We have seen that fields with arbitrary statistics may exist in D = 2, 3 and 4 spacetime dimensions. In D = 4, however, this is only possible for nonlocal fields, namely, field operators creating extended objects, such as strings and n-branes. In D = 2, 3, conversely, this is allowed for local fields, which are associated to point particles. Through the order-disorder duality procedure, we have been able to build such fields with arbitrary statistics out of bosonic fields. This method, known as bosonization in the case of fermion fields, consists in combining operator pairs each one of them carrying, respectively, the charge and the topological charge of the bosonic field) into a composite field that will replace the fermionic or anyonic field. The statistics of the latter is determined by the product of charge × topological charge of the bosonic field. In this chapter, we explore a general method, related to bosonization and known as statistical transmutation, by means of which we may continuously change the statistics of a field, and consequently of the objects it creates, in D = 3, 4. This is achieved by a mechanism, working at the Lagrangean level, through which a certain amount of topological charge is imparted on charged fields and on the objects they create. The resulting objects, consequently, change their amount of the product charge × topological charge, thereby modifying their statistics. We shall see that both bosonization and statistical transmutation have many interesting applications in condensed matter physics.

Generalized BF Theories

In the previous chapter, we introduced the topological current, in D = 2, 3, 4, expressed in terms of a generalized field Bij… (10.13), which was, respectively, a scalar, a vector and a rank-2 tensor (10.14) and (10.15). There, we have shown that Bij… is the bosonic field used both in the bosonization process and also for constructing fields with generalized statistics.

We now introduce a field theory involving the Bij…field, coupled to a vector U(1) abelian gauge field Aμ, where the topological charge of the former is the source of the latter. The magnetic field of the latter, conversely, is the “charge,” acting as the source of the former. This kind of theory is known as the BF-theory.