Coupling of Tachyons and Discrete States in $c = 1$ 2-D Gravity
  All the three point couplings involving tachyons and/or discrete states are obtained in $c=1$ two-dimensional (2-D) quantum gravity by means of the operator product expansion (OPE). Cocycle factors are found to be necessary in order to maintain the analytic structure of the OPE, and are constructed explicitly both for discrete states and for tachyons. The effective action involving tachyons and discrete states is worked out to summarize all of these three point couplings. 
Yoichiro Matsumura, Norisuke Sakai, Yoshiaki Tanii