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(N) Gross–Neveu and CP N−1 models for odd d≥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 Dd(z) functions analysed by Zagier. Since D2(z) gives the volume of ideal tetrahedra in 3 d 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 Cld(θ) 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 .