4,076 research outputs found
The Heisenberg group and conformal field theory
A mathematical construction of the conformal field theory (CFT) associated to
a compact torus, also called the "nonlinear Sigma-model" or "lattice-CFT", is
given. Underlying this approach to CFT is a unitary modular functor, the
construction of which follows from a "Quantization commutes with reduction"-
type of theorem for unitary quantizations of the moduli spaces of holomorphic
torus-bundles and actions of loop groups. This theorem in turn is a consequence
of general constructions in the category of affine symplectic manifolds and
their associated generalized Heisenberg groups.Comment: 45 pages, some parts have been rewritten. Version to appear in Quart.
J. Mat
The cyclic theory of Hopf algebroids
We give a systematic description of the cyclic cohomology theory of Hopf
algebroids in terms of its associated category of modules. Then we introduce a
dual cyclic homology theory by applying cyclic duality to the underlying
cocyclic object. We derive general structure theorems for these theories in the
special cases of commutative and cocommutative Hopf algebroids. Finally, we
compute the cyclic theory in examples associated to Lie-Rinehart algebras and
\'etale groupoids.Comment: 44 pages; to appear in Journal of Noncommutative Geometr
Higher genera for proper actions of Lie groups
Let G be a Lie group with finitely many connected components and let K be a
maximal compact subgroup. We assume that G satisfies the rapid decay (RD)
property and that G/K has non-positive sectional curvature. As an example, we
can take G to be a connected semisimple Lie group. Let M be a G-proper manifold
with compact quotient M/G. In this paper we establish index formulae for the
C^*-higher indices of a G-equivariant Dirac-type operator on M. We use these
formulae to investigate geometric properties of suitably defined higher genera
on M. In particular, we establish the G-homotopy invariance of the higher
signatures of a G-proper manifold and the vanishing of the A-hat genera of a
G-spin, G-proper manifold admitting a G-invariant metric of positive scalar
curvature.Comment: 20 pages, revised version, the main changes are in section 2.
Bihamiltonian cohomology of KdV brackets
Using spectral sequences techniques we compute the bihamiltonian cohomology
groups of the pencil of Poisson brackets of dispersionless KdV hierarchy. In
particular this proves a conjecture of Liu and Zhang about the vanishing of
such cohomology groups.Comment: 16 pages. v2: corrected typos, in particular formulas (28), (78
The bi-Hamiltonian cohomology of a scalar Poisson pencil
We compute the bi-Hamiltonian cohomology of an arbitrary dispersionless
Poisson pencil in a single dependent variable using a spectral sequence method.
As in the KdV case, we obtain that is isomorphic to
for , to for ,
, , , and vanishes otherwise
Cyclic cocycles on deformation quantizations and higher index theorems
We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic
vector space. Using this cyclic cocycle we construct an explicit, local,
quasi-isomorphism from the complex of differential forms on a symplectic
manifold to the complex of cyclic cochains of any formal deformation
quantization thereof. We give a new proof of Nest-Tsygan's algebraic higher
index theorem by computing the pairing between such cyclic cocycles and the
-theory of the formal deformation quantization. Furthermore, we extend this
approach to derive an algebraic higher index theorem on a symplectic orbifold.
As an application, we obtain the analytic higher index theorem of
Connes--Moscovici and its extension to orbifolds.Comment: 59 pages, this is a major revision, orbifold analytic higher index is
introduce
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