We construct and classify topological lattice field theories in three dimensions. After defining a general class of local lattice field theories, we impose invariance under arbitrary topology-preserving deformations of the underlying lattice, which are generated by two new local lattice moves. Invariant solutions are in one-to-one correspondence with Hopf algebras satisfying a certain constraint. As an example, we study in detail the topological lattice field theory corresponding to the Hopf algebra based on the group ring C[G], and show that it is equivalent to lattice gauge theory at zero coupling, and to the Ponzano-Regge theory for G =SU(2).
The Wess-Zumino-Witten (WZW) theory has a global symmetry denoted by G L ⊗ G R . In the standard gauged WZW theory, vector gauge fields (i.e. with vector gauge couplings) are in the adjoint representation of the subgroup H ⊂ G. In this paper, we show that, in the conformal limit in two dimensions, there is a gauged WZW theory where the gauge fields are chiral and belong to the subgroups H L and H R where H L and H R can be different groups. In the special case where H L = H R , the theory is equivalent to vector gauged WZW theory. For general groups H L and H R , an examination of the correlation functions (or more precisely, conformal blocks) shows that the chiral gauged WZW theory is equivalent to (G/H) L ⊗(G/H) R coset models in conformal field theory. The equivalence of the vector gauged WZW theory and the corresponding G/H coset theory then follows.
Starting from fractional supersymmetry as an extended Virasoro algebra, we generalize Felder's BRST cohomology method to construct the characters and branching functions — in terms of string functions — of the SU(2) and SU(3) coset models, for both the unitary and nonunitary cases.
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