arXiv subjects

S. Arthamonov

Publications and source records attributed to S. Arthamonov.

14 recordsLinked to original sources

arXiv2025

Cauchy identities for genus 2 Schur polynomials

Genus 2 Macdonald polynomials $\Psi^{(q,t)}_{j_1,j_2,j_3}$ generalize ordinary Macdonald polynomials in several aspects. First, they provide common eigenfunctions for commuting difference operators that generalize the Macdonald difference operators of type $A_1$. Second, the algebra generated by these difference operators together with multiplication operators admits an action of genus 2 mapping class group (MCG) that generalizes the well-known action of $SL(2,{\mathbb Z})$ for ordinary Macdonald polynomials. In this paper, one more important aspect of Macdonald theory is considered: the Cauchy identities. We construct a genus 2 generalization of Cauchy identities in the particular case when $t=q=1$, i.e. for genus 2 Schur polynomials.

math.RT↗

arXiv2023

Classical Limit of genus two DAHA

We show that one-parameter deformation $\mathcal A_{q,t}$ of the skein algebra $Sk_q(\Sigma_2)$ of a genus two surface suggested in [AS19] is flat. We solve the word problem in the algebra and describe monomial basis. In addition, we calculate the classical limit $\mathcal A_{q=1,t}$ of the algebra and prove that it is a one-parameter flat Poisson deformation of the coordinate ring $\mathcal A_{q=t=1}$ of an $SL(2,\mathbb C)$-charater variety of a genus two surface. As a byproduct, we obtain a remarkably simple presentation in terms of generators and relations for the coordinate ring $\mathcal A_{q=t=1}$ of a genus two character variety.

math.QA↗

arXiv2023

An Elliptic Generalization of $A_1$ Spherical DAHA at $K=2$

We construct an algebra that is an elliptic generalization of $A_1$ spherical DAHA acting on its finite-dimensional module at $t=-q^{-K/2}$ with $K=2$. We prove that $PSL(2,\mathbb Z)$ acts by automorphisms of the algebra we constructed, and provide an explicit representation of automorphisms and algebra operators alike by $3\times 3$ matrices of elliptic functions. A relation of this construction to the K-theory character of affine Laumon space is conjectured. We point out two potential applications, respectively to $SL(3,\mathbb Z)$ symmetry of Felder-Varchenko functions and to new elliptic invariants of torus knots and Seifert manifolds.

math.QA↗

arXiv2023

Lagrangian Grassmannians, CKP hierarchy and hyperdeterminantal relations

This work concerns the relation between the geometry of Lagrangian Grassmannians and the CKP integrable hierarchy. The Lagrange map from the Lagrangian Grassmannian of maximal isotropic (Lagrangian) subspaces of a finite dimensional symplectic vector space $V\oplus V^*$ into the projectivization of the exterior space $ΛV$ is defined by restricting the Plücker map on the full Grassmannian to the Lagrangian sub-Grassmannian and composing it with projection to the subspace of symmetric elements under dualization $V \leftrightarrow V^*$. In terms of the affine coordinate matrix on the big cell, this reduces to the principal minors map, whose image is cut out by the $2 \times 2 \times 2$ quartic {\em hyperdeterminantal} relations. To apply this to the CKP hierarchy, the Lagrangian Grassmannian framework is extended to infinite dimensions, with $V\oplus V^*$ replaced by a polarized Hilbert space $ {\mathcal H} ={\mathcal H}_+\oplus {\mathcal H}_-$, with symplectic form $ω$. The image of the Plucker map in the fermionic Fock space ${\mathcal F}= Λ^{\infty/2}{\mathcal H}$ is identified and the infinite dimensional Lagrangian map is defined. The linear constraints defining reduction to the CKP hierarchy are expressed as a fermionic null condition and the infinite analogue of the hyperdeterminantal relations is deduced. A multiparametric family of such relations is shown to be satisfied by the evaluation of the $τ$-function at translates of a point in the space of odd flow variables along the cubic lattices generated by power sums in the parameters.

math-ph↗

arXiv2022

Tau functions, infinite Grassmannians and lattice recurrences

The addition formulae for KP $τ$-functions, when evaluated at lattice points in the KP flow group orbits in the infinite dimensional Sato-Segal-Wilson Grassmannian, give infinite parametric families of solutions to discretizations of the KP hierarchy. The CKP hierarchy may similarly be viewed as commuting flows on the Lagrangian sub-Grassmannian of maximal isotropic subspaces with respect to a suitably defined symplectic form. Evaluating the $τ$-functions at a sublattice of points within the KP orbit, the resulting discretization gives solutions both to the hyperdeterminantal relations (or Kashaev recurrence) and the hexahedron (or Kenyon-Pemantle) recurrence.

math-ph↗

arXiv2022

Noncommutative Networks on a Cylinder

In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the space of noncommutative weights of arcs of a directed graph embedded in a disk or cylinder $Σ$, which gives rise to the quasi Poisson bracket of G.Massuyeau and V.Turaev on the group algebra $\mathbf kπ_1(Σ,p)$ of the fundamental group of a surface based at $p\in\partialΣ$. This bracket also induces a noncommutative Goldman Poisson bracket on the cyclic space $\mathcal C_\natural$, which is a $\mathbf k$-linear space of unbased loops. We show that the induced double quasi Poisson bracket between boundary measurements can be described via noncommutative $r$-matrix formalism. This gives a more conceptual proof of the result of N. Ovenhouse that traces of powers of Lax matrix form an infinite collection of noncommutative Hamiltonians in involution with respect to noncommutative Goldman bracket on $\mathcal C_\natural$.

math.QA↗

arXiv2019

Superintegrable Systems on Moduli Spaces of Flat Connections

The main result of this paper is the construction of a family of superintegrable Hamiltonian systems on moduli spaces of flat connections on a principle $G$-bundle on a surface. The moduli space is a Poisson variety with Atiyah-Bott Poisson structure. Among particular cases of such systems are spin generalizations of Ruijsenaars-Schneider models.

math-ph↗

arXiv2017

Genus Two Generalization of $A_1$ spherical DAHA

We consider a system of three commuting difference operators in three variables $x_{12},x_{13},x_{23}$ with two generic complex parameters $q,t$. This system and its eigenfunctions generalize the trigonometric $A_1$ Ruijsenaars-Schneider model and $A_1$ Macdonald polynomials, respectively. The principal object of study in this paper is the algebra generated by these difference operators together with operators of multiplication by $x_{ij} + x_{ij}^{-1}$. We represent the Dehn twists by outer automorphisms of this algebra and prove that these automorphisms satisfy all relations of the mapping class group of the closed genus two surface. Therefore we argue from topological perspective this algebra is a genus two generalization of $A_1$ spherical DAHA.

math.QA↗

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↗

arXiv2014

Rational Top and its Classical R-matrix

We construct a rational integrable system (the rational top) on a coadjoint orbit of ${\rm SL}_N$ Lie group. It is described by the Lax operator with spectral parameter and classical non-dynamical skew-symmetric $r$-matrix. In the case of the orbit of minimal dimension the model is gauge equivalent to the rational Calogero-Moser (CM) system. To obtain the results we represent the Lax operator of the CM model in two different factorized forms -- without spectral parameter (related to spinless case) and another one with the spectral parameter. The latter gives rise to the rational top while the first one is related to generalized Cremmer-Gervais $r$-matrices. The gauge transformation relating the rational top and CM model provides a classical rational version of the IRF-Vertex correspondence. From a geometrical point of view it describes the modification of ${\rm SL}(N,\mathbb C)$-bundles over degenerated elliptic curve. In view of Symplectic Hecke Correspondence the rational top is related to the rational spin CM model. Possible applications and generalizations of the suggested construction are discussed. In particular, the obtained $r$-matrix defines a class of KZB equations.

hep-th↗

arXiv2014

Link polynomial calculus and the AENV conjecture

Using the recently proposed differential hierarchy (Z-expansion) technique, we obtain a general expression for the HOMFLY polynomials in two arbitrary symmetric representations of link families, including Whitehead and Borromean links. Among other things, this allows us to check and confirm the recent conjecture of arXiv:1304.5778 that the large representation limit (the same as considered in the knot volume conjecture) of this quantity matches the prediction from mirror symmetry consideration. We also provide, using the evolution method, the HOMFLY polynomial in two arbitrary symmetric representations for an arbitrary member of the one-parametric family of 2-component 3-strand links, which includes the Hopf and Whitehead links.

hep-th↗

arXiv2013

Differential hierarchy and additional grading of knot polynomials

Colored knot polynomials possess a peculiar Z-expansion in certain combinations of differentials, which depends on the representation. The coefficients of this expansion are functions of the three variables (A,q,t) and can be considered as new distinguished coordinates on the space of knot polynomials, analogous to the coefficients of alternative character expansion. These new variables are decomposed in an especially simple way, when the representation is embedded into a product of the fundamental ones. The fourth grading recently proposed in arXiv:1304.3481, seems to be just a simple redefinition of these new coordinates, elegant but in no way distinguished. If so, it does not provide any new independent knot invariants, instead it can be considered as one more testimony of the hidden differential hierarchy (Z-expansion) structure behind the knot polynomials.

hep-th↗

arXiv2012

New 2x2-matrix linear problems for the Painleve equations

We construct 2x2-matrix linear problems with a spectral parameter for the Painleve equations I-V by means of the degeneration processes from the elliptic linear problem for the Painleve VI equation. These processes supplement the known degeneration relations between the Painleve equations with the degeneration scheme for the associated linear problems. The degeneration relations constructed in this paper are based on the trigonometric, rational, and Inozemtsev limits.

nlin.SI↗

arXiv2010

Reduction of the Elliptic SL(N,C) top

We propose a relation between the elliptic SL(N,C) top and Toda systems and obtain a new class of integrable systems in a specific limit of the elliptic SL(N,C) top. The relation is based on the Inozemtsev limit (IL) and a symplectic map from the elliptic Calogero-Moser system to the elliptic SL(N,C) top. In the case when N=2 we use an explicit form of a symplectic map from the phase space of the elliptic Calogero-Moser system to the phase space of the elliptic SL(2,C) top and show that the limiting tops are equivalent to the Toda chains. In the case when N>2 we generalize the above procedure using only the limiting behavior of Lax matrices. In a specific limit we also obtain a more general class of systems and prove the integrability in the Liouville sense of a certain subclass of these systems. This class is described by a classical r-matrix obtained from an elliptic r-matrix.

nlin.SI↗