(1, q = -1) model as a topological description of 2D string theory

We study the (1,q = -1) model coupled to topological gravity as a candidate to describing 2D string theory at the self-dual radius. We define the model by analytical continuation of q > 1 topological recursion relations to q = -1. We show that at genus zero the q = -1 recursion relations yield the W_1+∞ Ward identities for tachyon correlators on the sphere. A scheme for computing correlation functions of q = -1 gravitational descendants is proposed and applied for the computation of several correlators. It is suggested that the latter correspond to correlators of discrete states of the c = 1 string. In a similar manner to the q > 1 models, we shaw that there exist topological recursion relations for the correlators in the q = -1 theory that consist of only one and two splittings of the Riemann surface. Using a postulated regularized contact, we prove that the genus one q = -1 recursion relations for tachyon correlators coincide with the W_1+∞ Ward identities on the torus. We argue that the structure of these recursion relations coincides with that of the W_1+∞ Ward identities for any genus.