$3d$ fermion-boson map with imaginary chemical potential

Understand

We study the three-dimensional $U(N)$ Gross-Neveu and CP$^{N-1}$ models in the canonical formalism with fixed $U(1)$ charge.

• For large-$N$ this is closely related to coupling the models to abelian Chern-Simons in a monopole background.
• We show that the presence of the imaginary chemical potential for the $U(1)$ charge makes the phase structure of the models remarkably similar.
• We calculate their respective large-$N$ free energy densities and show that they are mapped into each other in a precise way.