Non-abelian self-duality from self-interaction

The non-abelian self-dual action in three dimensions is derived using the self-interaction mechanism. Self-duality in three dimensions was proposed initially by Townsend et. al. [1] as an alternative to the topologically massive theory[2]. In principle, they seem different descriptions of a locally massive spin 1 physical excitation: the self-dual theory is described by a non-gauge invariant first order action while the topologically massive action is written down in a gauge invariant second order formulation. Both actions have an abelian Chern-Simons term (ǫAm∂nAp). Despite these differences, Deser and Jackiw stablished that both theories are locally equivalent through the existence of a master action, even in the presence of external sources[3]. Moreover, both theories are dual equivalent[4] and the self-dual theory can be seen as a gauged fixed version of the topologically massive theory[5]. The self-dual theory for gravity and for higher spin in three dimensions was achieved in [6] and [7], respectively. If glogal properties are considered, the equivalence is modified, for instance, the partition functions of the self dual and topologically massive theories are not the same but they are related in the following way: ZSD = ZCSZTM [8] (where ZCS is the partition function of the abelian Chern-Simons action). The non-abelian generalization of the topologically massive theory was given in [2] while the non-abelian self-dual theory was formulated independently by McKeon [9] and Arias, et. al.[10], which has a structure of a Freedman-Townsend action[11]. In this letter, starting from an appropiate master action, we will derive the non-abelian self-dual action using the self-interaction mechanism[12].