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

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.Comment: 23 page

Anisotropy and effective dimensionality crossover of the fluctuation conductivity of hybrid superconductor/ferromagnet structures

We study the fluctuation conductivity of a superconducting film, which is placed to perpendicular non-uniform magnetic field with the amplitude $H_0$ induced by the ferromagnet with domain structure. The conductivity tensor is shown to be essentially anisotropic. The magnitude of this anisotropy is governed by the temperature and the typical width of magnetic domains $d$. For $d\ll L_{H_0}=\sqrt{\Phi_0/H_0}$ the difference between diagonal fluctuation conductivity components $\Delta\sigma_\parallel$ along the domain walls and $\Delta\sigma_\perp$ across them has the order of $(d/L_{H_0})^4$. In the opposite case for $d\gg L_{H_0}$ the fluctuation conductivity tensor reveals effective dimensionality crossover from standard two-dimensional $(T-T_c)^{-1}$ behavior well above the critical temperature $T_c$ to the one-dimensional $(T-T_c)^{-3/2}$ one close to $T_c$ for $\Delta\sigma_\parallel$ or to the $(T-T_c)^{-1/2}$ dependence for $\Delta\sigma_\perp$. In the intermediate case $d\approx L_{H_0}$ for a fixed temperature shift from $T_c$ the dependence $\Delta\sigma_\parallel(H_0)$ is shown to have a minimum at $H_0\sim\Phi_0/d^2$ while $\Delta\sigma_\perp(H_0)$ is a monotonically increasing function.Comment: 11 pages, 8 figure

Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals

The AGT relation identifies the Nekrasov functions for various N=2 SUSY gauge theories with the 2d conformal blocks, which possess explicit Dotsenko-Fateev matrix model (beta-ensemble) representations the latter being polylinear combinations of Selberg integrals. The "pure gauge" limit of these matrix models is, however, a non-trivial multiscaling large-N limit, which requires a separate investigation. We show that in this pure gauge limit the Selberg integrals turn into averages in a Brezin-Gross-Witten (BGW) model. Thus, the Nekrasov function for pure SU(2) theory acquires a form very much reminiscent of the AMM decomposition formula for some model X into a pair of the BGW models. At the same time, X, which still has to be found, is the pure gauge limit of the elliptic Selberg integral. Presumably, it is again a BGW model, only in the Dijkgraaf-Vafa double cut phase.Comment: 21 page