  Starting with the Ising model, statistical models with global symmetries
provide fruitful approaches to interesting physical systems, for example
percolation or polymers. These include the $O(n)$ model (symmetry group $O(n)$)
and the Potts model (symmetry group $S_Q$). Both models make sense for $n,Q\in
\mathbb{C}$ and not just $n,Q\in \mathbb{N}$, and both give rise to a conformal
field theory in the critical limit.
  Here, we study similar models based on the unitary group $U(n)$. We focus on
the two-dimensional case, where the models can be described either as gases of
non-intersecting orientable loops, or as alternating spin chains. This allows
us to determine their spectra either by computing a twisted torus partition
function, or by studying representations of the walled Brauer algebra.
  In the critical limit, our models give rise to a CFT with global $U(n)$
symmetry, which exists for any $n\in\mathbb{C}$. Its spectrum is similar to
those of the $O(n)$ and Potts CFTs, but a bit simpler. We conjecture that the
$O(n)$ CFT is a $\mathbb{Z}_2$ orbifold of the $U(n)$ CFT, where $\mathbb{Z}_2$
acts as complex conjugation.
