Universality classes of dense polymers 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 {\em 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 non-linear 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 \ell-leg exponents, h_\ell=g_\sigma^2 \ell(\ell-2)/8, which is shown numerically to hold very well.