Geometric intrinsic symmetries

The problem of geometric symmetries in the intrinsic frame of a many-body system (nucleus) is considered. An importance of symmetrization group notion is discussed. Ageneral structure of the intrinsic symmetry group structure is determined.     

The hidden symmetries of multi-centre metrics

We discuss the dynamical symmetries of the multi-centre metrics and apply our results to the scattering of B.P.S. monopoles and fluctuations around such monopoles. In particular, we give a detailed account of the hidden symmetries of the Taub-NUT metric.Groupe d'Astrophysique Relativiste — Observatoire de Paris-MeudonLaboratoire Propre du Centre National de la Recherche Scientifique, associé à l'Ecole Normale Supérieure et à l'Université de Paris-Sud, France     

Super-Landau models: Unitarity and hidden symmetries

We review a recent progress in constructing and studying superextensions of the Landau problem of a quantum particle moving on the two-sphere S ². These are the superspherical and superflag Landau models describing a nonrelativistic particle on the supermanifolds SU(2|1)/U(1|1) and SU(2|1)/[U(1) x U(1)]. Though in both models, under the “naive” definition of the inner product, there are negative-norm states, all norms can be made positive-definite by introducing a nontrivial “metric” in the space of states. So both models are unitary. The superflag model is shown to be quantum-equivalent to the supersphere model for the special choice of the intrinsic parameters. We also address the planar limit of the superspherical model, in which there arises the hidden world-line N =2 supersymmetry. An analog of this phenomenon in the superflag model is a dynamical enhancement of its SU(2|1) symmetry to SU(2|2).     

The hidden symmetries of Taub-Nut and monopole scattering

The recently discovered hidden symmetries in the large-distance interactions of BPS monopoles with each other and with fluctuations around them are traced to the existence on the self-dual Taub-NUT metric of a Killing-Yano tensor. The global action on classical phase space of these symmetries is discussed. The quantum picture involving the “dynamical groups” SO(4), SO(4, 1) and SO(4, 2) is also given.     

Type II hidden symmetries of the second heavenly equation

A Type II hidden symmetry of the non-linear second heavenly equation in gravitational physics is identified. Its provenance from other partial differential equations is studied. Two reductions of the second heavenly equation produce the Monge–Ampère equation in similarity variables and new analytic solutions are possible.     

A generating function for hidden symmetries of chiral models

By the introduction of a generating function for the class of transformations of two-dimensional chiral lagrangians recently shown to generate a Kac-Moody algebra, we show how structure of the transformations and the invariance proofs may be considerably clarified.     

Classical symmetries of some two-dimensional models

It is well-known that principal chiral models and symmetric space models in two-dimensional Minkowski space have an infinite-dimensional algebra of hidden symmetries. Because of the relevance of symmetric space models to duality symmetries in string theory, the hidden symmetries of these models are explored in some detail. The string theory application requires including coupling to gravity, supersymmetrization, and quantum effects. However, as a first step, this paper only considers classical bosonic theories in flat space-time. Even though the algebra of hidden symmetries of principal chiral models is confirmed to include a Kac-Moody algebra (or a current algebra on a circle), it is argued that a better interpretation is provided by a doubled current algebra on a semi-circle (or line segment). Neither the circle nor the semi-circle bears any apparent relationship to the physical space. For symmetric space models the line segment viewpoint is shown to be essential, and special boundary conditions need to be imposed at the ends. The algebra of hidden symmetries also includes Virasoro-like generators. For both principal chiral models and symmetric space models, the hidden symmetry stress tensor is singular at the ends of the line segment.     

Geometric aspects of ordering phenomena

A macroscopic system prepared in a disordered phase and quenched across a second-order phase transition into an ordered phase undergoes a coarsening process whereby it orders locally in one of the equilibrium states. The study of the evolution of the morphology of the ordered structures in two dimensions has recently unveiled two interesting and generic features. On the one hand, the dynamics first approach a critical percolating state via the growth of a new lengthscale and satisfying scaling properties with respect to it. The time needed to reach the critical percolating state diverges with the system size, though more weakly than the equilibration time. On the other hand, once the critical percolating structures established, the geometrical and statistical properties at larger scales than the one established by the usual dynamic growing length remain the ones of critical percolation. These observations are common to different microscopic dynamics (single spin flip, local and non-local spin exchange, voter) in pure or weakly disordered systems. We discuss these results and we refer to the relevant publications for details.     

On some geometrical consequences of physical symmetries

It is shown that there exists only one submanifold O^(4,m)₂ of the representation space C^4m of the group GL(4,C)×GL(m,C) which admits a unique projection onto Minkowski space, consistent with the group. We describe the decomposition of this manifold O^4,m)₂ when the group is restricted to the physical symmetry group SU (2,2)× ×SU(m) or P×SU(m). We consider also representations of SU(2,2)×SU(m) in the resulting submanifolds and in the Hilbert space of functions over these manifolds.     

New potential symmetries for some evolution equations

In this paper we derive new potential symmetries that seem not to be recorded in the literature. These potential symmetries are determined by considering a generalized potential system, rather than the natural potential system or a general integral variable. An inhomogeneous diffusion equation, a porous medium equation and the Fokker-Planck equation have been considered as application of this procedure.     

Geometric and analytical aspects of anyons

We consider quantum systems ofn indistinguishable spinless particles, constrained to closed compact surfaces and satisfying fractional statistics (anyons). We question the traditional choice of a configuration space, and show that a theory maintaining the diagonal is possible. Such a theory leads naturally to questions in algebraic geometry involving desingularizations of certain algebraic varieties. The desingularizations induce the possibility of an exotic exclusion principle for anyons, wherein multiple occupancy is not excluded in general.     

Geometric aspects of quantum spin states

A number of interesting features of the ground states of quantum spin chains are analyzed with the help of a functional integral representation of the system's equilibrium states. Methods of general applicability are introduced in the context of the SU(2S+1)-invariant quantum spin-S chains with the interaction –P ^(o), whereP ^(o) is the projection onto the singlet state of a pair of nearest neighbor spins. The phenomena discussed here include: the absence of Néel order, the possibility of dimerization, conditions for the existence of a spectral gap, and a dichotomy analogous to one found by Affleck and Lieb, stating that the systems exhibit either slow decay of correlations or translation symmetry breaking. Our representation elucidates the relation, evidence for which was found earlier, of the –P ^(o) spin-S systems with the Potts and the Fortuin-Kasteleyn random-cluster models in one more dimension. The method reveals the geometric aspects of the listed phenomena, and gives a precise sense to a picture of the ground state in which the spins are grouped into random clusters of zero total spin. E.g., within such structure the dichotomy is implied by a topological argument, and the alternatives correspond to whether, or not, the clusters are of finite mean length.Work supported in part by NSF Grant PHY-9214654.Also in the Mathematics Department.     

Geometric aspects of phonon polarization transport

We study the polarization transport of transverse phonons by adopting a new approach based on the quantum mechanics of spin-orbit interactions. This approach has the advantage of being apt for incorporating fluctuations in the system. The formalism gives rise to Berry effect terms manifested as the Rytov polarization rotation law and the polarization-dependent Hall effect. We derive the distribution of the Rytov rotation angle in the presence of thermal noise and show that the rotation angle is robust against fluctuations.     

Dynamics on the Virasoro group, 2D gravity and hidden symmetries

We derive the action of 2D gravity from the two-cocycle of the Virasoro group by using a previously introduced method for constructing a dynamical system from a Lie group. A natural construction and explanation of the hidden SL(2, )-Kac-Moody symmetry, of general applicability, is provided. We also show how the proper quantization goes beyond the ordinary co-adjoint orbit method, and the possible connection with the Poisson-Lie groups.     

Dynamical generation of gauge bosons of hidden local symmetries in nonlinear sigma models

We demonstrate how quantum corrections generate a kinetic term for the (at tree-level non-propagating) gauge fields of hidden local symmetries in nonlinear sigma models in four space-time dimensions.