SEARCH · Searcharxiv

Results for “dg-ga”

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 667 records · Page 37Linked to original sources

arXiv1996

Quantization of symplectic tori in a real polarization

We apply the geometric quantization method with real polarizations to the quantization of a symplectic torus. By quantizing with half-densities we canonically associate to the symplectic torus a projective Hilbert space and prove that the projective factor is expressible in terms of the Maslov-Kashiwara index. As in the quantization of a linear symplectic space, we have two ways of resolving the projective ambiguity: (i) by introducing a metaplectic structure and using half-forms in the definition of the Hilbert space; (ii) by choosing a 4-fold cover of the Lagrangian Grassmannian of the linear symplectic space covering the torus. We show that the Hilbert space constructed through either of these approaches realizes a unitary representation of the integer metaplectic group.

dg-ga↗

arXiv1996

Polyhedral representations of discrete differential manifolds

Any discrete differential manifold $M$ (finite set endowed with an algebraic differential calculus) can be represented by appropriate polyhedron ${\cal P}(M)$. This representation demonstrates the adequacy of the calculus of discrete differential manifolds and links this approach with that based on finitary substitutes of continuous spaces introduced by R.D.Sorkin.

dg-ga↗

arXiv1997

Space-time Averages in Macroscopic Gravity and Volume-preserving Coordinates

The definition of the covariant space-time averaging scheme for the objects (tensors, geometric objects, etc.) on differentiable metric manifolds with a volume n-form, which has been proposed for the formulation of macroscopic gravity, is analyzed. An overview of the space-time averaging procedure in Minkowski spacetime is given and comparison between this averaging scheme and that adopted in macroscopic gravity is carried out throughout the paper. Some new results concerning the algebraic structure of the averaging operator are precisely formulated and proved, the main one being that the averaging bilocal operator is idempotent iff it is factorized into a bilocal product of a matrix-valued function on the manifold, taken at a point, by its inverse at another point. The previously proved existence theorems for the averaging and coordination bilocal operators are revisited with more detailed proofs of related results. A number of new results concerning the structure of the volume-preserving averaging operators and the class of proper coordinate systems are given. It is shown, in particular, that such operators are defined on an arbitrary n-dimensional differentiable metric manifold with a volume n-form up to the freedom of (n-1) arbitrary functions of n arguments and 1 arbitrary function of (n-1) arguments. All the results given in this paper are also valid whenever appropriate for affine connection manifolds including (pseudo)-Riemannian manifolds.

dg-ga↗

arXiv1997

On 3+1 decompositions with respect to an observer field via differential forms

3+1 decompositions of differential forms on a Lorentzian manifold (M,g;+ - - -) with respect to arbitrary observer field and the decomposition of the standard operations acting on them are studied, making use of the ideas of the theory of connections on principal bundles. Simple explicit general formulas are given as well as their application to the Maxwell equations.

gr-qc↗

arXiv1997

Yang-Mills Flow and Uniformization Theorems

We consider a parabolic-like systems of differential equations involving geometrical quantities to examine uniformization theorems for two- and three-dimensional closed orientable manifolds. We find that in the two-dimensional case there is a simple gauge theoretic flow for a connection built from a Riemannian structure, and that the convergence of the flow to the fixed points is consistent with the Poincare Uniformization Theorem. We construct a similar system for the three-dimensional case. Here the connection is built from a Riemannian geometry, an SO(3) connection and two other 1-form fields which take their values in the SO(3) algebra. The flat connections include the eight homogeneous geometries relevant to the three-dimensional uniformization theorem conjectured by W. Thurston. The fixed points of the flow include, besides the flat connections (and their local deformations), non-flat solutions of the Yang-Mills equations. These latter "instanton" configurations may be relevant to the fact that generic 3-manifolds do not admit one of the homogeneous geometries, but may be decomposed into "simple 3-manifolds" which do.

hep-th↗

arXiv1996

Abelian Chern-Simons theory

We give a construction of the abelian Chern-Simons gauge theory from the point of view of a 2+1 dimensional topological quantum field theory. The definition of the quantum theory relies on geometric quantization ideas which have been previously explored in connection to the nonabelian Chern-Simons theory [JW,ADW]. We formulate the topological quantum field theory in terms of the category of extended 2- and 3-manifolds introduced by Walker [Wa] and prove that it satisfies the axioms of unitary topological quantum field theories formulated by Atiyah [A1].

dg-ga↗

arXiv1997

BRST Symmetries for the Tangent Gauge Group

For any principal bundle $P$, one can consider the subspace of the space of connections on its tangent bundle $TP$ given by the tangent bundle $T{\cal A}$ of the space of connections ${\cal A}$ on $P$. The tangent gauge group acts freely on $T{\cal A}$. Appropriate BRST operators are introduced for quantum field theories that include as fields elements of $T{\cal A}$, as well as tangent vectors to the space of curvatures. As the simplest application, the BRST symmetry of the so-called $BF$-Yang-Mills theory is described and the relevant gauge fixing conditions are analyzed. A brief account on the topological $BF$ theories is also included and the relevant Batalin-Vilkovisky operator is described.

hep-th↗

arXiv1998

Multivector Fields and Connections. Setting Lagrangian Equations in Field Theories

The integrability of multivector fields in a differentiable manifold is studied. Then, given a jet bundle $J^1E\to E\to M$, it is shown that integrable multivector fields in $E$ are equivalent to integrable connections in the bundle $E\to M$ (that is, integrable jet fields in $J^1E$). This result is applied to the particular case of multivector fields in the manifold $J^1E$ and connections in the bundle $J^1E\to M$ (that is, jet fields in the repeated jet bundle $J^1J^1E$), in order to characterize integrable multivector fields and connections whose integral manifolds are canonical lifting of sections. These results allow us to set the Lagrangian evolution equations for first-order classical field theories in three equivalent geometrical ways (in a form similar to that in which the Lagrangian dynamical equations of non-autonomous mechanical systems are usually given). Then, using multivector fields; we discuss several aspects of these evolution equations (both for the regular and singular cases); namely: the existence and non-uniqueness of solutions, the integrability problem and Noether's theorem; giving insights into the differences between mechanics and field theories.

dg-ga↗

arXiv1996

Almost Complex and Almost Product Einstein Manifolds from a Variational Principle

It is shown that the first order (Palatini) variational principle for a generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almost-product Einstein structure or to an almost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-Hermitian metric on a complex manifold satisfies the Kähler condition on the same manifold treated as a real manifold if and only if the metric is the real part of a holomorphic metric. A characterisation of anti-Kähler Einstein manifolds and almost-product Einstein manifolds is obtained. Examples of such manifolds are considered.

dg-ga↗

arXiv2004

N-Black Hole Stationary and Axially Symmetric Solutions of the Einstein-Maxwell Equations

The Einstein/Maxwell equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities phi: R^3Σ-> H^2_C, where Sigma is a subset of the axis of symmetry, and H^2_C is the complex hyperbolic plane. Motivated by this problem, we prove the existence and uniqueness of harmonic maps with prescribed singularities phi: R^nΣ-> H, where Sigma is a submanifold of R^n of co-dimension at least 2, and H is a classical Riemannian globally symmetric space of noncompact type and rank one. This result, when applied to the black hole problem, yields solutions which can be interpreted as equilibrium configurations of multiple co-axially rotating charged black holes held apart by singular struts.

gr-qc↗

arXiv1995

Existence of maximal hypersurfaces in some spherically symmetric spacetimes

We prove that the maximal development of any spherically symmetric spacetime with collisionless matter (obeying the Vlasov equation) or a massless scalar field (obeying the massless wave equation) and possessing a constant mean curvature $S^1 \times S^2$ Cauchy surface also contains a maximal Cauchy surface. Combining this with previous results establishes that the spacetime can be foliated by constant mean curvature Cauchy surfaces with the mean curvature taking on all real values, thereby showing that these spacetimes satisfy the closed-universe recollapse conjecture. A key element of the proof, of interest in itself, is a bound for the volume of any Cauchy surface $Σ$ in any spacetime satisfying the timelike convergence condition in terms of the volume and mean curvature of a fixed Cauchy surface $Σ_0$ and the maximal distance between $Σ$ and $Σ_0$. In particular, this shows that any globally hyperbolic spacetime having a finite lifetime and obeying the timelike-convergence condition cannot attain an arbitrarily large spatial volume.

gr-qc↗

arXiv1997

The Cosmological Time Function

Let $(M,g)$ be a time oriented Lorentzian manifold and $d$ the Lorentzian distance on $M$. The function $τ(q):=\sup_{p< q} d(p,q)$ is the cosmological time function of $M$, where as usual $p< q$ means that $p$ is in the causal past of $q$. This function is called regular iff $τ(q) < \infty$ for all $q$ and also $τ\to 0$ along every past inextendible causal curve. If the cosmological time function $τ$ of a space time $(M,g)$ is regular it has several pleasant consequences: (1) It forces $(M,g)$ to be globally hyperbolic, (2) every point of $(M,g)$ can be connected to the initial singularity by a rest curve (i.e., a timelike geodesic ray that maximizes the distance to the singularity), (3) the function $τ$ is a time function in the usual sense, in particular (4) $τ$ is continuous, in fact locally Lipschitz and the second derivatives of $τ$ exist almost everywhere.

gr-qc↗

arXiv1995

Non-standard connections in classical mechanics

In the jet-bundle description of first-order classical field theories there are some elements, such as the lagrangian energy and the construction of the hamiltonian formalism, which require the prior choice of a connection. Bearing these facts in mind, we analyze the situation in the jet-bundle description of time-dependent classical mechanics. So we prove that this connection-dependence also occurs in this case, although it is usually hidden by the use of the ``natural'' connection given by the trivial bundle structure of the phase spaces in consideration. However, we also prove that this dependence is dynamically irrelevant, except where the dynamical variation of the energy is concerned. In addition, the relationship between first integrals and connections is shown for a large enough class of lagrangians.

dg-ga↗

arXiv1995

Constant mean curvature surfaces via integrable dynamical system

It is shown that the equation which describes constant mean curvature surface via the generalized Weierstrass-Enneper inducing has Hamiltonian form. Its simplest finite-dimensional reduction has two degrees of freedom, integrable and its trajectories correspond to well-known Delaunay and do Carmo-Dajzcer surfaces (i.e., helicoidal constant mean curvature surfaces).

dg-ga↗

arXiv1996

New Generalized Poisson Structures

New generalized Poisson structures are introduced by using suitable skew-symmetric contravariant tensors of even order. The corresponding `Jacobi identities' are provided by conditions on these tensors, which may be understood as cocycle conditions. As an example, we provide the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras.

q-alg↗

arXiv1997

On the generalizations of Poisson structures

The characterization of the Nambu-Poisson n-tensors as a subfamily of the Generalized-Poisson ones recently introduced (and here extended to the odd order case) is discussed. The homology and cohomology complexes of both structures are compared, and some physical considerations are made.

hep-th↗

arXiv1995

Symetric Monopoles

We discuss $SU(2)$ Bogomolny monopoles of arbitrary charge $k$ invariant under various symmetry groups. The analysis is largely in terms of the spectral curves, the rational maps, and the Nahm equations associated with monopoles. We consider monopoles invariant under inversion in a plane, monopoles with cyclic symmetry, and monopoles having the symmetry of a regular solid. We introduce the notion of a strongly centred monopole and show that the space of such monopoles is a geodesic submanifold of the monopole moduli space. By solving Nahm's equations we prove the existence of a tetrahedrally symmetric monopole of charge $3$ and an octahedrally symmetric monopole of charge $4$, and determine their spectral curves. Using the geodesic approximation to analyse the scattering of monopoles with cyclic symmetry, we discover a novel type of non-planar $k$-monopole scattering process.

dg-ga↗

arXiv1994

A ``stable'' version of the Gromov-Lawson conjecture

We discuss a conjecture of Gromov and Lawson, later modified by Rosenberg, concerning the existence of metrics of positive scalar curvature. It says that a closed spin manifold $M$ of dimension $n\ge 5$ has such a metric if and only if the index of a suitable ``Dirac" operator in $KO_n(C^* (π_1(M)))$, the real $K$-theory of the group $C^*$-algebra of the fundamental group of $M$, vanishes. It is known that the vanishing of the index is necessary for existence of a positive scalar curvature metric on $M$, but this is known to be a sufficient condition only if $π_1(M)$ is the trivial group, $\Bbb Z/2$, an odd order cyclic group, or one of a fairly small class of torsion-free groups. \par We note that the groups $KO_n(C^*(π))$ are periodic in $n$ with period $8$, whereas there is no obvious periodicity in the original geometric problem. This leads us to introduce a ``stable'' version of the Gromov-Lawson conjecture, which makes the weaker statement that the product of $M$ with enough copies of the ``Bott manifold" $B$ has a positive scalar curvature metric if and only if the index of the Dirac operator on $M$ vanishes. (Here $B$ is a simply connected $8$-manifold which represents the periodicity element in $KO_8(pt)$.) We prove the stable Gromov-Lawson conjecture for all spin manifolds with finite fundamental group and for many spin manifolds with infinite fundamental group.

dg-ga↗