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We define a new correspondence for pairs $(\mathbb{F},H)$ formed by a flag manifold $\mathbb{F}$ together with an $H$-flux on $\mathbb{F}$. Given its role within our correspondence, infinitesimal $T$-duality may be viewed as a source of $H$-flux, in the sense that it contributes towards taking fluxless pairs $(\mathbb{F},0)$ to pairs $(\mathbb{F}^\vee, H^\vee)$ carrying nontrivial flux $H^\vee\neq 0$. We also illustrate how our correspondence exchanges complex structures with symplectic ones up to $B$-transformations.

In this paper we describe all invariant complex Dirac structures with constant real index on a maximal flag manifold in terms of the roots of the Lie algebra which defines the flag manifold. We also completely classify these structures under the action of $B$-transformations.

We characterize those real flag manifolds that can be endowed with invariant generalized almost complex structures. We show that no $GM_2$-maximal real flag manifolds admit integrable invariant generalized almost complex structures. We give a concrete description of the generalized complex geometry on the maximal real flags of type $B_2$, $G_2$, $A_3$, and $D_l$ with $l\geq 5$, where we prove that the space of invariant generalized almost complex structures under invariant $B$-transformations is homotopy equivalent to a torus and we classify all invariant generalized almost Hermitian structures on them., Comment: 34 pages. Minor changes have been made. Final version to appear in The Journal of Geometric Analysis

We describe moduli spaces of invariant generalized complex structures and moduli spaces of invariant generalized K\"ahler structures on maximal flag manifolds under $B$-transformations. We give an alternative description of the moduli space of generalized complex structures using pure spinors, and describe a cell decomposition of these moduli spaces induced by the action of the Weyl group., Comment: 29 pages. The title has been changed. Final version to appear in Journal of Geometry and Physics

The aim of this paper is to classify all invariant generalized complex structure on a partial flag manifold $\mathbb{F}_\Theta$ with at most four isotropy summands. To classify them all we proved that an invariant generalized almost complex structure on $\mathbb{F}_\Theta$ is `constant' in each component of the isotropy representation.

Let $G$ be a complex semi-simple Lie group and form its maximal flag manifold $\mathbb{F}=G/P=U/T$ where $P$ is a minimal parabolic subgroup, $U$ a compact real form and $T=U\cap P$ a maximal torus of $U$. The aim of this paper is to study invariant generalized complex structures on $\mathbb{F}$. We describe the invariant generalized almost complex structures on $\mathbb{F}$ and classify which one is integrable. The problem reduces to the study of invariant $4$-dimensional generalized almost complex structures restricted to each root space, and for integrability we analyse the Nijenhuis operator for a triple of roots such that its sum is zero. We also conducted a study about twisted generalized complex structures. We define a new bracket `twisted' by a closed $3$-form $\Omega $ and also define the Nijenhuis operator twisted by $\Omega $. We classify the $\Omega $-integrable generalized complex structure.

We show that the partition function on a generalised conifold $C_{m,n}$ with ${m+n \choose m}$ crepant resolutions can be equivalently computed on the compound du Val singularity $A_{m+n-1}\times \mathbb C$ with a unique crepant resolution., Comment: v2 has small corrections, has improved exposition. Revised version to appear in: "Journal of Mathematical Physics"