$(1,q=-1)$ Model as a Topological Description of $2d$ String Theory

Details $(1,q=-1)$ Model as a Topological Description of $2d$ String Theory $(1,q=-1)$ Model as a Topological Description of $2d$ String Theory

1994-06-09

arxiv.org/abs/hep-th/9406056v1

Nucl.Phys.B431:223-257,1994

Physics

High Energy Physics

High Energy Physics - Theory

39 pages,latex,taup-2170 -94

Scientific paper

10.1016/0550-3213(94)90105-8

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 $W_{1+\infty}$ 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 show 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 $W_{1+\infty}$ Ward identities on the torus. We argue that the structure of these recursion relations coincides with that of the $W_{1+\infty}$ Ward identities for any genus.

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