Generalized Kähler geometry and the pluriclosed flow

In Streets and Tian (2010) [1] the authors introduced a parabolic flow for pluriclosed metrics, referred to as pluriclosed flow. We also demonstrated in Streets and Tian (2010) (preprint) [2] that this flow, after certain gauge transformations, gives a class of solutions to the renormalization group flow of the nonlinear sigma model with B-field. Using these transformations, we show that our pluriclosed flow preserves generalized Kähler structures in a natural way. Equivalently, when coupled with a nontrivial evolution equation for the two complex structures, the B-field renormalization group flow also preserves generalized Kähler structure. We emphasize that it is crucial to evolve the complex structures in the right way to establish this fact.