Algebraic Structures and Differential Geometry in 2D String Theory

A careful treatment of closed string BRST cohomology shows that there are more discrete states and associated symmetries in $D=2$ string theory than has been recognized hitherto. The full structure, at the $SU(2)$ radius, has a natural description in terms of abelian gauge theory on a certain three dimensional cone $Q$. We describe precisely how symmetry currents are constructed from the discrete states, explaining the role of the ``descent equations.'' In the uncompactified theory, we compute the action of the symmetries on the tachyon field, and isolate the features that lead to nonlinear terms in this action. The resulting symmetry structure is interpreted in terms of a homotopy Lie algebra.