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 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 Cl 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.

Strong coupling universality at large N for pure CFT thermodynamics in 2+1 dimensions

“…Extensions of this work to higher odd dimensions is feasible and could be considered along the lines of Ref. [10]. Examples of solvable field theories with simple known gravity duals are rare, cf.…”

The fermion - boson map at imaginary chemical potential in odd dimensions

“…This hypothesis will be enhanced by the analysis of the next section. Finally, in [12] we have also suggested a geometric interpretation for this result. Namely, that the bosonic and fermionic free energies at imaginary chemical potential correspond to partial volumes of an ideal tetrahedron.…”

“…We have studied those systems in the canonical formalism by introducing an imaginary chemical potential for the U (1) charge or equivalently by introducing a constraint on the particle number. In such a setup the large-N canonical partition functions are intimately related to the partition functions of the same systems coupled to an abelian Chern-Simons gauge field expanded around a monopole background in a suitable mean field approximation [12,13]. Also, the imaginary U (1) charge density is related to the Chern-Simons level.…”

“…Encouraged by the above 3d results we move on to study the Gross-Neveu and CP N −1 models for odd dimensions d > 3 [12]. To begin thinking about a fermion-boson map for higher dimensions we need to take into account that a single Dirac fermion "weights" more as we increase the spacetime dimension.…”

“…We also introduce the first conjecture of the Generalized Bragg Law in 3 dimensions, with the 2 dimensional polygon model, as mentioned before. In Chapter 3 we present our results for the gap equations and the free energies of the higher-dimensional models of [12], we discuss the relevance of the D d (z) functions in our calculations and we give a formula for the approximate positions of the zeros and the extrema of the Clausen functions. At the same time, we extend our thinking on Generalized Bragg Law to more odd dimensions, resulting in an interesting specific limit.…”