Deformation classes in generalized Kähler geometry

We describe natural deformation classes of generalized Kähler structures using the Courant symmetry group, which determine natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. We show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.

U -duality extension of Drinfel’d double

A family of algebras $\mathcal{E}_n$ that extends the Lie algebra of the Drinfel’d double is proposed. This allows us to systematically construct the generalized frame fields $E_A{}^I$ which realize the proposed algebra by means of the generalized Lie derivative, i.e., $\hat{\pounds}_{E_A}E_B{}^I =-\mathcal{F}_{AB}{}^C\,E_C{}^I$. By construction, the generalized frame fields include a twist by a Nambu–Poisson tensor. A possible application to the non-Abelian extension of $U$-duality and a generalization of the Yang–Baxter deformation are also discussed.

The Diffeomorphism Type of Canonical Integrations of Poisson Tensors on Surfaces

A surface ∑ endowed with a Poisson tensor π is known to admit a canonical integration, 𝒢(π), which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if π is not an area form on the 2-sphere, then 𝒢(π) is diffeomorphic to the cotangent bundle T*∑. This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.