74 research outputs found
Towards matrix model representation of HOMFLY polynomials
We investigate possibilities of generalizing the TBEM eigenvalue matrix
model, which represents the non-normalized colored HOMFLY polynomials for torus
knots as averages of the corresponding characters. We look for a model of the
same type, which is a usual Chern-Simons mixture of the Gaussian potential,
typical for Hermitean models, and the sine Vandermonde factors, typical for the
unitary ones. We mostly concentrate on the family of twist knots, which
contains a single torus knot, the trefoil. It turns out that for the trefoil
the TBEM measure is provided by an action of Laplace exponential on the Jones
polynomial. This procedure can be applied to arbitrary knots and provides a
TBEM-like integral representation for the N=2 case. However, beyond the torus
family, both the measure and its lifting to larger N contain non-trivial
corrections in \hbar=\log q. A possibility could be to absorb these corrections
into a deformation of the Laplace evolution by higher Casimir and/or
cut-and-join operators, in the spirit of Hurwitz tau-function approach to knot
theory, but this remains a subject for future investigation.Comment: 10 page
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.Comment: 21 page
Cut-and-Join operator representation for Kontsevich-Witten tau-function
In this short note we construct a simple cut-and-join operator representation
for Kontsevich-Witten tau-function that is the partition function of the
two-dimensional topological gravity. Our derivation is based on the Virasoro
constraints. Possible applications of the obtained expression are discussed.Comment: 5 pages, minor correction
The Power of Nekrasov Functions
The recent AGT suggestion to use the set of Nekrasov functions as a basis for
a linear decomposition of generic conformal blocks works very well not only in
the case of Virasoro symmetry, but also for conformal theories with extended
chiral algebra. This is rather natural, because Nekrasov functions are
introduced as expansion basis for generalized hypergeometric integrals, very
similar to those which arise in expansion of Dotsenko-Fateev integrals in
powers of alpha-parameters. Thus, the AGT conjecture is closely related to the
old belief that conformal theory can be effectively described in the free field
formalism, and it can actually be a key to clear formulating and proof this
long-standing hypothesis. As an application of this kind of reasoning we use
knowledge of the exact hypergeometric conformal block for complete proof of the
AGT relation for a restricted class of external states.Comment: 8 page
From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators
In this letter,we present our conjecture on the connection between the
Kontsevich--Witten and the Hurwitz tau-functions. The conjectural formula
connects these two tau-functions by means of the group element. An
important feature of this group element is its simplicity: this is a group
element of the Virasoro subalgebra of . If proved, this conjecture
would allow to derive the Virasoro constraints for the Hurwitz tau-function,
which remain unknown in spite of existence of several matrix model
representations, as well as to give an integrable operator description of the
Kontsevich--Witten tau-function.Comment: 13 page
Racah coefficients and extended HOMFLY polynomials for all 5-, 6- and 7-strand braids
Basing on evaluation of the Racah coefficients for SU_q(3) (which supported
the earlier conjecture of their universal form) we derive explicit formulas for
all the 5-, 6- and 7-strand Wilson averages in the fundamental representation
of arbitrary SU(N) group (the HOMFLY polynomials). As an application, we list
the answers for all 5-strand knots with 9 crossings. In fact, the 7-strand
formulas are sufficient to reproduce all the HOMFLY polynomials from the
katlas.org: they are all described at once by a simple explicit formula with a
very transparent structure. Moreover, would the formulas for the relevant
SU_q(3) Racah coefficients remain true for all other quantum groups, the paper
provides a complete description of the fundamental HOMFLY polynomials for all
braids with any number of strands.Comment: 16 pages + Tables and Appendice
Resolvents and Seiberg-Witten representation for Gaussian beta-ensemble
The exact free energy of matrix model always obeys the Seiberg-Witten (SW)
equations on a complex curve defined by singularities of the quasiclassical
resolvent. The role of SW differential is played by the exact one-point
resolvent. We show that these properties are preserved in generalization of
matrix models to beta-ensembles. However, since the integrability and
Harer-Zagier topological recursion are still unavailable for beta-ensembles, we
need to rely upon the ordinary AMM/EO recursion to evaluate the first terms of
the genus expansion. Consideration in this paper is restricted to the Gaussian
model.Comment: 15 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
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