Structures and diagrammatics of four dimensional topological lattice field theories

Abstract

Crane and Frenkel proposed a state sum invariant for triangulated 4-manifolds. They sketched the definition of a Hopf category that was to be used in their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double of a finite group. In this paper we define a state sum invariant of triangulated 4-manifolds using Crane-Yetter cocycles as Boltzmann weights. Our invariant is analogous to the 3-dimensional invariants defined by Dijkgraaf and Witten and the invariants that are defined via Hopf algebras. We present diagrammatic methods for the study of such invariants that illustrate connections between Hopf categories and moves to triangulations. © 1999 Academic Press.