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.
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Author comments upon resubmission
We have changed the title of the paper and the name of the model to reflect its $PSU(n)$ symmetry. We have also included minor corrections as suggested by the referees.
