New $AdS_4\times X_7$ Geometries with $\mathcal{N}=6$ in M Theory

TL;DR

This work constructs a family of AdS$_4\times X_7$ solutions in M-theory where $X_7=S(t_1,t_2,t_3)$ is a tri-Sasakian Eschenburg space, obtained from an 8D hyperkähler cone via a $U(1)$ quotient. Using equivariant localization, it derives explicit, $t$-independent volume ratios for $X_7$ and its supersymmetric 5-cycles, and shows these feed into a proposed dual 3D ${\cal N}=3$ SCFT with gauge group $SU(N)_1\times SU(N)_2$ and baryonic operators with dimensions $\Delta = N/2$. The key results include a closed form for $\frac{\mathrm{vol}(X_7({\bf t}))}{\mathrm{vol}(S^7)}$ and a universal ratio $\frac{\mathrm{vol}(\Sigma_5)}{\mathrm{vol}(X_7)} = \frac{3}{\pi}$, both of which support matching operator dimensions via holography and provide a concrete geometric handle on the dual SCFT structure. These findings advance understanding of ${\cal N}=3$ holographic theories arising from M2-branes at Eschenburg-type singularities.

Abstract

We study supersymmetric AdS_4 x X_7 solutions of 11-dim supergravity where the tri-Sasakian space X_7 has generically U(1)^2\times SU(2)_R isometry. The compact and regular 7-dim spaces X_7=S(t_1,t_2,t_3) is originated from 8-dim hyperkahler quotient of a 12-dim flat hyperkahler space by U(1) and belongs to the class of the Eschenburg space. We calculate the volume of X_7 and that of the supersymmetric five cycle via localization. From this we discuss the 3-dim dual superconformal field theories with \CN=3 supersymmetry.