Flip-Moves and Graded Associative Algebras

The relation between discrete topological field theories on triangulations of two-dimensional manifolds and associative algebras was worked out recently. The starting point for this development was the graphical interpretation of the associativity as flip of triangles. We show that there is a more general relation between flip-moves with two n-gons and Zn−2-graded associative algebras. A detailed examination shows that flip-invariant models on a lattice of n-gons can be constructed from Z2or Z1-graded algebras, reducing in the second case to triangulations of the two-dimensional manifolds. Related problems occure naturally in three-dimensional topological lattice theories. University of Freiburg July 1994 THEP 94/6 Various aspects of topological lattice theories had been considered in the last years. First models had been constructed as discrete analogies of continuous topological field theories. The invariance of the continuous theorie under the diffeomorphism group was discretised to the invariance under flip moves of the lattice [1], see fig. 1. The field variables were located on the vertices of the triangulation. Another type of models grow out of matrix models of two-dimensional quantum gravity [2], where one wants to couple a topological action to the model to control the topologydependence of the series-expansion. These models have the field variables on the edges of the triangles and could be classified by associative algebras [3, 4]. The approach to topological lattice theories from the matrix models poses the problem to handle ‘ topological’ actions coupled to models which not only contain a cubic but higher polynoms in the potential. This was solved in [4] for monoms of degree