# nLab symplectic dual pair

Contents

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Idea

A kind of correspondence space between Poisson manifolds.

## Definition

###### Definition

For $(X_1, \pi_1)$ and $(X_2, \pi_2)$ two Poisson manifolds, a symplectic dual pair between them is a correspondence symplectic manifold $(Z,\pi_Z = \omega^{-1})$

$(X_1, \pi_1) \stackrel{i_1}{\leftarrow} (Z,\omega) \stackrel{i_2}{\to} (X_2, \pi_2)$

such that for all $f_1 \in C^\infty(X_1)$ and $f_2 \in C^\infty(X_2)$ the Poisson bracket on $Z$ vanishes:

$\{i_1^\ast f_1, i_2^\ast f_2\}_{\omega} = 0 \,.$

## Examples

###### Example

A Lagrangian correspondence between symplectic manifolds, regarded as Poisson manifolds, is a symplectic dual pair.

## References

The notion was introduced in

• M.V. Karasev, (1989), The Maslov quantization conditions in higher cohomology and analogs of notions developed in Lie theory for canonical fibre bundles of symplectic manifolds I, II Selecta Math. Soviet. 8, 213–234, 235–258.

and

• Alan Weinstein, The local structure of Poisson manifolds, J. Diff. Geom. 18, 523–557 (1983)

A review with an eye towards geometric quantization with codomain KK-theory (geometric quantization by push-forward) is in section 3 of

Last revised on December 13, 2017 at 05:47:37. See the history of this page for a list of all contributions to it.