arXiv subjects

Leonid Chekhov

Publications and source records attributed to Leonid Chekhov.

At least 19 recordsLinked to original sources

arXiv2024

Cluster structure on genus 2 spherical DAHA: seven-colored flower

We construct an embedding of the Arthamonov-Shakirov algebra of genus 2 knot operators into the quantized coordinate ring of the cluster Poisson variety of exceptional finite mutation type $X_7$. The embedding is equivariant with respect to the action of the mapping class group of the closed surface of genus 2. The cluster realization of the mapping class group action leads to a formula for the coefficient of each monomial in the genus 2 Macdonald polynomial of type $A_1$ as sum over lattice points in a convex polyhedron in 7-dimensional space.

math.RT↗

arXiv2023

Symplectic groupoid and cluster algebras

We consider the symplectic groupoid of pairs $(B,\mathbb{A})$ with $\mathbb A$ unipotent upper-triangular matrices and $B\in GL_n$ being such that $\widetilde {\mathbb A}=B{\mathbb A} B^{\text{T}}$ are also unipotent upper-triangular matrices. We explicitly solve this groupoid condition using Fock--Goncharov--Shen cluster variables and show that for $B$ satisfying the standard semiclassical Lie--Poisson algebra, the matrices $B$, $\mathbb A$, and $\widetilde{\mathbb A}$ satisfy the closed Poisson algebra relations expressible in the $r$-matrix form. Identifying entries of $\mathbb A$ and $\widetilde {\mathbb A}$ with geodesic functions for geodesics on the two halves of a closed Riemann surface of genus $g=n-1$ separated by the Markov element, we are able to construct the geodesic function $G_B$ ``dual'' to the Markov element. We thus obtain the complete cluster algebra description of Teichmüller space $\mathcal T_{2,0}$ of genus two. We discuss also the generalization of our construction for higher genera. For genus larger than three we need a Hamiltonian reduction based on the rank condition $\hbox{rank\,}({\mathbb A}+{\mathbb A}^{\text{T}})\le 4$; we present the example of such a reduction for $\mathcal T_{4,0}$.

math.QA↗

arXiv2021

Quantised Painlevé monodromy manifolds, Sklyanin and Calabi-Yau algebras

In this paper we study quantum del Pezzo surfaces belonging to a certain class. In particular we introduce the generalised Sklyanin-Painlevé algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalised Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painlevé equations.

math.QA↗

arXiv2018

Topological recursion with hard edges

We prove a Givental type decomposition for partition functions that arise out of topological recursion applied to spectral curves. Copies of the Konstevich-Witten KdV tau function arise out of regular spectral curves and copies of the Brezin-Gross-Witten KdV tau function arise out of irregular spectral curves. We present the example of this decomposition for the matrix model with two hard edges and spectral curve $(x^2-4)y^2=1$

math.AG↗

arXiv2017

Algebras of quantum monodromy data and decorated character varieties

The Riemann-Hilbert correspondence is an isomorphism between the de Rham moduli space and the Betti moduli space, defined by associating to each Fuchsian system its monodromy representation class. In 1997 Hitchin proved that this map is a symplectomorphism. In this paper, we address the question of what happens to this theory if we extend the de Rham moduli space by allowing connections with higher order poles. In our previous paper arXiv:1511.03851, based on the idea of interpreting higher order poles in the connection as boundary components with bordered cusps (vertices of ideal triangles in the Poincaré metric) on the Riemann surface, we introduced the notion of decorated character variety to generalize the Betti moduli space. This decorated character variety is the quotient of the space of representations of the fundamental groupid of arcs by a product of unipotent Borel sub-groups (one per bordered cusp). Here we prove that this representation space is endowed with a Poisson structure induced by the Fock--Rosly bracket and show that the quotient by unipotent Borel subgroups giving rise to the decorated character variety is a Poisson reduction. We deal with the Poisson bracket and its quantization simultaneously, thus providing a quantisation of the decorated character variety. In the case of dimension 2, we also endow the representation space with explicit Darboux coordinates. We conclude with a conjecture on the extended Riemann-Hilbert correspondence in the case of higher order poles.

math.AG↗

arXiv2016

Colliding holes in Riemann surfaces and quantum cluster algebras

In this paper, we describe a new type of surgery for non-compact Riemann surfaces that naturally appear when colliding two holes or two sides of the same hole in an orientable Riemann surface with boundary (and possibly orbifold points). As a result of this surgery, bordered cusps appear on the boundary components of the Riemann surface. In Poincaré uniformization, these bordered cusps correspond to ideal triangles in the fundamental domain. We introduce the notion of bordered cusped Teichmüller space and endow it with a Poisson structure, quantization of which is achieved with a canonical quantum ordering. We give a complete combinatorial description of the bordered cusped Teichmüller space by introducing the notion of maximal cusped lamination, a lamination consisting of geodesics arcs between bordered cusps and closed geodesics homotopic to the boundaries such that it triangulates the Riemann surface. We show that each bordered cusp carries a natural decoration, i.e. a choice of a horocycle, so that the lengths of the arcs in the maximal cusped lamination are defined as $λ$-lengths in Thurston--Penner terminology. We compute the Goldman bracket explicitly in terms of these $λ$-lengths and show that the Mapping Class Group acts as a generalized cluster algebra mutation. From the physical point of view, our construction provides an explicit coordinatization of moduli spaces of open/closed string worldsheets and their quantization.

math-ph↗

arXiv2016

Painlevé monodromy manifolds, decorated character varieties and cluster algebras

In this paper we introduce the concept of decorated character variety for the Riemann surfaces arising in the theory of the Painlevé differential equations. Since all Painlevé differential equations (apart from the sixth one) exhibit Stokes phenomenon, it is natural to consider Riemann spheres with holes and bordered cusps on such holes. The decorated character is defined as complexification of the bordered cusped Teichmüller space introduced in arXiv:1509.07044. We show that the decorated character variety of a Riemann sphere with s holes and n>1 bordered cusps is a Poisson manifold of dimension 3 s+ 2 n-6 and we explicitly compute the Poisson brackets which are naturally of cluster type. We also show how to obtain the confluence procedure of the Painlevé differential equations in geometric terms.

math-ph↗

arXiv2015

On a Poisson space of bilinear forms with a Poisson Lie action

We consider the space of bilinear forms on a complex N-dimensional vector space endowed with the quadratic Poisson bracket studied in our previous paper arXiv:1012.5251. We classify all possible quadratic brackets on the set of pairs of matrices A and B with the property that the natural action of B on the defining matrix A of a bilinear form is a Poisson action of a Poisson-Lie group, thus endowing this space of bilinear forms with the structure of Poisson homogeneous space. Beside the product Poisson structure we find two more (dual to each other) structures for which (in contrast to the product Poisson structure) we can implement the reduction to the space of bilinear forms with block upper triangular defining matrices by Dirac procedure. We consider the generalisation of the above construction to triples and show that the space of bilinear forms then acquires the structure of Poisson symmetric space. We study also the generalisation to chains of transformations and to the quantum and quantum affine algebras and the relation between the construction of Poisson symmetric spaces and that of the Poisson groupoid.

math.QA↗

arXiv2014

The matrix model for hypergeometric Hurwitz numbers

We present the multi-matrix models that are the generating functions for branched covers of the complex projective line ramified over $n$ fixed points $z_i$, $i=1,\dots,n$, (generalized Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profiles at two points, $z_1$ and $z_n$. We take a sum over all possible ramifications at other $n-2$ points with the fixed length of the profile at $z_2$ and with the fixed total length of profiles at the remaining $n-3$ points. All these models belong to a class of hypergeometric Hurwitz models thus being tau functions of the Kadomtsev--Petviashvili (KP) hierarchy. In the case described above, we can present the obtained model as a chain of matrices with a (nonstandard) nearest-neighbor interaction of the type $\tr M_iM_{i+1}^{-1}$. We describe the technique for evaluating spectral curves of such models, which opens the possibility of applying the topological recursion for developing $1/N^2$-expansions of these model. These spectral curves turn out to be of an algebraic type.

hep-th↗

arXiv2014

Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables

We generalize a new class of cluster type mutations for which exchange transformations are given by reciprocal polynomials. In the case of second-order polynomials of the form $x+2\cos{π/n_o}+x^{-1}$ these transformations are related to triangulations of Riemann surfaces of arbitrary genus with at least one hole/puncture and with an arbitrary number of orbifold points of arbitrary integer orders $n_o$. We propose the dual graph description of the corresponding Teichmüller spaces, construct the Poisson algebra of the Teichmüller space coordinates, propose the combinatorial description of the corresponding geodesic functions and find the mapping class group transformations.

math-ph↗

arXiv2014

The matrix model for dessins d'enfants

We present the matrix models that are the generating functions for branched covers of the complex projective line ramified over $0$, $1$, and $\infty$ (Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profile at infinity. For general ramifications at other points, the model is the two-logarithm matrix model with the external field studied previously by one of the authors (L.Ch.) and K.Palamarchuk. It lies in the class of the generalised Kontsevich models (GKM) thus being the Kadomtsev--Petviashvili (KP) hierarchy $τ$-function and, upon the shift of times, this model is equivalent to a Hermitian one-matrix model with a general potential whose coefficients are related to the KP times by a Miwa-type transformation. The original model therefore enjoys a topological recursion and can be solved in terms of shifted moments of the standard Hermitian one-matrix model at all genera of the topological expansion. We also derive the matrix model for clean Belyi morphisms, which turns out to be the Kontsevich--Penner model introduced by the authors and Yu. Makeenko. Its partition function is also a KP hierarchy tau function, and this model is in turn equivalent to a Hermitian one-matrix model with a general potential. Finally we prove that the generating function for general two-profile Belyi morphisms is a GKM thus proving that it is also a KP hierarchy tau function in proper times.

math.AG↗

arXiv2013

Quantum ordering for quantum geodesic functions of orbifold Riemann surfaces

We determine the explicit quantum ordering for a special class of quantum geodesic functions corresponding to geodesics joining exactly two orbifold points or holes on a non-compact Riemann surface. We discuss some special cases in which these quantum geodesic functions form sub--algebras of some abstract algebras defined by the reflection equation and we extend our results to the quantisation of matrix elements of the Fuchsian group associated to the Riemann surface in Poincaré uniformization. In particular we explore an interesting relation between the deformed $U_q(\mathfrak{sl}_2)$ and the Zhedanov algebra AW(3).

math.QA↗

arXiv2013

Seiberg-Witten equations and non-commutative spectral curves in Liouville theory

We propose that there exist generalized Seiberg-Witten equations in the Liouville conformal field theory, which allow the computation of correlation functions from the resolution of certain Ward identities. These identities involve a multivalued spin one chiral field, which is built from the stress-energy tensor. We solve the Ward identities perturbatively in an expansion around the heavy asymptotic limit, and check that the first two terms of the Liouville three-point function agree with the known result of Dorn, Otto, Zamolodchikov and Zamolodchikov. We argue that such calculations can be interpreted in terms of the geometry of non-commutative spectral curves.

hep-th↗

arXiv2011

Poisson algebras of block-upper-triangular bilinear forms and braid group action

In this paper we study a quadratic Poisson algebra structure on the space of bilinear forms on $C^{N}$ with the property that for any $n,m\in N$ such that $n m =N$, the restriction of the Poisson algebra to the space of bilinear forms with block-upper-triangular matrix composed from blocks of size $m\times m$ is Poisson. We classify all central elements and characterise the Lie algebroid structure compatible with the Poisson algebra. We integrate this algebroid obtaining the corresponding groupoid of morphisms of block-upper-triangular bilinear forms. The groupoid elements automatically preserve the Poisson algebra. We then obtain the braid group action on the Poisson algebra as elementary generators within the groupoid. We discuss the affinisation and quantisation of this Poisson algebra, showing that in the case $m=1$ the quantum affine algebra is the twisted $q$-Yangian for ${o}_n$ and for $m=2$ is the twisted $q$-Yangian for ${sp}_{2n}$. We describe the quantum braid group action in these two examples and conjecture the form of this action for any $m>2$.

math-ph↗

arXiv2011

Teichmüller spaces as degenerated symplectic leaves in Dubrovin--Ugaglia Poisson manifolds

In this paper we study the Goldman bracket between geodesic length functions both on a Riemann surface $Σ_{g,s,0}$ of genus $g$ with $s=1,2$ holes and on a Riemann sphere $Σ_{0,1,n}$ with one hole and $n$ orbifold points of order two. We show that the corresponding Teichmüller spaces $\mathcal T_{g,s,0}$ and $\mathcal T_{0,1,n}$ are realised as real slices of degenerated symplectic leaves in the Dubrovin--Ugaglia Poisson algebra of upper--triangular matrices $S$ with 1 on the diagonal.

math-ph↗

arXiv2010

Shear coordinate description of the quantised versal unfolding of D_4 singularity

In this paper by using Teichmuller theory of a sphere with four holes/orbifold points, we obtain a system of flat coordinates on the general affine cubic surface having a D_4 singularity at the origin. We show that the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere coincides with the Etingof-Ginzburg Poisson bracket on the affine D_4 cubic. We prove that this bracket is the image under the Riemann-Hilbert map of the Poisson Lie bracket on the direct sum of three copies of sl_2. We realise the action of the mapping class group by the action of the braid group on the geodesic functions . This action coincides with the procedure of analytic continuation of solutions of the sixth Painlevé equation. Finally, we produce the explicit quantisation of the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere and of the braid group action.

math-ph↗

arXiv2009

Isomonodromic deformations and twisted Yangians arising in Teichmüller theory

In this paper we build a link between the Teichmuller theory of hyperbolic Riemann surfaces and isomonodromic deformations of linear systems whose monodromy group is the Fuchsian group associated to the given hyperbolic Riemann surface by the Poincare' uniformization. In the case of a one-sheeted hyperboloid with n orbifold points we show that the Poisson algebra D_n of geodesic length functions is the semiclassical limit of the twisted q-Yangian for the orthogonal Lie algebra defined by Molev, Ragoucy and Sorba. We give a representation of the braid group action on this algebra in terms of an adjoint matrix action. We characterize two types of finite-dimensional Poissonian reductions and give an explicit expression for the generating function of their central elements. Finally, we interpret the algebra D_n as the Poisson algebra of monodromy data of a Frobenius manifold in the vicinity of a non-semisimple point.

math.AG↗

arXiv2007

Lecture Notes on Quantum Teichmuller Theory

These notes are based on a lecture course by L. Chekhov held at the University of Manchester in May 2006 and February-March 2007. They are divulgative in character, and instead of containing rigorous mathematical proofs, they illustrate statements giving an intuitive insight. We intentionally remove most bibliographic references from the body of the text devoting a special section to the history of the subject at the end.

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