Möbius Transformations in Noncommutative Conformal Geometry
نویسنده
چکیده
We study the projective linear group PGL2(A) associated with an arbitrary algebra A, and its subgroups from the point of view of their action on the space of involutions in A. This action formally resembles Möbius transformations known from complex geometry. By specifying A to be an algebra of bounded operators in a Hilbert space H, we rediscover the Möbius group μev(M) defined by Connes and study its action on the space of Fredholm modules over the algebra A. There is an induced action on the K-homology of A, which turns out to be trivial. Moreover, this action leads naturally to a simpler object, the polarized module underlying a given Fredholm module, and we discuss this relation in detail. Any polarized module can be lifted to a Fredholm module, and the set of different lifts forms a category, whose morphisms are given by generalized Möbius tranformations. We present an example of a polarized module canonically associated with the differentiable structure of a smooth manifold V . Using our lifting procedure we obtain a class of Fredholm modules characterizing the conformal structures on V . Fredholm modules obtained in this way are a special case of those constructed by Connes, Sullivan and Teleman. Supported in part by the Visitor Fund of the Department of Mathematics, University of Exeter. Supported in part by an LMS Scheme 4 grant and a grant from the Exeter University Research Fund.
منابع مشابه
On Infinitesimal Conformal Transformations of the Tangent Bundles with the Generalized Metric
Let be an n-dimensional Riemannian manifold, and be its tangent bundle with the lift metric. Then every infinitesimal fiber-preserving conformal transformation induces an infinitesimal homothetic transformation on . Furthermore, the correspondence gives a homomorphism of the Lie algebra of infinitesimal fiber-preserving conformal transformations on onto the Lie algebra of infinitesimal ...
متن کاملNoncommutative Conformal Field Theory in the Twist - deformed context
We discuss conformal symmetry on the two dimensional noncommutative plane equipped with Moyal product in the twist deformed context. We show that the consistent use of the twist procedure leads to results which are free from ambiguities. This lends support to the importance of the use of twist symmetries in noncommutative geometry.
متن کاملOn conformal transformation of special curvature of Kropina metrics
An important class of Finsler metric is named Kropina metrics which is defined by Riemannian metric α and 1-form β which have many applications in physic, magnetic field and dynamic systems. In this paper, conformal transformations of χ-curvature and H-curvature of Kropina metrics are studied and the conditions that preserve this quantities are investigated. Also it is shown that in the ...
متن کاملConformal Coupling and Invariance in Different Dimensions
Conformal transformations of the following kinds are compared: (1) conformal coordinate transformations, (2) conformal transformations of Lagrangian models for a D-dimensional geometry, given by a Riemannian manifold M with metric g of arbitrary signature, and (3) conformal transformations of (mini-)superspace geometry. For conformal invariance under this transformations the following applicati...
متن کامل