New topological theories and conjugacy classes of the Weyl group

The problem of interpreting a set of ${\cal W}$-algebra constraints constructed in terms of an arbitrarily twisted scalar field as the recursion relations of a topological theory is addressed. In this picture, the conventional models of topological gravity coupled to $A$, $D$ or $E$ topological matter, correspond to taking the scalar field twisted by the Coxeter element of the Weyl group. It turns out that not all conjugacy classes of the Weyl group lead to a topological model. For example, it is shown that for the $A$ algebras there are two possible choices for the conjugacy class, giving both the conventional and a new series of topological models. Furthermore, it is shown how the new series of theories contains the conventional series as a subsector. A tentative interpretation of this new series in terms of intersection theory is presented.