Self-intersecting fuzzy extra dimensions from squashed coadjoint orbits in N = 4 $$ \mathcal{N}=4 $$ SYM and matrix models

Abstract

We find new vacuum solutions of $$ \mathcal{N}=4 $$ super-Yang-Mills with totally antisymmetric cubic soft SUSY breaking terms, or equivalently solutions of the IKKT matrix model of type $$ {\mathbb{R}}_{\theta}^4\times {\mathcal{K}}_N $$ with flux terms. The solutions can be understood in terms of 4- and 6-dimensional fuzzy branes $$ {\mathcal{K}}_N $$ in extra dimensions, describing self-intersecting projections of compact flag manifolds of SU(3). The 6-dimensional solutions provide a 6-fold covering of the internal space near the origin, while the 4-dimensional branes have a triple self-intersection spanning all 6 internal directions. The solutions have lower energy than the trivial vacuum, and we prove that there are no negative modes. The massless modes are identified explicitly. In particular there are chiral fermionic zero modes, linking the coincident sheets with opposite flux at the origin. They have a $$ {\mathbb{Z}}_3 $$ family symmetry, originating from the Weyl group rotations.