The N3=3→N3=4 enhancement of super Chern–Simons theories in D=3, Calabi HyperKähler metrics and M2-branes on the C(N0,1,0) conifold

Abstract Considering matter coupled supersymmetric Chern–Simons theories in three dimensions we extend the Gaiotto-Witten mechanism of supersymmetry enhancement N 3 = 3 → N 3 = 4 from the case where the hypermultiplets span a flat HyperKahler 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 HyperKahler metrics compatible with the enhancement condition is provided by the Calabi metrics on T ⋆ P n . 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 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 HyperKahler manifolds in terms of gauged fixed supergroup Chern Simons theories.