The fermion-boson map for large d

TL;DR

This paper generalizes the fermion-boson duality map to all odd dimensions greater than three, revealing connections to hyperbolic geometry, complex Chern-Simons theories, and special functions through analysis of models with imaginary chemical potential.

Contribution

It extends the fermion-boson map to higher odd dimensions and links it to geometric and special function structures, providing new insights into large dimension limits.

Findings

The gap equations and free energies are expressed via Bloch-Wigner-Ramakrishnan functions.

Connections established between 3D results and hyperbolic geometry.

Identification of saddle points related to zeros and extrema of Clausen functions.

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 $d\geq 3$ 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. 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 $Cl_d(\theta)$ 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 $\pi$.