AdS/CFT with Tri-Sasakian Manifolds

TL;DR

This work analyzes AdS4/CFT with M2 branes at the tip of toric hyper‑Kähler cones, whose gravity dual is AdS4×X7 with X7 a Tri‑Sasakian manifold. It develops a localization–equivariant cohomology framework to compute volumes of X7 and its supersymmetric 5‑cycles from toric data, yielding the protected conformal dimension relation $\Delta = {\pi N\over 6}{\rm Vol}(\Sigma_5)\over {\rm Vol}(X_7)$ and, remarkably, $\Delta = N/2$ for baryonic operators independent of the toric charges. The paper also proposes UV quiver gauge theories whose Higgs branches flow to the ${\cal N}=3$ SCFT in the IR, with operator spectra matched to wrapped M5‑brane configurations. Overall, it provides a tractable geometric route to protected data in 3D ${\cal N}=3$ SCFTs and a concrete UV/IR bridging proposal via toric hyper‑Kähler quotients.

Abstract

We consider generic toric Tri-Sasakian 7-manifolds X_7 in the context of M-theory on AdS_4 X X_7 and study their AdS/CFT correspondence to N=3 SCFT in 3D spacetime. We obtain volumes of Tri-Sasakian manifolds and their supersymmetric 5-cycles via cohomological integration technique, and use this to calculate conformal dimensions of baryonic operators in the SCFT side. We also propose quiver-type gauge theories for UV description of the corresponding N=3 SCFT.