Louis Marchildon

The formalism of spontaneous symmetry breaking in gauge theories and the theory of nonlinear invariant Lagrangians are reviewed, with an emphasis on the relationships between the two. Having in mind building Lagrangian models which ex­hibit nonlinear invariance under supergroups, we first para­metrize general elements of the supergroups of interest. The graded conformal group SU(2,2/1) is introduced as a set of transformations leaving a given bilinear form invariant, and it is shown that a general element of SU(2,2/1) can be written as the product of an odd transformation (i.e., one that changes Bose into Fermi coordinates) times an even one. Two subgroups of SU(2,2/1) are then extracted, namely, the graded Poincare group GP and the graded de Sitter group OSp(1/4). Having characterized these groups, we consider small-dimensional lin­ ear representations of SU(2,2/1) and OSp(1/4), and the spaces of some of these representations are endowed with the struc­ture of a Jordan or a wedge algebra. Then we turn to nonlin­ear representations, defined on various coset spaces of these groups with respect to some of their subgroups. The explicit finite transformation laws of the parameters of each coset space under the action of the full group are derived, and in a special parametrization they have the form of fractional linear transformations. Use is then made of the explicit form of general elements of GP and OSpCl/A) to write down the locally gauge covariant derivatives associated with the nonlinear realizations of these two groups on their coset spaces with respect to the Lorentz group. The finite transformation properties of the covariant derivatives are also obtained from the nonlinear representations constructed previously. The covariant derivatives and so-called modified fields are then used to build Lagrangians which exhibit local nonlinear gauge invariance under GP or 0Sp(1/4). It turns out that the Lagrangian of supergravity can be expressed in terms of these invariants.