Faddeev–Jackiw quantization of an Abelian and non-Abelian exotic action for gravity in three dimensions

• A detailed Faddeev–Jackiw analysis for exotic action of gravity is performed.
• We show that Dirac’s brackets and Generalized [FJ] brackets are equivalent.
• Without fixing the gauge exotic action is a non-commutative theory.
• The fundamental gauge transformations of the theory are found.
• Dirac and Faddeev–Jackiw approaches are compared.

A detailed Faddeev–Jackiw quantization of an Abelian and non-Abelian exotic action for gravity in three dimensions is performed. We obtain for the theories under study the constraints, the gauge transformations, the generalized Faddeev–Jackiw brackets and we perform the counting of physical degrees of freedom. In addition, we compare our results with those found in the literature where the canonical analysis is developed, in particular, we show that both the generalized Faddeev–Jackiw brackets and Dirac’s brackets coincide to each other. Finally we discuss some remarks and prospects.