Invariant Effective Actions and Cohomology

We review the correspondence between effective actions resulting from non-invariant Lagrangian densities, for Goldstone bosons arising from spontaneous breakdown of a symmetry group G to a subgroup H, and non-trivial generators of the de Rham cohomology of G/H. We summarize the construction of cohomology generators in terms of symmetric tensors with certain invariance and vanishing properties with respect to G and H. The resulting actions in four dimensions arise either from products of generators of lower degree such as the Goldstone-Wilczek current, or are of the Wess-Zumino-Witten type. Actions in three dimensions arise as Chern-Simons terms evaluated on composite gauge fields and may induce fractional spin on solitons. {Contribution to the Proceedings of STRINGS 95, held at University of Southern California, March 13 - 18, 1995.}