English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Family Puzzle or Generation Problem demands an explanation of why there are 3families or generations of quarks and leptons in the Standard Model of particlephysics. Here we propose a novel solution -- the multiple of 3 families of 16Weyl fermions (namely $(N_f=3) \times 16$) in the 3+1d spacetime dimensions aretopologically robust due to constraints rooted in profound mathematics (such asHirzebruch signature and Rokhlin theorems, and cobordism) and derivable inphysics (such as chiral edge states, quantized thermal Hall conductance, andgravitational Chern-Simons theory), which holds true even forgetting or gettingrid of any global symmetry or gauge structure of the Standard Model. By thedimensional reduction through a sequence of sign-reversing mass domain wall ofdomain wall and so on, we reduce the Standard Model fermions to obtain the$(N_f=3) \times 16$ multiple of 1+1d Majorana-Weyl fermion with a total chiralcentral charge $c_-=24$. Effectively via the fermionization-bosonization, the1+1d theory becomes 3 copies of $c_-=8$ of (E$_8)_1$ conformal field theory,living on the boundary of 3 copies of 2+1d E$_8$ quantum Hall states. Based onthe framing anomaly-free $c_- = 0 \mod 24$ modular invariance, the framedbordism and string bordism $\mathbb{Z}_{24}$ class, the 2-framing and$p_1$-structure, the $w_1$-$p_1$ bordism $\mathbb{Z}_3$ class constraints, wederive the family number constraint $N_f \in (\frac{48}{16} =\frac{24}{8}=3)\mathbb{Z}$. The dimensional reduction process, although not necessary, issufficiently supported by the $\mathbb{Z}_{16}$ class Smith homomorphism. Wealso comment on the $\frac{45}{15}=3$ relation: the 3 families of 15Weyl-fermion Standard Model vacuum where the absence of some sterileright-handed neutrinos is fulfilled by additional topological field theories orconformal field theories in Ultra Unification.

Family Puzzle, Framing Topology, $c_-=24$ and 3(E8)$_1$ Conformal Field Theories: 48/16 = 45/15 = 24/8 =3