2,618 research outputs found
Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets
In this paper we derive the most general first-order symmetry operator
commuting with the Dirac operator in all dimensions and signatures. Such an
operator splits into Clifford even and Clifford odd parts which are given in
terms of odd Killing-Yano and even closed conformal Killing-Yano inhomogeneous
forms respectively. We study commutators of these symmetry operators and give
necessary and sufficient conditions under which they remain of the first-order.
In this specific setting we can introduce a Killing-Yano bracket, a bilinear
operation acting on odd Killing-Yano and even closed conformal Killing-Yano
forms, and demonstrate that it is closely related to the Schouten-Nijenhuis
bracket. An important non-trivial example of vanishing Killing-Yano brackets is
given by Dirac symmetry operators generated from the principal conformal
Killing-Yano tensor [hep-th/0612029]. We show that among these operators one
can find a complete subset of mutually commuting operators. These operators
underlie separability of the Dirac equation in Kerr-NUT-(A)dS spacetimes in all
dimensions [arXiv:0711.0078].Comment: 37 pages, no figure
Algebraic theories of brackets and related (co)homologies
A general theory of the Frolicher-Nijenhuis and Schouten-Nijenhuis brackets
in the category of modules over a commutative algebra is described. Some
related structures and (co)homology invariants are discussed, as well as
applications to geometry.Comment: 14 pages; v2: minor correction
Do Killing-Yano tensors form a Lie Algebra?
Killing-Yano tensors are natural generalizations of Killing vectors. We
investigate whether Killing-Yano tensors form a graded Lie algebra with respect
to the Schouten-Nijenhuis bracket. We find that this proposition does not hold
in general, but that it does hold for constant curvature spacetimes. We also
show that Minkowski and (anti)-deSitter spacetimes have the maximal number of
Killing-Yano tensors of each rank and that the algebras of these tensors under
the SN bracket are relatively simple extensions of the Poincare and (A)dS
symmetry algebras.Comment: 17 page
Novel Topological Invariant in the U(1) Gauge Field Theory
Based on the decomposition of U(1) gauge potential theory and the
-mapping topological current theory, the three-dimensional knot invariant
and a four-dimensional new topological invariant are discussed in the U(1)
gauge field.Comment: 10 pages, 0 figures accepted by MPL
The constitutive tensor of linear elasticity: its decompositions, Cauchy relations, null Lagrangians, and wave propagation
In linear anisotropic elasticity, the elastic properties of a medium are
described by the fourth rank elasticity tensor C. The decomposition of C into a
partially symmetric tensor M and a partially antisymmetric tensors N is often
used in the literature. An alternative, less well-known decomposition, into the
completely symmetric part S of C plus the reminder A, turns out to be
irreducible under the 3-dimensional general linear group. We show that the
SA-decomposition is unique, irreducible, and preserves the symmetries of the
elasticity tensor. The MN-decomposition fails to have these desirable
properties and is such inferior from a physical point of view. Various
applications of the SA-decomposition are discussed: the Cauchy relations
(vanishing of A), the non-existence of elastic null Lagrangians, the
decomposition of the elastic energy and of the acoustic wave propagation. The
acoustic or Christoffel tensor is split in a Cauchy and a non-Cauchy part. The
Cauchy part governs the longitudinal wave propagation. We provide explicit
examples of the effectiveness of the SA-decomposition. A complete class of
anisotropic media is proposed that allows pure polarizations in arbitrary
directions, similarly as in an isotropic medium.Comment: 1 figur
Is the Quantum Hall Effect influenced by the gravitational field?
Most of the experiments on the quantum Hall effect (QHE) were made at
approximately the same height above sea level. A future international
comparison will determine whether the gravitational field
influences the QHE. In the realm of (1 + 2)-dimensional phenomenological
macroscopic electrodynamics, the Ohm-Hall law is metric independent
(`topological'). This suggests that it does not couple to . We
corroborate this result by a microscopic calculation of the Hall conductance in
the presence of a post-Newtonian gravitational field.Comment: 4 page
Covariants,joint invariants and the problem of equivalence in the invariant theory of Killing tensors defined in pseudo-Riemannian spaces of constant curvature
The invariant theory of Killing tensors (ITKT) is extended by introducing the
new concepts of covariants and joint invariants of (product) vector spaces of
Killing tensors defined in pseudo-Riemannian spaces of constant curvature. The
covariants are employed to solve the problem of classification of the
orthogonal coordinate webs generated by non-trivial Killing tensors of valence
two defined in the Euclidean and Minkowski planes. Illustrative examples are
provided.Comment: 60 pages. to appear in J. Math. Phy
A Unified Algebraic Approach to Classical Yang-Baxter Equation
In this paper, the different operator forms of classical Yang-Baxter equation
are given in the tensor expression through a unified algebraic method. It is
closely related to left-symmetric algebras which play an important role in many
fields in mathematics and mathematical physics. By studying the relations
between left-symmetric algebras and classical Yang-Baxter equation, we can
construct left-symmetric algebras from certain classical r-matrices and
conversely, there is a natural classical r-matrix constructed from a
left-symmetric algebra which corresponds to a parak\"ahler structure in
geometry. Moreover, the former in a special case gives an algebraic
interpretation of the ``left-symmetry'' as a Lie bracket ``left-twisted'' by a
classical r-matrix.Comment: To appear in Journal of Physics A: Mathematical and Theoretica
Antisymmetric tensor coupling and conformal invariance in sigma models corresponding to gauged WZNW theories
String backgrounds associated with gauged WZNW models generically
depend on or . The exact expressions for the corresponding
metric G_{\m\n}, antisymmetric tensor B_{\m\n}, and dilaton can be
obtained by eliminating the gauge field from the local part of the
effective action of the gauged WZNW model. We show that there exists a
manifestly gauge-invariant prescription for the derivation of the antisymmetric
tensor coupling. When the subgroup is one-dimensional and is simple the
antisymmetric tensor is given by the semiclassical (-independent)
expression. We consider in detail the simplest non-trivial example with
non-trivial B_{\m\n} -- the D=3 sigma model corresponding to the gauged WZNW theory (`charged black string') and show that the exact
expressions for G_{\m\n}, B_{\m\n} and solve the Weyl invariance
conditions in the two-loop approximation. Similar conclusion is reached for the
closely related chiral gauged WZNW model. We find that there exists
a scheme in which the semiclassical background is also a solution of the
two-loop conformal invariance equations (but the tachyon equation takes a
non-canonical form). We discuss in detail the role of field redefinitions
(scheme dependence) in establishing a correspondence between the sigma model
and conformal field theory results.Comment: 55 pages, harvmac, CERN-TH.6969/93, THU-93/25, Imperial/TP/92-93/59.
(Another prescription for extracting the exact antisymmetric tensor is
described leading to a purely semiclassical expression for it
A covariant formalism for Chern-Simons gravity
Chern--Simons type Lagrangians in dimensions are analyzed from the
point of view of their covariance and globality. We use the transgression
formula to find out a new fully covariant and global Lagrangian for
Chern--Simons gravity: the price for establishing globality is hidden in a
bimetric (or biconnection) structure. Such a formulation allows to calculate
from a global and simpler viewpoint the energy-momentum complex and the
superpotential both for Yang--Mills and gravitational examples.Comment: 12 pages,LaTeX, to appear in Journal of Physics
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