Large N and double scaling limits in two dimensions

Recently, the author has constructed a series of four-dimensional non-critical string theories with eight supercharges, dual to theories of light electric and magnetic charges, for which exact formulas for the central charge of the space-time supersymmetry algebra as a function of the world-sheet couplings were obtained. The basic idea was to generalize the old matrix model approach, replacing the simple matrix integrals by the four-dimensional matrix path integrals of N=2 supersymmetric Yang-Mills theory, and the Kazakov critical points by the Argyres-Douglas critical points. In the present paper, we study qualitatively similar toy path integrals corresponding to the two-dimensional N=2 supersymmetric non-linear σ model with target space CPN and twisted mass terms. This theory has some very strong similarities with N=2 super Yang-Mills, including the presence of critical points in the vicinity of which the large-N expansion is IR divergent. The model being exactly solvable at large N, we can study non-BPS observables and give full proofs that double scaling limits exist and correspond to universal continuum limits. A complete characterization of the double scaled theories is given. We find evidence for dimensional transmutation of the string coupling in some non-critical string theories. We also identify en passant some non-BPS particles that become massless at the singularities in addition to the usual BPS states. (authors)