4,980 research outputs found
Equivariant -theory of GKM bundles
Given a fiber bundle of GKM spaces, , we analyze the
structure of the equivariant -ring of as a module over the equivariant
-ring of by translating the fiber bundle, , into a fiber bundle of
GKM graphs and constructing, by combinatorial techniques, a basis of this
module consisting of -classes which are invariant under the natural holonomy
action on the -ring of of the fundamental group of the GKM graph of .
We also discuss the implications of this result for fiber bundles where and are generalized partial flag varieties and show how
our GKM description of the equivariant -ring of a homogeneous GKM space is
related to the Kostant-Kumar description of this ring.Comment: 15 page
Coherent states and the quantization of 1+1-dimensional Yang-Mills theory
This paper discusses the canonical quantization of 1+1-dimensional Yang-Mills
theory on a spacetime cylinder, from the point of view of coherent states, or
equivalently, the Segal-Bargmann transform. Before gauge symmetry is imposed,
the coherent states are simply ordinary coherent states labeled by points in an
infinite-dimensional linear phase space. Gauge symmetry is imposed by
projecting the original coherent states onto the gauge-invariant subspace,
using a suitable regularization procedure. We obtain in this way a new family
of "reduced" coherent states labeled by points in the reduced phase space,
which in this case is simply the cotangent bundle of the structure group K.
The main result explained here, obtained originally in a joint work of the
author with B. Driver, is this: The reduced coherent states are precisely those
associated to the generalized Segal-Bargmann transform for K, as introduced by
the author from a different point of view. This result agrees with that of K.
Wren, who uses a different method of implementing the gauge symmetry. The
coherent states also provide a rigorous way of making sense out of the quantum
Hamiltonian for the unreduced system.
Various related issues are discussed, including the complex structure on the
reduced phase space and the question of whether quantization commutes with
reduction
Non-commutative integrable systems on -symplectic manifolds
In this paper we study non-commutative integrable systems on -Poisson
manifolds. One important source of examples (and motivation) of such systems
comes from considering non-commutative systems on manifolds with boundary
having the right asymptotics on the boundary. In this paper we describe this
and other examples and we prove an action-angle theorem for non-commutative
integrable systems on a -symplectic manifold in a neighbourhood of a
Liouville torus inside the critical set of the Poisson structure associated to
the -symplectic structure
Toric moment mappings and Riemannian structures
Coadjoint orbits for the group SO(6) parametrize Riemannian G-reductions in
six dimensions, and we use this correspondence to interpret symplectic
fibrations between these orbits, and to analyse moment polytopes associated to
the standard Hamiltonian torus action on the coadjoint orbits. The theory is
then applied to describe so-called intrinsic torsion varieties of Riemannian
structures on the Iwasawa manifold.Comment: 25 pages, 14 figures; Geometriae Dedicata 2012, Toric moment mappings
and Riemannian structures, available at
http://www.springerlink.com/content/yn86k22mv18p8ku2
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