On dual pairs in Dirac Geometry
arXiv:1602.02700 · doi:10.1007/s00209-017-1947-3
Abstract
In this note we discuss dual pairs in Dirac geometry. We show that this notion appears naturally when studying the problem of pushing forward a Dirac structure along a surjective submersion, and we prove a Dirac-theoretic version of Libermann's theorem from Poisson geometry. Our main result is an explicit construction of strong self-dual pairs for Dirac structures. This theorem not only recovers the global construction of symplectic realizations from [Crainic-Marcut 2011], but allows for a more conceptual understanding of it, yielding a simpler and more natural proof. As an application of the main theorem, we present a different approach to the recent normal form theorem around Dirac transversals from [Bursztyn-Lima-Meinrenken 2016].
27 pages. *Published version of the article* (as in https://doi.org/10.1007/s00209-017-1947-3), Math. Z. (2017)