MASSLESS FLOWS .1. THE SINE-GORDON AND O(N) MODELS

The massless flow between successive minimal models of conformal field theory is related to a flow within the sine-Gordon model when the coe fficient of the cosine potential is imaginary. This how is studied, pa rtly numerically, from three different points of view. First we work o ut the expansion close to the Kosterlitz-Thouless point, and obtain ro aming behavior, with the central charge going up and down in between t he UV and IR values of c = 1. Next we analytically continue the Casimi r energy of the massive how (i.e. with real cosine term). Finally we c onsider the lattice regularization provided by the O(n) model in which massive and massless flows correspond to high- and low-temperature ph ases. A detailed discussion of the case n = 0 is then given using the underlying N = 2 supersymmetry, which is spontaneously broken in the l ow-temperature phase. The ''index'' trF(-1)(F) follows from the Painle ve III differential equation, and is shown to have simple poles in thi s phase. These poles are interpreted as occurring from level crossing (one-dimensional phase transitions for polymers). As an application, n ew exact results for the connectivity constants of polymer graphs on c ylinders are obtained. These results and points of view are used in th e following paper to discuss the appropriate exact S matrices and the resulting Casimir energies.