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A. Yu. Orlov

Publications and source records attributed to A. Yu. Orlov.

At least 19 recordsLinked to original sources
arXiv2025

Products of random Hermitian matrices and brickwork Hurwitz numbers. Products of normal matrices

We consider products of $n$ random Hermitian matrices which generalize the one-matrix model and show its relation to Hurwitz numbers which count ramified coverings of certain type. Namely, these Hurwitz numbers count $2k$-fold ramified coverings of the Riemann sphere with arbitrary ramification type over $0$ and $\infty$ and ramifications related to the partition $(2^k)$ (``brickworks'' - involution without fixed points) elsewhere. Products of normal random matrices are also considered.

math-ph↗
arXiv2025

SU(N) integrals and tau functions

We present a family of solvable multi-matrix models associated with an arbitrary embedded graph $\Gamma$ with a single vertex. The graph with $n$ edges is equipped with $2n$ corner matrices. The partition function of each member of the family depends on the set of eigenvalues of monodromies of corner matrices around the vertices of the dual graph $\Gamma^*$ and sets of parameters attached to each vertex of $\Gamma$. We select the cases where the partition function of a model is a tau function of KP, 2KP and BKP hiearachies. We compare integrals over ${U}(N)$ and over ${SU}(N)$ groups. In $U(N)$ case there is no restriction on the number of vertices of $\Gamma$. We also consider mixed ensembles of matrices from $GL(N),U(N)$ and $SU(N)$.

math-ph↗
arXiv2024

Hopf link invariants and integrable hierarchies

The goal of this note is to study integrable properties of a generating function of the HOMFLY-PT invariants of the Hopf link colored with different representations. We demonstrate that such a generating function is a $\tau$-function of the KP hierarchy. Furthermore, this Hopf generating function in the case of composite representations, which is a generating function of the 4-point functions in topological string (corresponding to the resolved conifold with branes on the four external legs), is a $\tau$-function of the universal character(UC) hierarchy put on the topological locus. We also briefly discuss a simple matrix model associated with the UC hierarchy.

hep-th↗
arXiv2023

$W_{1+\infty}$ flows and multi-component hierarchy (KP case)

We show that abelian subalgebras of generalized $W_{1+\infty}$ ($GW_{1+\infty}$) algebra gives rise to the multicomponent KP flows. The matrix elements of the related group elements in the fermionic Fock space is expressed as a product of a certain factor (generalized content product) and of a number of the Schur functions and the skew Schur functions.

nlin.SI↗
arXiv2023

Series over fat partitions: matrix models and discrete ensembles

We consider series over Young diagrams of products of Schur functions $s_{\lambda\cup\lambda}$, marked with ``fat partitions'' $\lambda\cup\lambda$, which appear in matrix models associated with ensembles of symplectic and orthogonal matrices and quaternion Ginibre ensembles. We consider mixed matrix models that also contain complex Ginibre ensembles labeled by graphs with corner matrices and the three ensembles mentioned above. Cases are identified when a series of perturbations in coupling constants turn out to be tau functions of the DKP hierarchy introduced by the Kyoto school. This topic relates matrix models to random partitions - discrete symplectic ensemble and its modifications.

math-ph↗
arXiv2023

Coupling of different solvable ensembles of random matrices

Explicit expressions for multimatrix models with complex and unitary matrices allows to couple these models with well-known unitary, orthogonsl and sympletic ensembles. We consider examples of such mixed ensembles which are solvable in the sense that the partition functions of such ensembles can be considered as tau functions of the classical integrable equations.

hep-th↗
arXiv2023

Sigma model instantons and singular tau function

The generating series for the instanton contribution to Green functions of the $2D$ sigma model was found in the works of Schwarz, Fateev and Frolov. We show that this series can be written as a formal tau function of the two-sided two-component KP hierarchy. We call it formal singular tau function because this tau function is a sum where each term is the infrared and ultraviolet divergent one exactly as the series found by the mentioned authors. However one can regularize this singluar tau function and to obtain regular observables. This is because observables contains ratious of mentioned divergent expressions. Thus, we enladge the families of tau functions to work with.

nlin.SI↗
arXiv2023

New solvable two-matrix model and BKP tau function

We present exactly solvable modifications of the two-matrix Zinn-Justin-Zuber model and write it as a tau function. The grand partition function of these matrix integrals is written as the fermion expectation value. The perturbation theory series is written out explicitly in terms of series in strict partitions. The related string equations are presented.

hep-th↗
arXiv2023

Polygon gluing and commuting bosonic operators

We construct two series of commuting Hamiltonians parametrized by a constant matrix. The first series was my guess and the second one was know, and in our approach follows from the first series. For the proof we use familier facts known from our previous consideration of the links between random matrices and Hurwitz numbers, however the text is self-consistent.

math-ph↗
arXiv2022

Calogero model revisited, commuting Hamiltonians, Hurwitz numbers

The generalized Mironov-Morozov-Natanson (MMN) equation includes a set of commuting operators, which can be considered as Hamiltonians for the quantum Calogero-Sutherland problem with a special value of the coupling constant (free fermion point). These Hamiltonians can be considered as the center of the enveloping algebra of the group $GL_N(C)$. Another commuting series of Hamiltonians is presented, parametrized by an arbitrary matrix $A\in GL_N$. These Hamiltonians are related to the Hurwitz numbers in the same way as in the case of the MMN equation and generate a generalized variant of the Calogero-Surtheland model.

math-ph↗
arXiv2021

Bilinear expansions of lattices of KP $τ$-functions in BKP $τ$-functions: a fermionic approach

We derive a bilinear expansion expressing elements of a lattice of KP $τ$-functions, labelled by partitions, as a sum over products of pairs of elements of an associated lattice of BKP $τ$-functions, labelled by strict partitions. This generalizes earlier results relating determinants and Pfaffians of minors of skew symmetric matrices, with applications to Schur functions and Schur $Q$-functions. It is deduced using the representations of KP and BKP $τ$-functions as vacuum expectation values (VEV's) of products of fermionic operators of charged and neutral type, respectively. The lattice is generated by insertion of products of pairs of charged creation and annihilation operators. The result follows from expanding the product as a sum of monomials in the neutral fermionic generators and applying a factorization theorem for VEV's of products of operators in the mutually commuting subalgebras. Applications include the case of inhomogeneous polynomial $τ$-functions of KP and BKP type.

math-ph↗
arXiv2021

Polynomial KP and BKP $τ$-functions and correlators

Lattices of polynomial KP and BKP $τ$-functions labelled by partitions, with the flow variables equated to finite power sums, as well as associated multipair KP and multipoint BKP correlation functions are expressed via generalizations of Jacobi's bialternant formula for Schur functions and Nimmo's Pfaffian ratio formula for Schur $Q$-functions. These are obtained by applying Wick's theorem to fermionic vacuum expectation value representations in which the infinite group element acting on the lattice of basis states stabilizes the vacuum.

math-ph↗
arXiv2021

Bilinear expansion of Schur functions in Schur $Q$-functions: a fermionic approach

An identity is derived expressing Schur functions as sums over products of pairs of Schur $Q$-functions, generalizing previously known special cases. This is shown to follow from their representations as vacuum expectation values (VEV's) of products of either charged or neutral fermionic creation and annihilation operators, Wick's theorem and a factorization identity for VEV's of products of two mutually anticommuting sets of neutral fermionic operators.

math-ph↗
arXiv2019

Integrals of tau functions

We consider integrals of tau functions of Zakharov-Shabat systems whose higher times are related to the eigenvalues of products of random matrices. Apart of random matrices there is the set of $n$ pairs of given matrices which play the role of parameters. In terms of these matrices we introduce the notions of words, dressed words and dual words, these notions are related to the graphs and dual graphs drawn on a Riemann surface $Σ$. The integrals of tau functions over independent ensembles of random matrices can be computed in form of series over partitions of products of the Schur polynomials with a multiplier which depends on the Euler characteristic of the Riemann surface. This form allows to compare the integrals of tau functions with correlation functions of certain quantum models. We present a tau function whose integral is equal to the correlation function of the Wilson loops of the two-dimensional Yang-Mills model on $Σ$.

nlin.SI↗
arXiv2019

Instantons in $σ$ model and tau functions

We show that a number of multiple integrals may viewed as tau functions of various integrable hierarchies. The instanton contributions in the two-dimensional O(3)$\ σ$ model is an example of such an approach.

nlin.SI↗
arXiv2018

Hurwitz numbers and matrix integrals labeled with chord diagrams

We shall consider the product of complex random matrices from the independent complex Ginibre ensembles. The product includes complex matrices $Z_i, Z_i^\dagger, \, i = 1, \ldots, n$ and $2n$ sources (complex matrices $C_i$ and $C_i^*$). Any such product can be represented by a chord diagram that encodes the order of the matrices in the product. We introduce the Euler characteristic E$^*$ of the chord diagram and show that the spectral correlation functions of the product generate Hurwitz numbers that enumerate nonequivalent branched coverings of Riemann surfaces of genus $g^*$. The role of sources is the generation of branching profiles in critical points which are assigned to the vertices of the graph drawn on the base surface obtained as a result of gluing of the $2n$-gon related to the chord diagram in a standard way. Hurwitz numbers for Klein surfaces may also be obtained by a slight modification of the model. Namely, we consider $2n+1$ polygon and consider pairing of the extra matrix $Z_{2n+1}$ with a "Mobius" tau function. Thus, the presented matrix models labelled by chord diagrams generate Hurwitz numbers for any given Euler characteristic of the base surface and for any given set of ramification profiles.

math-ph↗