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Self-dual generalized metrics for pure $\mathcal{N}=1$ six-dimensional Supergravity

We geometrize six-dimensional pure $\mathcal{N}=1$ Supergravity by means of an exact Courant algebroid, whose Severa class is defined through the Supergravity three-form $H$, equipped with a generalized metric and a compatible, torsion-free, generalized connection. The Supergravity equations of motion follow from the vanishing of the Ricci curvature of the generalized metric, satisfying a natural notion of self-duality. This way, we interpret the solutions of six-dimensional pure, $\mathcal{N}=1$, Supergravity as generalized self-dual gravitational monopoles. For the D1-D5 black string solution, we explore the possibility of controlling space-time singularities by using $B$-field transformations.