Superpolynomials for toric knots from evolution induced by cut-and-join operators

The colored HOMFLY polynomials, which describe Wilson loop averages in Chern-Simons theory, possess an especially simple representation for torus knots, which begins from quantum R-matrix and ends up with a trivially-looking split W representation familiar from character calculus applications to matrix models and Hurwitz theory. Substitution of MacDonald polynomials for characters in these formulas provides a very simple description of "superpolynomials", much simpler than the recently studied alternative which deforms relation to the WZNW theory and explicitly involves the Littlewood-Richardson coefficients. A lot of explicit expressions are presented for different representations (Young diagrams), many of them new. In particular, we provide the superpolynomial P_[1]^[m,km\pm 1] for arbitrary m and k. The procedure is not restricted to the fundamental (all antisymmetric) representations and the torus knots, still in these cases some subtleties persist.Comment: 23 pages + Tables (51 pages

Liouville Type Models in Group Theory Framework. I. Finite-Dimensional Algebras

In the series of papers we represent the ``Whittaker'' wave functional of $d+1$-dimensional Liouville model as a correlator in $d+0$-dimensional theory of the sine-Gordon type (for $d=0$ and $1$). Asypmtotics of this wave function is characterized by the Harish-Chandra function, which is shown to be a product of simple $\Gamma$-function factors over all positive roots of the corresponding algebras (finite-dimensional for $d=0$ and affine for $d=1$). This is in nice correspondence with the recent results on 2- and 3-point correlators in $1+1$ Liouville model, where emergence of peculiar double-periodicity is observed. The Whittaker wave functions of $d+1$-dimensional non-affine ("conformal") Toda type models are given by simple averages in the $d+0$ dimensional theories of the affine Toda type. This phenomenon is in obvious parallel with representation of the free-field wave functional, which is originally a Gaussian integral over interior of a $d+1$-dimensional disk with given boundary conditions, as a (non-local) quadratic integral over the $d$-dimensional boundary itself. In the present paper we mostly concentrate on the finite-dimensional case. The results for finite-dimensional "Iwasawa" Whittaker functions were known, and we present their survey. We also construct new "Gauss" Whittaker functions.Comment: 47 pages, LaTe

New method of verifying cryptographic protocols based on the process model

A cryptographic protocol (CP) is a distributed algorithm designed to provide a secure communication in an insecure environment. CPs are used, for example, in electronic payments, electronic voting procedures, database access systems, etc. Errors in the CPs can lead to great financial and social damage, therefore it is necessary to use mathematical methods to justify the correctness and safety of the CPs. In this paper, a new mathematical model of a CP is introduced, which allows one to describe both the CPs and their properties. It is shown how, on the base of this model, it is possible to solve the problems of verification of CPs

Octonic Electrodynamics

In this paper we present eight-component values "octons", generating associative noncommutative algebra. It is shown that the electromagnetic field in a vacuum can be described by a generalized octonic equation, which leads both to the wave equations for potentials and fields and to the system of Maxwell's equations. The octonic algebra allows one to perform compact combined calculations simultaneously with scalars, vectors, pseudoscalars and pseudovectors. Examples of such calculations are demonstrated by deriving the relations for energy, momentum and Lorentz invariants of the electromagnetic field. The generalized octonic equation for electromagnetic field in a matter is formulated.Comment: 12 pages, 1 figur

Is Strong Gravitational Radiation predicted by TeV-Gravity?

In TeV-gravity models the gravitational coupling to particles with energies E\sim m_{Pl} \sim 10 TeV is not suppressed by powers of ultra-small ratio E/M_{Pl} with M_{Pl} \sim 10^{19} GeV. Therefore one could imagine strong synchrotron radiation of gravitons by the accelerating particles to become the most pronounced manifestation of TeV-gravity at LHC. However, this turns out to be not true: considerable damping continues to exist, only the place of E/M_{Pl} it taken by a power of a ratio \theta\omega/E, where the typical frequency \omega of emitted radiation, while increased by a number of \gamma-factors, can not reach E/\vartheta unless particles are accelerated by nearly critical fields. Moreover, for currently available magnetic fields B \sim 10 Tesla, multi-dimensionality does not enhance gravitational radiation at all even if TeV-gravity is correct.Comment: 7 pages, LaTe

Parafermionic Liouville field theory and instantons on ALE spaces

In this paper we study the correspondence between the $\hat{\textrm{su}}(n)_{k}\oplus \hat{\textrm{su}}(n)_{p}/\hat{\textrm{su}}(n)_{k+p}$ coset conformal field theories and $\mathcal{N}=2$ SU(n) gauge theories on $\mathbb{R}^{4}/\mathbb{Z}_{p}$. Namely we check the correspondence between the SU(2) Nekrasov partition function on $\mathbb{R}^{4}/\mathbb{Z}_{4}$ and the conformal blocks of the $S_{3}$ parafermion algebra (in $S$ and $D$ modules). We find that they are equal up to the U(1)-factor as it was in all cases of AGT-like relations. Studying the structure of the instanton partition function on $\mathbb{R}^4/\mathbb{Z}_p$ we also find some evidence that this correspondence with arbitrary $p$ takes place up to the U(1)-factor.Comment: 21 pages, 6 figures, misprints corrected, references added, version to appear in JHE

On some algebraic examples of Frobenius manifolds

We construct some explicit quasihomogeneous algebraic solutions to the associativity (WDVV) equations by using analytical methods of the finite gap integration theory. These solutions are expanded in the uniform way to non-semisimple Frobenius manifolds.Comment: 14 page

Nekrasov Functions and Exact Bohr-Sommerfeld Integrals

In the case of SU(2), associated by the AGT relation to the 2d Liouville theory, the Seiberg-Witten prepotential is constructed from the Bohr-Sommerfeld periods of 1d sine-Gordon model. If the same construction is literally applied to monodromies of exact wave functions, the prepotential turns into the one-parametric Nekrasov prepotential F(a,\epsilon_1) with the other epsilon parameter vanishing, \epsilon_2=0, and \epsilon_1 playing the role of the Planck constant in the sine-Gordon Shroedinger equation, \hbar=\epsilon_1. This seems to be in accordance with the recent claim in arXiv:0908.4052 and poses a problem of describing the full Nekrasov function as a seemingly straightforward double-parametric quantization of sine-Gordon model. This also provides a new link between the Liouville and sine-Gordon theories.Comment: 10 page