The fermion–boson map for large d

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 $U\left(N\right)$ Gross–Neveu and CP${}^{N-1}$ models for odd $d\ge 3$ 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 $D_d\left(z\right)$ functions analysed by Zagier. Since $D_2\left(z\right)$ gives the volume of ideal tetrahedra in 3d hyperbolic space our three-dimensional results are related to resent studies of complex Chern–Simons theories, while for $d>3$ they yield corresponding higher dimensional generalizations. As a spinoff, we observe that particular complex saddles of the partition functions correspond to the zeros and the extrema of the Clausen functions $C{l}_d\left(\theta \right)$ with odd and even index d respectively. These saddles lie on the unit circle at positions remarkably well approximated by a sequence of rational multiples of π.