Dirac Geometry

General Idea

Dirac structures simultaneously generalize Poisson structures and presymplectic structures. Intuitively a Dirac structure should be thought of as a smooth manifold foliated (possibly singular) by presymplectic manifolds.

Dirac structure

Definition(Dirac structure) . A Dirac structure on a smooth manifold $M$ is a subbundle of the standard Courant algebroid $L\leq \mathbb{T}M:= TM\oplus T^\ast M$ with the properties
  1. (Lagrangian) Maximally isotropic with respect to the symmetric pairing $$ \langle V\oplus \eta,W\oplus\xi\rangle_+ = \iota_V\xi + \iota_W\eta = \xi(V)+\eta(W) $$
  2. (Integrability) the sections are involutive (closed) under the Courant bracket $$ [V\oplus\eta,W\oplus\xi]:= [V,W]\oplus\mathcal{L}_V\xi - \mathcal{L}_W\eta + d(\iota_W \eta) $$
for any vector $v\in TM$ and covector $\eta\in T^\ast M$

There is also a twisted version of Dirac structure

Definition(Twisted Dirac structure) . A $\phi$-twisted Dirac structure on a smooth manifold $M$ is a subbundle of the $\phi$-twisted Courant algebroid $L\leq \mathbb{T}M:= TM\oplus T^\ast M$ which is
  1. (Lagrangian) Maximally isotropic with respect to the symmetric pairing $$ \langle V\oplus \eta,W\oplus\xi\rangle_+ = \iota_V\xi + \iota_W\eta = \xi(V)+\eta(W) $$
  2. (Integrability) The sections are involutive (closed) under the $\phi$-twisted Courant bracket $$ [V\oplus\eta,W\oplus\xi]^\phi:= [V,W]\oplus\mathcal{L}_V\xi - \mathcal{L}_W\eta + d(\iota_W \eta) + \iota_X\iota_Y\phi $$
for any vector $v\in TM$ and covector $\eta\in T^\ast M$

There are several nice examples that illustrates how the Dirac structure unifies various structures

Examples

Poisson and presymplectic structures

The Dirac structures are related to Poisson structure via the sharp map $\pi^\sharp:T^\ast M\to TM$ of a bivector field $\pi\in\Gamma(\wedge^2 TM)$ on $M$. Let $L_\pi$ be the graph of $\pi^\sharp$, then $L_{\pi}$ is a Dirac structure if $\pi$ is a Poisson bivector. It is useful to define Poisson structure in this way, so if the graph $L_\pi$ is a $\phi$-twisted Dirac structure, then $\pi$ is a $\phi$-twisted Poisson structure. This is ‘iff’ when $TM\cap L_\pi = \{0\}$.

The integrality condition for the Dirac structure thus defined translates to the Jacobi identity for the bivector field: $$ [\pi,\pi]=0, $$ or the twisted Jacobi identity $$ [\pi,\pi]=\wedge^3\pi^\sharp(\phi). $$

Similarly given a presymplectic structure $\omega$, via the flat map $\omega_\flat: TM \to T^\ast M$, the graph $L_\omega$ defines a Dirac structure iff $TM\cap L_\omega = \{0\}$. The integrality becomes $$ d\omega =0 $$ and for twisted case $$ d\omega + \phi =0, $$ where $\omega$ is now called a $\phi$-twisted presymplectic structure.

Twisting and Cartan-Dirac structures

In the context of Lie theory $\phi$ appear as the Cartan 3-form on some Lie group $G$ whose integral over a pullback $W\to G$ is the WZW Lagrangian.

Let $G$ be a Lie group whose Lie algebra $\mathfrak{g}$ is equipped with a nondegenerate $\operatorname{ad}$-invariant symmetric bilinear form $(\cdot,\cdot)_{\mathfrak{g}}$ (e.g. a compact Lie group) that identifies $TG$ and $T^\ast$ in $TG\oplus TG\sim TG\oplus T^\ast G$. Consider the maximal isotropic subbundle $$ L_G:=((v_r-v_l,\frac{1}{2}(v_r+v_l))| v\in \mathfrak{g}) $$ where $v_r$ and $v_l$ are respectively right and left invariant vector fields corresponding to $v$ (obtained by contraction with the Maurer-Cartan form). This is a $\phi^G$-twisted Dirac structure, called the Cartan-Dirac structure on $G$ associated with $(\cdot,\cdot)_\mathfrak{g}$. The twisting is the bi-invariant Cartan 3-form on $G$ defined on Lie algebra elements by $$ \phi^G(u,v,w) = \frac{1}{2}(u,[v,w])_{\mathfrak{g}}. $$

The Cartan-structure on $G$ is foloated by twisted presymplectic leaves which are the connected components of the conjugacy classes of $G$. Let $v_G = v_r - v_l$, the corresponding twisted presymplectic forms can be written as $$ \theta_g(v_G,w_G) :=\frac{1}{2}((\operatorname{Ad}_{g^{-1}}-\operatorname{Ad}_g)v,w)_{\mathfrak{g}} $$ at $g\in G$.

These are closely related to the theory of quasi-Hamiltonian spaces and group-valued momentum maps and to quasi-Poisson manifolds.

Momentum level sets

Let $M$ be a Poisson manifold and $\mu:M\to \mathfrak{g}^\ast$ be the moment map for a Hamiltonian action of a Lie group $G$. Let $\xi\in\mathfrak{g}^\ast$ be a regular value for $\mu$ and let $G_\xi$ be the stabilizer at $\xi$ w.r.t. to the coadjoint action.

Consider $Q = \mu^{-1}(\xi) \hookrightarrow M$. Let $L_Q$ be pointwisely given by $$ (L_Q)_x := \frac{L_x \cap (T_x Q\oplus T^\ast_x M)}{L_x \cap T_x Q^\circ} $$ for $x\in Q$. The stabilizer groups of the $G_\xi$-action on $Q$ have constant dimensions (this is the case whenever the $G_\xi$-orbits on $Q$ have constant dimension) iff $L_Q$ is a Dirac structure on $Q$.

This means for good enough $G$-actions on the level sets (“constrained submanifolds”) of a momentum map there are Dirac structures canonically defined.

Foliation theory

Given a twisted Dirac structure $L\subset TM\oplus T^\ast M$ on $M$, the image $\operatorname{pr}_1(L)$ is a integrable singular distribution, thus determines a decomposition of $M$ into leaves $\mathcal{O}$, satisfying $$ T_x \mathcal{O} = \operatorname{pr}_1 (L)_x $$ for all $x\in M$, where $\operatorname{pr}_1: TM\oplus T^\ast M\to TM$.

Each leaf of a twisted Dirac manifold $M$ is naturally equipped with a twisted presymplectic 2-form $\theta$ which is pointwisely $\theta_x$. The 2-form is defined for, untwisted Dirac structures, by $\theta_x(v,w) = \alpha(w)$ where $v,w \in T_x\mathcal{O}$ and $\alpha \in T^\ast_x\mathcal{O}$ s.t. $(v,\alpha)\in L$. With a twist $\phi$, $\theta$ is twisted by the pullback of $\phi$ to each leaf.

The topologically closed family of subspaces $TP\cap L =\operatorname{Ker}(\theta) \subset TP$ is called the characteristic distribution of $L$ and is denoted by $\operatorname{Ker}(\theta)$. It is always contained in $\operatorname{pr}_1(L)$, and when it has constant fiber dimension, it is integrable iff at each point of $M$ $$ \phi(X,Y,Z) = 0 \quad \forall X,Y\in \operatorname{Ker}(\theta), Z\in \operatorname{pr}_1(L). $$ The leaves of the corresponding characteristic foliations are then the null spaces of the presymplectic forms along the leaves (“Lagrangian subspaces”).

On each leaf $\iota:\mathcal{O}\hookrightarrow M$, the 2-form $\theta$ is basic w.r.t the characteristic foliation (i.e. is a pullback of a differential form on the base manifold) iff $$ \operatorname{Ker}(\theta)\subset \operatorname{Ker}(\iota^\ast\phi) $$ at all points of $\mathcal{O}$. In this case, forming the leaf space of the foliation locally produces a quotient manifold bearing a singualr foliation by twisted symplectic leaves, i.e. it is a twisted Poisson manifold.

Remark. If $L=L_\pi$ is determined by a Poisson structure $\pi$, then $\operatorname{pr}_1(L) = \pi^\sharp(T^\ast M)$, the characteristic distribution of the Poisson manifold $(M,\pi)$.