Coisotropic submanifolds and dual pairs

TL;DR

This paper explores the relationship between boundary conditions in the Poisson sigma model, coisotropic submanifolds, and dual pairs, establishing conditions under which the reduced phase space forms a dual pair and generalizing to tensor fields.

Contribution

It demonstrates that boundary conditions correspond to coisotropic submanifolds and that the reduced phase space forms a dual pair, extending the theory to general tensor fields and characterizing when evolution relations are Lagrangian.

Findings

Boundary conditions relate to coisotropic submanifolds.

Reduced phase space forms a dual pair under certain conditions.

Lagrangian evolution relations occur if and only if the tensor field is Poisson.

Abstract

The Poisson sigma model is a widely studied two-dimensional topological field theory. This note shows that boundary conditions for the Poisson sigma model are related to coisotropic submanifolds (a result announced in [math.QA/0309180]) and that the corresponding reduced phase space is a (possibly singular) dual pair between the reduced spaces of the given two coisotropic submanifolds. In addition the generalization to a more general tensor field is considered and it is shown that the theory produces Lagrangian evolution relations if and only if the tensor field is Poisson.