Universality classes of polymer melts and conformal sigma models

In the usual statistical model of a dense polymer (a single space-filling loop on a lattice) in two dimensions the loop does not cross itself. We modify this by including intersections in which three lines can cross at the same point, with some statistical weight w per crossing. We show that our model describes a line of critical theories with continuously varying exponents depending on w, described by a conformally invariant nonlinear sigma model with varying coupling constant g(sigma)(2) >= 0. For the boundary critical behavior, or the model defined in a strip, we propose an exact formula for the l-leg exponents, h(l) = g(sigma)(2)l (l - 2)/8, which is shown numerically to hold very well.