coisotropic submanifold

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

$\pi \;\colon \; T^* X \to T X$

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

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

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

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.

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

(adapted from Ševera 00)

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)

Revised on March 27, 2013 19:13:51
by Urs Schreiber
(89.204.154.37)