arXiv subjects

A. Mironov

Publications and source records attributed to A. Mironov.

At least 91 records · Page 5Linked to original sources

arXiv2019

On generalized Macdonald polynomials

Generalized Macdonald polynomials (GMP) are eigenfunctions of specifically-deformed Ruijsenaars Hamiltonians and are built as triangular polylinear combinations of Macdonald polynomials. They are orthogonal with respect to a modified scalar product, which could be constructed with the help of an increasingly important triangular perturbation theory, showing up in a variety of applications. A peculiar feature of GMP is that denominators in this expansion are fully factorized, which is a consequence of a hidden symmetry resulting from the special choice of the Hamiltonian deformation. We introduce also a simplified but deformed version of GMP, which we call generalized Schur functions. Our basic examples are bilinear in Macdonald polynomials.

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arXiv2019

Can tangle calculus be applicable to hyperpolynomials?

We make a new attempt at the recently suggested program to express knot polynomials through topological vertices, which can be considered as a possible approach to the tangle calculus: we discuss the Macdonald deformation of the relation between the convolution of two topological vertices and the HOMFLY-PT invariant of the 4-component link $L_{8n8}$, which both depend on four arbitrary representations. The key point is that both of these are related to the Hopf polynomials in composite representations, which are in turn expressed through composite Schur functions. The latter are further expressed through the skew Schur polynomials via the remarkable Koike formula. It is this decomposition that breaks under the Macdonald deformation and gets restored only in the (large $N$) limit of $A^{\pm 1}\longrightarrow 0$. Another problem is that the Hopf polynomials in the composite representations in the refined case are "chiral bilinears" of Macdonald polynomials, while convolutions of topological vertices involve "non-chiral combinations" with one of the Macdonald polynomials entering with permuted $t$ and $q$. There are also other mismatches between the Hopf polynomials in the composite representation and the topological 4-point function in the refined case, which we discuss.

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arXiv2019

The MacMahon R-matrix

We introduce an $R$-matrix acting on the tensor product of MacMahon representations of Ding-Iohara-Miki (DIM) algebra $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$. This $R$-matrix acts on pairs of $3d$ Young diagrams and retains the nice symmetry of the DIM algebra under the permutation of three deformation parameters $q$, $t^{-1}$ and $\frac{t}{q}$. We construct the intertwining operator for a tensor product of the horizontal Fock representation and the vertical MacMahon representation and show that the intertwiners are permuted using the MacMahon $R$-matrix.

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arXiv2019

Kerov functions for composite representations and Macdonald ideal

Kerov functions provide an infinite-parametric deformation of the set of Schur functions, which is a far-going generalization of the 2-parametric Macdonald deformation. In this paper, we concentrate on a particular subject: on Kerov functions labeled by the Young diagrams associated with the conjugate and, more generally, composite representations. Our description highlights peculiarities of the Macdonald locus (ideal) in the space of the Kerov parameters, where some formulas and relations get drastically simplified. However, even in this case, they substantially deviate from the Schur case, which illustrates the problems encountered in the theory of link hyperpolynomials. An important additional feature of the Macdonald case is uniformization, a possibility of capturing the dependence on $N$ for symmetric polynomials of $N$ variables into a single variable $A=t^N$, while in the generic Kerov case the $N$-dependence looks considerably more involved.

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arXiv2018

Kerov functions revisited

The Schur functions play a crucial role in the modern description of HOMFLY polynomials for knots and of topological vertices in DIM-based network theories, which could merge into a unified theory still to be developed. The Macdonald functions do the same for hyperpolynomials and refined vertices, but merging appears to be more problematic. For a detailed study of this problem, more knowledge is needed about the Macdonald polynomials than is usually available. As a preparation for the discussion of the knot/vertices relation, we summarize the relevant facts and open problems about the Macdonald and, more generally, Kerov functions. Like Macdonald polynomials, they are triangular combinations of Schur functions, but orthogonal in a more general scalar product. We explain that the parameters of the measure can be considered as a set of new time variables, and the Kerov functions are actually expressed through the Schur functions of these variables as well. Despite they provide an infinite-parametric extension of the Schur and Macdonald polynomials, the Kerov functions, and even the skew Kerov functions continue to satisfy the most important relations, like Cauchy summation formula, the transposition identity for reflection of the Young diagram and expression of the skew functions through the generalized Littlewood-Richardson structure constants. Since {\it these} are the properties important in most applications, one can expect that the Kerov extension exists for most of them, from the superintegrable matrix and tensor models to knot theory.

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arXiv2018

From Topological to Quantum Entanglement

Entanglement is a special feature of the quantum world that reflects the existence of subtle, often non-local, correlations between local degrees of freedom. In topological theories such non-local correlations can be given a very intuitive interpretation: quantum entanglement of subsystems means that there are "strings" connecting them. More generally, an entangled state, or similarly, the density matrix of a mixed state, can be represented by cobordisms of topological spaces. Using a formal mathematical definition of TQFT we construct basic examples of entangled states and compute their von Neumann entropy.

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arXiv2018

From Kronecker to tableau pseudo-characters in tensor models

We present a brief summary of the recent discovery of direct tensorial analogue of characters. We distinguish three degrees of generalization: (1) $c$-number Kronecker characters made with the help of symmetric group characters and inheriting most of the nice properties of conventional Schur functions, except for forming a complete basis for the case of rank $r>2$ tensors: they are orthogonal, are eigenfunctions of appropriate cut-and-join operators and form a complete basis for the operators with non-zero Gaussian averages; (2) genuine matrix-valued tensorial quantities, forming an over-complete basis but difficult to deal with; and (3) intermediate tableau pseudo-characters, depending on Young tables rather than on just Young diagrams, in the Kronecker case, and on entire representation matrices, in the genuine one.

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arXiv2018

A non-torus link from topological vertex

The recently suggested tangle calculus for knot polynomials is intimately related to topological string considerations and can help to build the HOMFLY-PT invariants from the topological vertices. We discuss this interplay in the simplest example of the Hopf link and link $L_{8n8}$. It turns out that the resolved conifold with four different representations on the four external legs, on the topological string side, is described by a special projection of the four-component link $L_{8n8}$, which reduces to the Hopf link colored with two composite representations. Thus, this provides the first explicit example of non-torus link description through the topological vertex. It is not a real breakthrough, because $L_{8n8}$ is just a cable of the Hopf link, still, it can help to intensify the development of the formalism towards more interesting examples.

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arXiv2018

Sum rules for characters from character-preservation property of matrix models

One of the main features of eigenvalue matrix models is that the averages of characters are again characters, what can be considered as a far-going generalization of the Fourier transform property of Gaussian exponential. This is true for the standard Hermitian and unitary (trigonometric) matrix models and for their various deformations, classical and quantum ones. Arising explicit formulas for the partition functions are very efficient for practical computer calculations. However, to handle them theoretically, one needs to tame the remaining finite sums over representations of a given size, which turns into an interesting conceptual problem. Already the semicircle distribution in the large-$N$ limit implies interesting combinatorial sum rules for characters. We describe also implications to $W$-representations, including a character decomposition of cut-and-join operators, which unexpectedly involves only single-hook diagrams and also requires non-trivial summation identities.

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arXiv2018

Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations

Obtaining colored HOMFLY-PT polynomials for knots from 3-strand braid carrying arbitrary $SU(N)$ representation is still tedious. For a class of rank $r$ symmetric representations, $[r]$-colored HOMFLY-PT $H_{[r]}$ evaluation becomes simpler. Recently it was shown that $H_{[r]}$, for such knots from 3-strand braid, can be constructed using the quantum Racah coefficients (6j-symbols) of $U_q(sl_2)$. In this paper, we generalise it to links whose components carry different symmetric representations. We illustrate the technique by evaluating multi-colored link polynomials $H_{[r_1],[r_2]}$ for the two-component link L7a3 whose components carry $[r_1]$ and $[r_2]$ colors.

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arXiv2018

Colored Alexander polynomials and KP hierarchy

We discuss the relation between knot polynomials and the KP hierarchy. Mainly, we study the scaling 1-hook property of the coloured Alexander polynomial: $\mathcal{A}^\mathcal{K}_R(q)=\mathcal{A}^\mathcal{K}_{[1]}(q^{\vert R\vert})$ for all 1-hook Young diagrams $R$. Via the Kontsevich construction, it is reformulated as a system of linear equations. It appears that the solutions of this system induce the KP equations in the Hirota form. The Alexander polynomial is a specialization of the HOMFLY polynomial, and it is a kind of a dual to the double scaling limit, which gives the special polynomial, in the sense that, while the special polynomials provide solutions to the KP hierarchy, the Alexander polynomials provide the equations of this hierarchy. This gives a new connection with integrable properties of knot polynomials and puts an interesting question about the way the KP hierarchy is encoded in the full HOMFLY polynomial.

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arXiv2018

Tangle blocks in the theory of link invariants

The central discovery of $2d$ conformal theory was holomorphic factorization, which expressed correlation functions through bilinear combinations of conformal blocks, which are easily cut and joined without a need to sum over the entire huge Hilbert space of states. Somewhat similar, when a link diagram is glued from tangles, the link polynomial is a multilinear combination of {\it tangle blocks} summed over just a few representations of intermediate states. This turns to be a powerful approach because the same tangles appear as constituents of very different knots so that they can be extracted from simpler cases and used in more complicated ones. So far this method has been technically developed only in the case of arborescent knots, but, in fact, it is much more general. We begin a systematic study of tangle blocks by detailed consideration of some archetypical examples, which actually lead to non-trivial results, far beyond the reach of other techniques. At the next level, the tangle calculus is about gluing of tangles, i.e. functorial mappings from ${\rm Hom}({\rm tangles})$, and its main advantage is an explicit realization of multiplicative composition structure, which is partly obscured in traditional knot theory.

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arXiv2018

Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model

Even though matrix model partition functions do not exhaust the entire set of tau-functions relevant for string theory, they seem to be elementary building blocks for many others and they seem to properly capture the fundamental symplicial nature of quantum gravity and string theory. We propose to consider matrix model partition functions as new special functions. This means they should be investigated and put into some standard form, with no reference to particular applications. At the same time, the tables and lists of properties should be full enough to avoid discoveries of unexpected peculiarities in new applications. This is a big job, and the present paper is just a step in this direction. Here we restrict our consideration to the finite-size Hermitean 1-matrix model and concentrate mostly on its phase/branch structure arising when the partition function is considered as a D-module. We discuss the role of the CIV-DV prepotential (as generating a possible basis in the linear space of solutions to the Virasoro constraints, but with a lack of understanding of why and how this basis is distinguished) and evaluate first few multiloop correlators, which generalize semicircular distribution to the case of multitrace and non-planar correlators.

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arXiv2018

On Hopf-induced deformation of topological locus

We provide a very brief review of the description of colored invariants for the Hopf link in terms of characters, which need to be taken at a peculiar deformation of the topological locus, depending on one of the two representations associated with the two components of the link. Most important, we extend the description of this locus to conjugate and, generically, to composite representations and also define the "adjoint" Schur functions emerging in the dual description.

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arXiv2018

$(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces

We describe the general strategy for lifting the Wess-Zumino-Witten model from the level of one-loop Kac-Moody $U_q(\widehat{\mathfrak{g}})_k$ to generic quantum toroidal algebras. A nearly exhaustive presentation is given for the two series $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$ and $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_n)$, when screenings do not exist and thus all the correlators are purely algebraic, i.e. do not include additional hypergeometric type integrations/summations. Generalizing the construction of the intertwiner (refined topological vertex) of the Ding-Iohara-Miki (DIM) algebra, we obtain the intertwining operators of the Fock representations of the quantum toroidal algebra of type $A_n$. The correlation functions of these operators satisfy the $(q,t)$-Knizhnik-Zamolodchikov (KZ) equation, which features the ${\cal R}$-matrix. Matching with the Nekrasov function for the instanton counting on the ALE space is worked out explicitly. We also present an important application of the DIM formalism to the study of $6d$ gauge theories described by the double elliptic integrable systems. We show that the modular and periodicity properties of the gauge theories are neatly explained by the network matrix models providing solutions to the elliptic $(q,t)$-KZ equations.

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arXiv2018

Quantum Racah matrices up to level 3 and multicolored link invariants

This paper is a next step in the project of systematic description of colored knot and link invariants started in previous papers. In this paper, we managed to explicitly find the inclusive Racah matrices, i.e. the whole set of mixing matrices in channels $R_1\otimes R_2\otimes R_3\longrightarrow Q$ with all possible $Q$, for $|R|\leq 3$. The calculation is made possible by use of the highest weight method. The result allows one to evaluate and investigate colored polynomials for arbitrary 3-strand knots and links and to check the corresponding eigenvalue conjecture. Explicit answers for Racah matrices and colored polynomials for 3-strand knots up to 10 crossings are available at http://knotebook.org. Using the obtained inclusive Racah matrices, we also calculated the exclusive Racah matrices with the help of trick earlier suggested in the case of knots. This method is proved to be effective and gives the exclusive Racah matrices earlier obtained by another method.

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arXiv2018

Eigenvalue hypothesis for multi-strand braids

Computing polynomial form of the colored HOMFLY-PT for non-arborescent knots obtained from three or more strand braids is still an open problem. One of the efficient methods suggested for the three-strand braids relies on the eigenvalue hypothesis which uses the Yang-Baxter equation to express the answer through the eigenvalues of the ${\cal R}$-matrix. In this paper, we generalize the hypothesis to higher number of strands in the braid where commuting relations of non-neighbouring $\mathcal{R}$ matrices are also incorporated. By solving these equations, we determine the explicit form for $\mathcal{R}$-matrices and the inclusive Racah matrices in terms of braiding eigenvalues (for matrices of size up to 6 by 6). For comparison, we briefly discuss the highest weight method for four-strand braids carrying fundamental and symmetric rank two $SU_q(N)$ representation. Specifically, we present all the inclusive Racah matrices for representation $[2]$ and compare with the matrices obtained from eigenvalue hypothesis.

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arXiv2018

Checks of integrality properties in topological strings

Tests of the integrality properties of a scalar operator in topological strings on a resolved conifold background or orientifold of conifold backgrounds have been performed for arborescent knots and some non-arborescent knots. The recent results on polynomials for those knots colored by SU(N) and SO(N) adjoint representations are useful to verify Marino's integrality conjecture up to two boxes in the Young diagram. In this paper, we review the salient aspects of the integrality properties and tabulate explicitly for an arborescent knot and a link. In our knotebook website, we have put these results for over 100 prime knots available in Rolfsen table and some links. The first application of the obtained results, an observation of the Gaussian distribution of the LMOV invariants is also reported.

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