Interactions of Discrete States with Nonzero Ghost Number \\ in $c=1$   $2D$ Gravity
  We study the interactions of the discrete states with nonzero ghost number in $c=1$ two-dimensional ($2D$) quantum gravity. By using the vertex operator representations, it is shown that their interactions are given by the structure constants of the group of the area preserving diffeomorphism similar to those of vanishing ghost number. The effective action for these states is also worked out. The result suggests the whole system has a BRST-like symmetry. 
Nobuyoshi Ohta and Hisao Suzuki