State-Sum TQFTs

Abstract

Abstract Manifolds can be decomposed into simplicies (triangles in 2D or tetrahedra in 3D). TQFTs can be constructed as state sums over discrete quantum numbers on the on these discretized manifolds. Such sums appear like partition functions, statistical mechanics sums of Boltzmann weights. In order to for these sums to yield manifold invariants the sum must be independent on the particular simplicial decomposition (or triangulation) of the manifold. The so-called “Pachner Moves” describe all possible changes of the decomposition, so a sum which is unchanged under Pachner moves gives a manifold invariant. One such state sum is the the Turaev-Viro state sum, which takes as an input a (spherical) fusion category (F matrices satisfying the pentagon equation) and then allows one to assign a (scalar) manifold invariant to a 3D manifold. The corresponding TQFT is known as the quantum double, or Drinfeld double of the fusion category. Much of the study of the Turaev-Viro model has been in the context of so-called spin-network models of quantum gravity. A very similar state sum TQFT is the Dijkgraaf-Witten model, which (in 3D) takes as an input a group and a 3-cocyle (acting as F-matrices). The Dijkgraaf-Witten model is generalizable to any dimension.