A Sojourn Through $c=1$ String Theory

We revisit the $S$-matrix, the thermal $S$-matrix and the thermal free energy of $c=1$ string theory from the worldsheet perspective. By numerically integrating Virasoro conformal blocks with DOZZ structure constants, we compute the scattering amplitudes and thermal amplitudes up to 6-point at tree level and up to 3-point at one loop, in addition to the genus two thermal free energy. All results are in agreement with the predicted answers from the $c=1$ matrix model.