Non-toric Cones and Chern-Simons Quivers

We introduce an integral formula for the volume of non-toric tri-Sasaki Einstein manifolds arising from nonabelian hyperkähler quotients. The formula is particularly valuable in the context of AdS₄×Y₇ vacua of M-theory and their field theory duals. As its application, we consider 3d N=3 Chern-Simons theories with extended D quivers. We compute the volumes of the corresponding 7-manifolds and compare to the prediction from supersymmetric localization in field theory, finding perfect agreement. Reference: arXiv:1702.05486 [hep-th]