37 research outputs found
Torus HOMFLY as the Hall-Littlewood Polynomials
We show that the HOMFLY polynomials for torus knots T[m,n] in all fundamental
representations are equal to the Hall-Littlewood polynomials in representation
which depends on m, and with quantum parameter, which depends on n. This makes
the long-anticipated interpretation of Wilson averages in 3d Chern-Simons
theory as characters precise, at least for the torus knots, and calls for
further studies in this direction. This fact is deeply related to
Hall-Littlewood-MacDonald duality of character expansion of superpolynomials
found in arXiv:1201.3339. In fact, the relation continues to hold for extended
polynomials, but the symmetry between m and n is broken, then m is the number
of strands in the braid. Besides the HOMFLY case with q=t, the torus
superpolynomials are reduced to the single Hall-Littlewood characters in the
two other distinguished cases: q=0 and t=0.Comment: 9 page
A direct proof of AGT conjecture at beta = 1
The AGT conjecture claims an equivalence of conformal blocks in 2d CFT and
sums of Nekrasov functions (instantonic sums in 4d SUSY gauge theory). The
conformal blocks can be presented as Dotsenko-Fateev beta-ensembles, hence, the
AGT conjecture implies the equality between Dotsenko-Fateev beta-ensembles and
the Nekrasov functions. In this paper, we prove it in a particular case of
beta=1 (which corresponds to c = 1 at the conformal side and to epsilon_1 +
epsilon_2 = 0 at the gauge theory side) in a very direct way. The central role
is played by representation of the Nekrasov functions through correlators of
characters (Schur polynomials) in the Selberg matrix models. We mostly
concentrate on the case of SU(2) with 4 fundamentals, the extension to other
cases being straightforward. The most obscure part is extending to an arbitrary
beta: for beta \neq 1, the Selberg integrals that we use do not reproduce
single Nekrasov functions, but only sums of them.Comment: 26 pages, 16 figures, 8 table
On Equivalence of two Hurwitz Matrix Models
In arXiv:0902.2627 a matrix model representation was found for the simplest
Hurwitz partition function, which has Lambert curve phi e^{-phi} = psi as a
classical equation of motion. We demonstrate that Fourier-Laplace transform in
the logarithm of external field Psi converts it into a more sophisticated form,
recently suggested in arXiv:0906.1206
New and Old Results in Resultant Theory
Resultants are getting increasingly important in modern theoretical physics:
they appear whenever one deals with non-linear (polynomial) equations, with
non-quadratic forms or with non-Gaussian integrals. Being a subject of more
than three-hundred-year research, resultants are of course rather well studied:
a lot of explicit formulas, beautiful properties and intriguing relationships
are known in this field. We present a brief overview of these results,
including both recent and already classical. Emphasis is made on explicit
formulas for resultants, which could be practically useful in a future physics
research.Comment: 50 pages, 15 figure