# String Theory Seminar

### (Para-)Quaternion-Kähler geometry from Supergravity (Part 2)

We review some recent results on quaternion-Kähler geometry which were obtained in the context supergravity and string theory.

1. By proving that the supergravity $r$-map and $c$-map preserve geodesic completeness, we established a method to construct non-homogeneous geodesically complete quaternion-Kähler manifolds starting from complete special Kähler or special real manifolds.
2. Deriving the hyper-Kähler/quaternion-Kähler correspondence using conification, we have obtained a new proof that spaces in the image of the $c$-map are quaternion-Kähler, which extends to include the one-loop quantum deformation.
3. Time permitting we will also discuss how some of the results can be adapted to para-quaternion-Kähler manifolds.