# The $\mathcal{N}_3=3\to \mathcal{N}_3=4$ enhancement of Super Chern-Simons theories in $D=3$, Calabi HyperK\"ahler metrics and M2-branes on the $\mathcal{C}(\mathrm{N^{0,1,0}})$ conifold

## TL;DR

This paper extends supersymmetry enhancement in 3D super Chern-Simons theories from flat to curved HyperK"ahler manifolds, exploring geometric conditions, specific metrics, and implications for M2-branes and dual theories.

## Contribution

It introduces conditions for supersymmetry enhancement on curved HyperK"ahler manifolds and identifies Calabi metrics, including the N^{0,1,0} cone resolution, linking geometry to M-theory compactifications.

## Key findings

- Derived conditions for supersymmetry enhancement on curved HyperK"ahler manifolds.
- Identified Calabi metrics compatible with enhancement, including the N^{0,1,0} cone resolution.
- Connected geometric structures to M2-brane solutions and dual supergroup Chern-Simons theories.

## Abstract

Considering matter coupled supersymmetric Chern-Simons theories in three dimensions we extend the Gaiotto-Witten mechanism of supersymmetry enhancement $\mathcal{N}_3=3\to \mathcal{N}_3=4$ from the case where the hypermultiplets span a flat HyperK\"ahler manifold to that where they live on a curved one. We derive the precise conditions of this enhancement in terms of generalized Gaiotto-Witten identities to be satisfied by the tri-holomorphic moment maps. An infinite class of HyperK\"ahler metrics compatible with the enhancement condition is provided by the Calabi metrics on $T^\star \mathbb{P}^{n}$. In this list we find, for $n=2$ the resolution of the metric cone on $\mathrm{N}^{0,1,0}$ which is the unique homogeneous Sasaki Einstein 7-manifold leading to an $\mathcal{N}_4=3$ compactification of M-theory. This leads to challenging perspectives for the discovery of new relations between the enhancement mechanism in $D=3$, the geometry of M2-brane solutions and also for the dual description of super Chern Simons theories on curved HyperK\"ahler manifolds in terms of gauged fixed supergroup Chern Simons theories. The relevant supergroup is in this case $\mathrm{SU(3|N)}$ where $\mathrm{SU(3)}$ is the flavor group and $\mathrm{U(N)}$ is the color group.