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

Juven Wang

TL;DR

The paper addresses the Family Puzzle by proposing a topological mechanism that fixes the SM family number to be a multiple of 3. It maps 48 Weyl fermions in 3+1D to a 1+1D boundary theory with total chiral central charge $c_-=24$, which decomposes into three copies of the bosonic $(E_8)_1$ CFT on the boundary of 2+1D $E_8$ quantum Hall states. This construction relies on a chain of domain-wall reductions and cobordism structures (framing, String, 2-Framing, and $w_1$-$p_1$) together with the Hirzebruch signature and Rokhlin theorems to derive the constraint $N_f \in 3\mathbb{Z}$ and to ensure anomaly cancellation via modular invariance. Through fermionization-bosonization, the 2D boundary theory is realized as $3({\rm E}_8)_1$ CFTs, consistent with bulk-boundary topological data. The framework is argued to be robust against the specifics of SM gauge structure and suggests connections to Ultra Unification and potential hidden sectors described by Leech lattice or related CFTs.

Abstract

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 $(N_f=3) \times 16$) 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 $(N_f=3) \times 16$ 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 (E$_8)_1$ conformal field theory, living on the boundary of 3 copies of 2+1d E$_8$ quantum Hall states. Based on the framing anomaly-free $c_- = 0 \mod 24$ modular invariance, the framed bordism 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, we derive the family number constraint $N_f \in (\frac{48}{16} =\frac{24}{8}=3) \mathbb{Z}$. The dimensional reduction process, although not necessary, is sufficiently supported by the $\mathbb{Z}_{16}$ class Smith homomorphism. We also comment on the $\frac{45}{15}=3$ 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.

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

TL;DR

The paper addresses the Family Puzzle by proposing a topological mechanism that fixes the SM family number to be a multiple of 3. It maps 48 Weyl fermions in 3+1D to a 1+1D boundary theory with total chiral central charge , which decomposes into three copies of the bosonic CFT on the boundary of 2+1D quantum Hall states. This construction relies on a chain of domain-wall reductions and cobordism structures (framing, String, 2-Framing, and -) together with the Hirzebruch signature and Rokhlin theorems to derive the constraint and to ensure anomaly cancellation via modular invariance. Through fermionization-bosonization, the 2D boundary theory is realized as CFTs, consistent with bulk-boundary topological data. The framework is argued to be robust against the specifics of SM gauge structure and suggests connections to Ultra Unification and potential hidden sectors described by Leech lattice or related CFTs.

Abstract

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 ) 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 multiple of 1+1d Majorana-Weyl fermion with a total chiral central charge . Effectively via the fermionization-bosonization, the 1+1d theory becomes 3 copies of of (E conformal field theory, living on the boundary of 3 copies of 2+1d E quantum Hall states. Based on the framing anomaly-free modular invariance, the framed bordism and string bordism class, the 2-framing and -structure, the - bordism class constraints, we derive the family number constraint . The dimensional reduction process, although not necessary, is sufficiently supported by the class Smith homomorphism. We also comment on the 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.
Paper Structure (12 sections, 20 equations, 1 figure)

This paper contains 12 sections, 20 equations, 1 figure.

Table of Contents

  1. Introduction and Summary
  2. Family Puzzle
  3. Background Information
  4. Physics Model Setup
  5. General Arguments: $c_-=24$ and 3$({\rm E}_8)_1$ Conformal Field Theories
  6. Mass Domain Wall of Domain Wall Reduction Toy Model
  7. From 6d Anomaly Polynomial, 5d Convertible Field Theory, 4d Anomaly of the SM to 3d Gravitational Chern-Simons Term
  8. ${\rm Spin} \times_{\mathbb{Z}_2^{\rm F}} {\rm U}(1)_{\mathbf{Q}-N_c \mathbf{L}} \equiv {\rm Spin}^c$ and ${\rm Spin} \times_{\mathbb{Z}_2^{\rm F}} {\mathbb{Z}_{4,X}}$ with a vector ${\rm U}(1)_{\mathbf{Q}-N_c \mathbf{L}}$ and a chiral $X$
  9. Another ${\rm Spin} \times_{\mathbb{Z}_2^{\rm F}} {\rm U}(1) \equiv {\rm Spin}^c$ with a chiral ${\rm U}(1)$: 3$({\rm E}_8)_1$ quantum Hall states
  10. Index Theorem and (Gravitational) Chern-Simons: Framing, String, 2-Framing, and $w_1$-$p_1$ structures
  11. Cobordism constraints of Framing, String, 2-Framing, and $w_1$-$p_1$ structures
  12. Conclusion and Comparison

Figures (1)

  • Figure 2: Domain wall of domain wall reduction and so on. The extra discrete symmetries (namely, the extra $\mathbb{Z}_2$ quotient in ${{\rm Spin} \times_{\mathbb{Z}_2} \mathbb{Z}_4}, {{\rm Pin}^+}, {{\rm Spin} \times {\mathbb{Z}_2}}, {{\rm Pin}^-}$) are not necessary for solving the Family Puzzle, but they are supportive to realize not only dimensional reduction of Fig. \ref{['fig1:fermion-d-reduction-Smith']}(a) but also the Smith map of Fig. \ref{['fig1:fermion-d-reduction-Smith']}(b) simultaneously in this model. The colors (purple, blue, green, red) are intentionally meant to match the colors of the dimensional reduction path in Fig. \ref{['fig1:fermion-d-reduction-Smith']}(a).