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

TL;DR

The paper discovers new vacuum solutions in ${ m N}=4$ super-Yang-Mills and matrix models involving self-intersecting fuzzy branes, revealing novel geometric structures with chiral zero modes and family symmetries.

Contribution

It introduces new self-intersecting fuzzy brane solutions in super-Yang-Mills and matrix models, with detailed analysis of their geometry, stability, and zero mode spectrum.

Findings

Solutions have lower energy than trivial vacuum

Existence of chiral fermionic zero modes

Solutions exhibit a $ ext{Z}_3$ family symmetry

Abstract

We find new vacuum solutions of ${\cal N}=4$ super-Yang-Mills with totally anti-symmetric cubic soft SUSY breaking terms, or equivalently solutions of the IKKT matrix model of type $\mathbb{R}^4_\theta \times {\cal K}_N$ with flux terms. The solutions can be understood in terms of 4- and 6- dimensional fuzzy branes ${\cal K}_N$ 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-intersections 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.