Reduction of Twisted Generalized Kähler Structure

We show that our earlierwork in [LT05] extends to the twisted case, that is, we defined a notion of moment map and reduction in both twisted generalized complex geometry and twisted generalized Kähler geometry. This is a short note to show that our results in our previously posted paper, Symmetries in generalized Kähler geometry [LT05], can be easily extended to the case of twisted generalized geometries with only minor modifications. In particular, we define twisted generalized complex reduction and twisted generalized Kähler reduction. We hope that this goes some way towards providing the framework which Kapustin and Li suggested would be useful [KL]. This note is not intended as a completely separate paper, but should rather be read in conjunction with [LT05]. In particular, we will not repeat the brief history of this subject which can be found there, except to reiterate that it was first introduced in [H02] and developed much further in [Gua04]. However, it is important tomention here that closely relatedworkswhich have appeared since we first posted. We have not had an opportunity to carefully read these papers but would like to at least attempt to describe some basic similarities and differences. Shengda Hu wrote a related paper [Hu05], which was partially inspired by an early version of our paper [LT05] which we gave him in early June. His paper includes a notion of twisted complex reduction which is very similar to ours in the untwisted case. More generally, he considers twisted complex structures in the framework of Hamiltonian symmetry. The work in the current note was completed after his paper appeared, so also cannot be considered as fully independent. Ourmain motivation is to demonstrate twisted generalized Kähler reduction, which is not in [Hu05]. However, it is worth noting that even in the twisted generalized complex case our results are slightly different. In [SX05], Stiénon and Ping independently develop notions of generalized complex and Kähler reduction which seems rather different from ours in both perspective and techniques. In particular, instead of working with generalized moment maps, they consider quotients of arbitrary subsets; so Date: February 2, 2008.