Smooth non-Abelian bosonization

Abstract

We present an extension of "smooth bosonization" to the non-Abelian case. We construct an enlarged theory containing both bosonic and fermionic fields which exhibits a local chiral gauge symmetry. A gauge fixing function depending on one real parameter allows us to interpolate smoothly between a purely fermionic and a purely bosonic representation. The procedure is, in the special case of bosonization, complementary to the approach based on duality.

Non-abelian bosonization in two and three dimensions

“…Strikingly, we found that in the three dimensional case, a BRST symmetry structure underlying the bosonic version of the fermionic generating functional plays a similar role and allows to end, at least in the large mass limit, with a simple bosonization rule. This BRST symmetry is highly related to that used in [5]- [7], [11]- [13], and is analogous to that arising in topological field theories [27]- [29], its origin being related to the way the originally "trivial" bosonic field enters into play.…”

“…In respect with the bosonization recipe for the fermion action, we considered the case of very massive fermions for which the fermion determinant is related to the non-Abelian Chern-Simons action. In this case the factorization of the auxiliary and Lagrange multiplier fields is achieved after discovering a BRST invariance reminiscent of that at the root of topological models and related to that exploited in the smooth bosonization approach [5]- [7], [11]- [12]. Addition of BRST exact terms allows us to extract the partition function for the boson counterpart of the original fermion fields.…”

“…Afterwards, different approaches rederived and discussed the bosonization recipe [30]- [35] but in general they were not constructive in the sense that the bosonic theory was not obtained from the fermionic one by following a series of steps that could be generalized to other cases, in particular to a possible higher-dimensional bosonization. More recently, using the the duality technique [8]- [9] which has as starting point the smooth bosonization approach [5]- [7], [11]- [12] the recipe for non-abelian bosonization was obtained by Burgess and Quevedo [10] in a way which is more adaptable to generalizations to higher dimensions. The approach to two-dimensional bosonization that we present in this section is related to that in [10] and we think that it is worthwhile to describe it here in detail since it provides many of the clues which allow us to derive the non-abelian bosonization recipe for 3-dimensional fermions.…”

“…This BRST transformations are related to those employed in the smooth bosonization [5]- [7], [11]- [12] approach and resemblant of those arising in topological field theories [27]- [29]. For example, in d = 4 topological YangMills theory the invariance of the starting classical action (the Chern-Pontryagin topological charge) under the most general transformation of the gauge field, b µ → b µ + ǫ µ , leads to a BRST transformation for b µ of the form δb µ = c µ , which corresponds to that in formula (52) [26]- [27].…”

“…However, as we shall see, once one introduces the auxiliary field b µ , a BRST invariance of the kind arising in topological field theories [27]- [29] can be unraveled. The use of BRST technique for bosonization of fermion models was initiated in the developement of the smooth bosonization approach [5]- [7], [11]- [12], closely related to bosonization duality and to the present treatment. In the present case, it allows to factor out the auxiliary field in the same way Polyakov-Wiegmann identity did the job in d = 2.…”