9 research outputs found
On the complete perturbative solution of one-matrix models
We summarize the recent results about complete solvability of Hermitian and
rectangular complex matrix models. Partition functions have very simple
character expansions with coefficients made from dimensions of representation
of the linear group , and arbitrary correlators in the Gaussian phase
are given by finite sums over Young diagrams of a given size, which involve
also the well known characters of symmetric group. The previously known
integrability and Virasoro constraints are simple corollaries, but no vice
versa: complete solvability is a peculiar property of the matrix model
(hypergeometric) -functions, which is actually a combination of these two
complementary requirements.Comment: 8 page
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 -matrix. In this paper, we generalize
the hypothesis to higher number of strands in the braid where commuting
relations of non-neighbouring matrices are also incorporated. By
solving these equations, we determine the explicit form for
-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 representation. Specifically, we present all
the inclusive Racah matrices for representation and compare with the
matrices obtained from eigenvalue hypothesis.Comment: 23 page
Tangle blocks in the theory of link invariants
The central discovery of conformal theory was holomorphic factorization,
which expressed correlation functions through bilinear combinations of
conformal blocks, which are easily cut and joined without a need to sum over
the entire huge Hilbert space of states. Somewhat similar, when a link diagram
is glued from tangles, the link polynomial is a multilinear combination of {\it
tangle blocks} summed over just a few representations of intermediate states.
This turns to be a powerful approach because the same tangles appear as
constituents of very different knots so that they can be extracted from simpler
cases and used in more complicated ones. So far this method has been
technically developed only in the case of arborescent knots, but, in fact, it
is much more general. We begin a systematic study of tangle blocks by detailed
consideration of some archetypical examples, which actually lead to non-trivial
results, far beyond the reach of other techniques. At the next level, the
tangle calculus is about gluing of tangles, i.e. functorial mappings from , and its main advantage is an explicit realization of
multiplicative composition structure, which is partly obscured in traditional
knot theory.Comment: 34 page
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Quadruply-graded colored homology of knots
We conjecture the existence of four independent gradings in colored HOMFLYPT homology, and make qualitative predictions of various interesting structures and symmetries in the colored homology of arbitrary knots. We propose an explicit conjectural description for the rectangular colored homology of torus knots, and identify the new gradings in this context. While some of these structures have a natural interpretation in the physical realization of knot homologies based on counting supersymmetric configurations (BPS states, instantons, and vortices), others are completely new. They suggest new geometric and physical realizations of colored HOMFLYPT homology as the Hochschild homology of the category of branes in a Landau–Ginzburg B-model or, equivalently, in the mirror A-model. Supergroups and supermanifolds are surprisingly ubiquitous in all aspects of this work
Challenges of beta-deformation
A brief review of problems, arising in the study of the beta-deformation,
also known as "refinement", which appears as a central difficult element in a
number of related modern subjects: beta \neq 1 is responsible for deviation
from free fermions in 2d conformal theories, from symmetric omega-backgrounds
with epsilon_2 = - epsilon_1 in instanton sums in 4d SYM theories, from
eigenvalue matrix models to beta-ensembles, from HOMFLY to super-polynomials in
Chern-Simons theory, from quantum groups to elliptic and hyperbolic algebras
etc. The main attention is paid to the context of AGT relation and its possible
generalizations.Comment: 20 page