286 research outputs found
The Physics Inside Topological Quantum Field Theories
We show that the equations of motion defined over a specific field space are
realizable as operator conditions in the physical sector of a generalized Floer
theory defined over that field space. The ghosts associated with such a
construction are found not to be dynamical. This construction is applied to
gravity on a four dimensional manifold, ; whereupon, we obtain Einstein's
equations via surgery, along , in a five-dimensional topological quantum
field theory.Comment: LaTeX, 7 page
Morse homology for the heat flow
We use the heat flow on the loop space of a closed Riemannian manifold to
construct an algebraic chain complex. The chain groups are generated by
perturbed closed geodesics. The boundary operator is defined in the spirit of
Floer theory by counting, modulo time shift, heat flow trajectories that
converge asymptotically to nondegenerate closed geodesics of Morse index
difference one.Comment: 89 pages, 3 figure
Unstable geodesics and topological field theory
A topological field theory is used to study the cohomology of mapping space.
The cohomology is identified with the BRST cohomology realizing the physical
Hilbert space and the coboundary operator given by the calculations of
tunneling between the perturbative vacua. Our method is illustrated by a simple
example.Comment: 28 pages, OCU-15
Quantum cohomology of flag manifolds and Toda lattices
We discuss relations of Vafa's quantum cohomology with Floer's homology
theory, introduce equivariant quantum cohomology, formulate some conjectures
about its general properties and, on the basis of these conjectures, compute
quantum cohomology algebras of the flag manifolds. The answer turns out to
coincide with the algebra of regular functions on an invariant lagrangian
variety of a Toda lattice.Comment: 35 page
Contact orderability up to conjugation
We study in this paper the remnants of the contact partial order on the
orbits of the adjoint action of contactomorphism groups on their Lie algebras.
Our main interest is a class of non-compact contact manifolds, called convex at
infinity.Comment: 28 pages, 1 figur
Symplectic cohomology and q-intersection numbers
Given a symplectic cohomology class of degree 1, we define the notion of an
equivariant Lagrangian submanifold. The Floer cohomology of equivariant
Lagrangian submanifolds has a natural endomorphism, which induces a grading by
generalized eigenspaces. Taking Euler characteristics with respect to the
induced grading yields a deformation of the intersection number. Dehn twists
act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz
fibrations give fully computable examples. A key step in computations is to
impose the "dilation" condition stipulating that the BV operator applied to the
symplectic cohomology class gives the identity. Equivariant Lagrangians mirror
equivariant objects of the derived category of coherent sheaves.Comment: 32 pages, 9 figures, expanded introduction, added details of example
7.5, added discussion of sign
Fast soliton scattering by delta impurities
We study the Gross-Pitaevskii equation (nonlinear Schroedinger equation) with
a repulsive delta function potential. We show that a high velocity incoming
soliton is split into a transmitted component and a reflected component. The
transmitted mass (L^2 norm squared) is shown to be in good agreement with the
quantum transmission rate of the delta function potential. We further show that
the transmitted and reflected components resolve into solitons plus dispersive
radiation, and quantify the mass and phase of these solitons.Comment: 32 pages, 3 figure
A beginner's introduction to Fukaya categories
The goal of these notes is to give a short introduction to Fukaya categories
and some of their applications. The first half of the text is devoted to a
brief review of Lagrangian Floer (co)homology and product structures. Then we
introduce the Fukaya category (informally and without a lot of the necessary
technical detail), and briefly discuss algebraic concepts such as exact
triangles and generators. Finally, we mention wrapped Fukaya categories and
outline a few applications to symplectic topology, mirror symmetry and
low-dimensional topology. This text is based on a series of lectures given at a
Summer School on Contact and Symplectic Topology at Universit\'e de Nantes in
June 2011.Comment: 42 pages, 13 figure
Vitamin K2 and its Impact on Tooth Epigenetics
The impact of nutritional signals plays an important role in systemic-based «models» of dental caries. Present hypotheses now focus both on the oral environment and other organs, like the nervous system and brain. The tooth is subjected to shear forces, nourishing and cleansing, and its present “support system” (the hypothalamus/parotid axis) relays endocrine signaling to the parotid gland. Sugar consumption enhances hypothalamic oxidative stress (ROS), reversing dentinal fluid flow, thus creating an enhanced vulnerability to the oral bacterial flora. The acid, produced by the oral bacterial flora, then leads to erosion of the dentine, and an irreversible loss of dental enamel layers. This attack brings about inflammatory responses, yielding metalloproteinase-based “dissolution”. However, vitamin K2 (i.e. MK-4/MK-7) may come to the rescue with its antioxidant property, locally (mouth cavity) or systemically (via the brain), thus sustaining/preserving hormone-induced dentinal fluid flow (encompassing oxidative stress) and boosting/magnifying bodily inflammatory responses. However, sugars may also reduce the tooth’s natural defences through endocrine signaling, thus enhancing acid-supported enamel dentine erosion. Vitamin K2 sustains and improves the salivary buffering capacity via its impact on the secretion/flow of calcium and inorganic phosphates. Interestingly, primitive cultures’ diets (low-sugar and high-K2 diets) preserve dental health
Khovanov homology is an unknot-detector
We prove that a knot is the unknot if and only if its reduced Khovanov
cohomology has rank 1. The proof has two steps. We show first that there is a
spectral sequence beginning with the reduced Khovanov cohomology and abutting
to a knot homology defined using singular instantons. We then show that the
latter homology is isomorphic to the instanton Floer homology of the sutured
knot complement: an invariant that is already known to detect the unknot.Comment: 124 pages, 13 figure
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