Cartan Equations Research Articles

Two-Dimensional Noncommutative Analysis, Maurer–Cartan Equation and Riemann–Hilbert Problem

In this paper a new analytic method is given, in which the quantum group theory and the classical analysis are fused into a noncommutative analysis. In this noncommutative analysis the Maurer–Cartan equation on the quantum plane can be solved. The classical Riemann–Hilbert problem can be extended to the noncommutative analysis, and by using this Riemann–Hilbert problem, some nontrivial new solutions of the quantum Maurer–Cartan equation can be generated.

The Extended Maurer-Cartan Equations for the BRST Operators on Coadjoint Orbits

Algebraic and geometric features of the BRST operators are studied on coadjoint orbits of some infinite dimensional Lie groups. An extended form of the Maurer·Cartan equation for the BRST operator Q is derived and it is found that the equation plays a fundamental role in field theories constructed on infinite·dimensional group manifolds. ' Recently a powerful geometric approach to D=2 conformal field theories has been proposed. It is the method of coadjoint orbits of infinite-dimensional Lie groups for derivation of geometric D=2 field theory actions.l),2) Surprisingly enough, using the standard Kirillov-Kostant symplectic structure on the coadjoint orbits of the Virasoro group one naturally finds the geometric action of the Polyakov-Liouville gravity, whereas for the Kac-Moody group one obtains the Wess-Zumino-Novikov­ Witten (WZNW) action. This approach has been extended also to the super­ Virasoro and super-Kac-Moody groups.3) According to Alekseev and Shatashvili (AS),l) the symplectic structure can be written in terms of the I-form Y with values in the Lie algebra g. The I-form Y satisfies the equation l )

Maurer–Cartan forms and equations for two‐dimensional superdiffeomorphisms

Explicit expressions for the Maurer–Cartan forms of the superdiffeomorphism group associated to a super‐Riemann surface are presented. As an application to superconformal field theory, these forms are used to evaluate the effective action for the factorized superdiffeomorphism anomaly.

The local embedding problem for optical structures

In this work we examine optical structures and shear-free geometries of rays. Motivated by a paper of Robinson and Trautman and proceeding in analogy with the Chern-Moser construction of the canonical Cartan connection on Cauchy-Riemann manifolds, we obtain special framings adapted to a geometry of rays satisfying a set of equations which are similar to Cartan's equations for Cauchy-Riemann spaces. We show that, given a hypersurface M in the five-dimensional real projective space which is tangent to the vertical distribution of the Hopf fibration, then a shear-free geometry of rays is determined over M. Thereupon, as an application of our construction, we obtain a proof of the following fact: from a local point of view, every real-analytic twisting shear-free geometry of rays can be constructed in this way. We note that the proof of this last assertion can be also obtained by combining a theorem due to Robinson and Trautman and the local embedding theorem for Cauchy-Riemann structures.

Material and spatial gauge theories of solids — I. Gauge constructs, geometry, and kinematics

A gauge-theoretic approach for modeling continuous distributions of defects in solids is undertaken, based on local internal and external symmetries of an appropriate Lagrangian for those materials. The invariance of the Lagrangian under global action of internal (spatial) and external (material) symmetry groups is extended to invariance under local action of both groups of symmetries through an application of the minimal replacement construct of gauge theory. This is achieved by introduction of nontrivial spatial and material gauge connections. Both connections are used for the construction of two sets of Cartan equations of structure with two different types of torsion and curvature (spatial and material). The spatial torsion and curvature are shown to model “microcracks” and “microrotations”, while the material torsion and curvature model dislocations, disclinations, and energy dissipation mechanisms.

The hyperspin structure of unitary groups

A sequence of homogeneous, isotropic, and compact solutions to the empty space hypergravity equations [D. Finkelstein, S. R. Finkelstein, and C. Holm, Phys. Rev. Lett. 59, 1265 (1987); S. R. Finkelstein, Ph.D. thesis, Georgia Institute of Technology, 1987 (unpublished)] is studied. These solutions are hyperspin manifolds constructed from unitary groups UN:=U(N,C) and describe static compact Einstein universes of the Kaluza–Klein type with Finslerian geometry. By taking the universal covering group of UN an N2-dimensional manifold with topology R×SUN is obtained, where the Abelian group R is the time axis and SUN provides a compact homogeneous spatial part. To describe the Finslerian geometry of these manifolds the Cartan–Penrose exterior calculus is extended from spinors to hyperspinors and the Maurer–Cartan equations are applied to obtain the hyperspin structure of UN. Hyperspin geometry seems to be a consistent alternative to the usual Riemannian geometry for spaces with internal dimensions, which deserves further study.

The geometry of minimal replacement for the Poincaré group

The constructs of this paper rest on two elementary facts: (1) the Poincaré group P 10 is the maximal group of isometries of Minkowski space-time M 4; (2) P 10 has a faithful matrix representation as a subgroup of GL(5, R) that maps an affine set into itself. Local action of P 10 and Yang-Mills minimal replacement are shown to induce a well-defined minimal replacement operator that maps the tensor algebra over M 4 onto the tensor algebra over a new space-time U 4. The natural frame and coframe fields of M 4 go over into a canonical system of frame and coframe fields of U 4 with both translation and Lorentz-rotation parts. The coframe fields define soldering 1-form fields for U 4 that give rise to the standard geometric quantities through the Cartan equations of structure. This leads to unique determinations of all relevant connection coefficients and the associated 2-forms of curvature and torsion that involve the compensating 1-forms for local action of both the translation and the Lorentz-rotation sectors. The metric tensor of U 4, that is induced by the minimal replacement operator, is shown to satisfy the Ricci lemma; U 4 is necessarily a Riemann-Cartan space. This space admits gauge covariant constant basis fields for the Lie algebra of the Lorentz group and for the Dirac algebra. The induced basis for the Dirac algebra evaluates the images of Dirac operators under minimal replacement, while the induced basis for the Lie algebra of L(4, R) serves to show that the holonomy group of U 4 is the Lorentz group. The minimal replacement operator is extended to include the case of a total gauge group that is the direct product of the Poincaré group and a Lie group of internal symmetries of matter fields. This provides a precise method of lifting any action integral of the matter fields from M 4 up to U 4 so that invariance properties are retained when the total group acts locally. The natural representations afforded by minimal replacement result in curvature being evaluated in terms of first order derivatives of the compensating fields that share many properties in common with the Dirac derivation algebra for spin fields. Direct interpretations of the compensating fields are obtained from the geodesic equations.

Canonical Cartan equations for higher order variational problems

Canonical Cartan equations for higher order variational problems

Pseudoscalar action in a Cartan spacetime

The Einstein-Cartan theory of gravitation is generalized by the addition of the most general action which is bilinear in the antisymmetric part of the connection and linear in the Levi-Civita density and hence contains both scalar and pseudoscalar contributions. For arbitrary-spin fields new tensor and pseudotensor interactions arise while for spin-1/2 fields the theory reduces to Cartan's equations, apart from a reduction in the spin-spin coupling constant.