Two Dimensional String Theory and the Topological Torus

We analyze topological string theory on a two dimensional torus, focusing on symmetries in the matter sector. Even before coupling to gravity, the topological torus has an infinite number of point-like physical observables, which give rise via the BRST descent equations to an infinite symmetry algebra of the model. The point-like observables of ghost number zero form a topological ground ring, whose generators span a spacetime manifold; the symmetry algebra represents all (ground ring valued) diffeomorphisms of the spacetime. At nonzero ghost numbers, the topological ground ring is extended to a superring, the spacetime manifold becomes a supermanifold, and the symmetry algebra preserves a symplectic form on it. In a decompactified limit of cylindrical target topology, we find a nilpotent charge which behaves like a spacetime topological BRST operator. After coupling to topological gravity, this model might represent a topological phase of c = 1 string theory. We also point out some analogies to two dimensional superstrings with the chiral GSO projection, and to string theory with c = −2. ⋆ e-mail addresses: [email protected] or [email protected] † Robert R. McCormick Fellow; research also supported by the NSF under Grant No. PHY90-00386; the DOE under Grant No. DEFG02-90ER40560; the Czechoslovak Chart 77 Foundation; and the ČSAV under Grant No. 91-11045.