14 research outputs found
Bispectral KP Solutions and Linearization of Calogero-Moser Particle Systems
A new construction using finite dimensional dual grassmannians is developed
to study rational and soliton solutions of the KP hierarchy. In the rational
case, properties of the tau function which are equivalent to bispectrality of
the associated wave function are identified. In particular, it is shown that
there exists a bound on the degree of all time variables in tau if and only if
the wave function is rank one and bispectral. The action of the bispectral
involution, beta, in the generic rational case is determined explicitly in
terms of dual grassmannian parameters. Using the correspondence between
rational solutions and particle systems, it is demonstrated that beta is a
linearizing map of the Calogero-Moser particle system and is essentially the
map sigma introduced by Airault, McKean and Moser in 1977.Comment: LaTeX, 24 page
Quantum Pieri rules for isotropic Grassmannians
We study the three point genus zero Gromov-Witten invariants on the
Grassmannians which parametrize non-maximal isotropic subspaces in a vector
space equipped with a nondegenerate symmetric or skew-symmetric form. We
establish Pieri rules for the classical cohomology and the small quantum
cohomology ring of these varieties, which give a combinatorial formula for the
product of any Schubert class with certain special Schubert classes. We also
give presentations of these rings, with integer coefficients, in terms of
special Schubert class generators and relations.Comment: 59 pages, LaTeX, 6 figure
Charges and fluxes in Maxwell theory on compact manifolds with boundary
We investigate the charges and fluxes that can occur in higher-order Abelian
gauge theories defined on compact space-time manifolds with boundary. The
boundary is necessary to supply a destination to the electric lines of force
emanating from brane sources, thus allowing non-zero net electric charges, but
it also introduces new types of electric and magnetic flux. The resulting
structure of currents, charges, and fluxes is studied and expressed in the
language of relative homology and de Rham cohomology and the corresponding
abelian groups. These can be organised in terms of a pair of exact sequences
related by the Poincar\'e-Lefschetz isomorphism and by a weaker flip symmetry
exchanging the ends of the sequences. It is shown how all this structure is
brought into play by the imposition of the appropriately generalised Maxwell's
equations. The requirement that these equations be integrable restricts the
world-volume of a permitted brane (assumed closed) to be homologous to a cycle
on the boundary of space-time. All electric charges and magnetic fluxes are
quantised and satisfy the Dirac quantisation condition. But through some
boundary cycles there may be unquantised electric fluxes associated with
quantised magnetic fluxes and so dyonic in nature.Comment: 28 pages, plain Te
Hodge Theory on Metric Spaces
Hodge theory is a beautiful synthesis of geometry, topology, and analysis,
which has been developed in the setting of Riemannian manifolds. On the other
hand, spaces of images, which are important in the mathematical foundations of
vision and pattern recognition, do not fit this framework. This motivates us to
develop a version of Hodge theory on metric spaces with a probability measure.
We believe that this constitutes a step towards understanding the geometry of
vision.
The appendix by Anthony Baker provides a separable, compact metric space with
infinite dimensional \alpha-scale homology.Comment: appendix by Anthony W. Baker, 48 pages, AMS-LaTeX. v2: final version,
to appear in Foundations of Computational Mathematics. Minor changes and
addition