Dirac manifold

∞-Lie theory (higher geometry) Background Smooth structure Higher groupoids Lie theory ∞-Lie groupoids ∞-Lie algebroids Formal Lie groupoids Cohomology Homotopy Related topics Examples -Lie groupoids -Lie groups -Lie algebroids -Lie algebras A Dirac manifold is a smooth manifold with a Dirac structure in the sense of (Courant 90). (Beware that sometimes “Dirac structure” is used for Dirac brackets in earlier contexts which are related but not formalized in this way.) An almost Dirac structure on a manifold is a subbundle of the tangent Courant algebroid, which is isotropic under a certain symmetric pairing. An almost Dirac structure is a Dirac structure if it satisfies an integrability condition. A Courant Lie 2-algebroid is a symplectic Lie n-algebroid for . Dirac structures are related to the Lagrangian dg-submanifolds (see there) of the dg-manifold formally dual to its Chevalley-Eilenberg algebra. With suitable identifications Dirac structures characterize D-branes. This is argued generally in (Asakawa-Sasa-Watamura), and later in (Demulder & Raml 22). An example is the canonical Cartan-Dirac structure on a Lie group, which yields the conjugacy classes of the Lie group as leaves. These are indeed known to be the D-branes of the WZW model on that Lie group. The original articles are Ted Courant, Alan Weinstein, Beyond Poisson structures, preprint, Berkeley 1986 pdf Irene Dorfman, Dirac structures of integrable evolution equations, Phys. Lett. A 125 (1987), no. 5, 240–246 doi MR89b:58088 Ted Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), no. 2, 631–661 MR90m:58065 doi; Tangent Dirac structures, J. Phys. A 23 (1990), no. 22, 5153–5168 MR92d:58064 iop Lecture notes include section 2 of The generalization of Dirac structures from base manifolds to base Lie groupoids (“multiplicative Dirac structure”) is discussed in Further references include Relating Dirac structures to D-branes: Tsuguhiko Asakawa, Shuhei Sasa, Satoshi Watamura, D-branes in...