Interacting Topological Superconductors and possible Origin of $16n$ Chiral Fermions in the Standard Model

Abstract

Motivated by the observation that the Standard Model of particle physics (plus a right-handed neutrino) has precisely 16 Weyl fermions per generation, we search for $(3+1)$-dimensional chiral fermionic theories and chiral gauge theories that can be regularized on a 3 dimensional spatial lattice when and only when the number of flavors is an integral multiple of 16. All these results are based on the observation that local interactions reduce the classification of certain $(4+1)$-dimensional topological superconductors from $\mathbb{Z}$ to $\mathbb{Z}_{8}$, which means that one of their $(3+1)$-dimensional boundaries can be gapped out by interactions without breaking any symmetry when and only when the number of boundary chiral fermions is an integral multiple of $16$.

Figures (2)

• Figure 1: Our proposal of regularizing SM (or other anomaly free chiral gauge theories with $16n$ chiral fermions) on the lattice. The chiral gauge theory will be realized on the $3d$ boundary of a $4d$ topological superconductor with a thin fourth dimension (which makes the bulk a $3d$ system), and the mirror sector is realized on the opposite boundary. Without interaction, the symmetry forbids any fermion bilinear mass at the boundary; but interaction can gap out the mirror sector without generating a fermion bilinear mass term. Thus the only low energy degree of freedom is the chiral gauge theory on the top boundary. However, this is only possible with $16n$ chiral fermions on each boundary.
• Figure 2: Illustration of topological defects: (a) monopole, (b) vortex line, (c) domain wall. Without interaction, all these defects are nontrivial, i.e. they have degenerate/gapless spectra. However, interactions make all these defects gapped and nondegerate, thus after proliferating these defects, the $3d$ boundary enters a symmetric, fully gapped and nondegenerate phase.