nLab
coisotropic submanifold

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

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

π:T *XTX \pi \;\colon \; T^* X \to T X

to the conormal bundle N *ST *SN^* S \hookrightarrow T^* S factors through the tangent bundle TST S

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

Equivalently, SXS\hookrightarrow X is coisotropic if the subalgebra of C (X)C^\infty(X) of functions vanishing on SS 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=1n = 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

nn \in \mathbb{N}symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of (n+1)(n+1)-d sigma-modelhigher symplectic geometry(n+1)(n+1)d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension (n+1)(n+1)discussed in:
0symplectic manifoldsymplectic manifoldsymplectic geometryLagrangian submanifoldordinary space of states (in geometric quantization)geometric quantization
1Poisson Lie algebroidsymplectic groupoid2-plectic geometryPoisson sigma-modelcoisotropic submanifold (of underlying Poisson manifold)brane of Poisson sigma-model2-module = category of modules over strict deformation quantiized algebra of observablesextended geometric quantization of 2d Chern-Simons theory
2Courant Lie 2-algebroidsymplectic 2-groupoid3-plectic geometryCourant sigma-modelDirac structureD-brane in type II geometry
nnsymplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometryd=n+1d = 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

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

and discussed in more detail in section 7.2 of

Comments on higher algebra aspects are in the slides

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