Lattice topological field theory in two dimensions

Abstract

The lattice definition of a two-dimensional topological field theory (TFT) is given generically, and the exact solution is obtained explicitly. In particular, the set of all lattice topological field theories is shown to be in one-to-one correspondence with the set of all associative algebras R, and the physical Hilbert space is identified with the center Z(R) of the associative algebra R. Perturbations of TFT's are also considered in this approach, showing that the form of topological perturbations is automatically determined, and that all TFT's are obtained from one TFT by such perturbations. Several examples are presented, including twisted N = 2 minimal topological matter and the case where R is a group ring. *

Quandle cohomology and state-sum invariants of knotted curves and surfaces

“…Let G be a finite group, and assign elements of G on the edges, such that if the two edges of the same directions receive x and y in this direction, then the other edge receives xy. The value α(x, y) of a 2-cocycle α is assigned to such a triangle [9,16] (see also [5]). With this convention, two ways of triangulating a square correspond to the 2-cocycle condition as depicted in Figure 9. 7.1.…”

Lattice topological field theory and first order phase transition

“…The condition (1) guarantees the invariance of the partition function (2) under the lation of the surface [1,4,13,14] .…”

“…And we assign g ij to each edge and connect the tensors on the two triangles which have the common edge. Then we will obtain a numerical value for the triangulated surface and call it the partition function [1,4,13,14] of the surface:…”

“…where the operator P ijk (x) denotes the puncture operator [14] of the triangle at x with indices i, j, k on its edges, and the one-point function P ijk (x) g is that of the original lattice topological field theory on the surface with genus g. Because of the topological nature, the one point function P ijk (x) g is independent of the position x, and hence the summation over the position x can be trivially done. Moreover all the puncture operators P ijk are not linearly independent in the correlation functions, and they are equal to certain linear combinations of the physical puncture [14] .…”

“…The calculation rules in the two-dimensional lattice topological field theories are very simple in the standard basis [14] . The physical puncture operators satisfy the following rules :…”