nLab
coisotropic submanifold

Definition

For $(X,\pi )$ a Poisson manifold, a submanifold $S\hookrightarrow X$ is called coisotropic if the restriction of the contraction map with the Poisson tensor

$$\pi :T^{*}X\to TX$$\pi \;\colon \; T^* X \to T X

to the conormal bundle $N^{*}S\hookrightarrow T^{*}S$ factors through the tangent bundle $TS$

$$\pi {\mid }_{N^{*}S}:N^{*}S\to TS\hookrightarrow TX.$$\pi|_{N^* S} \;\colon\; {N^* S} \to T S \hookrightarrow T X \,.

Equivalently, $S\hookrightarrow X$ is coisotropic if the subalgebra of $C^{\mathrm{\infty }}(X)$ of functions vanishing on $S$ is closed under the Poisson bracket.

Properties

Relation to Poisson Lie algebroids

A Poisson manifold induces a Poisson Lie algebroid, which is a symplectic Lie n-algebroid for $n=1$. Its coisotropic submanifolds correspond to the Lagrangian dg-submanifolds (see there) of this Poisson Lie algebroid.

Related concepts

• isotropic submanifold

• Dirac structure

• coisotropic calculus

∞-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

Surveys include

• Aïssa Wade, On the geometry of coisotropic submanifolds of Poisson manifolds (pdf)

The relation to the Poisson sigma-model is discussed in

• Alberto Cattaneo, Giovanni Felder, Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model, Lett. Math. Phys. 69 (2004) 157-175 (arXiv:math/0309180)

Characterization in terms of leaves of Lagrangian foliation of the Poisson Lie algebroid is mentioned in

• Pavol Ševera, Some title containing the words “homotopy” and “symplectic”, e.g. this one (arXiv:0105080)

and discussed in more detail in section 7.2 of

• Eli Hawkins, A groupoid approach to quantization, J. Symplectic Geom. Volume 6, Number 1 (2008), 61-125. (arXiv:math.SG/0612363)

Comments on higher algebra aspects are in the slides

• Florian Schätz, Homotopical algebra of coisotropic submanifolds (pdf)