Exactly soluble diffeomorphism invariant theories

A class of diffeomorphism invariant theories is described for which the Hilbert space of quantum states can be explicitly constructed. These theories can be formulated in any dimension and include Witten's solution to 2+1 dimensional gravity as a special case. Higher dimensional generalizations exist which start with an action similar to the Einstein action inn dimensions. Many of these theories do not involve a spacetime metric and provide examples of topological quantum field theories. One is a version of Yang-Mills theory in which the only quantum states onS ³×R are the vacua. Finally it is shown that the three dimensional Chern-Simons theory (which Witten has shown is intimately connected with knot theory) arises naturally from a four dimensional topological gauge theory.On leave from the Department of Physics, University of California, Santa Barbara, CA, USA