Duality and an operator realization for the Fermi-Bose transmutation in 3+1 dimensions

Abstract

We consider the Maxwell-Higgs system in the broken phase, described in terms of a Kalb-Ramond field interacting with the electromagnetic field through a topological coupling. We then study the creation operators of states which respectively carry a point charge and a closed magnetic string in the electromagnetic language or a point topological charge and a closed Kalb-Ramond charged string in the Kalb-Ramond dual language. Their commutation relation is evaluated, implying they satisfy a dual algebra and their composite possesses generalized statistics. In the local limit where the radius of the string vanishes, only Fermi or Bose statistics are allowed. This provides an explicit operator realization for statistical transmutation in 3+1D.

Bosonization of free Weyl fermions

“…The natural bosonic field in D = 4 is a 2-tensor B µν [19,22]. Assuming the fermionic current j ψγ ψ = µ µ is expressed as…”

“…We have seen that the natural bosonic field in D = 4 is the antisymmetric tensor gauge-field B µν , also known as Kalb-Ramond field. The relevant dual operators were introduced before [22], namely, C ( ) σ and x ( ) µ . These are expressed in terms of this field by…”

“…The C ( ) σ operator is nonlocal and creates, out of the vacuum, a quantum state which is an eigenstate of the (vector) source of the Kalb-Ramond field Q i , with eigenvalue a along the closed curve C [22]. Since we are going to bosonize a local field, https://doi.org/10.1088/1742-5468/aa60cc J. Stat.…”

“…where μ and ε are, respectively, infrared and ultraviolet regulators. Now, inserting (22) into (35), we get…”

Duality in non-linear B∧F models: equivalence between self-dual and topologically massive Born–Infeld B∧F models

“…In arbitrary dimensions duality will relate tensors of different ranks and D = 3 is a special case where vectors are dualized into vectors. In particular for D = 4 the inclusion of the so called B ∧ F term has been responsible for new and interesting features such as topological mass generation [34] and statistical transmutation [35]. In this respect we have examined recently [26], using the gauge embedding procedure, the duality between a gauge non invariant B ∧ F model,…”