I will explain a version of the Polyakov-Alvarez formula on non-smooth surfaces in terms of Brownian loops. This was motivated by earlier attempts to understand Liouville quantum gravity (LQG) surfaces from a heuristic viewpoint in physics. The result implies that weighting certain regularized LQG surfaces by the Brownian loop soup partition function changes the central charges as expected. Based on joint work with Joshua Pfeffer and Scott Sheffield.