∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
$\infty$-Lie groupoids
$\infty$-Lie groups
$\infty$-Lie algebroids
$\infty$-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 $X$ is a subbundle $L\subset T X \oplus T^*X$ 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 $n = 2$. 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).
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.
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
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:
Related observations for D-branes in the WZW model had long been made (unpublished) for the Cartan-Dirac structure over a Lie group.
Last revised on July 3, 2023 at 17:39:54. See the history of this page for a list of all contributions to it.