  The nature of the theta point for a polymer in two dimensions has long been
debated, with a variety of candidates put forward for the critical exponents.
This includes those derived by Duplantier and Saleur (DS) for an exactly
solvable model. We use a representation of the problem via the $CP^{N-1}$ sigma
model in the limit $N \rightarrow 1$ to determine the stability of this
critical point. First we prove that the DS critical exponents are robust, so
long as the polymer does not cross itself: they can arise in a generic lattice
model, and do not require fine tuning. This resolves a longstanding theoretical
question. However there is an apparent paradox: two different lattice models,
apparently both in the DS universality class, show different numbers of
relevant perturbations, apparently leading to contradictory conclusions about
the stability of the DS exponents. We explain this in terms of subtle
differences between the two models, one of which is fine-tuned (and not
strictly in the DS universality class). Next, we allow the polymer to cross
itself, as appropriate e.g. to the quasi-2D case. This introduces an additional
independent relevant perturbation, so we do not expect the DS exponents to
apply. The exponents in the case with crossings will be those of the generic
tricritical $O(n)$ model at $n=0$, and different to the case without crossings.
We also discuss interesting features of the operator content of the $CP^{N-1}$
model. Simple geometrical arguments show that two operators in this field
theory, with very different symmetry properties, have the same scaling
dimension for any value of $N$ (equivalently, any value of the loop fugacity).
Also we argue that for any value of $N$ the $CP^{N-1}$ model has a marginal
parity-odd operator which is related to the loops' winding angle.
