We present theories ofgravitation based, respectively, on the general linear group GL(n, R) and its inhomoIgeneous extension IGL(n, R) [SO(n-l, 1) and ISO(n -1, 1) for torsion-free manifolds]. Noting that the geometry of the conventional gauge theories can be described in terms of a fiber bundle, and that their action is a scalar in such a superspace, we construct principal fiber bundles based on the above gauge groups and propose to describe gravitation in terms of their corresponding scalar curvatures. To ensure that these manifolds do indeed have close ties with the space-time of general relativity, we make use of the notion of the parallel transport of vector fields in space-time to uniquely relate the connections in space-time to the gauge potentials in fiber bundles. The relations turn out to be similar to that suggested earlier by Yang. The actions we obtain are related to those of Einstein and Yang but are distinct from both and have an Einstein limit. The inclusion of internal symmetry leads to the analogs of Einstein-Yang-Mills equations. A number of variations and less attractive alternatives based on the above groups or their subgroups are also discussed,
Based on the geometry of local (super-) gauge invariance, a theoretical framework for constructing superunified theories is given. The main ingredients of this approach are (a) a method of constructing invariants associated with unconstrained gauge theories and (b) the concept of "constrained" geometries for the description of gravlty and supergravity as well as the choice of their gauge groups. It is argued that any unified theory must contain gravity, and then, to retain the invariances of pure gravity theory, such a theory must be a superunified one. The formalism is then applied, respectively, to pure gravity, gravity coupled to Yang-Mills fields, simple supergravity, and SO(2)-extended supergravity. Important properties of these theories are discussed in detail.
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