SL(2,C)-gauge theory andN=1 supergravity

It is shown that as Riemannian space may be taken to give rise to a Poincaré gauge theory of gravitation, the superspace where the coordinates are given by (X, ), being a spinorial variable gives rise to anSL(2, C)-gauge theory and corresponds toN= 1 supergravity. It leads to a conserved current and the conserved quantity here corresponds to isospin, where the latter is taken to be generated from conformal reflection. Thus, supergravity plays a predominant role in the microlocal space-time.     

SL(2,C) gravitational conserved current and Noether's theorem

Noether's theorem is applied to Hilbert's Lagrangian written as a functional of spinorial variables. The associatedSL(2,C) conserved current is obtained, and its expression for the Tolman metric is given explicitly.     

SL(2C) invariance of the gravitational field

The gravitational field equations of general relativity theory are cast into a Yang-Mills-type theory by use of the group SL(2,C). The spin coefficients take the rôle of the Yang-Mills-like potentials, whereas the Riemann tensor takes the rôle of the fields. Comparison of this formalism with that of Utiyama and Kibble who related invariance under the Lorentz and the Poincaré groups to the existence of the gravitational field, is discussedBased on a lecture given at the International Conference on Relativity and Gravitation, Copenhagen, July 1971.     

SL(2,C) gauge theory of gravitation: Conservation laws

Conservation laws in the SL(2,C) gauge theory of gravitation are reviewed and their relation to the ordinary conservation laws in general relativity theory is discussed. The vector currents that were proposed by different authors along the lines of the Yang-Mills conserved vector current are discussed and their interrelation is given. Likewise Lagrangian densitiies, from which one obtains the SL(2,C) gauge theory gravitational field equations, are discussed and related to the conservation laws through Noether's theorem.     

SL(2,C)-gauge theory,N=1 supergravity and torsion

It is shown that the SL(2,C)-gauge theory of gravitation may be considered to correspond toN = 1 supergravity and the conserved current gives rise to the Einstein-Cartan action. The torsion term here appears due to the spinorial variable, which is associated with the internal helicity giving rise to the isospin algebra from the conformal reflection group. In this sense, the internal symmetry of hadrons is found to take a dominant role in gravitational phenomena in the microlocal space-time region where the Einstein-Cartan action becomes significant.     

Solitons and SL(2, R)

The known relationship between non-linear partial differential equations which have soliton solutions, and SL (2, R), is developed to the point where it provides a framework for discussing Bäcklund transformations, and equations for the inverse scattering method.     

Analysis of the basic matrix representation ofGL q(2,C)

We investigate the quantum deformation of the group of 2×2 matrices. We show that then-th power of a quantum matrix corresponds to then-th power of the deformation parameter. We also prove that a quantum matrix can be expressed as the exponential of a matrix with suitable non-commuting matrix elements.     

SL(2,C) Gauge Theory of Gravitation and the Quantization of the Gravitational Field

A new approach to quantize the gavitationalfield is presented. It is based on the observation thatthe quantum character of matter becomes more significantas one gets closer to the big bang. As the metric loses its meaning, it makes sense to considerSchrodinger's three generic types of manifolds —unconnected differentiable, affinely connected, andmetrically connected — as a temporal sequencefollowing the big bang. Hence one should quantize thegravitational field on general differentiable manifoldsor on affinely connected manifolds. The SL(2,C) gaugetheory of gravitation is employed to explore thispossibility. Within this framework, the quantization itselfmay well be canonical.     

Star products for SL(2)

The deformation program (the use of star products in harmonic analysis) leads to the definition of an adapted Fourier transform, unitary transformation between spaces of square integrable functions of the group G and on the dual of its Lie algebra, describing the unitary dual of G and its Plancherel transform. This Letter is an application of this program to the universal covering group of SL(2).     

Tensor representation of the quantum groupSL q (2,C) and quantum Minkowski space

We investigate the structure of the tensor product representation of the quantum groupSL _q (2,C) by using the 2-dimensional quantum plane as a building block. Two types of 4-dimensional spaces are constructed applying the methods used in twistor theory. We show that the 4-dimensional real representation ofSL _q (2,C) generates a consistent non-commutative algebra, and thus it provides a quantum deformation of Minkowski space. The transformation of this 4-dimensional space gives the quantum Lorentz groupSO _q(3, 1).     

SO(n,1) Wigner rotation as an SL(2,R) problem

We show that the composition of not only two SO(3,1) boosts, but also that of two SO(n,1) boosts for anyn 2, is basically an SO(2,1) problem and hence can be analysed completely using SL(2,R) matrices. By computing the expression for the Thomas/Wigner angle directly using SL(2,R) matrices we show that this approach results in considerable economy of algebra.     

Unitarity of SL(2)-conformal blocks in genus zero

It has been conjectured for quite some time that a bundle of conformal blocks carries a unitary structure that is (projectively) flat for the Hitchin connection. This was recently established by T.R. Ramadas in the simplest nontrivial case, namely where the genus is zero and the group is SL(2). In this paper we present a shorter and more direct version of his proof. We also identify the conformal block space with the bidegree (N$N$, 0)-part of an eigenspace of a finite group acting on a Hodge structure of weight N$N$.