The M theory lift of two O6 planes and four D6 branes

TL;DR

This work constructs the M-theory lift of a brane configuration with two O6$^-$ planes and four D6 branes, realizing a class of 4D $ ext{N}=2$ theories via holomorphic embeddings of M5 branes in a hyperkähler background $Q_M$. It provides explicit finite-genus Seiberg–Witten curves for the Coulomb branch by developing the background $Q_0$ (and its mass deformations) and incorporating the shift $M$ that corresponds to the antisymmetric mass; the M5 curves are extended to the shifted background $Q_M$ and organized to respect the $ ext{Z}_2$ orbifold. Consistency checks—including parameter counting, weak-coupling limits, and decoupling limits—confirm the curves reproduce known SP and SU theories and their mass/ coupling identifications. In an application, the paper embeds the $ ext{N}=4$ SU$(n)$ theory within an asymptotically free $ ext{N}=2$ framework to derive a subgroup of the S-duality group, providing exact duality-invariant structure for the coupling space and illustrating the power of geometric engineering in capturing nonperturbative dualities.

Abstract

We solve for the effective actions on the Coulomb branches of a class of N=2 supersymmetric theories by finding the complex structure of an M5 brane in an appropriate background hyperkahler geometry corresponding to the lift of two O6^- orientifolds and four D6 branes to M theory. The resulting Seiberg-Witten curves are of finite genus, unlike other solutions proposed in the literature. The simplest theories in this class are the scale invariant Sp(k) theory with one antisymmetric and four fundamental hypermultiplets and the SU(k) theory with two antisymmetric and four fundamental hypermultiplets. Infinite classes of related theories are obtained by adding extra SU(k) factors with bifundamental matter and by turning on masses to flow down to various asymptotically free theories. The N=4 supersymmetric SU(k) theory can be embedded in these asymptotically free theories, allowing a derivation of a subgroup of its S duality group as an exact equivalence of quantum field theories.