Disk Amplitudes in Two-Dimensional Open String Theories

Disk amplitudes of tachyons in two-dimensional open string theories (two-dimensional quantum gravity coupled to $c \leq 1$ conformal field theories) are obtained using the continuum Liouville field approach. The structure of momentum singularities is different from that of sphere amplitudes and is more complicated. It can be understood by factorizations of the amplitudes with the tachyon and the discrete states as intermediate states.