nLab
Dirac manifold

Contents

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Symplectic geometry

Contents

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 XX is a subbundle LTXT *XL\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=2n = 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).

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

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

nn \in \mathbb{N}symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of (n+1)(n+1)-d sigma-modelhigher symplectic geometry(n+1)(n+1)d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension (n+1)(n+1)discussed in:
0symplectic manifoldsymplectic manifoldsymplectic geometryLagrangian submanifoldordinary space of states (in geometric quantization)geometric quantization
1Poisson Lie algebroidsymplectic groupoid2-plectic geometryPoisson sigma-modelcoisotropic submanifold (of underlying Poisson manifold)brane of Poisson sigma-model2-module = category of modules over strict deformation quantiized algebra of observablesextended geometric quantization of 2d Chern-Simons theory
2Courant Lie 2-algebroidsymplectic 2-groupoid3-plectic geometryCourant sigma-modelDirac structureD-brane in type II geometry
nnsymplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometryd=n+1d = n+1 AKSZ sigma-model

(adapted from Ševera 00)

References

General

The original articles are

  • 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

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

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

A relation between Dirac structures and D-branes is discussed in

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

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 March 20, 2013 at 14:00:46. See the history of this page for a list of all contributions to it.