Fermion Families and Pontryagin Class: Topological Field Theory via Colour Symmetry Extension

Abstract

Family puzzle asks why the Standard Model (SM) features exactly 3 families of quarks and leptons. Motivated by topological constraints, we study 4-dimensional fermionic anomalies with discrete $Z_n$ symmetry, classified by the 5d spin bordism group. We show that only the group-cohomology subclass H$^5(Z_n,U(1))\cong Z_n$ can be canceled by an anomalous $Z_n$-symmetric 4d $Z_n$-gauge topological quantum field theory (TQFT), while beyond-group-cohomology $A_{Z_n} p_1$ involving the Pontryagin class $p_1$ cannot (except $n=2,3$). More generally, we prove that any cocycle $α_d\in$H$^d(Z_n,U(1))$ in odd spacetime dimension $d\ge3$ is trivialised by the symmetry extension $1\to Z_n\to Z_{n^2}\to Z_n\to 1,$ and we construct the corresponding symmetric anomalous boundary TQFT. For $d=5$ and $n=3$, this yields a Spin$\times Z_3$-symmetric 4d $Z_3$-gauge TQFT that cancels the mixed discrete $(\bf B+L)$-gauge-gravitational anomaly of the SM in the absence of 3 "sterile" right-handed neutrinos $ν_R$. We further analyze a generalized SM with $N_c$ colors and $N_f$ families and argue that missing $N_f$ copies of the $ν_R$ can be naturally replaced by that 4d anomalous $Spin\times_{Z_2^F} Z_{2 N_f,{{\bf B} + {\bf L}}}$ symmetric $Z_{N_c}$-gauge TQFT under the anomaly cancellation, via an appropriate $Z_{N_c}$-color symmetry extension construction $1\to Z_{N_c}\to Spin\times Z_{N_cN_f}\to Spin\times_{Z_2^F} Z_{2N_f}\to1$ of anomalous topological order. For minimal nonzero positive integers $N_c$ and $N_f$, we find the minimal color extensions: $N_c=3, N_f \ge 3$; $N_c=4, N_f \ge 2$; and $N_c=12, N_f \ge 6$. If we further require that an SM baryon is a fermion so $N_c$ is odd, then $Z_{N_c}=Z($SU$(N_c))$ color center, we prove 3 families and 3 colors, $N_c=N_f=3$, is the unique case that stands out. We also prove that $A_{Z_3}p_1= 0\mod3$ for the mod 3 cohomology class in an appropriate context.