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M5-branes on S^2 x M_4: Nahm's Equations and 4d Topological Sigma-models

Benjamin Assel, Sakura Schafer-Nameki, Jin-Mann Wong

TL;DR

This work analyzes the 6d $N=(0,2)$ theory on $S^2 imes M_4$ and derives a 4d topological sigma-model on $M_4$ whose target is the Nahm monopole moduli space, equivalently the moduli space of solutions to Nahm's equations with Nahm poles. The reduction is implemented via a 5d SYM framework on an interval, followed by a dimensional reduction to 4d, yielding a sigma-model into $oldsymbol{ rak M}_k$ with couplings controlled by the $S^2$ area, and, in Hyper-Kähler $M_4$, a tri-holomorphic map structure. For $k=2$, the target is the Atiyah–Hitchin manifold, producing a 4d theory with scalars and self-dual two-forms; in the abelian limit the theory reduces to a free topological sigma-model, while the non-abelian cases connect to Bagger–Witten–type descriptions. The paper further establishes a 4d–2d correspondence between the 4d topological sigma-model and a 2d half-twisted $(0,2)$ theory on $S^2$, and discusses various twists, hyper-Kähler geometries, and potential applications to localization and dualities.

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.

M5-branes on S^2 x M_4: Nahm's Equations and 4d Topological Sigma-models

TL;DR

This work analyzes the 6d theory on and derives a 4d topological sigma-model on whose target is the Nahm monopole moduli space, equivalently the moduli space of solutions to Nahm's equations with Nahm poles. The reduction is implemented via a 5d SYM framework on an interval, followed by a dimensional reduction to 4d, yielding a sigma-model into with couplings controlled by the area, and, in Hyper-Kähler , a tri-holomorphic map structure. For , the target is the Atiyah–Hitchin manifold, producing a 4d theory with scalars and self-dual two-forms; in the abelian limit the theory reduces to a free topological sigma-model, while the non-abelian cases connect to Bagger–Witten–type descriptions. The paper further establishes a 4d–2d correspondence between the 4d topological sigma-model and a 2d half-twisted theory on , and discusses various twists, hyper-Kähler geometries, and potential applications to localization and dualities.

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.

Paper Structure

This paper contains 46 sections, 311 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Topological Twists and Supergravity Backgrounds
  3. Twists of the M5-brane on $M_4$
  4. Twisting on $S^2$
  5. Supergravity Background Fields
  6. Killing spinors
  7. From 6d $(0,2)$ on $S^2$ to 5d SYM
  8. The 6d $(0,2)$ Theory
  9. 5d SYM in the Supergravity Background
  10. Cylinder Limit
  11. Nahm's Equations and Boundary Considerations
  12. Nahm's Equations and 4d Sigma-Model
  13. Poles and Monopoles
  14. Reduction to the 4d Sigma-Model
  15. Scalars
  16. ...and 31 more sections

Figures (2)

  • Figure 1: 4d-2d correspondence between the reduction of the 6d $(0,2)$ theory on $M_4$ to a 2d $(0,2)$ SCFT on $S^2$, and the 'dual' 4d topological sigma-model from $M_4$ into the Nahm or monopole moduli space, which is obtained in this paper by reducing the 6d theory on a two-sphere.
  • Figure 2: The dimensional reduction of the 6d $N=(0,2)$ theory on an $S^2$, viewed as a circle-fibration along an interval $I$, is determined by dimensional reduction via 5d SYM. The scalars of the 5d theory satisfy the Nahm equations, with Nahm pole boundary conditions at the endpoints of the interval. The 4d theory is a topological sigma-model into the moduli space of solutions to these Nahm equations, or equivalently the moduli space of monopoles.