Group actions and non-linear sigma models

Abstract

The Yang-Mills sector of the non-linear sigma models is considered in D dimensions and its symmetries are studied. It is found that a diffeomorphism ƒ of the sigma model manifold M is a symmetry provided that it can be lifted to a map ƒ ↑ on the principal bundle P associated with the theory and the corresponding connection of P is an invariant Lie-algebra valued one-form with respect to the lifted map. If the group G of transformations of M is compact, connected and the action of G can be lifted to P, it is shown that there is at least one invariant connection. The lifting of the group actions of M to P are studied for U (1) and trivial principal bundles, and for sigma model manifolds which are homogeneous spaces. The number of independent parameters of sigma models on G/H spaces is also examined using Lie-algebra cohomology with applications in the (1, 0) supersymmetric sigma models in two dimensions. Finally, it is shown that there are topological obstructions to gauging the rigid symmetries of a generic bosonic non-linear sigma model.