Dimer statistics on the Möbius strip and the Klein bottle

Closed-form expressions are obtained for the generating function of close-packed dimers on a 2M×2N simple quartic lattice embedded on a Möbius strip and a Klein bottle. Finite-size corrections are also analyzed and compared with those under cylindrical and free boundary conditions. Particularly, it is found that, for large lattices of the same size and with a square symmetry, the number of dimer configurations on a Möbius strip is 70.2% of that on a cylinder. We also establish two identities relating dimer generating functions for Möbius strips and cylinders.     

On two-dimensional supergravity and the quantum super-Möbius group

We describe the quantum-group structure of two-dimensionalN=1 supergravity in the conformal gauge. The operator-algebra of this (super-Liouville) theory is shown to correspond to a quantum deformation of the super-Möbius group, which provides a new solution of Yang and Baxter's equation. This structure is used to investigate the strong-coupling regime of the theory (1d9). Ford=5, a unitary truncation theorem-similar to the one of the bosonic case-is derived for the fundamental representation.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud     

Finite-size scaling for the ising model on the Möbius strip and the klein bottle

We study the finite-size scaling properties of the Ising model on the M?bius strip and the Klein bottle. The results are compared with those of the Ising model under different boundary conditions, that is, the free, cylindrical, and toroidal boundary conditions. The difference in the magnetization distribution function p(m) for various boundary conditions is discussed in terms of the number of the percolating clusters and the cluster size. We also find interesting aspect-ratio dependence of the value of the Binder parameter at T = T(c) for various boundary conditions. We discuss the relation to the finite-size correction calculations for the dimer statistics.     

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.     

Optical Möbius symmetry in metamaterials

We experimentally observed a new topological symmetry in optical composites, namely, metamaterials. While it is not found yet in nature materials, the electromagnetic M?bius symmetry discovered in metamaterials is equivalent to the structural symmetry of a M?bius strip, with the number of twists controlled by the sign change of the electromagnetic coupling between the meta-atoms. We further demonstrate that metamaterials with different coupling signs exhibit resonance frequencies that depend only on the number but not the locations of the "twists," thus confirming its topological nature. The new topological symmetry found in metamaterials may enable unique functionalities in optical materials.     

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.     

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.     

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     

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.     

Renormalization of Möbius infinities and partition function representation for the string theory effective action

We show that subtraction of the Möbius group volume in the open string massless amplitudes can be realized as a renormalization of linear 2D ultraviolet divergences in the generating functional (“partition function”). This implies that the vector field effective action can be represented as a renormalized partition function (i.e. as a path integral of the “Wilson factor”). We check this by computing several leading terms in the non-abelian effective action.     

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.     

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.     

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.