Abstract: The vacuum-to-vacuum genus-zero string graph is expected to be related to the thermodynamics of the target spacetime, and is thus an interesting quantity to compute. However, this quantity is very confusing in critical string theory. In order to understand it better, we compute the vacuum sphere partition function in noncritical string theory in the semiclassical limit, where a saddle point approximation becomes available.
The main conceptual point is that the unfixed PSL(2,C) gauge group is spontaneously broken by the saddle point configuration of the Liouville field, which removes the usual obstacles in computing this path integral in critical string theory.
Furthermore, the quantity sphere/disk^2 is free of all multiplicative ambiguities. After computing the FZZT disk path integral using similar techniques to fix the PSL(2,R) gauge freedom, we find a precise numerical match between the string theory answer and the answer from the dual double-scaled one-matrix integral.
This talk is based on work with Douglas Stanford and Cynthia Yan.