nLab Dirac manifold

Idea

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.

Properties

Relation to Lagrangian dg-submanifolds

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.

Relation to D-branes

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.

Examples

• Cartan-Dirac structure on a Lie group

Related concepts

• polarization

• Poisson structure

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

$n \in \mathbb{N}$	symplectic Lie n-algebroid	Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-model	higher symplectic geometry	$(n+1)$d sigma-model	dg-Lagrangian submanifold/ real polarization leaf	= brane	(n+1)-module of quantum states in codimension $(n+1)$	discussed in:
0	symplectic manifold	symplectic manifold	symplectic geometry		Lagrangian submanifold	–	ordinary space of states (in geometric quantization)	geometric quantization
1	Poisson Lie algebroid	symplectic groupoid	2-plectic geometry	Poisson sigma-model	coisotropic submanifold (of underlying Poisson manifold)	brane of Poisson sigma-model	2-module = category of modules over strict deformation quantiized algebra of observables	extended geometric quantization of 2d Chern-Simons theory
2	Courant Lie 2-algebroid	symplectic 2-groupoid	3-plectic geometry	Courant sigma-model	Dirac structure	D-brane in type II geometry
$n$	symplectic Lie n-algebroid	symplectic n-groupoid	(n+1)-plectic geometry	$d = n+1$ AKSZ sigma-model

(adapted from Ševera 00)

References

General

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

• Irene Dorfman, Dirac structures and integrability of non-linear evolution equations, John Wiley and sons, 1993. xii+176 pp. MR94j:58081

Lecture notes include section 2 of

• Henrique Bursztyn, Alan Weinstein, Poisson geometry and Morita equivalence (arXiv:0402347)

The generalization of Dirac structures from base manifolds to base Lie groupoids (“multiplicative Dirac structure”) is discussed in

• Cristian Ortiz, Multiplicative Dirac structures (arXiv:1212.0176)

Further references include

• Anton Alekseev, Eric Meinrenken, Dirac structures and Dixmier-Douady bundles, Int. Math. Res. Not. IMRN 2012, no. 4, 904–956. MR2889163

Relation to D-branes

Relating Dirac structures to D-branes:

• Tsuguhiko Asakawa, Shuhei Sasa, Satoshi Watamura, D-branes in Generalized Geometry and Dirac-Born-Infeld Action [arXiv:1206.6964]

• Saskia Demulder, Thomas Raml. Poisson-Lie T-duality defects and target space fusion (2022). (arXiv:2208.04662).

Related observations for D-branes in the WZW model had long been made (unpublished) for the Cartan-Dirac structure over a Lie group.