arXiv subjects

A. Mironov

Publications and source records attributed to A. Mironov.

At least 145 records · Page 8Linked to original sources

arXiv2015

On the Nonrelativistic 2D Purely Magnetic Supersymmetric Pauli Operator

The Complete Manifold of Ground State Eigenfunctions for the Purely Magnetic 2D Pauli Operator is considered as a by-product of the new reduction found by the present authors few years ago for the Algebrogeometric Inverse Spectral Data (i.e. Riemann Surfaces and Divisors). This reduction is associated with the (2+1) Soliton Hierarhy containing a 2D analog of the famous "Burgers System". This article contains also exposition of the previous works made since 1980 including the first topological ideas in the space of quasimomenta. We present here also new results dedicated to the self-adjoint boundary problems for Pauli Operator. The 2D zero level "nonspectral" Bloch-Floquet functions give discrete points of additional spectrum similar to the "boundary states" of finite-gap 1D potentials in the gaps.

math-ph↗

arXiv2014

Colored knot polynomials for Pretzel knots and links of arbitrary genus

A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich $(g+1)$-parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed through the Racah matrix of U_q(SU_N), and looks related to a modular transformation of toric conformal block.

hep-th↗

arXiv2014

On KP-integrable Hurwitz functions

There is now a renewed interest to the Hurwitz tau-function, counting the isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and Grothiendicks's dessins d'enfant. It is distinguished by belonging to a particular family of Hurwitz tau-functions, possessing conventional Toda/KP integrability properties. We explain how the variety of recent observations about this function fits into the general theory of matrix model tau-functions. All such quantities possess a number of different descriptions, related in a standard way: these include Toda/KP integrability, several kinds of W-representations (we describe four), two kinds of integral (multi-matrix model) descriptions (of Hermitian and Kontsevich types), Virasoro constraints, character expansion, embedding into generic set of Hurwitz tau-functions and relation to knot theory. When approached in this way, the family of models in the literature has a natural extension, and additional integrability with respect to associated new time-variables. Another member of this extended family is the Itsykson-Zuber integral.

hep-th↗

arXiv2014

Wall Crossing Invariants: from quantum mechanics to knots

We offer a pedestrian level review of the wall-crossing invariants. The story begins from the scattering theory in quantum mechanics where the spectrum reshuffling can be related to permutations of S-matrices. In non-trivial situations, starting from spin chains and matrix models, the S-matrices are operator-valued and their algebra is described in terms of R- and mixing (Racah) U-matrices. Then, the Kontsevich-Soibelman invariants are nothing but the standard knot invariants made out of these data within the Reshetikhin-Turaev-Witten approach. The R- and Racah matrices acquire a relatively universal form in the quasiclassical limit, where the basic reshufflings with the change of moduli are those of the Stokes line. Natural from this point of view are matrices provided by the modular transformations of conformal blocks (with the usual identification R=T and U=S), and in the simplest case of the first degenerate field (2,1), when the conformal blocks satisfy a second order Shroedinger-like equation, the invariants coincide with the Jones (N=2) invariants of the associated knots. Another possibility to construct knot invariants is to realize the cluster coordinates associated with reshufflings of the Stokes lines immediately in terms of check-operators acting on the solutions to the Knizhnik-Zamolodchikov equations. Then, the R-matrices are realized as products of successive mutations in the cluster algebra and are manifestly described in terms of quantum dilogarithms ultimately leading to the Hikami construction of knot invariants.

hep-th↗

arXiv2014

Seiberg-Witten curves and double-elliptic integrable systems

An old conjecture claims that commuting Hamiltonians of the double-elliptic integrable system are constructed from the theta-functions associated with Riemann surfaces from the Seiberg-Witten family, with moduli treated as dynamical variables and the Seiberg-Witten differential providing the pre-symplectic structure. We describe a number of theta-constant equations needed to prove this conjecture for the $N$-particle system. These equations provide an alternative method to derive the Seiberg-Witten prepotential and we illustrate this by calculating the perturbative contribution. We provide evidence that the solutions to the commutativity equations are exhausted by the double-elliptic system and its degenerations (Calogero and Ruijsenaars systems). Further, the theta-function identities that lie behind the Poisson commutativity of the three-particle Hamiltonians are proven.

hep-th↗

arXiv2014

Is the Standard Model saved asymptotically by conformal symmetry?

It is pointed out that the top-quark and Higgs masses and the Higgs VEV satisfy with great accuracy the relations 4m_H^2=2m_T^2=v^2, which are very special and reminiscent of analogous ones at Argyres - Douglas points with enhanced conformal symmetry. Furthermore, the RG evolution of the corresponding Higgs self-interaction and Yukawa couplings λ(0)=1/8 and y(0)=1 leads to the free-field stable point λ(M_Pl)= \dot λ(M_Pl)=0 in the pure scalar sector at the Planck scale, also suggesting enhanced conformal symmetry. Thus, it is conceivable that the Standard Model is the low-energy limit of a distinct special theory with (super?) conformal symmetry at the Planck scale. In the context of such a "scenario" one may further speculate that the Higgs particle is the Goldstone boson of (partly) spontaneously broken conformal symmetry. This would simultaneously resolve the hierarchy and Landau pole problems in the scalar sector and would provide a nearly flat potential with two almost degenerate minima at the electroweak and Planck scales.

hep-ph↗

arXiv2014

On genus expansion of superpolynomials

Recently it was shown that the (Ooguri-Vafa) generating function of HOMFLY polynomials is the Hurwitz partition function, i.e. that the dependence of the HOMFLY polynomials on representation R is naturally captured by symmetric group characters (cut-and-join eigenvalues). The genus expansion and expansion through Vassiliev invariants explicitly demonstrate this phenomenon. In the present letter we claim that the superpolynomials are not functions of such a type: symmetric group characters do not provide an adequate linear basis for their expansions. Deformation to superpolynomials is, however, straightforward in the multiplicative basis: the Casimir operators are β-deformed to Hamiltonians of the Calogero-Moser-Sutherland system. Applying this trick to the genus and Vassiliev expansions, we observe that the deformation is fully straightforward only for the thin knots. Beyond the family of thin knots additional algebraically independent terms appear in the Vassiliev and genus expansions. This can suggest that the superpolynomials do in fact contain more information about knots than the colored HOMFLY and Kauffman polynomials. However, even for the thin knots the beta-deformation is non-innocent: already in the simplest examples it seems inconsistent with the postivity of colored superpolynomials in non-(anti)symmetric representations, which also happens in I.Cherednik's (DAHA-based) approach to the torus knots.

hep-th↗

arXiv2014

On colored HOMFLY polynomials for twist knots

Recent results of J.Gu and H.Jockers provide the lacking initial conditions for the evolution method in the case of the first non-trivially colored HOMFLY polynomials H_{[21]} for the family of twist knots. We describe this application of the evolution method, which finally allows one to penetrate through the wall between (anti)symmetric and non-rectangular representations for a whole family. We reveal the necessary deformation of the differential expansion, what, together with the recently suggested matrix model approach gives new opportunities to guess what it could be for a generic representation, at least for the family of twist knots.

hep-th↗

arXiv2014

Matching branches of non-perturbative conformal block at its singularity divisor

Conformal block is a function of many variables, usually represented as a formal series, with coefficients which are certain matrix elements in the chiral (e.g. Virasoro) algebra. Non-perturbative conformal block is a multi-valued function, defined globally over the space of dimensions, with many branches and, perhaps, additional free parameters, not seen at the perturbative level. We discuss additional complications of non-perturbative description, caused by the fact that all the best studied examples of conformal blocks lie at the singularity locus in the moduli space (at divisors of the coefficients or, simply, at zeroes of the Kac determinant). A typical example is the Ashkin-Teller point, where at least two naive non-perturbative expressions are provided by elliptic Dotsenko-Fateev integral and by the celebrated Zamolodchikov formula in terms of theta-constants, and they are different. The situation is somewhat similar at the Ising and other minimal model points.

hep-th↗

arXiv2014

Link polynomial calculus and the AENV conjecture

Using the recently proposed differential hierarchy (Z-expansion) technique, we obtain a general expression for the HOMFLY polynomials in two arbitrary symmetric representations of link families, including Whitehead and Borromean links. Among other things, this allows us to check and confirm the recent conjecture of arXiv:1304.5778 that the large representation limit (the same as considered in the knot volume conjecture) of this quantity matches the prediction from mirror symmetry consideration. We also provide, using the evolution method, the HOMFLY polynomial in two arbitrary symmetric representations for an arbitrary member of the one-parametric family of 2-component 3-strand links, which includes the Hopf and Whitehead links.

hep-th↗

arXiv2013

S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality

A recent claim that the S-duality between 4d SUSY gauge theories, which is AGT related to the modular transformations of 2d conformal blocks, is no more than an ordinary Fourier transform at the perturbative level, is further traced down to the commutation relation [P,Q]=-i\hbar between the check-operator monodromies of the exponential resolvent operator in the underlying Dotsenko-Fateev matrix models and beta-ensembles. To this end, we treat the conformal blocks as eigenfunctions of the monodromy check operators, what is especially simple in the case of one-point toric block. The kernel of the modular transformation is then defined as the intertwiner of the two monodromies, and can be obtained straightforwardly, even when the eigenfunction interpretation of the blocks themselves is technically tedious. In this way, we provide an elementary derivation of the old expression for the modular kernel for the one-point toric conformal block.

hep-th↗

arXiv2013

Spectral dualities in XXZ spin chains and five dimensional gauge theories

Motivated by recent progress in the study of supersymmetric gauge theories we propose a very compact formulation of spectral duality between XXZ spin chains. The action of the quantum duality is given by the Fourier transform in the spectral parameter. We investigate the duality in various limits and, in particular, prove it for q-->1, i.e. when it reduces to the XXX/Gaudin duality. We also show that the universal difference operators are given by the normal ordering of the classical spectral curves.

hep-th↗

arXiv2013

Three-particle Integrable Systems with Elliptic Dependence on Momenta and Theta Function Identities

We claim that some non-trivial theta-function identities at higher genus can stand behind the Poisson commutativity of the Hamiltonians of elliptic integrable systems, which are made from the theta-functions on Jacobians of the Seiberg-Witten curves. For the case of three-particle systems the genus-2 identities are found and presented in the paper. The connection with the Macdonald identities is established. The genus-2 theta-function identities provide the direct way to construct the Poisson structure in terms of the coordinates on the Jacobian of the spectral curve and the elements of its period matrix. The Lax representations for the two-particle systems are also obtained.

hep-th↗

arXiv2013

Differential hierarchy and additional grading of knot polynomials

Colored knot polynomials possess a peculiar Z-expansion in certain combinations of differentials, which depends on the representation. The coefficients of this expansion are functions of the three variables (A,q,t) and can be considered as new distinguished coordinates on the space of knot polynomials, analogous to the coefficients of alternative character expansion. These new variables are decomposed in an especially simple way, when the representation is embedded into a product of the fundamental ones. The fourth grading recently proposed in arXiv:1304.3481, seems to be just a simple redefinition of these new coordinates, elegant but in no way distinguished. If so, it does not provide any new independent knot invariants, instead it can be considered as one more testimony of the hidden differential hierarchy (Z-expansion) structure behind the knot polynomials.

hep-th↗

arXiv2013

Evolution method and "differential hierarchy" of colored knot polynomials

We consider braids with repeating patterns inside arbitrary knots which provides a multi-parametric family of knots, depending on the "evolution" parameter, which controls the number of repetitions. The dependence of knot (super)polynomials on such evolution parameters is very easy to find. We apply this evolution method to study of the families of knots and links which include the cases with just two parallel and anti-parallel strands in the braid, like the ordinary twist and 2-strand torus knots/links and counter-oriented 2-strand links. When the answers were available before, they are immediately reproduced, and an essentially new example is added of the "double braid", which is a combination of parallel and anti-parallel 2-strand braids. This study helps us to reveal with the full clarity and partly investigate a mysterious hierarchical structure of the colored HOMFLY polynomials, at least, in (anti)symmetric representations, which extends the original observation for the figure-eight knot to many (presumably all) knots. We demonstrate that this structure is typically respected by the t-deformation to the superpolynomials.

hep-th↗

arXiv2013

On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions

In the genus expansion of the HOMFLY polynomials their representation dependence is naturally captured by symmetric group characters. This immediately implies that the Ooguri-Vafa partition function (OVPF) is a Hurwitz tau-function. In the planar limit involving factorizable special polynomials, it is actually a trivial exponential tau-function. In fact, in the double scaling Kashaev limit (the one associated with the volume conjecture) dominant in the genus expansion are terms associated with the symmetric representations and with the integrability preserving Casimir operators, though we stop one step from converting this fact into a clear statement about the OVPF behavior in the vicinity of q=1. Instead, we explain that the genus expansion provides a hierarchical decomposition of the Hurwitz tau-function, similar to the Takasaki-Takebe expansion of the KP tau-functions. This analogy can be helpful to develop a substitute for the universal Grassmannian description in the Hurwitz tau-functions.

hep-th↗

arXiv2013

Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux

If a knot is represented by an m-strand braid, then HOMFLY polynomial in representation R is a sum over characters in all representations Q\in R^{\otimes m}. Coefficients in this sum are traces of products of quantum R-matrices along the braid, but these matrices act in the space of intertwiners, and their size is equal to the multiplicity M_{RQ} of Q in R^{\otimes m}. If R is the fundamental representation R=[1], then M_{[1] Q} is equal to the number of paths in representation graph, which lead from the fundamental vertex [1] to the vertex Q. In the basis of paths the entries of the m-1 relevant R-matrices are associated with the pairs of paths and are non-vanishing only when the two paths either coincide or differ by at most one vertex; as a corollary R-matrices consist of just 1x1 and 2x2 blocks, given by very simple explicit expressions. If cabling method is used to color the knot with the representation R, then the braid has m|R| strands, Q have a bigger size m|R|, but only paths passing through the vertex R are included into the sums over paths which define the products and traces of the m|R|-1 relevant R-matrices. In the case of SU(N) this path sum formula can also be interpreted as a multiple sum over the standard Young tableaux. By now it provides the most effective way for evaluation of the colored HOMFLY polynomials, conventional or extended, for arbitrary braids.

hep-th↗

arXiv2013

Genus expansion of HOMFLY polynomials

In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus corrections to this formula in the form of expansion in powers of z = q-q^{-1}. Expansion coefficients are expressed through the eigenvalues of the cut-and-join operators, i.e. symmetric group characters. Moreover, the z-expansion is naturally exponentiated. Representation through cut-and-join operators makes contact with Hurwitz theory and its sophisticated integrability properties. Our formulas describe the shape of genus expansion for the HOMFLY polynomials, which for their matrix model counterparts is usually controlled by Virasoro like constraints and AMM/EO topological recursion. The genus expansion differs from the better studied weak coupling expansion at finite number of colors N, which is described in terms of the Vassiliev invariants and Kontsevich integral.

hep-th↗