Cobordism and Deformation Class of the Standard Model

Abstract

't Hooft anomalies of quantum field theories (QFTs) with an invertible global symmetry G (including spacetime and internal symmetries) in a $d$d spacetime are known to be classified by a $d+1$d cobordism group TP$_{d+1}$(G), whose group generator is a $d+1$d cobordism invariant written as an invertible topological field theory (iTFT) Z$_{d+1}$. The deformation class of QFT is recently proposed to be specified by its symmetry G and an iTFT Z$_{d+1}$. Seemingly different QFTs of the same deformation class can be deformed to each other via quantum phase transitions. In this work, we ask which deformation class controls the 4d ungauged or gauged (SU(3)$\times$SU(2)$\times$U(1))/$\mathbb{Z}_q$ Standard Model (SM) for $q=1,2,3,6$ with a continuous or discrete $(\bf{B}-\bf{L})$ symmetry. We show that the answer contains some combination of 5d iTFTs: two $\mathbb{Z}$ classes associated with $(\bf{B}-\bf{L})^3$ and $(\bf{B}-\bf{L})$-(gravity)$^2$ 4d perturbative local anomalies, a mod 16 class Atiyah-Patodi-Singer $\eta$ invariant and a mod 2 class Stiefel-Whitney $w_2w_3$ invariant associated with 4d nonperturbative global anomalies, and additional $\mathbb{Z}_3\times\mathbb{Z}_2$ classes involving higher symmetries whose charged objects are Wilson electric or 't Hooft magnetic line operators. Out of $\mathbb{Z}$ classes of local anomalies and 24576 classes of global anomalies, we pin down a deformation class of SM labeled by $(N_f,n_{\nu_{R}},$ p$',q)$, the family and "right-handed sterile" neutrino numbers, magnetic monopole datum, and mod $q$ relation. Grand Unifications and Ultra Unification that replaces sterile neutrinos with new exotic gapped/gapless sectors (e.g., topological or conformal field theory) or gravitational sectors with topological or cobordism constraints, all reside in an SM deformation class. Neighbor phases/transitions/critical regions near SM exhibit beyond SM phenomena.