arXiv subjects

A. Morozov

Publications and source records attributed to A. Morozov.

At least 55 records · Page 3Linked to original sources

arXiv2024

Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models

We extend our consideration of commutative subalgebras (rays) in different representations of the $W_{1+\infty}$ algebra to the elliptic Hall algebra (or, equivalently, to the Ding-Iohara-Miki (DIM) algebra $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$). Its advantage is that it possesses the Miki automorphism, which makes all commutative rays equivalent. Integrable systems associated with these rays become finite-difference and, apart from the trigonometric Ruijsenaars system not too much familiar. We concentrate on the simplest many-body and Fock representations, and derive explicit formulas for all generators of the elliptic Hall algebra $e_{n,m}$. In the one-body representation, they differ just by normalization from $z^nq^{m\hat D}$ of the $W_{1+\infty}$ Lie algebra, and, in the $N$-body case, they are non-trivially generalized to monomials of the Cherednik operators with action restricted to symmetric polynomials. In the Fock representation, the resulting operators are expressed through auxiliary polynomials of $n$ variables, which define weights in the residues formulas. We also discuss $q,t$-deformation of matrix models associated with constructed commutative subalgebras.

hep-th↗

arXiv2024

Planar decomposition of the HOMFLY polynomial for bipartite knots and links

The theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from sl(2) to sl(N) at arbitrary N -- but for a special class of bipartite diagrams made entirely from the anitparallel lock tangle. Many amusing and important knots and links can be described in this way, from twist and double braid knots to the celebrated Kanenobu knots for even parameters -- and for all of them the entire HOMFLY polynomials possess planar decomposition. This provides an approach to evaluation of HOMFLY polynomials, which is complementary to the arborescent calculus, and this opens a new direction to homological techniques, parallel to Khovanov-Rozansky generalisations of the Kauffman calculus. Moreover, this planar calculus is also applicable to other symmetric representations beyond the fundamental one, and to links which are not fully bipartite what is illustrated by examples of Kanenobu-like links.

hep-th↗

arXiv2024

Banana diagrams as functions of geodesic distance

We extend the study of banana diagrams in coordinate representation to the case of curved space-times. If the space is harmonic, the Green functions continue to depend on a single variable -- the geodesic distance. But now this dependence can be somewhat non-trivial. We demonstrate that, like in the flat case, the coordinate differential equations for powers of Green functions can still be expressed as determinants of certain operators. Therefore, not-surprisingly, the coordinate equations remain straightforward -- while their reformulation in terms of momentum integrals and Picard-Fuchs equations can seem problematic. However we show that the Feynman parameter representation can also be generalized, at least for banana diagrams in simple harmonic spaces, so that the Picard-Fuchs equations retain their Euclidean form with just a minor modification. A separate story is the transfer to the case when the Green function essentially depends on several rather than a single argument. In this case, we provide just one example, that the equations are still there, but conceptual issues in the more general case will be discussed elsewhere.

hep-th↗

arXiv2024

Deformation of superintegrability in the Miwa-deformed Gaussian matrix model

We consider an arbitrary deformation of the Gaussian matrix model parameterized by Miwa variables $z_a$. One can look at it as a mixture of the Gaussian and logarithmic (Selberg) potentials, which are both superintegrable. The mixture is not, still one can find an explicit expression for an arbitrary Schur average as a linear transform of a {\it finite degree} polynomial made from the values of skew Schur functions at the Gaussian locus $p_k=δ_{k,2}$. This linear operation includes multiplication with an exponential $ e^{z_a^2/2}$ and a kind of Borel transform of the resulting product, which we call multiple and enhanced. The existence of such remarkable formulas appears intimately related to the theory of auxiliary $K$-polynomials, which appeared in {\it bilinear} superintegrable correlators at the Gaussian point (strict superintegrability). We also consider in the very detail the generating function of correlators $<(\Tr X)^k>$ in this model, and discuss its integrable determinant representation. At last, we describe deformation of all results to the Gaussian $β$-ensemble.

hep-th↗

arXiv2024

Position space equations for generic Feynman graphs

We propose the extension of the position space approach to Feynman integrals from the banana family to generic Feynman diagrams. Our approach is based on getting rid of integration in position space and then writing differential equations for the products of propagators defined for any graph. We employ the so-called ''bananization'' to start with simple Feynman graphs and further substituting each edge with a multiple one. We explain how the previously developed theory of banana diagrams can be used to describe what happens to the differential equations (Ward identities) on Feynman diagrams after this transformation. Our approach works for generic enough (large enough) dimension and masses. We expect that after Fourier transform our equations should be related to the Picard-Fuchs equations. Therefore, we describe the challenges of Fourier transform that arise in our approach.

hep-th↗

arXiv2024

On character expansion and Gaussian regularization of Itzykson-Zuber measure

Character expansions are among the most important approaches to modern quantum field theory, which substitute integrals by combinations of peculiar special functions from the Schur-Macdonald family. These formulas allow various deformations, which are not transparent in integral formulation. We analyze from this point of view the Itzykson-Zuber integral over unitary matrices which is exactly solvable, but difficult to deform in $β$ and $(q,t)$ directions. Character expansion straightforwardly resolves this problem. However, taking averages with the so defined measure can look problematic, because integrals of individual expansion terms often diverge and well defined is only the sum of them. We explain a way to overcome this problem by Gaussian regularization, which can have a broad range of further applications.

hep-th↗

arXiv2024

From equations in coordinate space to Picard-Fuchs and back

We continue the development of a position space approach to equations for Feynman multi-loop integrals. The key idea of the approach is that unintegrated products of Greens functions in position space are still loop integral in momentum space. The natural place to start are the famous banana diagrams, which we explore in this paper. In position space, these are just products of $n$ propagators. Firstly, we explain that these functions satisfy an equation of order $2^n$. These should be compared with Picard-Fuchs equations derived for the momentum space integral. We find that the Fourier transform of the position space operator contains the Picard-Fuchs one as a rightmost factor. The order of these operators is a special issue, especially since the order in momentum space is governed by degree in $x$ in position space. For the generic mass case this factorization pattern is complicated and it seems like the order of the Fourier transformed position space operators is much bigger than that of the Picard-Fuchs. Furthermore, one may ask what happens if after factorization we take the Picard-Fuchs operators back into position space. We discover that the result is again factorized, with the rightmost factor being the original position space equation. We demonstrate how this works in examples and discuss implications for more sophisticated Feynman integrals.

hep-th↗

arXiv2024

In-beam test results of an RPC-based module for position-sensitive neutron detectors with timing readout

Recently we have proposed a new concept of a thermal neutron detector based on resistive plate chambers and 10B4C solid neutron converters, enabling to readout with high resolution in both the 3D position of neutron capture and the neutron time of flight (ToF). In this paper, we report the results of the first beam tests conducted with a new neutron RPC detection module, coupled to the position readout units of a new design. The main focus is on the measurements of the neutron ToF and identification of the converter layer where the neutron is captured, giving the position along the beam direction.

physics.ins-det↗

arXiv2024

Summing up perturbation series around superintegrable point

We work out explicit formulas for correlators in the Gaussian matrix model perturbed by a logarithmic potential, i.e. by inserting Miwa variables. In this paper, we concentrate on the example of a single Miwa variable. The ordinary Gaussian model is superintegrable, i.e. the average of the Schur functions $S_Q$ is an explicit function of the Young diagram $Q$. The question is what happens to this property after perturbation. We show that the entire perturbation series can be nicely summed up into a kind of Borel transform of a universal exponential function, while the dependence on $R$ enters through a polynomial factor in front of this exponential. Moreover, these polynomials can be described explicitly through a single additional structure, which we call ``truncation'' of the Young diagram $Q$. It is unclear if one can call this an extended superintegrability, but at least it is a tremendously simple deformation of it. Moreover, the vanishing Gaussian correlators remain vanishing and, hence, are not deformed at all.

hep-th↗

arXiv2023

On the status of DELL systems

A detailed review of the $p,q$-duality for Calogero system and its generalizations is given. For the first time, we present some of elliptic-trigonometric Hamiltonians dual to the elliptic Ruijsenaars Hamiltonians (i.e. trigonometric-elliptic ones), and explain their relations to the bi-elliptic Koroteev-Shakirov (KS) model. The most interesting self-dual double-elliptic (DELL) system remains a mystery, but we provide a clearer formulation of the problem and describe the steps that are still to be done.

hep-th↗

arXiv2023

Tau-functions beyond the group elements

Matrix elements in different representations are connected by quadratic relations. If matrix elements are those of a $\textit{group element}$, i.e. satisfying the property $Δ(X) = X\otimes X$, then their generating functions obey bilinear Hirota equations and hence are named $τ$-functions. However, dealing with group elements is not always easy, especially for non-commutative algebras of functions, and this slows down the development of $τ$-function theory and the study of integrability properties of non-perturbative functional integrals. A simple way out is to use arbitrary elements of the universal enveloping algebra, and not just the group elements. Then the Hirota equations appear to interrelate a whole system of generating functions, which one may call $\textit{generalized}$ $τ$-functions. It was recently demonstrated that this idea can be applicable even to a somewhat sophisticated case of the quantum toroidal algebra. We consider a number of simpler examples, including ordinary and quantum groups, to explain how the method works and what kind of solutions one can obtain.

hep-th↗

arXiv2023

On factorization hierarchy of equations for banana Feynman amplitudes

We present a review of the relations between various equations for maximal cut banana Feynman diagrams, i.e. amplitudes with propagators substituted with $δ$-functions. We consider both equal and generic masses. There are three types of equation to consider: those in coordinate space, their Fourier transform and Picard-Fuchs equations originating from the parametric representation. First, we review the properties of the corresponding differential operators themselves, mainly their factorization properties at the equal mass locus and their form at special values of the dimension. Then we study the relation between the Fourier transform of the coordinate space equations and the Picard-Fuchs equations and show that they are related by factorization as well. The equations in question are the counterparts of the Virasoro constraints in the much-better studied theory of eigenvalue matrix models and are the first step towards building a full-fledged theory of Feynman integrals, which will reveal their hidden integrable structure.

hep-th↗

arXiv2023

Commutative families in $W_\infty$, integrable many-body systems and hypergeometric $τ$-functions

We explain that the set of new integrable systems generalizing the Calogero family and implied by the study of WLZZ models, which was described in arXiv:2303.05273, is only the tip of the iceberg. We provide its wide generalization and explain that it is related to commutative subalgebras (Hamiltonians) of the $W_{1+\infty}$ algebra. We construct many such subalgebras and explain how they look in various representations. We start from the even simpler $w_\infty$ contraction, then proceed to the one-body representation in terms of differential operators on a circle, further generalizing to matrices and in their eigenvalues, in finally to the bosonic representation in terms of time-variables. Moreover, we explain that some of the subalgebras survive the $β$-deformation, an intermediate step from $W_{1+\infty}$ to the affine Yangian. The very explicit formulas for the corresponding Hamiltonians in these cases are provided. Integrable many-body systems generalizing the rational Calogero model arise in the representation in terms of eigenvalues. Each element of $W_{1+\infty}$ algebra gives rise to KP/Toda $τ$-functions. The hidden symmetry given by the families of commuting Hamiltonians is in charge of the special, (skew) hypergeometric $τ$-functions among these.

hep-th↗

arXiv2023

Towards tangle calculus for Khovanov polynomials

We provide new evidence that the tangle calculus and "evolution" are applicable to the Khovanov polynomials for families of long braids inside the knot diagram. We show that jumps in evolution, peculiar for superpolynomials, are much less abundant than it was originally expected. Namely, for torus and twist satellites of a fixed companion knot, the main (most complicated) contribution does not jump, all jumps are concentrated in the torus and twist part correspondingly, where these jumps are necessary to make the Khovanov polynomial positive. Among other things, this opens a way to define a jump-free part of the colored Khovanov polynomials, which differs from the naive colored polynomial just "infinitesimally". The separation between jumping and smooth parts involves a combination of Rasmussen index and a new knot invariant, which we call "Thickness".

hep-th↗

arXiv2023

Commutative subalgebras from Serre relations

We demonstrate that commutativity of numerous one-dimensional subalgebras in $W_{1+\infty}$ algebra, i.e. the existence of many non-trivial integrable systems described in recent arXiv:2303.05273 follows from the subset of relations in algebra known as Serre relations. No other relations are needed for commutativity. The Serre relations survive the deformation to the affine Yangian $Y(\hat{\mathfrak{gl}}_1)$, hence the commutative subalgebras do as well. A special case of the Yangian parameters corresponds to the $β$-deformation. The preservation of Serre relations can be thought of a selection rule for proper systems of commuting $β$-deformed Hamiltonians. On the contrary, commutativity in the extended family associated with ``rational (non-integer) rays" is {\it not} reduced to the Serre relations, and uses also other relations in the $W_{1+\infty}$ algebra. Thus their $β$-deformation is less straightforward.

hep-th↗

arXiv2023

$(q,t)$-deformed (skew) Hurwitz $τ$-functions

We follow the general recipe for constructing commutative families of $W$-operators, which provides Hurwitz-like expansions in symmetric functions (Macdonald polynomials), in order to obtain a difference operator example that gives rise to a $(q,t)$-deformation of the earlier studied models. As before, a key role is played by an appropriate deformation of the cut-and-join rotation operator. We outline its expression both in terms of generators of the quantum toroidal algebra and in terms of the Macdonald difference operators.

hep-th↗

arXiv2023

Averaging method in combinatorics of symmetric polynomials

We elaborate on the recent suggestion to consider averaging of Cauchy identities for the Schur functions over power sum variables. This procedure has apparent parallels with the Borel transform, only it changes the number of combinatorial factors like $d_R$ in the sums over Young diagrams instead of just factorials in ordinary sums over numbers. It provides a universal view on a number of previously known, but seemingly random identities.

hep-th↗

arXiv2023

Production of $π^+$ and $K^+$ mesons in argon-nucleus interactions at 3.2 AGeV

First physics results of the BM@N experiment at the Nuclotron/NICA complex are presented on π+ and K+ meson production in interactions of an argon beam with fixed targets of C, Al, Cu, Sn and Pb at 3.2 AGeV. Transverse momentum distributions, rapidity spectra and multiplicities of $π^+$ and $K^+$ mesons are measured. The results are compared with predictions of theoretical models and with other measurements at lower energies.

hep-ex↗