The supersymmetric extension of local Lorentz symmetry in the vierbein formalism is presented in detail. In contrast to supergravity, this new supersymmetry is purely internal, that is, totally irrelevant to general coordinate transformations. Because of its non-Poincaré nature, the degrees of freedom of bosons and fermions need not be equal and it is unnecessary to introduce gravitino. Instead, it is important to introduce a new type of nonlinear realization, called “ξ-field realization”. The Einstein Lagrangian density is trivially superinvariant and the Dirac theory can be supersymmetrized. A globally superinvariant gauge-fixing plus FP-ghost Lagrangian density is constructed in the BRS-invariant way. In the simplest case, canonical quantization is explicitly carried out and unitarity is proved. It is shown that the manifestly covariant canonical formalism of quantum gravity is extended to the supersymmetric form without spoiling its beauty in contrast to supergravity. This supersymmetric theory provides a natural link between quantum gravity and gauge theory; its implies that the chiral gauge group should be SO (N).
Supersymmetric Extension of the Three-Dimensional Local Lorentz Symmetry and the Chern-Simons Term
