The N₃=3→N₃=4 enhancement of super Chern–Simons theories in D=3, Calabi HyperKähler metrics and M2-branes on the C(N^0,1,0) conifold

Considering matter coupled supersymmetric Chern–Simons theories in three dimensions we extend the Gaiotto–Witten mechanism of supersymmetry enhancement N₃=3→N₃=4 from the case where the hypermultiplets span a flat HyperKähler 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 HyperKähler metrics compatible with the enhancement condition is provided by the Calabi metrics on T^⋆Pⁿ. In this list we find, for n=2 the resolution of the metric cone on N^0,1,0 which is the unique homogeneous Sasaki–Einstein 7-manifold leading to an N₄=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 HyperKähler manifolds in terms of gauged fixed supergroup Chern–Simons theories.