Non-toric cones and Chern-Simons quivers

We obtain an integral formula for the volume of non-toric tri-Sasaki Einstein manifolds arising from nonabelian hyperkähler quotients. The derivation is based on equivariant localization and generalizes existing formulas for Abelian quotients, which lead to toric manifolds. The formula is particularly valuable in the context of AdS₄ ×Y₇ vacua of M-theory and their field theory duals. As an application, we consider 3d N= 3 Chern-Simons theories with affine ADE quivers. While the  series corresponds to toric Y₇, the D^ and Ê series are non-toric. We compute the volumes of the corresponding seven-manifolds and compare to the prediction from supersymmetric localization in field theory, finding perfect agreement. This is the first test of an infinite number of non-toric AdS₄/CFT₃ dualities.