2023/12/22 by Juven Wang, Wang, Juven
Physics and Astronomy · #Crystallography and Radiation Phenomena #FOS: Physical sciences #High Energy Physics - Lattice (hep-lat) #High Energy Physics - Phenomenology (hep-ph) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Particle physics theoretical and experimental studies #Strongly Correlated Electrons (cond-mat.str-el) #Topological Materials and Phenomena
paper · pdf · doi:10.48550/arxiv.2312.14928
openalex publication_date 2023/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Family Puzzle or Generation Problem demands an explanation of why there are 3 families or generations of quarks and leptons in the Standard Model of particle physics. Here we propose a novel solution -- the multiple of 3 families of 16 Weyl fermions (namely (Nf=3) × 16) in the 3+1d spacetime dimensions are topologically robust due to constraints rooted in profound mathematics (such as Hirzebruch signature and Rokhlin theorems, and cobordism) and derivable in physics (such as chiral edge states, quantized thermal Hall conductance, and gravitational Chern-Simons theory), which holds true even forgetting or getting rid of any global symmetry or gauge structure of the Standard Model. By the dimensional reduction through a sequence of sign-reversing mass domain wall of domain wall and so on, we reduce the Standard Model fermions to obtain the (Nf=3) × 16 multiple of 1+1d Majorana-Weyl fermion with a total chiral central charge c-=24. Effectively via the fermionization-bosonization, the 1+1d theory becomes 3 copies of c-=8 of (E8)1 conformal field theory, living on the boundary of 3 copies of 2+1d E8 quantum Hall states. Based on the framing anomaly-free c- = 0 \mod 24 modular invariance, the framed bordism and string bordism ℤ24 class, the 2-framing and p1-structure, the w1-p1 bordism ℤ3 class constraints, we derive the family number constraint Nf ∈ ((48)/(16) =(24)/(8)=3) ℤ. The dimensional reduction process, although not necessary, is sufficiently supported by the ℤ16 class Smith homomorphism. We also comment on the (45)/(15)=3 relation: the 3 families of 15 Weyl-fermion Standard Model vacuum where the absence of some sterile right-handed neutrinos is fulfilled by additional topological field theories or conformal field theories in Ultra Unification.