Möbius Transformations in Noncommutative Conformal Geometry

We study the projective linear group PGL ₂(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 M?bius transformations known from complex geometry. By specifying A to be an algebra of bounded operators in a Hilbert space H, we rediscover the M?bius group μ _ev (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 M?bius 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. Received: 2 October 1997 / Accepted: 11 August 1998     

Optical Möbius strips, twisted ribbons, and the index theorem

The twist numbers of circular optical M?bius strips and twisted ribbons are shown to obey the index theorem under rotation of the plane of observation, and under change in the radius of the path.     

Optical Mbius Cubes:Integrated Packaging Configurations

l.IntroductionoptiCalinterconnectionnetworks[1-5Jcanmeetthevastdemandforparallelcompu-tationandfastcommunication-Inthedesignofopticalinterconnectionnetworks,thechoiceofnetworkstructuresisvatical.TherearemanyconsiderationsinchoosingaPar-ticularinterconnectionnetwork,whichinclude:networkdiameter,numberofconnectionspernode,nodelayout,symmetryofnodeconnections.Fordecreasingthenetworkdiameter,manykindsoftwistedcubeswithtwistedcommuniCationsaresuggested,theMdbiuscubes[6J,thevariantSofhyPercube,gi…     

Möbius transformation and conformal relativity

The Möbius transformation (MT) was analyzed as a coordinate transformation in the Minkowski form. The transformation function contains three separate light cones. The Weyl spheres were interpreted as basic constituents of local light cones. These cones are related to the denominators of the MT and its inverse, and their apexes define an axis with the top of the global light cone as the centerpoint. That axis represents the local part of the world-line of a moving frame of reference. On the world-line, the scale factor of the MT is proportional to the ratio of the radii of the initial Weyl sphere and the equivalent transformed one. The projection centers, defining the transformation of the world-line, were determined graphically. There are two types of such MT's. Inner transformations have their projetion centers on the axes of the frame at rest, outer transformations on the axes of the observed moving frame. The signature of x ₀ ² –r² is conserved during inner transformations. All possible directions of the world-line of an inertial frame form a timelike mass cone around the time axis of the frame at rest. The mass cone and the related spacelike phase cone may be seen as projections of a lightlike central motion on the surface of a Weyl sphere. Conformal transformations leave both the mass cone and the phase cone invariant: the MT locally, and the Lorentz transformation globally. The moving frame of any lightlike particle rotates by /2 radians, thus exchanging the time axis with one of the space axes. The measurable mass of such a particle is considered to be zero because only the central plane of the particle is space-oriented. The real mass is invariant. To test this hypothesis, the two -quanta produced by the electron pair annihilation should be led to a recollision, for recreating the initial electron-position pair.     

Möbius strips and twisted ribbons in intersecting Gauss-Laguerre beams

Circularly polarized, intersecting Gauss-Laguerre beams are shown to give rise to a rich array of unusual polarization structures whose properties can be tuned by varying the angle between the beams. These structures include Möbius strips with arbitrarily large, odd numbers of half twists, twisted ribbons with arbitrarily large, even numbers of half twists, and coronets (crowns). Six topological/geometrical indices are used to characterize these structures. Analytical expressions are given for some indices; others, which still present a considerable theoretical challenge, are treated numerically and illustrated graphically. It is argued that the polarization structures described here are, or soon will be, amenable to experiment.     

Möbius and super-Möbius gauge fixing for the closed string amplitudes on the desk

We systematically study the gauge fixing of the Möbius and the super-Möbius transformations for the N-point closed-string amplitudes on the disk. Using the Faddeev-Popov method, we obtain the explicit formulas for the Koba-Nielsen factors for these amplitudes.     

Representations of Algebras Associated with a Möbius Transformation

Abstract

The Hilbert space representations of a class of commutation relations associated with a Möbius transformation is studied using results on convergence of continued fractions.     

Theoretical investigation on optical properties of Möbius carbon nanobelts in one- and two-photon absorption

The first successful synthesis of fully fused and fully conjugated Möbius carbon nanobelts (CNBs) has attracted considerable attention. However, theoretical calculations based on such π-conjugated Möbius CNB are still insufficient. Herein, we theoretically investigated molecular spectroscopy of Möbius CNBs without and with n-butoxy groups via visualization methods. The results show that the presence of n-butoxy groups can significantly affect Möbius CNBs’ optical performance, changing electron-hole coherence and enhancing two-photon absorption cross-sections. Our work provides a deeper understanding of photophysical mechanisms of Möbius CNBs in one- and two-photon absorption and reveals possible applications on optoelectronic devices.     

The Klein-Gordon Operator on Möbius Strip Domains and the Klein Bottle in ℝ n

In this paper we present explicit formulas for the fundamental solution to the Klein-Gordon operator on some higher dimensional generalizations of the Möbius strip and the Klein bottle with values in distinct pinor bundles. The fundamental solution is described in terms of generalizations of the Weierstraß $\wp $ -function that are adapted to the context of these geometries. The explicit formulas for the kernel then allow us to express all solutions to the homogeneous and inhomogeneous Klein-Gordon problem with given boundary data in the context of these manifolds. In the case of the Klein bottle we are able to describe all null solutions of the Klein-Gordon equation in terms of finite linear combinations of the fundamental solution and its partial derivatives.     

Möbius function for the set of acyclic directed backbone graphs

Consider a two-rooted graphG, the edges of which are directed in such a way that there are no cycles and every edge belongs to some self-avoiding walk from rootu to rootv which follows the direction of the edges. Au-v backbone ofG is a subgraph formed by taking the union of any subset of directed self-avoiding walks fromu tov. Let _uv be the set of all such backbones ofG partially ordered by set-inclusion. We prove the conjecture of Bhatti and Essam that the Möbius function of this set is given, for acyclicb,b _uv withbb, by(b,b)=(–1) ^c-c , wherec andc are the respective cycle ranks ofb andb. The significance of this result in percolation theory is reviewed together with previous results for other sets of subgraphs.     

Coherent states for a quantum particle on a Möbius strip

The coherent states for a quantum particle on a Möbius strip are constructed and the relation with the natural phase space for fermionic fields is shown. The explicit comparison of the obtained states with previous works where the cylinder quantization was used and the spin 1/2 was introduced by hand is given.