  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 $c=-2$.
