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A. Mironov

Publications and source records attributed to A. Mironov.

At least 55 records · Page 3Linked to original sources
arXiv2023

Interpolating Matrix Models for WLZZ series

We suggest a two-matrix model depending on three (infinite) sets of parameters which interpolates between all the models proposed in arXiv:2206.13038, and defined there through $W$-representations. We also discuss further generalizations of these WLZZ models realized by $W$-representations associated with infinite commutative families of generators of $w_\infty$-algebra which are presumably related to more sophisticated multi-matrix models. Integrable properties of these generalizations are described by what we call the skew hypergeometric $τ$-functions.

hep-th↗
arXiv2022

Relaxation electrodynamics of superinsulators

Superinsulators offer a unique laboratory realizing strong interaction phenomena like confinement and asymptotic freedom in quantum materials. Recent experiments evidenced that superinsulators are the mirror-twins of superconductors with reversed electric and magnetic field effects. Cooper pairs and Cooper holes in the superinsulator are confined into neutral electric pions by electric strings, with the Cooper pairs playing the role of quarks. Here we report the non-equilibrium relaxation of the electric pions in superinsulating films. We find that the time delay $t_{\mathrm{sh}}$ of the current passage in the superinsulator is related to the applied voltage $V$ via the power law, $t_{\mathrm{sh}}\propto (V-V_{\mathrm p})^{-μ}$, where $V_{\mathrm p}$ is the effective threshold voltage. Two distinct critical exponents, $μ=1/2$ and $μ=3/4$, correspond to jumps from the electric Meissner state to the mixed state and to the superinsulating resistive state with broken charge confinement, respectively. The $μ=1/2$ value establishes a direct experimental evidence for the electric strings' linear potential confining the charges of opposite signs in the electric Meissner state and effectively rules out disorder-induced localization as a mechanism for superinsulation. We further report the memory effects and their corresponding dynamic critical exponents arising upon the sudden reversal of the applied voltage. Our observations open routes for exploring fundamental strong interaction charge confinement via desktop experiments.

cond-mat.supr-con↗
arXiv2022

AGT correspondence, (q-)Painlevè equations and matrix models

Painlevè equation for conformal blocks is a combined corollary of integrability and Ward identities, which can be explicitly revealed in the matrix model realization of AGT relations. We demonstrate this in some detail, both for $q$-Painlevè equations for the $q$-Virasoro conformal block, or AGT dual gauge theory in $5d$, and for ordinary Painlevè equations, or AGT dual gauge theory in $4d$. Especially interesting is the continuous limit from $5d$ to $4d$ and its description at the level of equations for eight $τ$-functions. Half of these equations are governed by integrability and another half by Ward identities.

hep-th↗
arXiv2022

Superintegrability as the hidden origin of Nekrasov calculus

Once famous and a little mysterious, AGT relations between Nekrasov functions and conformal blocks are now understood as the Hubbard-Stratanovich duality in the Dijkgraaf-Vafa (DV) phase of a peculiar Dotsenko-Fateev multi-logarithmic matrix model. However, it largely remains a collection of somewhat technical tricks, lacking a clear and generalizable conceptual interpretation. Our new claim is that the Nekrasov functions emerge in matrix models as a straightforward implication of superintegrability, factorization of peculiar matrix model averages. Recently, we demonstrated that, in the Gaussian Hermitian model, the factorization property can be extended from averages of single characters to their bilinear combinations. In this paper, we claim that this is true also for multi-logarithmic matrix models, where factorized are just the point-split products of two characters. It is this enhanced superintegrability that is responsible for existence of the Nekrasov functions and the AGT relations. This property can be generalized both to multi-matrix models, thus leading to AGT relations for multi-point conformal blocks, and to DV phases of other non-Gaussian models.

hep-th↗
arXiv2022

Towards matrix model representation of HOMFLY polynomials

We investigate possibilities of generalizing the TBEM eigenvalue matrix model, which represents the non-normalized colored HOMFLY polynomials for torus knots as averages of the corresponding characters. We look for a model of the same type, which is a usual Chern-Simons mixture of the Gaussian potential, typical for Hermitean models, and the sine Vandermonde factors, typical for the unitary ones. We mostly concentrate on the family of twist knots, which contains a single torus knot, the trefoil. It turns out that for the trefoil the TBEM measure is provided by an action of Laplace exponential on the Jones polynomial. This procedure can be applied to arbitrary knots and provides a TBEM-like integral representation for the N=2 case. However, beyond the torus family, both the measure and its lifting to larger N contain non-trivial corrections in \hbar=\log q. A possibility could be to absorb these corrections into a deformation of the Laplace evolution by higher Casimir and/or cut-and-join operators, in the spirit of Hurwitz tau-function approach to knot theory, but this remains a subject for future investigation.

hep-th↗
arXiv2022

Connection between cut-and-join and Casimir operators

We study cut-and-join operators for spin Hurwitz partition functions. We provide explicit expressions for these operators in terms of derivatives in $p$-variables without straightforward matrix realization, which is yet to be found. With the help of these expressions spin cut-and-join operators can be calculated directly and algorithmically. The reason why it is possible is the connection between mentioned operators and specially chosen Casimir operators that are easy to compute. An essential part of the connection involves shifted Q-Schur functions.

hep-th↗
arXiv2022

Spin Hurwitz theory and Miwa transform for the Schur Q-functions

Schur functions are the common eigenfunctions of generalized cut-and-join operators which form a closed algebra. They can be expressed as differential operators in time-variables and also through the eigenvalues of auxiliary $N\times N$ matrices $X$, known as Miwa variables. Relevant for the cubic Kontsevich model and also for spin Hurwitz theory is an alternative set of Schur Q-functions. They appear in representation theory of the Sergeev group, which is a substitute of the symmetric group, related to the queer Lie superalgebras $\mathfrak{q}(N)$.. The corresponding spin $\hat{\cal W}$-operators were recently found in terms of time-derivatives, but a substitute of the Miwa parametrization remained unknown, which is an essential complication for the matrix model technique and further developments. We demonstrate that the Miwa representation, in this case, involves a fermionic matrix $Ψ$ in addition to $X$, but its realization using supermatrices is {\it not} quite naive.

hep-th↗
arXiv2022

Bilinear character correlators in superintegrable theory

We continue investigating the superintegrability property of matrix models, i.e. factorization of the matrix model averages of characters. This paper focuses on the Gaussian Hermitian example, where the role of characters is played by the Schur functions. We find a new intriguing corollary of superintegrability: factorization of an infinite set of correlators bilinear in the Schur functions. More exactly, these are correlators of products of the Schur functions and polynomials $K_Δ$ that form a complete basis in the space of invariant matrix polynomials. Factorization of these correlators with a small subset of these $K_Δ$ follow from the fact that the Schur functions are eigenfunctions of the generalized cut-an-join operators, but the full set of $K_Δ$ is generated by another infinite commutative set of operators, which we manifestly describe.

hep-th↗
arXiv2022

Superintegrability summary

We enumerate generalizations of the superintegrability property $ \ \sim {\rm character}$ and illuminate possible general structures behind them. We collect variations of original formulas available up to date and emphasize the remaining difference between the cases of Hermitian and complex matrices, bosonic and fermionic ones. Especially important is that the story is in no way restricted to Gaussian potentials.

hep-th↗
arXiv2021

Matrix model partition function by a single constraint

In the recent study of Virasoro action on characters, we discovered that it gets especially simple for peculiar linear combinations of the Virasoro operators: particular harmonics of $\hat w$-operators. In this letter, we demonstrate that even more is true: a {\it single} $w$-constraint is sufficient to uniquely specify the partition functions provided one assumes that it is a power series in time-variables. This substitutes the previous specifications in terms of {\it two} requirements: either a string equation imposed on the KP/Toda $τ$-function or a pair of Virasoro generators. This mysterious {\it single}-entry definition holds for a variety of theories, including Hermitian and complex matrix models, and also matrix models with external matrix: the unitary and cubic Kontsevich models. In these cases, it is equivalent to W-representation and is closely related to {\it super}\,integrability. However, a similar single equation that completely determines the partition function exists also in the case of the generalized Kontsevich model (GKM) with the potential of higher degree, when the constraint algebra is a larger $W$-algebra, and neither W-representation nor superintegrability are understood well enough.

hep-th↗
arXiv2021

Natanzon-Orlov model and refined superintegrability

We reconsider the simple matrix model description of Hurwitz numbers proposed by S. Natanzon and A. Orlov, which uses the superintegrability property of the complex matrix model, and discuss a way of its possible supersymmetric extension to approach spin Hurwitz numbers.

hep-th↗
arXiv2021

From superintegrability to tridiagonal representation of $β$-ensembles

The wonderful formulas by I.Dumitriu and A.Edelman rewrite $β$-ensemble, with eigenvalue integrals containing Vandermonde factors in the power $2β$, through integrals over tridiagonal matrices, where $β$-dependent are the powers of individual matrix elements, not their differences. These potentially useful formulas are usually deduced from rather complicated and non-transparent combinatorics and are not as widely known as they deserve. We explain that the superintegrability property, i.e. a simple expression of the Gaussian averages of arbitrary Jack polynomials through the same Jack polynomials, is immediately consistent with this tridiagonal representation, which may serve as a clue to its simple and transparent interpretation. For a formal non-perturbative proof, we use the Virasoro constraints, which themselves acquire an interesting structure in the tridiagonal realization. We also attract attention to the surprising spontaneous breakdown of discrete invariance by the tridiagonal measure, which may signal a new interesting anomaly at the elementary level of the basic eigenvalue matrix model.

hep-th↗
arXiv2021

Superintegrability of matrix Student's distribution

For ordinary matrix models, the eigenvalue probability density decays rapidly as one goes to infinity, in other words, has "short tails". This ensures that all the multiple trace correlators (multipoint moments) are convergent and well-defined. Still, many critical phenomena are associated with an enhanced probability of seemingly rare effects, and one expects that they are better described by the "long tail" models. In absence of the exponential fall-off, the integrals for high moments diverge, and this could imply a loss of (super)integrability properties pertinent to matrix and eigenvalue models and, presumably, to the non-perturbative (exact) treatment of more general quantum systems. In this paper, we explain that this danger to modern understanding could be exaggerated. We consider a simple family of long-tail matrix models, which preserve the crucial feature of superintegrability: exact factorized expressions for a full set of basic averages. It turns out that superintegrability can survive after an appropriate (natural and obvious) analytical continuation even in the presence of divergencies, which opens new perspectives for the study of the long-tail matrix models.

hep-th↗
arXiv2021

Non-Abelian W-representation for GKM

$W$-representation is a miraculous possibility to define a non-perturbative (exact) partition function as an exponential action of somehow integrated Ward identities on unity. It is well known for numerous eigenvalue matrix models when the relevant operators are of a kind of $W$-operators: for the Hermitian matrix model with the Virasoro constraints, it is a $W_3$-like operator, and so on. We extend this statement to the monomial generalized Kontsevich models (GKM), where the new feature is the appearance of an ordered P-exponential for the set of non-commuting operators of different gradings.

hep-th↗
arXiv2021

Virasoro versus superintegrability. Gaussian Hermitian model

Relation between the Virasoro constraints and KP integrability (determinant formulas) for matrix models is a lasting mystery. We elaborate on the claim that the situation is improved when integrability is enhanced to super-integrability, i.e. to explicit formulas for Gaussian averages of characters. In this case, the Virasoro constraints are equivalent to simple recursive formulas, which have appropriate combinations of characters as their solutions. Moreover, one can easily separate dependence on the size of matrix, and deduce superintegrability from the Virasoro constraints. We describe one of the ways to do so for the Gaussian Hermitian matrix model. The result is a spectacularly elegant reformulation of Virasoro constraints as identities for the Schur functions evaluated at appropriate loci in the space of time-variables.

hep-th↗
arXiv2021

Duality in elliptic Ruijsenaars system and elliptic symmetric functions

We demonstrate that the symmetric elliptic polynomials $E_λ(x)$ originally discovered in the study of generalized Noumi-Shiraishi functions are eigenfunctions of the elliptic Ruijsenaars-Schneider (eRS) Hamiltonians that act on the mother function variable $y_i$ (substitute of the Young-diagram variable $λ$). This means they are eigenfunctions of the dual eRS system. At the same time, their orthogonal complements in the Schur scalar product, $P_R(x)$ are eigenfunctions of the elliptic reduction of the Koroteev-Shakirov (KS) Hamiltonians. This means that these latter are related to the dual eRS Hamiltonians by a somewhat mysterious orthogonality transformation, which is well defined only on the full space of time variables, while the coordinates $x_i$ appear only after the Miwa transform. This observation explains the difficulties with getting the apparently self-dual Hamiltonians from the double elliptic version of the KS Hamiltonians.

hep-th↗
arXiv2021

Superintegrability and Kontsevich-Hermitian relation

We analyze the well-known equivalence between the quadratic Kontsevich-Penner and Hermitian matrix models from the point of view of superintegrability relations, i.e. explicit formulas for character averages. This is not that trivial on the Kontsevich side and seems important for further studies of various deformations of Kontsevich models. In particular, the Brezin-Hikami extension of the above equivalence becomes straightforward.

hep-th↗