N = 2 supersymmetry, Painlevé III and exact scaling functions in 2D polymers

Abstract

We discuss in this paper various aspects of the off-critical O( n) model in two dimensions. We find the ground-state energy conjectured by Zamolodchikov for the unitary minimal models, and extend the results to some non-unitary minimal cases. These results are applied to the discussion of scaling functions for polymers on a cylinder. Using the underlying N = 2 supersymmetry, the scaling function for one non-contractible polymer loop around the cylinder is shown to be simply related to the solution of the Painlevé III differential equation. We also find the ground-state energy for a single polymer on the cylinder. These results are checked by numerically simulating the polymer system. The flow to the dense polymer phase is also analyzed numerically. We find there surprising results, with a c eff function that is not monotonically decreasing and seems to have a roaming behavior, getting very close to the value 81 70 and 7 10 between its UV and IR values of 1.