’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 $(\mathrm{SU}(3)\times \mathrm{SU}(2)\times \mathrm{U}(1))/{\mathbb{Z}}_q$ group for $q=1$, 2, 3, 6 with a continuous or discrete baryon minus lepton $(\mathbf{B}-\mathbf{L})$-like symmetry. We show that the answer contains some combination of 5d iTFTs; two $\mathbb{Z}$ classes associated with $(\mathbf{B}-\mathbf{L}{)}^3$ and $(\mathbf{B}-\mathbf{L})-{(\text{gravity})}^2$ 4d perturbative local anomalies, a ${\mathbb{Z}}_{16}$ class Atiyah-Patodi-Singer $\eta$ invariant, a ${\mathbb{Z}}_2$ class Stiefel-Whitney $w_2{w}_3$ invariant associated with 4d nonperturbative global anomalies, and additional ${\mathbb{Z}}_3\times {\mathbb{Z}}_2$ 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 ($N_f,n_{{\nu }_R},{\mathrm{p}}^{\prime },q$), 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 $su(4)\times su(2)\times su(2)$, 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.

中文翻译：

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标准模型的协调和变形等级

具有可逆全局对称性的量子场论 (QFT) 的 't Hooft 异常$G$（包括时空和内部对称性）在一个$d\mathrm{d}$已知时空被分类为$d+1\mathrm{d}$协调组${\mathrm{TP}}_{d+1}(G)$，其组生成器是$d+1\mathrm{d}$cobordism 不变量写为具有配分函数的可逆拓扑场论 (iTFT)${\mathbf{Z}}_{d+1}$. 最近有人提出，QFT 的变形类别由其对称性指定$G$和一个 iTFT${\mathbf{Z}}_{d+1}$. 看似相同变形类别的不同 QFT 可以通过量子相变相互变形。在这项工作中，我们询问哪个 cobordism 类和变形类控制未测量或测量的 4d 标准模型 (SM)$(苏(3)\times 苏(2)\times \mathrm{ü}(1))/{\mathbb{Z}}_q$组为$q=1$, 2, 3, 6 具有连续或离散的重子减轻子$(\mathbf{乙}-\mathbf{大号})$- 对称性。我们证明答案包含 5d iTFT 的某种组合；二$\mathbb{Z}$相关的类$(\mathbf{乙}-\mathbf{大号}{)}^3$和$(\mathbf{乙}-\mathbf{大号})-{(\text{重力})}^2$4d 扰动局部异常，a${\mathbb{Z}}_{16}$类 Atiyah-Patodi-Singer$\eta$不变的，一个${\mathbb{Z}}_2$Stiefel-Whitney 级$w_2{w}_3$与 4d 非微扰全局异常相关的不变量，以及额外的${\mathbb{Z}}_3\times {\mathbb{Z}}_2$涉及更高对称性的全局异常，其带电物体是威尔逊电或't Hooft磁线算子。在多重无限中$\mathbb{Z}$局部异常类和 24576 类全局异常，我们确定了标记为 (${ñ}_F,n_{{\nu }_R},{\mathrm{p}}^{\text{'}},q$)、族数、总“右手无菌”中微子数、磁单极子数据和 mod$q$关系。我们展示了像 Georgi-Glashow 这样的大统一$s你(5)$, 帕提-萨拉姆$s你(4)\times s你(2)\times s你(2)$, 巴尔的翻转$你(5)$，以及熟悉的或修改过的$s○(n)$自旋模型（$n$) 仪表组，例如，与$n=10$, 18 都可以驻留在适当的 SM 变形类中。我们展示了超统一，它用新的奇异有间隙/无间隙扇区（例如，拓扑或共形场论）或具有拓扑起源的引力扇区取代了一些无菌中微子，通过 cobordism 约束，也存在于 SM 变形类中。SM 附近的相邻量子相或其相变，以及相邻的无间隙量子临界区自然表现出超越 SM 的现象。