  We study the $2d$ chiral Gross-Neveu model at finite temperature $T$ and
chemical potential $\mu$. The analysis is performed by relating the theory to a
$SU(N)\times U(1)$ Wess-Zumino-Witten model with appropriate levels and global
identifications necessary to keep track of the fermion spin structures. At
$\mu=0$ we show that a certain $\mathbb{Z}_2$-valued 't Hooft anomaly forbids
the system to be trivially gapped when fermions are periodic along the thermal
circle for any $N$ and any $T>0$. We also study the two-point function of a
certain composite fermion operator which allows us to determine the remnants
for $T>0$ of the inhomogeneous chiral phase configuration found at $T=0$ for
any $N$ and any $\mu$. The inhomogeneous configuration decays exponentially at
large distances for anti-periodic fermions while it persists for $T>0$ and any
$\mu$ for periodic fermions, as expected from anomaly considerations. A large
$N$ analysis confirms the above findings.
