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 4d 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 ${\alpha }_d\in$H${}^d(Z_n,U(1))$ in odd spacetime dimension $d\ge 3$ 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 $(\mathbf{B}+\mathbf{L})$-gauge-gravitational anomaly of the SM in the absence of 3 "sterile" right-handed neutrinos ${\nu }_R$. We analyze a generalized SM with $N_c$ colors and $N_f$ families and argue that missing $N_f$ copies of the ${\nu }_R$ can be naturally replaced by a 4d anomalous $Spin{\times }_{Z_2^F}{Z}_{2N_f,\mathbf{B}+\mathbf{L}}$ symmetric $Z_N$-gauge TQFT under the anomaly cancellation, via a $Z_N$ symmetry extension construction $1\to Z_N\to Spin\times Z_{N{N}_f}\to Spin{\times }_{Z_2^F}{Z}_{2N_f}\to 1$ of anomalous topological order. For minimal nonzero $(N,N_f)$, the allowed minimal extensions are $N=1,3,4,12$, depending on divisibility of $N_f$ by 2 and 3. Combining Witten anomaly and other constraints, we prove that $N=N_c=N_f=3$, with 3 families and 3 colors, is the unique minimal solution to match with the color-center baryon-to-quark symmetry extension $1\to Z_{N_c}\to Spin{\times }_{Z_2^F}{Z}_{2N_c{N}_f,\mathbf{Q}+N_c\mathbf{L}}\to Spin{\times }_{Z_2^F}{Z}_{2N_f,\mathbf{B}+\mathbf{L}}^F\to 1$. We also prove that $A_{Z_3}{p}_1=0\mathrm{mod}3$ for the mod 3 cohomology class.

Comments: 49 pages. Sequel to arXiv:2312.14928, 2501.00607, 2502.21319, 2506.19710, 2512.25038. PRD compares the $Z_N$-gauge TQFT symmetry extension $1\to Z_N\to Spin\times Z_{N{N}_f}\to Spin{\times }_{Z_2^F}{Z}_{2N_f}^F\to 1$ and the color-center baryon $\mathbf{B}$ to quark $\mathbf{Q}$ symmetry extension $1\to Z_{N_c}\to Spin{\times }_{Z_2^F}{Z}_{2N_c{N}_f,\mathbf{Q}+N_c\mathbf{L}}\to Spin{\times }_{Z_2^F}{Z}_{2N_f,\mathbf{B}+\mathbf{L}}^F\to 1$