# 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 $\eta\in T^\ast M$

• 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.

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 $(M,\pi)$.