COMBINATORIAL INVARIANTS FROM FOUR-DIMENSIONAL LATTICE MODELS

Abstract

We study the subdivision properties of certain lattice gauge theories based on the discrete Abelian group Zp, in four dimensions. The Boltzmann weights are shown to be invariant under all type (k, l) subdivision moves, at certain discrete values of the coupling parameter. The partition function then provides a combinatorial invariant of closed fourmanifolds; however, we further prove that the invariant is unity for all manifolds in this class. Attention is then focused on a bounding three-manifold, and we show that type 2 and type 3 Alexander subdivision of a bounding simplex is equivalent to the insertion of an operator which is equal to a delta function on trivial bounding holonomies. The original four-dimensional model then gives rise to a topological model on its boundary.