  Using methods from the conformal bootstrap, we study the properties of
Noether currents in the critical $O(n)$ loop model. We confirm that they do not
give rise to a Kac-Moody algebra (for $n\neq 2$), a result expected from the
underlying lack of unitarity. By studying four-point functions in detail, we
fully determine the current-current OPEs, and thus obtain several structure
constants with physical meaning. We find in particular that the terms
$:\!J\bar{J}\!:$ in the identity and adjoint channels vanish exactly,
invalidating the argument made in \cite{car93-1} that adding
orientation-dependent interactions to the model should lead to continuously
varying exponents in self-avoiding walks. We also determine the residue of the
identity channel in the $JJ$ two-point function, finding that it coincides both
with the result of a transfer-matrix computation for an orientation-dependent
correlation function in the lattice model, and with an earlier Coulomb gas
computation of Cardy \cite{car93}. This is, to our knowledge, one of the first
instances where the Coulomb gas formalism and the bootstrap can be successfully
compared.
