M5-branes on S ² × M ₄: Nahm’s equations and 4d topological sigma-models

We study the 6d N = (0, 2) superconformal field theory, which describes multiple M5-branes, on the product space S² × M₄, and suggest a correspondence between a 2d N = (0, 2) half-twisted gauge theory on S² and a topological sigma-model on the four-manifold M₄. To set up this correspondence, we determine in this paper the dimensional reduction of the 6d N = (0, 2) theory on a two-sphere and derive that the four-dimensional theory is a sigma-model into the moduli space of solutions to Nahm’s equations, or equivalently the moduli space of k-centered SU(2) monopoles, where k is the number of M5-branes. We proceed in three steps: we reduce the 6d abelian theory to a 5d Super-Yang-Mills theory on I × M₄, with I an interval, then non-abelianize the 5d theory and finally reduce this to 4d. In the special case, when M₄ is a Hyper-Kähler manifold, we show that the dimensional reduction gives rise to a topological sigma-model based on tri-holomorphic maps. Deriving the theory on a general M₄ requires knowledge of the metric of the target space. For k = 2 the target space is the Atiyah-Hitchin manifold and we twist the theory to obtain a topological sigma-model, which has both scalar fields and self-dual two-forms.