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.

Figures (4)

• Figure 1: The figure illustrates the correspondence between the action of the group $H$ on the centres of the resolution of $\mathbb{R}^{4}/\mathbb{Z}_{N}$ with nilpotent Higgsing and particular elements of the Slodowy slice $\mathcal{S}_{e}$. These elements correspond to the geometric Higgs branch moduli.
• Figure 2: The nodes represents the two centres of the $\widetilde{\mathbb{R}^{4}/\mathbb{Z}_{2}}$ geometry. The line between the two centres represent a 2-sphere of the creapant resolution. The red arrow represent the $\mathbb{Z}_{2}^{H}$ action on the two centres, which can also be understood as $\vec{x}_{1}\sim\vec{x}_{2}$ .
• Figure 3: Here, the nodes represent the four centres of the $\widetilde{\mathbb{R}^{4}/\mathbb{Z}_{4}}$ geometry. From left to right, we label the centers by $\vec{x}_{1}$, $\vec{x}_{2}$, $\vec{x}_{3}$, and $\vec{x}_{4}$. The lines between these nodes represent the 2-spheres corresponding to the Cartan of $\mathfrak{su}(4)$. These centres are arranged in pairs under a $\mathbb{Z}_{2}$, say that generated by $A$ given in \ref{['eq:rotation-hol-B49']}; that we have $\vec{x}_{1}\sim \vec{x}_{4}$ and $\vec{x}_{2}\sim \vec{x}_{3}$. Upon taking the $H$ quotient, only one 2-sphere survive. There are another two equivalent configurations given as: $\vec{x}_{1}\sim \vec{x}_{2}$ and $\vec{x}_{3}\sim \vec{x}_{4}$ and $\vec{x}_{1}\sim \vec{x}_{3}$ and $\vec{x}_{2}\sim \vec{x}_{4}$. Their corresponding $P_{[2,2]}$ can be easily constructed.
• Figure 4: The figure illustrates a nilpotent element $e$ together with its orbit $\mathcal{O}_{e}$, generated by the adjoint action of the group on $e$. The tangent space at $e$, $T_{e}\mathcal{O}_{e} = \mathrm{im}(\mathrm{ad}_{e})$, admits a natural physical interpretation as the space of linearized gauge transformations. The transverse fluctuations at $e$ are captured by the Slodowy slice $\mathcal{S}_{e}$, which we interpret as the space of physical degrees of freedom. The figure is adapted from section 1.5 of slodowy1980four.