# Non-toric Cones and Chern-Simons Quivers

## TL;DR

This paper derives a new integral formula for the volume of non-toric tri-Sasaki Einstein manifolds from nonabelian hyperkahler quotients, enabling tests of non-toric AdS4/CFT3 dualities in M-theory.

## Contribution

It generalizes existing volume formulas from Abelian to nonabelian quotients, providing a tool for analyzing non-toric manifolds in AdS/CFT correspondence.

## Key findings

- Derived an integral volume formula for non-toric manifolds.
- Computed volumes for non-toric ADE quiver manifolds.
- Confirmed field theory predictions with perfect agreement.

## Abstract

We obtain an integral formula for the volume of non-toric tri-Sasaki Einstein manifolds arising from nonabelian hyperkahler 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 AdS$_{4}\times Y_{7}$ vacua of M-theory and their field theory duals. As an application, we consider 3d $\mathcal N=3$ Chern-Simons theories with affine ADE quivers. While the $\widehat A$ series corresponds to toric $Y_{7}$, the $\widehat D$ and $\widehat E$ 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 AdS$_4$/CFT$_3$ dualities.