The fermion-boson map for large d

Understand

We show that the three-dimensional map between fermions and bosons at finite temperature generalises for all odd dimensions $d>3$.

• We further argue that such a map has a nontrivial large $d$ limit.
• Evidence comes from studying the gap equations, the free energies and the partition functions of the $U(N)$ Gross-Neveu and CP$^{N-1}$ models for odd $d\geq 3$ in the presence of imaginary chemical potential.
• We find that the gap equations and the free energies can be written in terms of the Bloch-Wigner-Ramakrishnan $D_d(z)$ functions analysed by Zagier.