Searcharxiv⌕ Search

arXiv subjects

L. Chekhov

Publications and source records attributed to L. Chekhov.

At least 19 recordsLinked to original sources
arXiv2024

Geometric leaf of symplectic groupoid

We consider the symplectic groupoid of pairs $(B, A)$ with $A$ real unipotent upper-triangular matrix and $B\in GL_n$ being such that $\tilde A=BAB^T$ is also a unipotent upper-triangular matrix. Fock and Chekhov defined a Poisson map of Teichm\"uller space ${\mathcal T_{g,s}$ of genus $g$ surfaces with $s$ holes into the space of unipotent upper-triangular $n\times n$ matrices whose image forms the \emph{geometric locus}. The elements of geometric locus satisfy \emph{rank condition}. We describe the Hamiltonian reduction of the Poisson cluster variety of symplectic groupoid by the rank condition for $n=5$ and $6$. In both cases, we analyze the induced cluster structures on the results of Hamiltonian reduction and recover celebrated cluster structure on ${\mathcal T}_{2,1}$ for $n=5$ and ${\mathcal T}_{2,2}$ for $n=6$.

math.QA↗
arXiv2020

Darboux coordinates for symplectic groupoid and cluster algebras

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 $R$-matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for $\mathcal A_3$ and $\mathcal A_4$ in terms of geodesic functions for Riemann surfaces with holes. We represent braid-group transformations for $\mathcal A_n$ via sequences of cluster mutations in the special $\mathbb A_n$-quiver. We prove the groupoid relations for quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit.

math.QA↗
arXiv2017

Perturbed generalized multicritical one-matrix models

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 away from the generalized Kazakov model. Nevertheless it is possible to relate the perturbed partition function to the tau-function of a KdV hierarchy and solve the model by a genus expansion in the double scaling limit.

hep-th↗
arXiv2010

Logarithmic potential beta-ensembles and Feynman graphs

We present the diagrammatic technique for calculating the free energy of the matrix eigenvalue model (the model with arbitrary power beta by the Vandermonde) to all orders of 1/N expansion in the case where the limiting eigenvalue distribution spans arbitrary (but fixed) number of disjoint intervals (curves) and when logarithmic terms are present. This diagrammatic technique is corrected and refined as compared to our first paper with B.Eynard of year 2006.

math-ph↗
arXiv2009

Topological expansion of the Bethe ansatz, and quantum algebraic geometry

In this article, we solve the loop equations of the β-random matrix model, in a way similar to what was found for the case of hermitian matrices β=1. For β=1, the solution was expressed in terms of algebraic geometry properties of an algebraic spectral curve of equation y^2=U(x). For arbitrary β, the spectral curve is no longer algebraic, it is a Schroedinger equation ((\hbar\partial)^2-U(x)).ψ(x)=0 where \hbar\propto (\sqrtβ-1/\sqrtβ). In this article, we find a solution of loop equations, which takes the same form as the topological recursion found for β=1. This allows to define natural generalizations of all algebraic geometry properties, like the notions of genus, cycles, forms of 1st, 2nd and 3rd kind, Riemann bilinear identities, and spectral invariants F_g, for a quantum spectral curve, i.e. a D-module of the form y^2-U(x), where [y,x]=\hbar. Also, our method allows to enumerate non-oriented discrete surfaces.

math-ph↗
arXiv2006

Matrix models with hard walls: Geometry and solutions

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 unrestricted case. Derivatives of tau-functions for these solutions are associated with families of Riemann surfaces (with possible double points) and satisfy the Witten--Dijkgraaf--Verlinde--Verlinde equations. We then develop the diagrammatic technique for finding free energy of this model in all orders of the 't~Hooft expansion in the reciprocal matrix size generalizing the Feynman diagrammatic technique for the Hermitian one-matrix model due to Eynard.

hep-th↗
arXiv2005

Extension of geodesic algebras to continuous genus

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.

gr-qc↗
arXiv2005

Complex Geometry of Matrix Models

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=1 SUSY gauge theories. We study the derivatives of tau-functions for these solutions, associated with the families of Riemann surfaces (with possible double points), and relations for these derivatives imposed by complex geometry, including the WDVV equations. We also find the free energy in subleading order of the 't Hooft expansion and prove that it satisfies certain determinant relations.

hep-th↗
arXiv2004

AdS_3/CFT_2 on torus in the sum over geometries

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 both regular and $Z_N$ orbifold cases, in the sum over geometries, the two-point correlation function for massless modes factors, up to divergent terms proportional to the volume of the $SL(2,Z)/Z}$ group, into the finite sum of products of the conformal--anticonformal CFT Green's functions.

hep-th↗
arXiv2004

On Quantizing Teichmüller and Thurston theories

In earlier work, Chekhov and Fock have given a quantization of Teichmüller 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 Teichmüller theory. In particular, the earlier quantum ordering solution is found to essentially agree with an ``improved'' operator ordering given by serially traversing general edge-paths on a graph in the underlying surface. Now, insofar as Thurston's sphere of projectivized foliations of compact support provides a useful compactification for Teichmüller space in the classical case, it is natural to consider corresponding limits of appropriate operators to provide a framework for studying degenerations of quantum hyperbolic structures. After surveying the required background material on Thurston theory and ``train tracks'', the current paper continues to give a quantization of Thurston's boundary in the special case of the once-punctured torus, where there are already substantial analytical and combinatorial challenges. Indeed, an operatorial version of continued fractions as well as the improved quantum ordering are required to prove existence of these limits. Since Thurston's boundary for the once-punctured torus is a topological circle, the main new result may be regarded as a quantization of this circle. There is a discussion of quantizing Thurston's boundary spheres for higher genus surfaces in closing remarks.

math.AG↗
arXiv2004

Genus one correlation to multi-cut matrix model solutions

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

hep-th↗
arXiv2003

DV and WDVV

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 N=1 supersymmetric Yang-Mills theories, satisfies the Witten-Dijkgraaf-Verlinde-Verlinde equations.

hep-th↗
arXiv2002

Matrix models vs. Seiberg-Witten/Whitham theories

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 Hermitean one-matrix model in the planar (large $N$) limit coincides with the prepotential of the corresponding SW theory. This partition function is the logarithm of a Whitham $τ$-function. The corresponding Whitham hierarchy is explicitly constructed. The double-point problem is solved.

hep-th↗
arXiv1999

Spectral problem on graphs and L-functions

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-function is associated with the finite subgraph-the reduced graph containing all loops of the generalized tree. We study the spectral problem on these graphs, for which we introduce the notion of spherical functions-eigenfunctions of a discrete Laplace operator acting on the graph. We define the S-matrix and prove its unitarity. We present a proof of the Hashimoto-Bass theorem expressing L-function of any finite (reduced) graph via determinant of a local operator $\Delta(u)$ acting on this graph and relate the S-matrix determinant to this L-function thus obtaining the analogue of the Selberg trace formula. The discrete spectrum points are also determined and classified by the L-function. Numerous examples of L-function calculations are presented.

cond-mat.mes-hall↗
arXiv1999

Quantum Teichmüller space

We describe explicitly a noncommutative deformation of the *-algebra of functions on the Teichmüller space of Riemann surfaces with holes equivariant w.r.t. the mapping class group action.

math.QA↗
arXiv1999

AdS_3/CFT_2 correspondence at finite temperature

The AdS/CFT correspondence is established for the AdS_3 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 AdS_3 are T-dual to the solution in the AdS_3 space without singularity.

hep-th↗