Partition functions and physical states in two-dimensional quantum gravity and supergravity

We review the Liouville theory calculation of the genus-one path integral for c ≤ 1 conformal models coupled to two-dimensional gravity. From the modular integrand we derive the existence of an infinite number of physical operators which are in one-to-one correspondence with the conformal primary fields and null states of the matter theory. We also calculate the torus path integral and find the spectrum of physical operators for c ̂ ≤ 1 superconformal models coupled to supergravity. The amplitude in the odd spin structure requires a special treatment and is found to be proportional to the Witten index of the matter theory.