We review the dual transformation from pure lattice gauge theory to spin foammodels with an emphasis on a geometric viewpoint. This allows us to give asimple dual formulation of SU(N) Yang-Mills theory, where spin foam surfacesare weighted with the exponentiated area. In the case of gravity, we introducea symmetry condition which demands that the amplitude of an individual spinfoam depends only on its geometric properties and not on the lattice on whichit is defined. For models that have this property, we define a new sum overabstract spin foams that is independent of any choice of lattice ortriangulation. We show that a version of the Barrett-Crane model satisfies oursymmetry requirement.
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