Topological Field Theory Interpretations and LG Representation of c=1 String Theory

We analyze the topological nature of $c=1$ string theory at the self--dual radius. We find that it admits two distinct topological field theory structures characterized by two different puncture operators. We show it first in the unperturbed theory in which the only parameter is the cosmological constant, then in the presence of any infinitesimal tachyonic perturbation. We also discuss in detail a Landau--Ginzburg representation of one of the two topological field theory structures.