M-theory and T-geometry: Higgs branch moduli and charged matter

Abstract: M-theory geometric engineering on manifolds of special holonomy yields a rich class of novel field theories. In this paper, we construct new 3d $\mathcal{N}=2^{\ast}$ and $\mathcal{N}=4^{\ast}$ gauge theories, realized as mass-deformations of theories with 16 supercharges, within this framework. These arise from non-compact 8d geometries given by fibrations of $\mathbb{R}^{4}/Γ_{ADE}$ over Biberbach 4-manifolds. The existence of consistent $Spin(7)$-structures on the 8d spaces requires the rotational holonomy of the Biberbach spaces to act on the $Sp(1)$-structure of the fibers. Furthermore, we analyze Higgsing the 7d $\mathcal{N}=1$ $ADE$ gauge theories induced by the action of a permutation group on the centres of the corresponding $\mathbb{R}^{4}/Γ_{ADE}$ spaces. We show that this operation admits a natural interpretation in terms of nilpotent, upper-triangular, Higgsing, although it breaks supersymmetry. Supersymmetry is restored by fibering the singular geometry over a compact internal space, whose structure group is chosen to coincide with the permutation group to implement the nilpotent Higgsing. We refer to such backgrounds as T-geometries, where ``T'' denotes the triangular nature of the nilpotent Higgsing. Within this framework, we investigate the nilpotent Higgsing of the 3d $\mathcal{N}=2^{\ast}$ and 4d $\mathcal{N}=1^{\ast}$ theories, where the rotational holonomy groups of the Bieberbach spaces realize the permutation groups. We demonstrate that the Higgs branch moduli are encoded by specific elements of the Slodowy slices associated with nilpotent elements. Moreover, we demonstrate that additional elements of the same slice give rise to non-chiral charged matter under the unbroken gauge algebra. We establish that both the Higgs branch moduli and the charged matter are massless and admit a natural interpretation as localized matter.