Boson-fermion duality in three dimensions

We study the 2+1 dimensional boson-fermion duality in the presence of background curvature and electromagnetic fields. The main players are, on the one hand, free massive complex scalar fields coupled to U(1) Maxwell-Chern-Simons gauge fields at Chern-Simons levels $pm1$, representing relativistic composite bosons with one unit of attached flux, and on the other hand, free massive Dirac fermions. We prove, in a curved background and at the level of the partition function, that a doublet of relativistic composite bosons, in the infinite coupling limit, is dual to a doublet of Dirac fermions. The spin connection arises from the expectation value of the Wilson loop in the Chern-Simons theory, whereas a non-minimal coupling of bosons to the scalar curvature is necessary in order to obtain agreement between partition functions. Remarkably, we find that the correspondence does not hold in the presence of background electromagnetic fields, a pathology rooted to the coupling of electromagnetism to the spin angular momentum of the Dirac spinor, which can not be reproduced from minimal coupling in the bosonic side. The presence of framing and parity anomalies in the Chern-Simons and fermionic theories, respectively, poses a difficulty in realizing the duality as an exact mapping between partition functions. The existence of non matching anomalies is circumvented by the Dirac fermions coming in pairs, making the fermionic theory parity anomaly free, and by the inclusion of a Maxwell term in the bosonic side, acting as a regulator forcing the CS theory to be quantized in a non-topological way. The Coulomb interaction stemming from the Maxwell term is also of key importance to prevent intersections of worldlines in the path integral. An extension of the duality to the massless case fails if bosons and fermions are in a topological phase, but is possible when the mapping is between trivial theories.