41 research outputs found
Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
We continue to develop the tensor-algebra approach to knot polynomials with
the goal to present the story in elementary and comprehensible form. The
previously reviewed description of Khovanov cohomologies for the gauge group of
rank N-1=1 was based on the cut-and-join calculus of the planar cycles, which
are involved rather artificially. We substitute them by alternative and natural
set of cycles, not obligatory planar. Then the whole construction is
straightforwardly lifted from SL(2) to SL(N) and reproduces Khovanov-Rozansky
(KR) polynomials, simultaneously for all values of N. No matrix factorization
and related tedious calculations are needed in such approach, which can
therefore become not only conceptually, but also practically useful.Comment: 66 page
On the shapes of elementary domains or why Mandelbrot Set is made from almost ideal circles?
Direct look at the celebrated "chaotic" Mandelbrot Set in Fig..\ref{Mand2}
immediately reveals that it is a collection of almost ideal circles and
cardioids, unified in a specific {\it forest} structure. In /hep-th/9501235 a
systematic algebro-geometric approach was developed to the study of generic
Mandelbrot sets, but emergency of nearly ideal circles in the special case of
the family was not fully explained. In the present paper the shape of
the elementary constituents of Mandelbrot Set is explicitly {\it calculated},
and difference between the shapes of {\it root} and {\it descendant} domains
(cardioids and circles respectively) is explained. Such qualitative difference
persists for all other Mandelbrot sets: descendant domains always have one less
cusp than the root ones. Details of the phase transition between different
Mandelbrot sets are explicitly demonstrated, including overlaps between
elementary domains and dynamics of attraction/repulsion regions. Explicit
examples of 3-dimensional sections of Universal Mandelbrot Set are given. Also
a systematic small-size approximation is developed for evaluation of various
Feigenbaum indices.Comment: 65 pages, 30 figure
A-infinity structure on simplicial complexes
A discrete (finite-difference) analogue of differential forms is considered,
defined on simplicial complexes, including triangulations of continuous
manifolds. Various operations are explicitly defined on these forms, including
exterior derivative and exterior product. The latter one is non-associative.
Instead, as anticipated, it is a part of non-trivial A-infinity structure,
involving a chain of poly-linear operations, constrained by nilpotency
relation: (d + \wedge + m + ...)^n = 0 with n=2.Comment: final version. 29 page
Higher Nilpotent Analogues of A-infinity Structure
Higher nilpotent analogues of the -structure are explicitly defined
on arbitrary simplicial complexes, generalizing explicit construction of
/hep-th/0704.2609. These structures are associated with the higher nilpotent
differential , satisfying , which is naturally defined on
triangulated manifolds (tetrahedral lattices). The deformation is defined with the help of the
-versions of discrete exterior product and the -operator.Comment: preliminary version, essential corrections mad
Introduction to Integral Discriminants
The simplest partition function, associated with homogeneous symmetric forms
S of degree r in n variables, is integral discriminant J_{n|r}(S) = \int
e^{-S(x_1 ... x_n)} dx_1 ... dx_n. Actually, S-dependence remains the same if
e^{-S} in the integrand is substituted by arbitrary function f(S), i.e.
integral discriminant is a characteristic of the form S itself, and not of the
averaging procedure. The aim of the present paper is to calculate J_{n|r} in a
number of non-Gaussian cases. Using Ward identities -- linear differential
equations, satisfied by integral discriminants -- we calculate J_{2|3},
J_{2|4}, J_{2|5} and J_{3|3}. In all these examples, integral discriminant
appears to be a generalized hypergeometric function. It depends on several
SL(n) invariants of S, with essential singularities controlled by the ordinary
algebraic discriminant of S.Comment: 36 pages, 19 figure
Universal Mandelbrot Set as a Model of Phase Transition Theory
The study of Mandelbrot Sets (MS) is a promising new approach to the phase
transition theory. We suggest two improvements which drastically simplify the
construction of MS. They could be used to modify the existing computer programs
so that they start building MS properly not only for the simplest families.
This allows us to add one more parameter to the base function of MS and
demonstrate that this is not enough to make the phase diagram connectedComment: 5 pages, 3 figure
Non-Linear Algebra and Bogolubov's Recursion
Numerous examples are given of application of Bogolubov's forest formula to
iterative solutions of various non-linear equations: one and the same formula
describes everything, from ordinary quadratic equation to renormalization in
quantum field theory.Comment: LaTex, 21 page
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