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Advances in Mathematical Physics
Volume 2018, Article ID 3474760, 7 pages
Research Article

Symplectic Approach of Three-Dimensional Palatini Theory Plus a Chern-Simons Term

Instituto de Física Luis Rivera Terrazas, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, 72570 Puebla, PUE, Mexico

Correspondence should be addressed to Alberto Escalante; xm.paub.paufi@nalacsea

Received 20 December 2017; Revised 13 February 2018; Accepted 26 February 2018; Published 29 March 2018

Academic Editor: Dimitrios Tsimpis

Copyright © 2018 Alberto Escalante and Prihel Cavildo-Sánchez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Using the symplectic framework of Faddeev-Jackiw, the three-dimensional Palatini theory plus a Chern-Simons term [P-CS] is analyzed. We report the complete set of Faddeev-Jackiw constraints and we identify the corresponding generalized Faddeev-Jackiw brackets. With these results, we show that although P-CS produces Einstein’s equations, the generalized brackets depend on a Barbero-Immirzi-like parameter. In addition, we compare our results with those found in the canonical analysis showing that both formalisms lead to the same results.