10,758 research outputs found
Nijenhuis operator in contact homology and descendant recursion in symplectic field theory
In this paper we investigate the algebraic structure related to a new type of
correlator associated to the moduli spaces of -parametrized curves in
contact homology and rational symplectic field theory. Such correlators are the
natural generalization of the non-equivariant linearized contact homology
differential (after Bourgeois-Oancea) and give rise to an invariant Nijenhuis
(or hereditary) operator (\`a la Magri-Fuchssteiner) in contact homology which
recovers the descendant theory from the primaries. We also sketch how such
structure generalizes to the full SFT Poisson homology algebra to a (graded
symmetric) bivector. The descendant hamiltonians satisfy to recursion
relations, analogous to bihamiltonian recursion, with respect to the pair
formed by the natural Poisson structure in SFT and such bivector. In case the
target manifold is the product stable Hamiltonian structure , with
a symplectic manifold, the recursion coincides with genus topological
recursion relations in the Gromov-Witten theory of .Comment: 30 pages, 3 figure
On rational approximation of algebraic functions
We construct a new scheme of approximation of any multivalued algebraic
function by a sequence of rational
functions. The latter sequence is generated by a recurrence relation which is
completely determined by the algebraic equation satisfied by . Compared
to the usual Pad\'e approximation our scheme has a number of advantages, such
as simple computational procedures that allow us to prove natural analogs of
the Pad\'e Conjecture and Nuttall's Conjecture for the sequence
in the complement \mathbb{CP}^1\setminus
\D_{f}, where \D_{f} is the union of a finite number of segments of real
algebraic curves and finitely many isolated points. In particular, our
construction makes it possible to control the behavior of spurious poles and to
describe the asymptotic ratio distribution of the family . As an application we settle the so-called 3-conjecture of
Egecioglu {\em et al} dealing with a 4-term recursion related to a polynomial
Riemann Hypothesis.Comment: 25 pages, 8 figures, LaTeX2e, revised version to appear in Advances
in Mathematic
Loop equations from differential systems
To any differential system where belongs to a Lie
group (a fiber of a principal bundle) and is a Lie algebra
valued 1-form on a Riemann surface , is associated an infinite sequence
of "correlators" that are symmetric -forms on . The goal of
this article is to prove that these correlators always satisfy "loop
equations", the same equations satisfied by correlation functions in random
matrix models, or the same equations as Virasoro or W-algebra constraints in
CFT.Comment: 20 page
Quantum curves for Hitchin fibrations and the Eynard-Orantin theory
We generalize the topological recursion of Eynard-Orantin (2007) to the
family of spectral curves of Hitchin fibrations. A spectral curve in the
topological recursion, which is defined to be a complex plane curve, is
replaced with a generic curve in the cotangent bundle of an arbitrary
smooth base curve . We then prove that these spectral curves are
quantizable, using the new formalism. More precisely, we construct the
canonical generators of the formal -deformation family of -modules
over an arbitrary projective algebraic curve of genus greater than ,
from the geometry of a prescribed family of smooth Hitchin spectral curves
associated with the -character variety of the fundamental
group . We show that the semi-classical limit through the WKB
approximation of these -deformed -modules recovers the initial family
of Hitchin spectral curves.Comment: 34 page
Elliptic Gromov - Witten invariants and the generalized mirror conjecture
A conjecture expressing genus 1 Gromov-Witten invariants in mirror-theoretic
terms of semi-simple Frobenius structures and complex oscillating integrals is
formulated. The proof of the conjecture is given for torus-equivariant Gromov -
Witten invariants of compact K\"ahler manifolds with isolated fixed points and
for concave bundle spaces over such manifolds. Several results on genus 0
Gromov - Witten theory include: a non-linear Serre duality theorem, its
application to the genus 0 mirror conjecture, a mirror theorem for concave
bundle spaces over toric manifolds generalizing a recent result of B. Lian, K.
Liu and S.-T. Yau. We also establish a correspondence (see the extensive
footnote in section 4) between their new proof of the genus 0 mirror conjecture
for quintic 3-folds and our proof of the same conjecture given two years ago.Comment: 56 page
Critical properties of an aperiodic model for interacting polymers
We investigate the effects of aperiodic interactions on the critical behavior
of an interacting two-polymer model on hierarchical lattices (equivalent to the
Migadal-Kadanoff approximation for the model on Bravais lattices), via
renormalization-group and tranfer-matrix calculations. The exact
renormalization-group recursion relations always present a symmetric fixed
point, associated with the critical behavior of the underlying uniform model.
If the aperiodic interactions, defined by s ubstitution rules, lead to relevant
geometric fluctuations, this fixed point becomes fully unstable, giving rise to
novel attractors of different nature. We present an explicit example in which
this new attractor is a two-cycle, with critical indices different from the
uniform model. In case of the four-letter Rudin-Shapiro substitution rule, we
find a surprising closed curve whose points are attractors of period two,
associated with a marginal operator. Nevertheless, a scaling analysis indicates
that this attractor may lead to a new critical universality class. In order to
provide an independent confirmation of the scaling results, we turn to a direct
thermodynamic calculation of the specific-heat exponent. The thermodynamic free
energy is obtained from a transfer matrix formalism, which had been previously
introduced for spin systems, and is now extended to the two-polymer model with
aperiodic interactions.Comment: 19 pages, 6 eps figures, to appear in J. Phys A: Math. Ge
- …