2014 No. 7:12 P. 1-39

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 ...

Added: January 23, 2015

Journal of Geometry and Physics 2011 Vol. 61 P. 1203-1222

We present a summary of current knowledge about the AGT relations for conformal blocks with additional insertion of the simplest degenerate operator, and a special choice of the corresponding intermediate dimension, when the conformal blocks satisfy hypergeometric-type differential equations in position of the degenerate operator. A special attention is devoted to representation of conformal block ...

Added: February 28, 2013

Nuclear Physics B - Proceedings Supplements 2020 Vol. 952 No. 114931 P. 1-20

We establish a remarkable relationship between the quantum Gaudin models with boundary and the classical many-body integrable systems of Calogero-Moser type associated with the root systems of classical Lie algebras (B, C and D). We show that under identification of spectra of the Gaudin Hamiltonians HjG with particles velocities q˙j of the classical model all ...

Added: August 20, 2020

Journal of Physics A: Mathematical and Theoretical 2014 Vol. 47 P. 1-11

We consider the 'universal monodromy operators' for the Baxter Q-operators. They are given as images of the universal R-matrix in oscillator representation. We find related universal factorization formulas in the Uq(\hat{sl}(2)) case. ...

Added: December 8, 2014

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2015 Vol. 91 P. 022125

The generalized totally asymmetric exclusion process (TASEP) [J. Stat. Mech. (2012) P05014] is an integrable generalization of the TASEP equipped with an interaction, which enhances the clustering of particles. The process interpolates between two extremal cases: the TASEP with parallel update and the process with all particles irreversibly merging into a single cluster moving as ...

Added: February 19, 2015

Loop groups, Clusters, Dimers and Integrable systems / Cornell University. Series math "arxiv.org". 2014.

We describe a class of integrable systems on Poisson submanifolds of the affine Poisson-Lie groups PGLˆ(N), which can be enumerated by cyclically irreducible elements the co-extended affine Weyl groups (Wˆ×Wˆ)♯. Their phase spaces admit cluster coordinates, whereas the integrals of motion are cluster functions. We show, that this class of integrable systems coincides with the ...

Added: October 29, 2014

Gaiotto-Witten superpotential and Whittaker D-modules on monopoles / Cornell University. Series math "arxiv.org". 2014.

Let G be an almost simple simply connected group over complex numbers. For a positive element α of the coroot lattice of G let Z^α denote the space of based maps from the projective line to the flag variety of G of degree α. This space is known to be isomorphic to the space of ...

Added: February 3, 2015

International Mathematics Research Notices 2015 Vol. 18 P. 8925-8958

A Poisson structure is defined on the space {\mathcal {W}} of twisted polygons in {\mathbb {R}}^{\nu }. Poisson reductions with respect to two Poisson group actions on {\mathcal {W}} are described. The \nu =2 and \nu =3 cases are discussed in detail. Amongst the Poisson structures arising in examples are to be found the lattice ...

Added: November 28, 2014

Journal of High Energy Physics 2018 Vol. 2018 No. 2 P. 1-33

We discuss the relation between the cluster integrable systems and q-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlevé dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of ...

Added: October 14, 2018

Nuclear Physics B 2014 Vol. 887 P. 400-422

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 gl2 rational models. The corresponding top is equivalent to the 2-body Ruijsenaars–Schneider (RS) or the 2-body Calogero–Moser (CM) model depending ...

Added: January 22, 2015

Journal of Physics A: Mathematical and Theoretical 2013 Vol. 46 No. 46 P. 465205

The conditions of the integrability of general zero range chipping models with factorized steady states, which were proposed in Evans et al (2004 J. Phys. A: Math. Gen. 37 L275), are examined. We find a three-parametric family of hopping probabilities for the models solvable by the Bethe ansatz, which includes most of known integrable stochastic particle ...

Added: November 14, 2013

Journal of Mathematical Physics 2019 Vol. 60 No. 7 P. 071706-1-071706-22

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. ...

Added: September 19, 2019

Classification of Isomonodromy Problems on Elliptic Curves / Cornell University. Series math "arxiv.org". 2013.

We consider the isomonodromy problems for flat $G$-bundles over punctured
elliptic curves $\Sigma_\tau$ 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
$H^2(\Sigma_\tau,{\mathcal Z}(G))$, where ${\mathcal Z}(G)$ is the center of
$G$. For any complex simple Lie group $G$ and arbitrary class we define ...

Added: December 27, 2013

Journal of High Energy Physics 2019 Vol. 100 No. 10 P. 1-52

We discuss relation between the cluster integrable systems and spin chains in the context of their correspondence with 5d supersymmetric gauge theories. It is shown that glN XXZ-type spin chain on M sites is isomorphic to a cluster integrable system with N × M rectangular Newton polygon and N × M fundamental domain of a ...

Added: October 21, 2019

Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations / Cornell University. Series math "arxiv.org". 2014.

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 $\tau$. At the level of classical mechanics the deformation parameter
$\tau$ allows to extend the ...

Added: January 23, 2015

Relativistic Classical Integrable Tops and Quantum R-matrices / Cornell University. Series math "arxiv.org". 2014.

e 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 ...

Added: January 23, 2015

Instanton moduli spaces and W-algebras / Cornell University. Series math "arxiv.org". 2014.

We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincar\'e pairing) in terms of representation theory of some vertex operator algebras ("W-algebras"). ...

Added: January 30, 2015

Painleve Field Theory / Cornell University. Series math "arxiv.org". 2013.

We propose multidimensional versions of the Painleve VI equation and its degenerations. These field theories are related to the isomonodromy problems on 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 ...

Added: December 27, 2013

The ASEP and determinantal point processes / Cornell University. Series math "arxiv.org". 2016.

We introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line, with correlation kernels defined via spectral projections for the associated Jacobi matrices. For classical weights, we show how such ensembles arise as limits of various hypergeometric orthogonal polynomials ensembles.
We then prove that the q-Laplace transform of the height ...

Added: December 2, 2016