Abstract
We study the 6d N=(0,2) superconformal field theory, which describes multiple M5-branes, on the product space S^2 x M_4, and suggest a correspondence between a 2d N=(0,2) half-twisted gauge theory on S^2 and a topological sigma-model on the four-manifold M_4. 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 x M_4, with I an interval, then non-abelianize the 5d theory and finally reduce this to 4d. In the special case, when M_4 is a Hyper-Kahler 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 M_4 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.
Highlights
The six-dimensional N = (0, 2) superconformal theory (SCFT) with an ADE type gauge group is believed to describe the theory on multiple M5-branes
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
The particular background that we consider is a half-topological twist along the S2, together with a VafaWitten-like twist on M4, and we will find that the theory on M4 is a twisted version of a sigma-model into the moduli space of SU(2) monopoles with k centers, where k is the number of M5-branes, or equivalently, the moduli space of Nahm’s equations [3] with certain singular boundary conditions
Summary
The sigma-model into the one-monopole moduli space S1×R3, corresponding to the reduction of the abelian theory to a free 4d hypermultiplet, gives rise upon twisting to a (free) theory on M4 with a compact scalar and a self-dual two-form, and belongs to the class of 4d A-model of [30]. The reduction of the 6d theory on either T 2 or S2 with half-twist gives rather distinct 4d topological theories: in the former, the 4d N = 4 SYM theory with Vafa-Witten twist, in the latter, we find a four-dimensional topological sigma-model into the monopole moduli space, which for general M4 has both scalars as well as self-dual two-forms.
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