arXiv subjects

A. Morozov

Publications and source records attributed to A. Morozov.

At least 73 records · Page 4Linked to original sources

arXiv2023

3-Schurs from explicit representation of Yangian $Y(\hat{\mathfrak{gl}}_1)$. Levels 1-5

We suggest an ansatz for representation of affine Yangian $Y(\hat{ \mathfrak{gl}}_1)$ by differential operators in the triangular set of time-variables ${\bf P}_{a,i}$ with $1\leqslant i\leqslant a$, which saturates the MacMahon formula for the number of $3d$ Young diagrams/plane partitions. In this approach the 3-Schur polynomials are defined as the common eigenfunctions of an infinite set of commuting "cut-and-join" generators $ψ_n$ of the Yangian. We manage to push this tedious program through to the 3-Schur polynomials of level 5, and this provides a rather big sample set, which can be now investigated by other methods.

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arXiv2023

Hunt for 3-Schur polynomials

This paper describes our attempt to understand the recent success of Na Wang in constructing the 3-Schur polynomials, associated with the plane partitions. We provide a rather detailed review and try to figure out the new insights, which allowed to overcome the problems of the previous efforts. In result we provide a very simple definition of time-variables ${\bf P}_{i\geqslant j}$ and the cut-and-join operator $\hat W_2$, which generates the set of $3$-Schur functions. Some coefficients in $\hat W_2$ remain undefined and require more effort to be fixed.

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arXiv2023

On information paradox and the fate of black holes

A sketchy review of the "island" paradigm in black hole evaporation theory, which actually brings us back to the old idea that interior of black hole decouples from our universe after Page time, so that Hawking radiation is entangled with emerging new universe, thus leaving no room for the information paradox. Instead this provides a self-consistent description of multiverse, where every black hole in a parent universe is a white hole -- the origin -- of a new one.

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arXiv2023

Spectral curves and $W$-representations of matrix models

We explain how the spectral curve can be extracted from the ${\cal W}$-representation of a matrix model. It emerges from the part of the ${\cal W}$-operator, which is linear in time-variables. A possibility of extracting the spectral curve in this way is important because there are models where matrix integrals are not yet available, and still they possess all their important features. We apply this reasoning to the family of WLZZ models and discuss additional peculiarities which appear for the non-negative value of the family parameter $n$, when the model depends on additional couplings (dual times). In this case, the relation between topological and $1/N$ expansions is broken. On the other hand, all the WLZZ partition functions are $τ$-functions of the Toda lattice hierarchy, and these models also celebrate the superintegrability properties.

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arXiv2023

Many-body integrable systems implied by WLZZ models

We provide some details about the recently discovered integrable systems implied by commutativity of $W$ operators along the rays on the plane of roots of $w_\infty$-algebra. The simplest system of this type is the rational Calogero model, other systems escaped attention in the past. Existence of these systems is intimately tied to the very interesting WLZZ matrix models, which are now under intensive study.

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arXiv2023

Timing resistive plate chambers for thermal neutron detection with 3D position sensitivity

An optimized design of a neutron detector based on timing RPCs (Resistive Plate Chambers) with boron-10 neutron converters is presented. The detector is composed of a stack of ten double gap RPCs with aluminium cathode plates coated on both sides with $^{10}B_{4}C$. This design enables simultaneous determination with high accuracy of both the neutron time-of-flight (down to ns resolution) and the interaction position in 3D (down to 0.25 mm resolution across and ~1 mm along the beam). It is shown that the detection efficiency can approach 60% for neutrons with $λ$ = 4.7 Ȧ. A new geometry with less material budget is introduced for the signal pick-up strip arrays. The results of simulation-based optimization of the design are reported considering the trade-off between the detection efficiency, the count rate capability and the amount of elastic scattering on the detector components.

physics.ins-det↗

arXiv2023

On KP-integrable skew Hurwitz $τ$-functions and their $β$-deformations

We extend the old formalism of cut-and-join operators in the theory of Hurwitz $τ$-functions to description of a wide family of KP-integrable {\it skew} Hurwitz $τ$-functions, which include, in particular, the newly discovered interpolating WLZZ models. Recently, the simplest of them was related to a superintegrable two-matrix model with two potentials and one external matrix field. Now we provide detailed proofs, and a generalization to a multi-matrix representation, and propose the $β$ deformation of the matrix model as well. The general interpolating WLZZ model is generated by a $W$-representation given by a sum of operators from a one-parametric commutative sub-family (a commutative subalgebra of $w_\infty$). Different commutative families are related by cut-and-join rotations. Two of these sub-families (`vertical' and `45-degree') turn out to be nothing but the trigonometric and rational Calogero-Sutherland Hamiltonians, the `horizontal' family is represented by simple derivatives. Other families require an additional analysis.

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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.

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arXiv2022

Equating Schur Functions

We wonder if there is a way to make all Schur functions in all representations equal. This is impossible for fixed value of time variables, but can be achieved for averages. It appears that the corresponding measure is just Gaussian in times, which are all independent. The generating function for the number of Young diagrams does not straightforwardly appear as a product, but is reproduced in a non-trivial way.

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arXiv2022

Integrability and Matrix Models

A brief review of the eigenvalue matrix model integrability and superintegrability properties, focused on the simplest, still representative, Gaussian Hermitian case.

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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.

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arXiv2022

Defect and degree of the Alexander polynomial

Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be alternatively described as the degree in $q^{\pm 2}$ of the fundamental Alexander polynomial, which formally corresponds to the case of no colors. We also pose a question if these Alexander polynomials can be arbitrary integer polynomials of a given degree. A first attempt to answer the latter question is a preliminary analysis of antiparallel descendants of the 2-strand torus knots, which provide a nice set of examples for all values of the defect. The answer turns out to be positive in the case of defect zero knots, what can be observed already in the case of twist knots. This proved conjecture also allows us to provide a complete set of $C$-polynomials for the symmetrically colored Alexander polynomials for defect zero. In this case, we achieve a complete separation of representation and knot variables.

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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.

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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.

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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.

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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.

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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.

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arXiv2022

Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed

For a peculiar family of double braid knots there is a remarkable factorization formula for the coefficients of the differential (cyclotomic) expansion (DE), which nowadays is widely used to construct the exclusive Racah matrices $S$ and $\bar S$ in arbitrary representations. The origins of the factorization remain obscure and the special role of double braids remains a mystery. In an attempt to broaden the perspective, we extend the family of double braids to antiparallel triple pretzels, which are obtained by the defect-preserving deformation from the trefoil and all have defect zero. It turns out that factorization of DE coefficients is violated quite strongly, still remains described by an elegant formula, at least for all symmetric representations.

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