Critical potentials and correlation functions in the minus two dimensional matrix model

Abstract

We review the construction and exact solution of the critical points of the matrix model with a pair of anti-commuting coordinates. The generic critical point describes c = −2 matter coupled to two-dimensional quantum gravity, while other critical points do not obey KPZ-DDK scaling, and have not yet been identified. We define scaling operators which perturb each of the critical points towards others, and find their correlation functions to all orders in the genus expansion. We also discuss other operators, which perturb the c = −2 matrix model towards K = 2, 3,… critical points of the one-matrix model. Some of their correlation functions diverge, and these divergent amplitudes are proportional to the correlation functions of the topological gravity.