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 $d\mathrm{d}$ spacetime are known to be classified by a $d+1\mathrm{d}$ cobordism group ${\mathrm{TP}}_{d+1}(G)$, whose group generator is a $d+1\mathrm{d}$ cobordism invariant written as an invertible topological field theory (iTFT) with a partition function ${\mathbf{Z}}_{d+1}$. It has recently been proposed that the deformation class of QFTs is specified by its symmetry $G$ and an iTFT ${\mathbf{Z}}_{d+1}$. Seemingly different QFTs of the same deformation class can be deformed to each other via quantum phase transitions. In this work, we ask which cobordism class and deformation class control the 4d standard model (SM) of ungauged or gauged $(\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1))/{\mathbb{Z}}_{q}$ group for $q=1$, 2, 3, 6 with a continuous or discrete baryon minus lepton $(\mathbf{B}\ensuremath{-}\mathbf{L})$-like symmetry. We show that the answer contains some combination of 5d iTFTs; two $\mathbb{Z}$ classes associated with $(\mathbf{B}\ensuremath{-}\mathbf{L}{)}^{3}$ and $(\mathbf{B}\ensuremath{-}\mathbf{L})\ensuremath{-}{(\text{gravity})}^{2}$ 4d perturbative local anomalies, a ${\mathbb{Z}}_{16}$ class Atiyah-Patodi-Singer $\ensuremath{\eta}$ invariant, a ${\mathbb{Z}}_{2}$ class Stiefel-Whitney ${w}_{2}{w}_{3}$ invariant associated with 4d nonperturbative global anomalies, and additional ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ global anomalies involving higher symmetries whose charged objects are Wilson electric or 't Hooft magnetic line operators. Out of the multiple infinite $\mathbb{Z}$ classes of local anomalies and 24576 classes of global anomalies, we pin down the deformation class of the SM labeled by (${N}_{f},{n}_{{\ensuremath{\nu}}_{R}},\text{ }{\mathrm{p}}^{\ensuremath{'}},q$), the family number, the total ``right-handed sterile'' neutrino number, the magnetic monopole datum, and the mod $q$ relation. We show that grand unification such as Georgi-Glashow $su(5)$, Pati-Salam $su(4)\ifmmode\times\else\texttimes\fi{}su(2)\ifmmode\times\else\texttimes\fi{}su(2)$, Barr's flipped $u(5)$, and the familiar or modified $so(n)$ models of Spin($n$) gauge group, e.g., with $n=10$, 18 can all reside in an appropriate SM deformation class. We show that ultra unification, which replaces some of sterile neutrinos with new exotic gapped/gapless sectors (e.g., topological or conformal field theory) or gravitational sectors with topological origins via cobordism constraints, also resides in an SM deformation class. Neighbor quantum phases near SM or their phase transitions, and neighbor gapless quantum critical regions naturally exhibit beyond SM phenomena.