Duality and an Operator Realization for the Fermi-Bose Transmutation in 3+1 Dimensions

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. ∗ Work supported in part by CNPq-Brazilian National Research Council. E-Mail address: [email protected] 1) Introduction The mechanism of statistical transmutation occurring in 2+1D for the composite state of a point charge and a magnetic vortex [1] has been generalized for 3+1D a few years ago. It was shown that the composite of a closed Nambu charged string and a point Kalb-Ramond vortex presented generalized statistics [2] in four-dimensional spacetime. Also the Chern-Simons mechanism of statistical transmutation [3] was shown to hold for stringlike objects in 3+1D [4]. More recently, in the framework of the Abelian Higgs Model (AHM), it was demonstrated that a closed magnetic (NielsenOlesen) string in the presence of a point charge also displayed the phenomenon of Fermi-Bose transmutation [5]. In this work we provide an operator realization for this fact. We consider the AHM in the broken phase, which can be described in terms of an antisymmetric tensor (Kalb-Ramond) gauge field interacting with the electromagnetic field by means of a topological coupling. In this language the topological charge of the Kalb-Ramond field becomes the electric charge and the Kalb-Ramond charge becomes the magnetic flux. A creation operator, σ, for the magnetic string states of the AHM was recently introduced in [6] and a creation operator, μ, for the topologically charged states of the Kalb-Ramond field was constructed some time before in [7]. Here, we obtain the operator dual to μ, in the Kalb-Ramond framework and show that it is nothing but the magnetic string operator σ, introduced in [6] rewritten in terms of the antisymmetric tensor gauge field. We evaluate their commutation relation and thereby explicitly show that the composite operator ψ = μσ has generalized statistics. In the local limit where the radius of the closed string vanishes, we show that only Fermi or Bose statistics are allowed. In Kalb-Ramond language these composite states carry both a point topological charge (point Kalb-Ramond vortex) and a closed KalbRamond charged string. In electromagnetic language, they carry both a point electric charge and a closed magnetic string. Through the composite operator ψ, therefore, we obtain an explicit realization for both the mechanisms discovered in [2] and [5].