# Leonid ChekhovRussian Academy of Sciences | RAS · Steklov Mathematical Institute

Leonid Chekhov

PhD at Steklov Math. Inst., Moscow, 1987

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133

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Introduction

Additional affiliations

June 2014 - December 2015

May 2012 - November 2013

## Publications

Publications (133)

We use the Darboux coordinate representation found by two of the authors (L.Ch. and M.Sh.) for entries of general symplectic leaves of the $\mathcal A_n$-groupoid of upper-triangular matrices to express roots of the characteristic equation $\det(\mathbb A-\lambda \mathbb A^{\text{T}})=0$, with $\mathbb A\in \mathcal A_n$, in terms of Casimirs of th...

We show that having any planar (cyclic or acyclic) directed network on a disc with the only condition that all $n_1+m$ sources are separated from all $n_2+m$ sinks, we can construct a cluster-algebra realization of elements of an affine Lie--Poisson algebra $R(\lambda,\mu)T^{1}(\lambda)T^{2}(\mu)=T^{2}(\mu)T^{1}(\lambda)R(\lambda,\mu)$ with $(n_1\t...

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 m...

We explicitly show that the Poisson bracket on the set of shear coordinates introduced by V.V. Fock in 1997 induces the Fenchel--Nielsen bracket on the set of gluing parameters (length and twist parameters) for pairs of pants decomposition for Riemann surfaces with holes $\Sigma_{g,s}$. We generalize these structures to the case of Riemann surfaces...

Дается обзор описания с помощью ленточных графов римановых поверхностей $\Sigma _{g,s,n}$ и соответствующих пространств Тейхмюллера $\mathfrak T_{g,s,n}$ с $s>0$ дырками и $n>0$ граничными каспами в подходе гиперболической геометрии. В случае, когда $n>0$, имеет место взаимно однозначное соответствие между множеством тeрстоновских координат смещени...

We recall the fat-graph description of Riemann surfaces Σg,s,n and the corresponding Teichmüller spaces \({\mathfrak{T}_{g,s,n}}\) with s > 0 holes and n > 0 bordered cusps in the hyperbolic geometry setting. If n > 0, we have a bijection between the set of Thurston shear coordinates and Penner’s λ-lengths. Then we can define, on the one hand, a Po...

Using Fock--Goncharov higher Teichm\"uller space variables we derive Darboux coordinate representation for entries of general symplectic leaves of the $\mathcal A_n$ groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric $R...

Показано, что пуассонова скобка на множестве координат смещений, введенная В. В. Фоком в 1997 г., индуцирует скобку Фенхеля-Нильсена на множестве параметров склеек (параметров длин и скруток) для разрезаний на штаны римановых поверхностей с дырками $\Sigma_{g,s}$. Эти структуры обобщаются на случай римановых поверхностей $\Sigma_{g,s,n}$ с дырками...

We recall the fat-graph description of Riemann surfaces $\Sigma_{g,s,n}$ and the corresponding Teichm\"uller spaces $\mathfrak T_{g,s,n}$ with $s>0$ holes and $n>0$ bordered cusps in the hyperbolic geometry setting. If $n>0$, we have a bijection between the set of Thurston shear coordinates and Penner's $\lambda$-lengths and we can induce, on the o...

In this paper we study quantum del Pezzo surfaces belonging to a certain class. In particular we introduce the generalised Sklyanin-Painlev\'e 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...

This chapter examines the Poisson structure of the representation variety of the fundamental groupoid of a Riemann surface with punctures and cusps, and the associated decorated character variety.

We consider multi-matrix models that are generating functions for the numbers of branched covers of the complex projective line ramified over n fixed points zi, i=1,…,n, (generalized Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profiles at two points, z1 and zn. Ramifications at other n−2 points enter the sum with t...

We consider multi-matrix models that are generating functions for the numbers of 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 $z_n$. Ramifications at other $n-2$ points e...

Let A be the space of bilinear forms on CN with defining matrices A endowed with a quadratic Poisson structure of reflection equation type. The paper begins with a short description of previous studies of the structure, and then this structure is extended to systems of bilinear forms whose dynamics is governed by the natural action A→ BABT of the G...

We study perturbations around the generalized Kazakov multicritical one-matrix model. The multicritical matrix model has a potential where the coefficients of $z^n$ only fall off as a power $1/n^{s+1}$. This implies that the potential and its derivatives have a cut along the real axis, leading to technical problems when one performs perturbations a...

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 Rh...

Kontsevich and Soibelman reformulated and slightly generalised the topological recursion of math-ph/0702045, seeing it as a quantization of certain quadratic Lagrangians in $T^*V$ for some vector space $V$. KS topological recursion is a procedure which takes as initial data a quantum Airy structure -- a family of at most quadratic differential oper...

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 decompo...

The topological recursion is an ubiquitous structure in enumerative geometry of surfaces and topological quantum field theories. Since its invention in the context of matrix models, it has been found or conjectured to compute intersection numbers in the moduli space of curves, topological string amplitudes, asymptotics of knot invariants, and more...

We derive the Do and Norbury recursion formula for the one-loop mean of an
irregular spectral curve from a variant of replica method by Brez\'in and
Hikami. We express this recursion in special times in which all terms
$W_1^{(g)}$ of the genus expansion of the one-loop mean are polynomials. We
find a generalization of this recursion to the generali...

We introduce explicit relations between genus-filtrated s-loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich–Penner matrix model (KPMM), which is the generating function for volumes of discretized (open) moduli spaces M
g,s
disc (discrete volumes). Using these relations, we express Gaussian means in all order...

In this paper we introduce the concept of decorated character variety for the
Riemann surfaces arising in the theory of the Painlev\'e differential
equations. Since all Painlev\'e 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 de...

We introduce the notion of bordered cusped Teichm\"uller space, as the
Teichm\"uller space of Riemann surfaces with at least one hole and at least one
bordered cusp on the boundary. We propose a combinatorial graph description of
this bordered cusped Teichm\"uller space and endow it with a Poisson structure.
This new notion arises in the limit of c...

We prove combinatorially the explicit relation between genus filtrated
$s$-loop means of the Gaussian matrix model and terms of the genus expansion of
the Kontsevich--Penner matrix model (KPMM). The latter is the generating
function for volumes of discretized (open) moduli spaces
$M_{g,s}^{\mathrm{disc}}$ given by $N_{g,s}(P_1,\dots,P_s)$ for
$(P_1...

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...

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 ext...

We consider the space A of bilinear forms on C N with defining matrix A endowed with the quadratic Poisson structure studied by the authors in [3]. We classify all possible quadratic brackets on (B, A) ∈ GL N ×A with the property that the natural action A → BAB T of the GL N Poisson–Lie group on the space A is a Poisson action thus endowing A with...

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...

In the present article, we review a derivation of the numbers of RNA complexes of an arbitrary topology. These numbers are encoded in the free energy of the Hermitian matrix model with potential V(x)=x2/2-stx/(1-tx), where s and t are respective generating parameters for the number of RNA molecules and hydrogen bonds in a given complex. The free en...

In this paper, we study the Goldman bracket between geodesic length functions both on a Riemann surface Σg,s,0 of genus g with s=1,2 holes and on a Riemann sphere Σ0,1,n with one hole and n orbifold points of order two. We show that the corresponding Teichmüller spaces Tg,s,0 and T0,1,n are realised as real slices of degenerated symplectic leaves i...

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 pertur...

We introduce and study the Hermitian matrix model with potential
V(x)=x^2/2-stx/(1-tx), which enumerates the number of linear chord diagrams of
fixed genus with specified numbers of backbones generated by s and chords
generated by t. For the one-cut solution, the partition function, correlators
and free energies are convergent for small t and all s...

The effective scalar theory is constructed for the nonlinear Schrödinger (NS) matrix equation, which gives us the tool for investigating the two-point correlator and the partition function for some (classical) NS-like systems. The spatial dependence of the correlator coincides with the free correlator in all quantization schemes. For the classical...

The technique for finding correlation functions on homogeneous spaces of PGL(ℚp) groups (factorized Bruhat-Tits trees Tp/ΓN with finite number of cycles) is presented. It was shown in Refs. 5 and 6 that the homogeneous spaces Tp/ΓN are in fact the multiloop world sheets in p-adic string theory.

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 $x+2\cos{\pi/n_o}+x^{-1}$ these
transformations are related to triangulations of Riemann surfaces of arbitrary
genus with at least one hole/puncture and with an arbitrary num...

In this paper we study the Goldman bracket between geodesic length functions
both on a Riemann surface $\Sigma_{g,s,0}$ of genus $g$ with $s=1,2$ holes and
on a Riemann sphere $\Sigma_{0,1,n}$ with one hole and $n$ orbifold points of
order two. We show that the corresponding Teichm\"uller spaces $\mathcal
T_{g,s,0}$ and $\mathcal T_{0,1,n}$ are rea...

In this paper we build a link between the Teichmüller 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 Poincaré uniformization. In the case of a one-sheeted hyperboloid with n orbifold points we show that the P...

We present a diagrammatic technique for calculating the free energy of the matrix eigenvalue model (the model with an arbitrary power of the Vandermonde determinant) to all orders of the 1/N expansion in the case when the limiting eigenvalue distribution spans an arbitrary (but fixed) number of disjoint intervals (curves) and when logarithmic terms...

We construct the solution of the loop equations of the β-ensemble model in a form analogous to the solution in the case of
the Hermitian matrices β = 1. The solution for β = 1 is expressed in terms of the algebraic spectral curve given by y2 = U(x). The spectral curve for arbitrary β converts into the Schrödinger equation (ħ∂)2 − U(x) ψ(x) = 0, whe...

In this paper we study a quadratic Poisson algebra structure on the space of bilinear forms on \({\mathbb{C}^{N}}\) with the property that for any \({n, m \in \mathbb{N}}\) such that n
m = N, the restriction of the Poisson algebra to the space of bilinear forms with a block-upper-triangular matrix composed from blocks of size \({m \times m}\) is Po...

In this communication, by using Teichmüller theory of a sphere with four holes/orbifold points, we obtain a system of flat coordinates on the general affine cubic surface having a D4 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 bra...

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...

A fat graph description is given for Teichmüller spaces ofRiemann surfaces with holes and with - and -orbifold points (conical singularities) in the Poincaré uniformization. The corresponding mapping class group transformations are presented, geodesic functions are constructed, and the Poisson structure is introduced. The resulting Poisson algebras...

Using the structure of algebroid of block-upper-triangular matrices composed
from blocks of size $m\times m$ we obtain the Poisson brackets on the entries
of these matrices and construct the braid-group action that preserves the
Poisson algebra in the case of arbitrary $m$. We extend these algebras to
semiclassical twisted Yangian algebras and find...

In this article, we solve the loop equations of the \beta-random matrix model, in a way similar to what was found for the case of hermitian matrices \beta=1. For \beta=1, the solution was expressed in terms of algebraic geometry properties of an algebraic spectral curve of equation y^2=U(x). For arbitrary \beta, the spectral curve is no longer alge...

We study the Teichm\"uller theory of Riemann surfaces with orbifold points of
order two using the fat graph technique. The previously developed technique of
quantization, classical and quantum mapping-class group transformations, and
Poisson and quantum algebras of geodesic functions is applicable to the
surfaces with orbifold points. We describe c...

We interpret the previously developed Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces)
as the Teichmüller theory of Riemann surfaces with orbifold points of order 2. In the Poincaré uniformization pattern, we
describe necessary and sufficient conditions for the group generated by the Fuchsian group of the...

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 th...

A description is given of properties of the most general multisupport solutions of one-matrix models, beginning with the one-matrix model in the presence of hard walls, that is, the case where the eigenvalue support is confined to several fixed intervals of the real axis. The eigenvalue model, which generalizes the one-matrix model to the Dyson gas...

This is a survey of the theory of quantum Teichmüller and Thurston spaces. The Thurston (or train track) theory is described and quantized using the quantization of coordinates for Teichmüller spaces of Riemann surfaces with holes. These surfaces admit a description by means of the fat graph construction proposed by Penner and Fock. In both theorie...

We propose the graph description of Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of Thurston variables (foliation-shear coordinates),...

We propose the graph description of Teichm\"uller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of Thurston variables (foliation-shear coordinates...

We present the diagrammatic technique for calculating the free energy of the matrix eigenvalue model (the model with arbitrary power $\beta$ by the Vandermonde determinant) to all orders of 1/N expansion in the case where the limiting eigenvalue distribution spans arbitrary (but fixed) number of disjoint intervals (curves). Comment: Latex, 27 pages

We compute the complete topological expansion of the formal hermitian
two-matrix model. For this, we refine the previously formulated diagrammatic
rules for computing the 1/ N expansion of the nonmixed correlation functions
and give a new formulation of the spectral curve. We extend these rules
obtaining a closed formula for correlation functions i...

We present the diagrammatic technique for calculating the free energy of
the Hermitian one-matrix model to all orders of 1/N expansion in the
case where the limiting eigenvalue distribution spans arbitrary (but
fixed) number of disjoint intervals (curves).

We discuss various aspects of most general multisupport solutions to matrix models in the presence of hard walls, i.e., in the case where the eigenvalue support is confined to subdomains of the real axis. The structure of the solution at the leading order is described by semiclassical, or generalized Whitham--Krichever hierarchies as in the unrestr...

We investigate the AdS3/CFT2 correspondence for the Euclidean AdS3 space compactified on a solid torus with the CFT field on the regularizing boundary surface in the bulk. Correlation functions
corresponding to the bulk theory at a finite temperature tend to the standard CFT correlation functions in the limit of removed
regularization. In the sum o...

Using the Penner--Fock parameterization for Teichmuller spaces of Riemann surfaces with holes, we construct the string-like free-field representation of the Poisson and quantum algebras of geodesic functions in the continuous-genus limit. The mapping class group acts naturally in the obtained representation. Comment: 16 pages, submitted to Lett.Mat...

The paper contains some new results and a review of recent achievements, concerning the multisupport solutions to matrix models. In the leading order of the 't Hooft expansion for matrix integral, these solutions are described by quasiclassical or generalized Whitham hierarchies and are directly related to the superpotentials of four-dimensional N=...

We calculate genus-one corrections to the Hermitian one-matrix model solution with an arbitrary number of cuts based on the loop equation, confirming the answer previously obtained from algebro-geometric considerations and generalizing it to the case of arbitrary potentials.

We investigate the AdS$_3$/CFT$_2$ correspondence for the Euclidean AdS$_3$ space compactified on a solid torus with the CFT field on the regularizing boundary surface in the bulk. Correlation functions corresponding to the bulk theory at finite temperature tend to the standard CFT correlation functions in the limit of removed regularization. In bo...

In earlier work, Chekhov and Fock have given a quantization of Teichm\"uller space as a Poisson manifold, and the current paper first surveys this material adding further mathematical and other detail, including the underlying geometric work by Penner on classical Teichm\"uller theory. In particular, the earlier quantum ordering solution is found t...

We discuss the relation between matrix models and the Seiberg-Witten type (SW) theories, recently proposed by Dijkgraaf and Vafa. In particular, we prove that the partition function of the Hermitian one-matrix model in the planar (large N) limit coincides with the prepotential of the corresponding SW theory. This partition function is the logarithm...

We prove that the quasiclassical tau-function of the multi-support solutions to matrix models, proposed recently by Dijkgraaf and Vafa to be related to the Cachazo–Intrilligator–Vafa superpotentials of the supersymmetric Yang–Mills theories, satisfies the Witten–Dijkgraaf–Verlinde–Verlinde equations.

We find necessary and sufficient conditions for an operator of the fourth order on a graph with loops to admit the (L,A,B)-triple Krichever-Novikov deformation of one energy level.

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The algebra of quantum geodesics obtained by quantizing the coordinates of the Teichmller spaces is the Nelson–Regge quantum so
q(m) algebra of monodromies (Wilson loops) in the Chern–Simons theory, which provides an effective description of (2+1)-dimensional gravity.

We study the connection between characteristics of moduli spaces of Riemann surfaces with marked points and matrix models. The Kontsevich matrix model describes intersection indices on continuous moduli spaces, and the Kontsevich–Penner matrix model describes intersection indices on discretized moduli spaces. Analyzing the constraint algebras satis...

The explicit form of non-Abelian noncommutative supersymmetric (SUSY) chiral anomaly is calculated, the Wess-Zumino consistency condition is verified and the correspondence of the Yang-Mills sector to the previously obtained results is shown. We generalize the Seiberg-Witten map to the case of N=1 SUSY Yang-Mills theory and calculations up to the s...

The algebra of quantum geodesics obtained by quantizing the coordinates of the Teichmller spaces is the quantumso
q(m) algebra by Nelson and Regge.

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The scattering process on multiloop infinite p+1-valent graphs (generalized trees) is studied. These graphs are discrete spaces being quotients of the uniform tree over free acting discrete subgroups of the projective group $PGL(2, {\bf Q}_p)$. As the homogeneous spaces, they are, in fact, identical to p-adic multiloop surfaces. The Ihara-Selberg L...

The AdS/CFT correspondence is established for the AdS3 space compactified on a solid torus with the CFT field on the boundary. Correlation functions that correspond to the bulk theory at finite temperature are obtained in the regularization a la Gubser, Klebanov, and Polyakov. The BTZ black hole solutions in AdS3 are T-dual to the solution in the A...

We explicitly describe a noncommutative deformation of the *-algebra of functions on the Teichmüller space of Riemann surfaces
with holes that is equivariant with respect to the action of the mapping class group.

It is shown that the classical L-operator algebra of the elliptic Ruijsenaars-Schneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions. It is governed by two dynamical r- and r ¯-matrices satisfying a closed system of equations. The corresponding...

Possible scenario for quantizing the moduli spaces of Riemann curves is considered. The proper quantum observables are quantum geodesics that are invariant with respect to a quantum modular group and satisfy the quantum algebra. 1 Introduction A problem of constructing an appropriate quantum analogue