Hamiltonian symmetries and reduction in generalized geometry

TL;DR

This paper extends Hamiltonian symmetry concepts and reduction techniques from symplectic geometry to generalized complex geometry, incorporating twists by closed 3-forms and exploring topology changes during cutting.

Contribution

It introduces H-twisted Hamiltonian symmetries, defines Hamiltonian actions and moment maps in generalized complex geometry, and constructs a reduction procedure analogous to symplectic reduction.

Findings

Defined H-twisted Hamiltonian symmetries.

Constructed Hamiltonian reduction in generalized complex geometry.

Showed topology change during cutting involves twisting.

Abstract

A closed 3-form $H \in \Omega^3_0(M)$ defines an extension of $\Gamma(TM)$ by $\Omega^2_0(M)$. This fact leads to the definition of the group of $H$-twisted Hamiltonian symmetries $\Ham(M, \JJ; H)$ as well as Hamiltonian action of Lie group and moment map in the category of (twisted) generalized complex manifold. The Hamiltonian reduction in the category of generalized complex geometry is then constructed. The definitions and constructions are natural extensions of the corresponding ones in the symplectic geometry. We describe cutting in generalized complex geometry to show that it's a general phenomenon in generalized geometry that topology change is often accompanied by twisting (class) change.