A remark on the rigidity of Poincaré–Einstein manifolds

In this paper, we give an optimal inequality relating the relative Yamabe invariant of a certain compactification of a conformally compact Poincaré–Einstein manifold with the Yamabe invariant of its boundary at infinity. As an application, we obtain an elementary proof (without any additional assumption) of the rigidity of the hyperbolic space as the only conformally compact Poincaré–Einstein manifold with the round sphere as its conformal infinity.

     

A Construction of Critical GJMS Operators Using Wodzicki's Residue

For an even dimensional, compact, conformal manifold without boundary we construct a conformally invariant differential operator of order the dimension of the manifold. In the conformally flat case, this operator coincides with the critical GJMS operator of Graham-Jenne-Mason-Sparling. We use the Wodzicki residue of a pseudo-differential operator of order −2, originally defined by A. Connes, acting on middle dimension forms.     

Differential forms and the Wodzicki residue for manifolds with boundary

In [A. Connes, Quantized calculus and applications, XIth International Congress of Mathematical Physics (Paris,1994), 15–36, Internat Press, Cambridge, MA, 1995], Connes found a conformal invariant using Wodzicki’s 1-density and computed it in the case of 4-dimensional manifold without boundary. In [W. J. Ugalde, Differential forms and the Wodzicki residue, arXiv: Math, DG/0211361], Ugalde generalized the Connes’ result to n-dimensional manifold without boundary. In this paper, we generalize the results of [A. Connes, Quantized calculus and applications, XIth International Congress of Mathematical Physics (Paris,1994), 15–36, Internat Press, Cambridge, MA, 1995] and [W. J. Ugalde, Differential forms and the Wodzicki residue, arXiv: Math, DG/0211361] to the case of manifolds with boundary.     

Conformal boundary conditions and three-dimensional topological field theory

We present a general construction of all correlation functions of a two-dimensional rational conformal field theory, for an arbitrary number of bulk and boundary fields and arbitrary topologies. The correlators are expressed in terms of Wilson graphs in a certain three-manifold, the connecting manifold. The amplitudes constructed this way can be shown to be modular invariant and to obey the correct factorization rules.     

On a scale function for testing the conformality of a Finsler manifold to a Berwald manifold

In this paper, we are going to discuss the problem whether how we can check the conformality of a Finsler manifold to a Berwald manifold. The method is based on a differential 1-form constructing on the underlying manifold by the help of integral formulas such that its exterior derivative is conformally invariant. If the Finsler manifold is conformal to a Berwald manifold, then the exterior derivative vanishes. This gives the following necessary condition: the differential form is closed and, at least locally, it is exact as the exterior derivative of a scale function for testing the conformality. A necessary and sufficient condition is also given in terms of a distinguished linear connection on the underlying manifold – it is expressed by the help of canonical data. In order to illustrate how we can simplify the process in special cases Randers manifolds are considered with some explicit calculations.     

Quantum brownian motion on a triangular lattice and c=2 boundary conformal field theory

We study a single particle diffusing on a triangular lattice and interacting with a heat bath, using boundary conformal field theory (CFT) and exact integrability techniques. We derive a correspondence between the phase diagram of this problem and that recently obtained for the 2-dimensional 3-state Potts model with a boundary. Exact results are obtained on phases with intermediate mobilities. These correspond to nontrivial boundary states in a conformal field theory with 2 free bosons which we explicitly construct for the first time. These conformally invariant boundary conditions are not simply products of Dirichlet and Neumann ones and unlike those trivial boundary conditions, are not invariant under a Heisenberg algebra.     

Reduction of Vaisman structures in complex and quaternionic geometry

We consider locally conformal Kähler geometry as an equivariant (homothetic) Kähler geometry: a locally conformal Kähler manifold is, up to equivalence, a pair $(K,\Gamma )$, where $K$ is a Kähler manifold and $\Gamma$ is a discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kähler manifold $(K,\Gamma )$ as the rank of a natural quotient of $\Gamma$, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kähler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover, we define locally conformal hyperKähler reduction as an equivariant version of hyperKähler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally, we show that locally conformal hyperKähler reduction induces hyperKähler with torsion (HKT) reduction of the associated HKT structure and the two reductions are compatible, even though not every HKT reduction comes from a locally conformal hyperKähler reduction.     

Characteristics of in-out intermittency in delay-coupled FitzHugh–Nagumo oscillators

We analyze a pair of delay-coupled FitzHugh–Nagumo oscillators exhibiting in-out intermittency as a part of the generating mechanism of extreme events. We study in detail the characteristics of in-out intermittency and identify the invariant subsets involved – a saddle fixed point and a saddle periodic orbit – neither of which are chaotic as in the previously reported cases of in-out intermittency. Based on the analysis of a periodic attractor possessing in-out dynamics, we can characterize the approach to the invariant synchronization manifold and the spiralling out to the saddle periodic orbit with subsequent ejection from the manifold. Due to the striking similarities, this analysis of in-out dynamics also explains in-out intermittency     

Holographic dual of a boundary conformal field theory

We propose a holographic dual of a conformal field theory defined on a manifold with boundaries, i.e., boundary conformal field theory (BCFT). Our new holography, which may be called anti-de?Sitter BCFT, successfully calculates the boundary entropy or g function in two-dimensional BCFTs and it agrees with the finite part of the holographic entanglement entropy. Moreover, we can naturally derive a holographic g theorem. We also analyze the holographic dual of an interval at finite temperature and show that there is a first order phase transition.     

Conformally symmetric semi-Riemannian manifolds

In this paper we introduce the concept of conformal curvature-like tensor on a semi-Riemannian manifold, which is weaker than the notion of conformal curvature tensor defined on a Riemannian manifold. By such kind of conformal curvature-like tensor we give a complete classification of conformally symmetric semi-Riemannian manifolds with generalized non-null stress energy tensor.     

Conformal field theory on the horizon of a BTZ black hole

In a three-dimensional spacetime with negative cosmological constant, general relativity can be written as two copies of SO(2,1) Chern-Simons theory. On a manifold with a boundary, the Chern-Simons theory induces a conformal field theory—Wess-Zumino-Witten theory on the boundary. In this paper, it is shown that with suitable boundary conditions for a Banados-Teitelboim-Zanelli black hole, the Wess-Zumino-Witten theory can reduce to a chiral massless scalar field on the horizon.     

Stalactite basin structure of dynamical systems with transient chaos in an invariant manifold

Dynamical systems with invariant manifolds occur in a variety of situations (e.g., identical coupled oscillators, and systems with a symmetry). We consider the case where there is both a nonchaotic attractor (e.g., a periodic orbit) and a nonattracting chaotic set (or chaotic repeller) in the invariant manifold. We consider the character of the basins for the attracting nonchaotic set in the invariant manifold and another attractor not in the invariant manifold. It is found that the boundary separating these basins has an interesting structure: The basin of the attractor not in the invariant manifold is characterized by thin cusp shaped regions ("stalactites") extending down to touch the nonattracting chaotic set in the invariant manifold. We also develop theoretical scalings applicable to these systems, and compare with numerical experiments. (c) 2000 American Institute of Physics.     

Relativistic Gauge Invariant Potentials

A global method characterizing the invariant connections on an abelian principal bundle under a group of transformations is applied in order to get gauge invariant electromagnetic (elm.) potentials in a systematic way. So, we have classified all the elm. gauge invariant potentials under the Poincaré subgroups of dimensions 4, 5, and 6, up to conjugation. It is paid attention in particular to the situation where these subgroups do not act transitively on the space-time manifold. We have used the same procedure for some galilean subgroups to get nonrelativistic potentials and study the way they are related to their relativistic partners by means of contractions. Some conformal gauge invariant potentials have also been derived and considered when they are seen as consequence of an enlargement of the Poincaré symmetries.     

Revisiting Cardassian model and cosmic constraint

Boundary conformal field theory (BCFT) is the study of conformal field theory (CFT) in semi-infinite space-time. In the non-relativistic limit (x???x,t??t,???0), the boundary conformal algebra changes to boundary Galilean conformal algebra (BGCA). In this work, some aspects of AdS/BCFT in the non-relativistic limit were explored. We constrain correlation functions of Galilean conformal invariant fields with BGCA generators. For a situation with a boundary condition at surface x=0 ( $z=\overline{z}$ ), our result agrees with the non-relativistic limit of the BCFT two-point function. We also introduce the holographic dual of boundary Galilean conformal field theory.     

Group manifold analysis of the structure of relativistic quantum dynamics

We present a group law, derived as a contraction of the conformal group, from which we obtain by using a canonical procedure a relativistic quantum system with an invariant evolution parameter (the proper time) and where the position operator belongs to the Lie algebra of the group. The restriction of the theory to the mass shell breaks part of the symmetry; of the previous 15 generators, only 10 remain which generate an action of the Poincaré group defining an orbit in the former group manifold. Some comments on the relativistic position operator are also made.     

Black hole temperature: Minimal coupling vs conformal coupling

In this article, we discuss the propagation of scalar fields in conformally transformed spacetimes with either minimal or conformal coupling. The conformally coupled equation of motion is transformed into a one-dimensional Schrödinger-like equation with an invariant potential under conformal transformation. In a second stage, we argue that calculations based on conformal coupling yield the same Hawking temperature as those based on minimal coupling. Finally, it is conjectured that the quasi normal modes of black holes are invariant under conformal transformation.     

A note on the conformal quasi-invariance of the Laplacian on a pseudo-Riemannian manifold

We consider the Laplacian on a pseudo-Riemannian manifold with constant scalar curvature (e.g. Euclidian space with an arbitrary signed inner product or its conformal compactification and coverings of this) and show that for this minus a constant we have quasi-invariance with respect to an action of the conformal group on functions.     

Asymptotics of the Eta Invariant

We prove an asymptotic bound on the eta invariant of a family of coupled Dirac operators on an odd dimensional manifold. In the case when the manifold is the unit circle bundle of a positive line bundle over a complex manifold, we obtain precise formulas for the eta invariant.