29 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
Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed
We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson
brackets of hydrodynamic type vanishes for almost all degrees. This implies the
existence of a full dispersive deformation of a semisimple bihamiltonian
structure of hydrodynamic type starting from any infinitesimal deformation.Comment: 22 pages. v2: corrected typos. v3: small improvements of the
presentation. v4: typos, small improvements in the introduction and the
presentatio
Higher genera for proper actions of Lie groups, Part 2: the case of manifolds with boundary
Let G be a finitely connected Lie group and let K be a maximal compact
subgroup. Let M be a cocompact G-proper manifold with boundary, endowed with a
G-invariant metric which is of product type near the boundary. Under additional
assumptions on G, for example that it satisfies the Rapid Decay condition and
is such that G/K has nonpositive sectional curvature, we define higher
Atiyah-Patodi-Singer C^*-indices associated to smooth group cocycles on G and
to a generalized G-equivariant Dirac operator D on M with L^2-invertible
boundary operator D_\partial. We then establish a higher index formula for
these C^*-indices and use it in order to introduce higher genera for M, thus
generalizing to manifolds with boundary the results that we have established in
Part 1. Our results apply in particular to a semisimple Lie group G. We use
crucially the pairing between suitable relative cyclic cohomology groups and
relative K-theory groups.Comment: Updated version: small corrections. Additivity of higher genera adde
Hochschild cohomology of Lie-Rinehart algebras
We compute the Hochschild cohomology of universal enveloping algebras of
Lie-Rinehart algebras in terms of the Poisson cohomology of the associated
graded quotient algebras. Central in our approach are two cochain complexes of
"nonlinear Chevalley-Eilenberg" cochains whose origins lie in Lie-Rinehart
modules "up to homotopy", one on the Hochschild cochains of the base algebra,
another related to the adjoint representation. The Poincare-Birkhoff-Witt
isomorphism is then extended to a certain intertwiner between such modules.
Finally, exploiting the twisted Calabi-Yau structure, we obtain results for the
dual Hochschild and cyclic homology.Comment: 33 page