We introduce a model of two-dimensional compact polymers on the
square lattice taking into account self-avoidance, compactness,
and polydispersity. Special weights w_s and w_c are assigned to
vertices where the polymer does not bend (goes straight) or makes
a close contact with itself. The special case (w_s,w_c) = (1,0)
is exactly solvable, using Coulomb gas methods, and yields a
c=-1 conformal field theory of three bosonic fields coupled to
a background electric charge. In the Flory case w_c = 0, the
model stays critical when w_s is kept below some critical
value w_0, with the same central charge. One half of the critical
exponants depends continuously on w_s, while the other half
remains constant. In the field theory, this corresponds to one
of the bosons decoupling from the other two. Exact values of
the exponents at the transition w_0 are given. The generic case
of the model can be viewed as a generalisation of the 8-vertex
model. We numerically examine this case, and present results
for the phase diagram.
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