Abstract
We propose a definition of symplectic 2-groupoid which includes integrations of Courant algebroids that have been recently constructed. We study in detail the simple but illustrative case of constant symplectic 2-groupoids. We show that the constant symplectic 2-groupoids are, up to equivalence, in one-to-one correspondence with a simple class of Courant algebroids that we call constant Courant algebroids. Furthermore, we find a correspondence between certain Dirac structures and Lagrangian sub-2-groupoids.
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Notes
Recently, Ševera and Širaň [17] described a general construction for local integration and showed that the local integrations can be glued together up to coherent homotopy.
The extension of the proof to the general case \(\overline{W}ND\), where D is a symplectic double Lie groupoid, is similar.
We note that we are using the condition \(D^\perp = D\) in place of the usual requirement that D be maximally isotropic. In most cases of interest, E has signature (n, n), and the two conditions are equivalent. However, if E does not have signature (n, n), then, according to the definition we are using, there do not exist any Dirac structures in E.
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Acknowledgements
We would like to thank Ezra Getzler for inspiring discussions and explanations about symplectic structures on differentiable n-stacks. We would like to thank Damien Calaque for a discussion about the relation between integration of Courant algebroids and derived symplectic geometry. The research of the second author is partially supported by NSF Grant DMS 1363250.
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Mehta, R.A., Tang, X. Constant symplectic 2-groupoids. Lett Math Phys 108, 1203–1223 (2018). https://doi.org/10.1007/s11005-017-1026-z
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DOI: https://doi.org/10.1007/s11005-017-1026-z