9 research outputs found
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
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
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
The matrix model version of AGT conjecture and CIV-DV prepotential
Recently exact formulas were provided for partition function of conformal
(multi-Penner) beta-ensemble in the Dijkgraaf-Vafa phase, which, if interpreted
as Dotsenko-Fateev correlator of screenings and analytically continued in the
number of screening insertions, represents generic Virasoro conformal blocks.
Actually these formulas describe the lowest terms of the q_a-expansion, where
q_a parameterize the shape of the Penner potential, and are exact in the
filling numbers N_a. At the same time, the older theory of CIV-DV prepotential,
straightforwardly extended to arbitrary beta and to non-polynomial potentials,
provides an alternative expansion: in powers of N_a and exact in q_a. We check
that the two expansions coincide in the overlapping region, i.e. for the lowest
terms of expansions in both q_a and N_a. This coincidence is somewhat
non-trivial, since the two methods use different integration contours:
integrals in one case are of the B-function (Euler-Selberg) type, while in the
other case they are Gaussian integrals.Comment: 27 pages, 1 figur