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On dual pairs in Dirac geometry

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Abstract

In this note we discuss (weak) 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 self-dual pairs for Dirac structures. This theorem not only recovers the global construction of symplectic realizations from Crainic and Mărcuţ (J Symplectic Geom 9(4):435–444, 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 et al. (J für die reine und angewandte Mathematik (Crelles J), doi:10.1515/crelle-2017-0014, 2017).

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Notes

  1. In the literature [7, 10, 38], a symplectic realization in our sense, in which the Poisson map is required to be submersive, is called a full symplectic realization.

  2. Since the inception of Dirac geometry in the work of Courant [16], many people worked on the field, and there are several good sources available. For background material on Dirac structures particularly close in spirit to the present note, we refer the reader to e.g. [2, 11, 23, 31]. For the specific conventions and notations used in this paper, we refer the reader to Sect. 2.

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Acknowledgements

At a late stage of this Project, we learned that E. Meinrenken was aware that the argument we present in the proof of our main theorem would work. We wish to thank him for his interest and kind encouragement, as well as many interesting discussions. We also wish the thank the referee for the detailed comments and suggestions, which greatly improved the quality of the paper. The first author was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Vrije Competitie Grant ”Flexibility and Rigidity of Geometric Structures” 612.001.101) and by IMPA (CAPES-FORTAL project) and the second author by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Veni Grant 613.009.031) and the National Science Foundation (Grant DMS 14-05671).

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Authors and Affiliations

  1. Universidade Federal do Rio Grande do Sul, Campus Litoral Norte, Rodovia RS 030, 11.700, Km 92, Emboaba, Tramandaí, RS, CEP 95590-000, Brazil

    Pedro Frejlich

  2. IMAPP, Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands

    Ioan Mărcuț

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  1. Pedro Frejlich
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  2. Ioan Mărcuț
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Correspondence to Pedro Frejlich.

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Frejlich, P., Mărcuț, I. On dual pairs in Dirac geometry. Math. Z. 289, 171–200 (2018). https://doi.org/10.1007/s00209-017-1947-3

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