Relating conifold geometries to NS5-branes

We construct the first known example of a near horizon supergravity solution for a pair of NS5-branes, intersecting on R1,3${\mathbb{R}}^{1,3}$ and localised in all directions except a single transverse circle. We do this by establishing an explicit map between the conifold metric and the near horizon geometry of two intersecting NS5-branes, clarifying and correcting a number of open issues in the literature en route. Our technique is general in nature and may be applied to a whole class of 1/4-BPS five-brane webs and their geometric duals. These 1/4-BPS solutions may have an interesting holographic interpretation in terms of little string theory.