Noncompact Spaces Research Articles

Supersymmetry and cotangent bundle over non-compact exceptional Hermitian symmetric space

Abstract We construct $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over the non-compact exceptional Hermitian symmetric spaces $$ \mathrm{\mathcal{M}}={E}_{6\left(-14\right)}/\mathrm{SO}(10)\times \mathrm{U}(1) $$ ℳ = E 6 − 14 / SO 10 × U 1 and E 7(−25) /E 6 × U(1). In order to construct them we use the projective superspace formalism which is an $$ \mathcal{N}=2 $$ N = 2 off-shell superfield formulation in four-dimensional space-time. This formalism allows us to obtain the explicit expression of $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over any Hermitian symmetric spaces in terms of the $$ \mathcal{N}=1 $$ N = 1 superfields, once the Kähler potentials of the base manifolds are obtained. We derive the $$ \mathcal{N}=1 $$ N = 1 supersymmetric nonlinear sigma models on the Kähler manifolds $$ \mathrm{\mathcal{M}} $$ ℳ . Then we extend them into the $$ \mathcal{N}=2 $$ N = 2 supersymmetric models with the use of the result in arXiv:1211.1537 developed in the projective superspace formalism. The resultant models are the $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over the Hermitian symmetric spaces $$ \mathrm{\mathcal{M}} $$ ℳ . In this work we complete constructing the cotangent bundles over all the compact and non-compact Hermitian symmetric spaces.

Smeared antibranes polarise in AdS

AbstractIn the recent literature it has been questioned whether the local backreaction of antibranes in flux throats can induce a perturbative brane-flux decay. Most evidence for this can be gathered for D6 branes and Dp branes smeared over 6 − p compact directions, in line with the absence of finite temperature solutions for these cases. The solutions in the literature have flat worldvolume geometries and non-compact transversal spaces. In this paper we consider what happens when the worldvolume is AdS and the transversal space is compact. We show that in these circumstances brane polarisation smoothens out the flux singularity, which is an indication that brane-flux decay is prevented. This is consistent with the fact that the cosmological constant would be less negative after brane-flux decay. Our results extend recent results on AdS7 solutions from D6 branes to AdSp+1 solutions from Dp branes. We show that supersymmetry of the AdS solutions depend on p non-trivially.

Large time limit and local $L^2$-index theorems for families

We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut–Lott type superconnections in the L^2 -setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L^2 -index formulas. As applications, we prove a local L^2 -index theorem for families of signature operators and an L^2 -Bismut–Lott theorem, expressing the Becker–Gottlieb transfer of flat bundles in terms of Kamber–Tondeur classes. With slightly stronger regularity we obtain the respective refined versions: we construct L^2 -eta forms and L^2 -torsion forms as transgression forms.

Examples of hyperpolar actions of the automorphism groups of Lie algebras

We are studying isometric actions obtained by the automorphism groups of Lie algebras on the noncompact symmetric space GLn(R)/O(n). In this paper, we show that the actions which come from some solvable Lie algebras provide examples of hyperpolar actions with singular orbit and of higher cohomogeneity.

Analytically defined uniformly dense sequences

We introduce a notion of uniform density. This notion is an analogue of uniform distribution modulo 1, but is of interest mostly for non-compact spaces. Our main results show that various specific (families of) sequences or real numbers, defined by means of analytic formulas, are uniformly dense.

A generalised Gangolli–Lévy–Khintchine formula for infinitely divisible measures and Lévy processes on semi-simple Lie groups and symmetric spaces

In 1964 R.Gangolli published a Levy-Khintchine type formula which characterised K bi-invariant infinitely divisible probability measures on a symmetric space G=K. His main tool was Harish-Chandra's spherical functions which he used to construct a generalisation of the Fourier transform of a measure. In this paper we use generalised spherical functions (or Eisenstein integrals) and extensions of these which we construct using representation theory to obtain such a characterisation for arbitrary infinitely divisible probability measures on a non-compact symmetric space. We consider the example of hyperbolic space in some detail.

Admissible representations, multiplicity-free representations and visible actions on non-tube type Hermitian symmetric spaces

This paper presents new characterization for a non-compact Hermitian symmetric space $G/K$ to be of tube type (or non-tube type) by multiplicities in some branching laws and visible actions. The study in this paper gives an example of a kind of the Cartan decomposition for non-symmetric homogeneous spaces.

Quantum group perturbative formalism for affine Toda models

In the case of non-compact space we consider the quantum affine Toda systems in the operator approach. We introduce quantum analogues of the systems of equations as well as elements of quantum connections. For the solution of the quantum affine Toda systems we use the light-cone as well as Yang-Feldman formalisms of quantization. We then propose solutions to the quantum affine Toda equations based on the quantum-group approach.

Specification Property for Topological Spaces

We introduce and study here the notion of specification property termed as topological specification property for homeomorphisms on non-compact non-metrizable spaces. We prove that if a self homeomorphism on a totally bounded uniform space is mixing, topologically expansive and topologically shadowing then the map has topological specification property.

Hardy and Rellich type inequalities on metric measure spaces

In this paper, some Hardy and Rellich type inequalities on complete non-compact smooth metric measure spaces have been established. Moreover, based on the main results about the Hardy and Rellich type inequalities, several inequalities of the Hardy–Poincaré type or the Heisenberg–Pauli–Weyl type can also be given as byproducts.

The Caffarelli–Kohn–Nirenberg inequalities and manifolds with nonnegative weighted Ricci curvature

We prove that n-dimensional (n⩾3) complete and non-compact metric measure spaces with non-negative weighted Ricci curvature in which some Caffarelli–Kohn–Nirenberg type inequality holds are close to the model metric measure n-space (i.e., the Euclidean metric n-space).

Canonical Extension of Submanifolds and Foliations in Noncompact Symmetric Spaces

We propose a method to extend submanifolds, singular Riemannian foliations and isometric actions from a boundary component of a noncompact symmetric space to the whole space. This extension method preserves minimal submanifolds, isoparametric foliations and polar actions, among other properties. One of the several applications yields the first examples of inhomogeneous isoparametric hypersurfaces in noncompact symmetric spaces of rank at least two.

Random Weighted Sobolev Inequalities and Application to Quantum Ergodicity

This paper is a continuation of Poiret et al. (Ann Henri Poincaré 16:651–689, 2015), where we studied a randomisation method based on the Laplacian with harmonic potential. Here we extend our previous results to the case of any polynomial and confining potential V on \({\mathbb{R}^d}\). We construct measures, under concentration type assumptions, on the support of which we prove optimal weighted Sobolev estimates on \({\mathbb{R}^d}\). This construction relies on accurate estimates on the spectral function in a non-compact configuration space. Then we prove random quantum ergodicity results without specific assumption on the classical dynamics. Finally, we prove that almost all bases of Hermite functions are quantum uniquely ergodic.

COHOMOGENEITY ONE ACTIONS ON SOME NONCOMPACT SYMMETRIC SPACES OF RANK TWO

We classify, up to orbit equivalence, the cohomogeneity one actions on the noncompact Riemannian symmetric spaces of rank two G ℂ2 /G2, SL3(ℂ)/SU3 and SO 02,n + 2 /SO2SO n + 2, n ≥ 1.

ONE-POINT CONNECTIFICATIONS

A space $Y$ is called an extension of a space $X$ if $Y$ contains $X$ as a dense subspace. An extension $Y$ of $X$ is called a one-point extension if $Y\setminus X$ is a singleton. Compact extensions are called compactifications and connected extensions are called connectifications. It is well known that every locally compact noncompact space has a one-point compactification (known as the Alexandroff compactification) obtained by adding a point at infinity. A locally connected disconnected space, however, may fail to have a one-point connectification. It is indeed a long-standing question of Alexandroff to characterize spaces which have a one-point connectification. Here we prove that in the class of completely regular spaces, a locally connected space has a one-point connectification if and only if it contains no compact component.

Long-term stability of sequential Monte Carlo methods under verifiable conditions

This paper discusses particle filtering in general hidden Markov models (HMMs) and presents novel theoretical results on the long-term stability of bootstrap-type particle filters. More specifically, we establish that the asymptotic variance of the Monte Carlo estimates produced by the bootstrap filter is uniformly bounded in time. On the contrary to most previous results of this type, which in general presuppose that the state space of the hidden state process is compact (an assumption that is rarely satisfied in practice), our very mild assumptions are satisfied for a large class of HMMs with possibly noncompact state space. In addition, we derive a similar time uniform bound on the asymptotic $\mathsf{L}^p$ error. Importantly, our results hold for misspecified models; that is, we do not at all assume that the data entering into the particle filter originate from the model governing the dynamics of the particles or not even from an HMM.

Classical and quantum dynamics in an inverse square potential

The classical motion of a particle in a 3D inverse square potential with negative energy, E, is shown to be geodesic, i.e., equivalent to the particle's free motion on a non-compact phase space manifold irrespective of the sign of the coupling constant. We thus establish that all its classical orbits with E < 0 are unbounded. To analyse the corresponding quantum problem, the Schrödinger equation is solved in momentum space. No discrete energy levels exist in the unrenormalized case and the system shows a complete “fall-to-the-center” with an energy spectrum unbounded by below. Such behavior corresponds to the non-existence of bound classical orbits. The symmetry of the problem is SO(3) × SO(2, 1) corroborating previously obtained results.

A TOPOLOGICAL CHARACTERIZATION OF ­Ω-LIMIT SETS ON DYNAMICAL SYSTEMS

Abstract. In this article, we deal with the notion of ›-limit setsin dynamical systems. We show that the ›-limit set of a compactsubset of a phase space is quasi-attracting. 1. IntroductionThe theory for the notion of attractors is important for the classi-cal theory of dynamical systems. Conley [7] introduced a topologicaldeflnition of attractors for a °ow on a compact metric space. Hurley[9, 10] obtained results which is related to the correspondence betweenattractors and Lyapunov functions on noncompact spaces. Akin [1] andMcGehee [12] obtained many properties of attractors in set-valued dy-namics.The concept of omega-limit set, arising from their ubiquitous applica-tions in dynamical systems, is also an extremely used tool in the abstracttheory of dynamical systems. Especially, the notion of omega-limit setsis much related to the notion of attractors. These notions are used todescribe eventually the positive time behavior for dynamical systems.In recent years, Choy and Chu [4] described the characterizations ofomega-limit sets for analytic °ows on R

A reverse isoperimetric inequality for J-holomorphic curves

We prove that the length of the boundary of a $J$-holomorphic curve with Lagrangian boundary conditions is dominated by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary conditions and the genus. A similar result holds for the length of the real part of a real $J$-holomorphic curve. The infimum over $J$ of the constant properly normalized gives an invariant of Lagrangian submanifolds. We calculate this invariant to be $2\pi$ for the Lagrangian submanifold $\mathbb R P^n \subset \mathbb C P^n.$ We apply our result to prove compactness of moduli of $J$-holomorphic maps to non-compact target spaces that are asymptotically exact. In a different direction, our result implies the adic convergence of the superpotential.

Why are the effective equations of loop quantum cosmology so accurate?

We point out that the relative Heisenberg uncertainty relations vanish for noncompact spaces in homogeneous loop quantum cosmology. As a consequence, for sharply peaked states quantum fluctuations in the scale factor never become important, even near the bounce point. This shows why quantum backreaction effects remain negligible and explains the surprising accuracy of the effective equations in describing the dynamics of sharply peaked wave packets. This also underlines the fact that minisuperspace models---where it is global variables that are quantized---do not capture the local quantum fluctuations of the geometry.