Codimension reduction in symmetric spaces

In this paper we give a short geometric proof of a generalization of a well-known result about reduction of codimension for submanifolds of Riemannian symmetric spaces.     

Coherent states and symmetric spaces

Properties of system of the coherent states related to representations of the class I of principal series of the motion groups of symmetric spaces of rank 1 have been studied. It has been proved that such states are given by horospherical kernels and are the generalization of the plane waves for the case of symmetric spaces.     

Extrinsic symplectic symmetric spaces

We define the notion of extrinsic symplectic symmetric spaces and exhibit some of their properties. We construct large families of examples and show how they fit in the perspective of a complete classification of these manifolds. We also build a natural ?$?$-quantization on a class of examples.     

Bogomol'nyi solitons and Hermitian symmetric spaces

We apply the coadjoint orbit method to construct relativistic nonlinear sigma models (NLSM) on the target space of coadjoint orbits coupled with the Chern-Simons (CS) gauge field and we study self-dual solitons. When the target space is given by a Hermitian symmetric space (HSS), we find that the system admits self-dual solitons whose energy is Bogomol'nyi bounded from below by a topological charge. The Bogomol'nyi potential on the Hermitian symmetric space is obtained in the case when the maximal torus subgroup is gauged, and the self-dual equation in the CP(N − 1) case is explored. We also discuss the self-dual solitons in the case of noncompact SU(1, 1) and present a detailed analysis for the rotationally symmetric solutions.     

On weakly symmetric Finsler spaces

In this paper, we study weakly symmetric Finsler spaces. We first study an existence theorem of weakly symmetric Finsler spaces. Then we study some geometric properties of these spaces and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Finally we show that each weakly symmetric Finsler space is of Berwald type.     

Symmetric spaces and star representations: II. Causal symmetric spaces

We construct and identify star representations canonically associated with holonomy-reducible simple symplectic symmetric spaces. This leads a non-commutative geometric realization of the correspondence between causal symmetric spaces of Cayley-type and Hermitian symmetric spaces of tube-type.     

Spinor two-point functions in maximally symmetric spaces

The two-point function for spinors on maximally symmetric four-dimensional spaces is obtained in terms of intrinsic geometric objects. In the massless case, Weyl spinors in anti de Sitter space can not satisfy boundary conditions appropriate to the supersymmetric models. This is because these boundary conditions break chiral symmetry, which is proven by showing that the order parameter for a massless Dirac spinor is nonzero. We also give a coordinate-independent formula for the bispinor introduced by Breitenlohner and Freedman [1], and establish the precise connection between our results and those of Burges, Davis, Freedman and Gibbons [2].     

Vector two-point functions in maximally symmetric spaces

We obtain massive and massless vector two-point functions in maximally symmetric spaces (and vacua) of any number of dimensions. These include de Sitter space and anti-de Sitter space, and their Euclidean analogsS ⁿ andH ⁿ. Our method is based on a simple way of constructing every possible maximally symmetric bitensorT _a...bc...d(x, x) which carries tangent-space indicesa...b atx andc...d atx.     

Three graded exceptional algebras and symmetric spaces

The exceptional algebras of typeE ₇ are studied from the point of view of their three graded structure. The connection between three-grading and the Jordan Pair structure of such Lie algebras is analyzed. The Jordan Pair content is in turn related to the symmetric spaces and . Coset spaces of this type have been recently suggested as possible scalar manifolds in supergravity. We develop a way of representingE ₇ in a matrix form, which makes the Jordan Pair content of theE ₇ real and complex forms quite transparent and shows whether such forms admit a three graded structure.     

Integrability and Huygens' principle on symmetric spaces

The explicit formulas for fundamental solutions of the modified wave equations on certain symmetric spaces are found. These symmetric spaces have the following characteristic property: all multiplicities of their restricted roots are even. As a corollary in the odd-dimensional case one has that the Huygens' principle in Hadamard's sense for these equations is fulfilled. We consider also the heat and Laplace equations on such a symmetric space and give explicitly the corresponding fundamental solutions-heat kernel and Green's function. This continues our previous investigations [16] of the spherical functions on the same symmetric spaces based on the fact that the radial part of the Laplace-Beltrami operator on such a space is related to the algebraically integrable case of the generalised Calogero-Sutherland-Moser quantum system. In the last section of this paper we apply the methods of Heckman and Opdam to extend our results to some other symmetric spaces, in particular to complex and quaternian grassmannians.     

BFV quantization on hermitian symmetric spaces

Gauge-invariant BFV approach to geometric quantization is applied to the case of hermitian symmetric spaces G/H. In particular, gauge invariant quantization on the Lobachevski plane and sphere is carried out. Due to the presence of symmetry, master equations for the first-class constraints, quantum observables and physical quantum states are exactly solvable. BFV-BRST operator defines a flat G-connection in the Fock bundle over G/H. Physical quantum states are covariantly constant sections with respect to this connection and are shown to coincide with the generalized coherent states for the group G. Vacuum expectation values of the quantum observables commuting with the quantum first-class constraints reduce to the covariant symbols of Berezin. The gauge-invariant approach to quantization on symplectic manifolds synthesizes geometric, deformation and Berezin quantization approaches.     

Twistor spinors on Lorentzian symmetric spaces

An indecomposable Riemannian symmetric space which admits non-trivial twistor spinors has constant sectional curvature. Furthermore, each homogeneous Riemannian manifold with parallel spinors is flat. In the present paper we solve the twistor equation on all indecomposable Lorentzian symmetric spaces explicitly. In particular, we show that there are — in contrast to the Riemannian case — indecomposable Lorentzian symmetric spaces with twistor spinors, which have non-constant sectional curvature and non-flat and non-Ricci flat homogeneous Lorentzian manifolds with parallel spinors.     

Conformational completion of Lorentz symmetric spaces

Every simply connected Lorentz symmetric space of dimensionn3 has a symmetric quotient which can be conformally imbedded in a quadric of the projective space ⁿ⁺¹ (R).     

The spinor heat kernel in maximally symmetric spaces

The heat kernelK(x, x, t) of the iterated Dirac operator on anN-dimensional simply connected maximally symmetric Riemannian manifold is calculated. On the odd-dimesional hyperbolic spacesK is a Minakshisundaram-DeWitt expansion which terminates to the coefficienta _N–1)/2 and is exact. On the odd spheres the heat kernel may be written as an image sum of WKB kernels, each term corresponding to a classical path (geodesic). In the even dimensional case the WKB approximation is not exact, but a closed form ofK is derived both in terms of (spherical) eigenfunctions and of a sum over classical paths. The spinor Plancherel measure () and function in the hyperbolic case are also calculated. A simple relation between the analytic structure of onH ^N and the degeneracies of the Dirac operator onS ^N is found.     

Long range diffusion in ultrametric spaces

A model for diffusion in spaces with ultrametric topology is introduced. The parameters of the model are two sets of numbers: the branching numberm _l+1 /m _l describing the hierarchical structure of the metric and the transition ratesr _l between different levels of the hierarchy. This model is expected to be relevant to the spin glas problem. The exact solution of the dynamical problem can be given in the general case. In the special casem _l =m ^l andr _l R ^l anomalous long time behaviour of the autocorrelation function is found which decays with a power law multiplied by a periodically varying amplitude.Dedicated to B. Mühlschlegel on the occasion of his 60th birthdayThis work has been supported by the Sonderforschungsbereich 125, Aachen-Jülich-Köln     

Path integrals on symmetric spaces and group manifolds

Invariant path integrals on symmetric and group spaces are defined in terms of a sum over the paths formed by broken geodesic segments. Their evaluation proceeds by using the mean value properties of functions over the geodesic and complex radius spheres. It is shown that on symmetric spaces the invariant path integral gives a kernel of the Schrödinger equation in terms of the spectral resolution of the zonal functions of the space. On compact group spaces the invariant path integral reduces to a sum over powers of Gaussian-type integrals which, for a free particle, yields the standard Van Vleck-Pauli propagator. Explicit calculations are performed for the case ofSU(2) andU(N) group spaces.     

Non-compact symmetric spaces and the Toda molecule equations

It has been shown by Olshanetsky and Perelomov that the Toda molecule equations associated with any Lie groupG describe special geodesic motions on the Riemannian non-compact symmetric space which is the quotient of the normal real form ofG, G ^N, by its maximal compact subgroup. This is explained in more detail and it is shown that the fundamental Poisson bracket relation involving the Lax operatorA and leading to the Yang-Baxter equation and integrability properties is a direct consequence of the fact that the Iwasawa decomposition forG ^N endows the symmetric space with a hidden group theoretic structure.Supported by CNP _q (Brasil)