Algebra Of Diffeomorphisms Research Articles

Gauged diffeomorphisms and hidden symmetries in Kaluza–Klein theories

We analyse the symmetries that are realized on the massive Kaluza–Klein modes in generic D-dimensional backgrounds with three non-compact directions. For this, we construct the unbroken phase given by the decompactification limit, in which the higher Kaluza–Klein modes are massless. The latter admits an infinite-dimensional extension of the three-dimensional diffeomorphism group as local symmetry and, moreover, a current algebra associated with together with the diffeomorphism algebra of the internal manifold as global symmetries. It is shown that the ‘broken phase’ can be reconstructed by gauging a certain subgroup of the global symmetries. This deforms the three-dimensional diffeomorphisms to a gauged version, and it is shown that they can be governed by a Chern–Simons theory, which unifies the spin-2 modes with the Kaluza–Klein vectors. This provides a reformulation of D-dimensional Einstein gravity, in which the physical degrees of freedom are described by the scalars of a gauged nonlinear σ-model based on , while the metric appears in a purely topological Chern–Simons form.

*‐Lie derivative, *‐covariant derivative and *‐gravity

AbstractThe Lie algebra of infinitesimal diffeomorphisms on noncommutative space allows to develop differential and Riemannian noncommutative geometry. Noncommutative Einstein's gravity equations are formulated.

Irreducible representations for the abelian extension of the Lie algebra of diffeomorphisms of tori in dimensions greater than 1

We classify the irreducible weight modules of the abelian extension of the Lie algebra of diffeomorphisms of tori of dimension greater than 1, with finite-dimensional weight spaces.

Conservation laws in Skyrme-type models

The zero curvature representation of Zakharov and Shabat [V. E. Zakharov and A. B. Shabat, Soviet Phys. JETP 34, 62 (1972)] has been generalized recently to higher dimensions and has been used to construct nonlinear field theories which are integrable or contain integrable submodels. The Skyrme model, for instance, contains an integrable subsector with infinitely many conserved currents, and the simplest Skyrmion with baryon number 1 belongs to this subsector. Here we use a related method, based on the geometry of target space, to construct a whole class of theories which are integrable or contain integrable subsectors (where integrability means the existence of infinitely many conservation laws). These models have three-dimensional target space, like the Skyrme model, and their infinitely many conserved currents turn out to be Noether currents of the volume-preserving diffeomorphisms on target space. Specifically for the Skyrme model, we find both weak and strong integrability conditions, where the conserved currents form a subset of the algebra of volume-preserving diffeomorphisms in both cases, but this subset is a subalgebra only for the weak integrable submodel.

Einstein-Riemann Gravity on Deformed Spaces

A differential calculus, differential geometry and the E-R Gravity theory are studied on noncommutative spaces. Noncommutativity is formulated in the star product formalism. The basis for the gravity theory is the infinitesimal algebra of diffeomorphisms. Considering the corresponding Hopf algebra we find that the deformed gravity is based on a deformation of the Hopf algebra.

Interacting D2 branes in ten dimensions and non-Abelian Born–Infeld theory

In this paper we extend the bosonic D-brane action in D = 10 obtained by duality from the D = 11 membrane wrapped on S1 to an SU(2) non-Abelian system. This system presents only first class constraints, whose algebra closes off-shell and generalizes the algebra of diffeomorphisms of the D2 brane to include non-Abelian symmetry generators. From the SU(2)D-brane action, we also obtain the SU(2) Born–Infeld theory by performing a covariant reduction to a flat background. This calculation agrees up to fourth order with the result obtained from the superstring amplitudes and gives an alternative approach to analyse non-Abelian Born–Infeld theories.

Casimir effect in a two-dimensional signature changing space-time

We study the Casimir effect for free massless scalar fields propagating on a two-dimensional cylinder with a metric that admits a change of signature from Lorentzian to Euclidean. We obtain a nonzero pressure, on the hypersurfaces of signature change, which destabilizes the signature changing region and so alters the energy spectrum of scalar fields. The modified region and spectrum, themselves, back react on the pressure. Moreover, the central term of diffeomorphism algebra of corresponding infinite conserved charges changes correspondingly.

Conformal fluid dynamics

We present a conformal theory of a dissipationless relativistic fluid in 2 space–time dimensions. The theory carries with it a representation of the algebra of 2D area-preserving diffeomorphisms in the target space of the complex scalar potentials. A complete canonical description is given, and the central charge of the current algebra is calculated. The passage to the quantum theory is discussed in some detail; as a result of operator ordering problems, full quantization at the level of the fields is as yet an open problem.

A gravity theory on noncommutative spaces

A deformation of the algebra of diffeomorphisms is constructed for canonically deformed spaces with constant deformation parameter θ. The algebraic relations remain the same, whereas the comultiplication rule (Leibniz rule) is different from the undeformed one. Based on this deformed algebra, a covariant tensor calculus is constructed and all the concepts such as metric, covariant derivatives, curvature and torsion can be defined on the deformed space as well. The construction of these geometric quantities is presented in detail. This leads to an action invariant under the deformed diffeomorphism algebra and can be interpreted as a θ-deformed Einstein–Hilbert action. The metric or the vierbein field will be the dynamical variable as they are in the undeformed theory. The action and all relevant quantities are expanded up to second order in θ.

2D superconductivity: Classification of universality classes by infinite symmetry

I consider superconducting condensates which become incompressible in the infinite gap limit. Classical 2D incompressible fluids possess the dynamical symmetry of area-preserving diffeomorphisms. I show that the corresponding infinite dynamical symmetry of 2D superconducting fluids is the coset W 1 + ∞ ⊗ W ¯ 1 + ∞ U ( 1 ) diagonal , with W 1 + ∞ the chiral algebra of quantum area-preserving diffeomorphisms and I derive its minimal models. These define a discrete set of 2D superconductivity universality classes which fall into two main categories: conventional superconductors with their vortex excitations and unconventional superconductors. These are characterized by a broken U ( 1 ) vector ⊗ U ( 1 ) axial symmetry and are labeled by an integer level m. They possess neutral spinon excitations of fractional spin and statistics S = θ 2 π = m − 1 2 m , which carry also an SU ( m ) isospin quantum number; this hidden SU ( m ) symmetry implies that these anyon excitations are non-Abelian. The simplest unconventional superconductor is realized for m = 2 : in this case the spinon excitations are semions (half-fermions). My results show that spin–charge separation in 2D superconductivity is a universal consequence of the infinite symmetry of the ground state. This infinite symmetry and its superselection rules realize a quantum protectorate in which the neutral spinons can survive even as soft modes on a rigid, spinless charge condensate.

Non-commutative Hopf algebra of formal diffeomorphisms

This paper deals with two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second is the set of formal diffeomorphisms with the group law being a composition of series. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization procedure in quantum field theory, and to the renormalization functor given by the double-tensor algebra on a bi-algebra.

Generalized [formula omitted] higher-spin algebras and symbolic calculus on flag manifolds

We study a new class of infinite-dimensional Lie algebras W ∞ ( N + , N − ) generalizing the standard W ∞ algebra, viewed as a tensor operator algebra of SU ( 1 , 1 ) in a group-theoretic framework. Here we interpret W ∞ ( N + , N − ) either as an infinite continuation of the pseudo-unitary symmetry U ( N + , N − ) , or as a “higher- U ( N + , N − ) -spin extension” of the diffeomorphism algebra diff ( N + , N − ) of the N = N + + N − torus U ( 1 ) N . We highlight this higher-spin structure of W ∞ ( N + , N − ) by developing the representation theory of U ( N + , N − ) (discrete series), calculating higher-spin representations, coherent states and deriving Kähler structures on flag manifolds. They are essential ingredients to define operator symbols and to infer a geometric pathway between these generalized W ∞ symmetries and algebras of symbols of U ( N + , N − ) -tensor operators. Classical limits (Poisson brackets on flag manifolds) and quantum (Moyal) deformations are also discussed. As potential applications, we comment on the formulation of diffeomorphism-invariant gauge field theories, like gauge theories of higher-extended objects, and non-linear sigma models on flag manifolds.

Noncommutative coordinates on Riemann surfaces

We show that the definition of a projective coordinate frame within a Laguerre–Forsyth scheme, leads to the extension of the factorized diffeomorphism algebra. The quantum improvement of this symmetry can be performed only if these coordinates switch, at the quantum level, into a noncommutative regime.

Economic analysis of a transesophageal echocardiography-guided approach to cardioversion of patients with atrial fibrillation: The ACUTE economic data at eight weeks

A symmetric set operad is a monoid in the category of combinatorial species with respect to the operation of substitution. From a symmetric set operad, we give here a simple construction of a commutative and non-co-commutative Hopf algebra, that we call the natural Hopf algebra of the operad. We obtain a combinatorial formula for its antipode in terms of Schröder trees, generalizing the Haiman–Schmitt formula for the Faá di Bruno Hopf algebra. From there we derive antipode formulas for specific operads. The classical Lagrange inversion formula is obtained in this way from the set operad of pointed sets. We also derive antipodes formulas for the natural Hopf algebra corresponding to the operads of connected graphs, the NAP operad, and for its generalization, the set operad of trees enriched with a monoid. When the set operad is left cancellative, we can construct a family of posets. The natural Hopf algebra is then obtained as a reduced incidence Hopf algebra, by taking a suitable equivalence relation over the intervals of that family of posets. We also present a simple combinatorial construction of an epimorphism from the natural Hopf algebra corresponding to the NAP operad, to the Connes and Kreimer Hopf algebra. For non-symmetric operads a similar construction leads to the world of non-commutative Hopf algebras. We recover from our general formula the Novelli–Thibon combinatorial form of the antipode for the non-commutative Hopf algebra of formal diffeomorphisms.

EXTENDED DIFFEOMORPHISM ALGEBRAS IN (QUANTUM) GRAVITATIONAL PHYSICS

We construct an explicit representation of the algebra of local diffeomorphisms of a manifold with realistic dimensions. This is achieved in the setting of a general approach to the (quantum) dynamics of a physical system which is characterized by the fundamental role assigned to a basic underlying symmetry. The mathematical formalism developed addresses the relevant gravitational notions by means of the addition of some extra structure. The specific manners in which this is accomplished, together with their corresponding physical interpretation, lead to different gravitational models. Distinct strategies are in fact briefly outlined, showing the versatility of the present conceptual framework.

From Diffeomorphisms to Dark Energy?

There are two physical actions that have a natural setting in terms of the coadjoint representation of the algebra of diffeomorphisms and of affine Lie algebras. One is the usual geometric action that comes from coadjoint orbits. The other action lives on the phase space that is transverse to the orbits and are called transverse actions, where Yang-Mills theory in two dimensions is an example. Here we show that the transverse action associated with the Virasoro algebra might contain clues for a theory for dark energy. These actions might also suggests a mechanism for symmetry changing.

The horizon and its charges in the first order gravity

In this work the algebra of charges of diffeomorphisms at the horizon of generic black holes is analyzed within first order gravity. This algebra reproduces the algebra of diffeomorphisms at the horizon, (Diff(S^1)), without central extension.

Noether-charge realization of diffeomorphism algebra

It is shown in the covariant phase space formalism that the Noether charges with respect to the diffeomorphism generated by vector fields and their horizontal variations in general relativity form a diffeomorphism algebra. It is also shown with the help of the null tetrad which is well defined everywhere that the central term of the reduced diffeomorphism algebra on the Killing horizon for a large class of vector fields vanishes.

STOCHASTIC PROCESSES ON GROUPS OF DIFFEOMORPHISMS AND VISCOUS HYDRODYNAMICS

The viscous hydrodynamics is investigated via the studying diffusion processes on groups of Hs Sobolev diffeomorphisms of a flat n-dimensional torus (s > ½n + 1). A certain stochastic perturbation of the curve on the above groups, describing the motion of perfect incompressible fluid (or of the diffuse matter), is constructed such that it satisfies a certain stochastic analogue of the geodesic equation and the expectation of its "backward velocity" in the tangent space at identical diffeomorphism (algebra of the group) is a solution of Navier–Stokes (or Burgers, respectively) equations. In particular, the existence of solutions of those equations for lower s is proved. Some other approaches to stochastic presentation of viscous hydrodynamics, using the groups of diffeomorphisms, are also discussed.

Master canonical action and BRST charge of the M-theory bosonic five-brane

A complete analysis of the canonical structure for a gauge fixed PST bosonic five-brane action is performed. This canonical formulation is quadratic in the dependence on the antisymmetric field and it has second class constraints. We remove the second class constraints and a master canonical action with only first class constraints is proposed. The nilpotent BRST charge and its BRST invariant effective theory is constructed. The construction does not assume the existence of the inverse of the induced metric. Singular configurations are then physical ones. We obtain the physical Hamiltonian of the theory and analyze its stability properties. Finally, by studying the algebra of diffeomorphisms we find under mild assumptions the general structure for the Hamiltonian constraint for theories invariant under six-dimensional diffeomorphisms and we give an algebraic characterization of the constraint associated with the bosonic five-brane action. We also identify the constraint for the bosonic five-brane action upgraded with a cosmological term, it contains a Born–Infeld type term.