Towards Mass Deformed N=4 SO(n) and Sp(k) gauge configurations

TL;DR

The paper advances the brane realization of mass-deformed $N=4$ theories with orthogonal and symplectic gauge groups by incorporating orientifold six-planes into elliptic (M-theory) models, yielding Seiberg-Witten curves as spectral covers on $E_\tau$-based geometries. It analyzes two principal orientifold backgrounds, $O6^+$-$O6^-$ and $O6^-$-$O6^-$, to construct a wide class of $\beta=0$ theories and derives explicit SW curves for these models, including detailed discussions of flat directions and dualities. A geometric interpretation of Montonen-Olive duality emerges from the interplay of poles and zeros on the torus, providing insights into dual descriptions among $Sp(k)$ and $SO(2k+1)$ factors. The work also investigates the introduction of a nonvanishing hypermultiplet mass (global mass) within these frameworks, outlining potential schemes by which adjoint masses can be incorporated and matched to gauge-theory expectations. Together, these results extend the elliptic-model toolkit to finite and mass-deformed theories with $SO(n)$ and $Sp(k)$ gauge groups, clarifying dualities and Coulomb-branch structures in a broad class of four-dimensional $N=2$ theories.

Abstract

We study the introduction of orientifold six-planes in the type IIA brane configurations known as elliptic models. The N=4 SO(n) and $Sp(k)$ theories softly broken to N=2 through a mass for the adjoint hypermultiplet can be realized in this framework in the presence of two orientifold planes with opposite RR charge. A large class of $\b=0$ models is solved for vanishing sum of hypermultiplet masses by embedding the type IIA configuration into M-theory. We also find a geometric interpretation of Montonen-Olive duality based on the properties of the curves. We make a proposal for the introduction of non-vanishing sum of hypermultiplet masses in a sub-class of models. In the presence of two negatively charged orientifold planes and four D6-branes other interesting $β=0$ theories are constructed, e.g. $Sp(k)$ with four flavours and a massive antisymmetric hypermultiplet. We comment on the difficulties in obtaining the curves within our framework due to the arbitrary positions of the D6-branes.