The $H$-flux on flag manifolds generated by infinitesimal $T$-duality

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 $H^{\vee }\mathrm{\ne }0$. We also illustrate how our correspondence exchanges complex structures with symplectic ones up to $B$-transformations.