GL(N, R) AS A CANDIDATE FOR FUNDAMENTAL SYMMETRY IN FIELD-THEORY

We formulate generally covariant and GL(n, R)-invariant variational principles for the field of linear co-frames in a n-dimensional manifold. The internal GL(n,R)-symmetry is the main difference between presented models and the metric-teleparallel theories of gravitation (including Einstein theory), invariant under the global Lorentz group O(1, n - 1) (local in Einstein theory). There is a structural similarity between our Lagrangians and those used in Born-Infeld electrodynamics. We discuss field equations and Lagrangian constraints. Field equations are nonempty because the existence of certain <<vacuumlike>> solutions is explicitly shown. These solutions are isomorphic to canonical forms of semi-simple Lie groups (just as in certain field-theoretic models studied by Toller, D'Adda, Nelson and Regge), or to appropriately deformed canonical forms of trivial central extensions of semi-simple Lie groups. The latter applies, e.g., to four-dimensional space-times. The normal-hyperbolic signature is not a priori assumed, however, if n = 4, it turns out to be a property of the most natural solutions; roughly speaking, it is implied by differential equations. On the other hand, if we assume that the signature is to be normal-hyperbolic, then n = 4 is the lowest nontrivial dimension. We present certain arguments in favour of the hypothesis that GL(n, R) is a promising candidate for the symmetry group of generally covariant fundamental field theories. Our GL(n, R)-invariant Lagrangians for frames provide a first step towards developments in this direction.