arXiv : The fermion-boson map for large $d$

Abstract 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 D d ( z ) functions analysed by Zagier. Since D 2 ( 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 C l d ( 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 .