## 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

*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

- (Lagrangian) Maximally isotropic with respect to the symmetric pairing $$ \langle V\oplus \eta,W\oplus\xi\rangle_+ = \iota_V\xi + \iota_W\eta = \xi(V)+\eta(W) $$
- (Integrability) the sections are involutive (closed) under the
*Courant bracket*$$ [V\oplus\eta,W\oplus\xi]:= [V,W]\oplus\mathcal{L}_V\xi - \mathcal{L}_W\eta + d(\iota_W \eta) $$

There is also a twisted version of Dirac structure

*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

- (Lagrangian) Maximally isotropic with respect to the symmetric pairing $$ \langle V\oplus \eta,W\oplus\xi\rangle_+ = \iota_V\xi + \iota_W\eta = \xi(V)+\eta(W) $$
- (Integrability) The sections are involutive (closed) under the $\phi$-
*twisted Courant bracket*$$ [V\oplus\eta,W\oplus\xi]^\phi:= [V,W]\oplus\mathcal{L}_V\xi - \mathcal{L}_W\eta + d(\iota_W \eta) + \iota_X\iota_Y\phi $$

- Here it is the
*notion*of Dirac structure that is twisted rather than the Dirac structure. The terminology is consistent with terms such as “twisted sheaf” and “twisted cohomology”. - Without the integrability conditions these are
*almost Dirac structure*and $\phi$-*twisted almost Dirac structure*.

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.

*characteristic distribution*of the Poisson manifold $(M,\pi)$. ◈