3d Chern-Simons theory from M5-branes

A bstract We study 5d = 2 maximally supersymmetric Yang-Mills theory with a gauge group G on S 2 × M 3 , where M 3 is a 3-manifold. By explicit localization computation we show that the path-integral of the 5d = 2 theory reduces to that of the 3d Chern-Simons theory on M 3 , where is the complexification of G . This gives a direct derivation of the appearance of the Chern-Simons theory from the compactification of the 6d (2 , 0) theory, confirms the predictions from the 3d/3d correspondence for G = SU( N ), and suggests the generalization of the correspondence to more general gauge groups.