Categorical Symmetry of the Standard Model from Gravitational Anomaly

TL;DR

This work shows that the Standard Model’s invertible B-L-like symmetry can be nonperturbatively broken by mixed gravitational anomalies, yet a noninvertible, categorical B-L symmetry can survive in gravitational backgrounds via anomaly inflow and topological defects. By formulating the SM anomalies through a 6d anomaly polynomial $I_6$ and a 5d invertible phase $S_5$, the authors build topologically robust defect operators using 3d TQFT data and Spin^c/Spin$\times_{\mathbb{Z}_2^{F}}\mathbb{Z}_4$ structures, and extend the construction to the mixed $\mathbb{Z}_4$-gravity anomaly with a $\mathbb{Z}_{16}$ index. They present a detailed framework where noninvertible symmetry charges arise from combining invertible anomalous symmetries with bulk topological terms, including an abelian TQFT sector and lattice data, organized into fusion monoids that map to $\mathbb{Q}/\mathbb{Z}$. The results connect SM anomaly considerations with cobordism classifications and higher-form/symmetric TQFT data, suggesting new avenues for gravitational leptogenesis and BSM physics through noninvertible categorical symmetries.

Abstract

In the Standard Model, some combination of the baryon $\bf B$ and lepton $\bf L$ number symmetry is free of mixed anomalies with strong and electroweak $su(3) \times su(2) \times u(1)_{\tilde Y}$ gauge forces. However, it can still suffer from a mixed gravitational anomaly, hypothetically pertinent to leptogenesis in the very early universe. This happens when the total "sterile right-handed" neutrino number $n_{ν_R}$ is not equal to the family number $N_f$. Thus the invertible $\bf B - L$ symmetry current conservation can be violated quantum mechanically by gravitational backgrounds such as gravitational instantons. In specific, we show that a noninvertible categorical $\bf B - L$ generalized symmetry still survives in gravitational backgrounds. In general, we propose a construction of noninvertible symmetry charge operators as topological defects derived from invertible anomalous symmetries that suffer from mixed gravitational anomalies. Examples include the perturbative local and nonperturbative global anomalies classified by $\mathbb{Z}$ and $\mathbb{Z}_{16}$ respectively. For this construction, we utilize the anomaly inflow bulk-boundary correspondence, the 4d Pontryagin class and the gravitational Chern-Simons 3-form, the 3d Witten-Reshetikhin-Turaev-type topological quantum field theory corresponding to a 2d rational conformal field theory with an appropriate rational chiral central charge, and the 4d $\mathbb{Z}_4^{\rm TF}$-time-reversal symmetric topological superconductor with 3d boundary topological order.

Figures (3)

• Figure 1: A schematic drawing of a small deformation of a submanifold $\mathcal{Y}\subset M$ to $\mathcal{Y}' \subset M$. The shaded domain depicts $\mathcal{Z}$ such that $\partial{\mathcal{Z}}=\mathcal{Y}'-\mathcal{Y}$.
• Figure 2: A schematic drawing of how locally a $\mathbb{Z}_4$-cycle $\tilde{{\cal Y}}$ inside $M^4$ looks like. The cycle corresponds to a network of charge operators on the classical level. The numbers denote the $\mathbb{Z}_4$ charges of the operators in the network. Note that operators of charge $3=-1\mod 4$ is equivalent to operators of charge $+1$ but with reversed orientation. Therefore we can assume that there are only two types of nontrivail operators in the network: of charge $1$ and $2$. Since $2=-2\mod 4$, the operators of charge $2$ do not require a choice of orientation. In blue we depict ${\cal Y}$, the mod 2 reduction of $\tilde{{\cal Y}}$. It is realized by forgetting in the network all operators of charge $2$ and also forgetting the orientation of operators of charge $1$. It can always be deformed into a smooth unoriented 3-submanifold inside $M^4$, which we will denote by the same symbol, ${\cal Y}$.
• Figure 3: The 4d hypersurface ${\rm PD}(A)$ inside the 5d bulk, with the action $\pi \space\mathrm{i}\space \upnu\eta/8$ supported on it and ending on ${\cal Y}\subset M^4$, is needed to unambiguosly define the classical charge operator network $U(\tilde{{\cal Y}})$, with ${\cal Y}=\tilde{{\cal Y}}\mod 2$. The 4d hypersurface ${\cal Z}$, with the action $-\pi \space\mathrm{i}\space \eta/8$, and also ending on ${\cal Y}$, is needed to unambiguosly define the ${\mathbf{Z}}_{\mathbf T}[{\cal Y}]$, the partition function of an anomalous ${\rm Pin}^+$ TQFT ${\mathbf T}$. The union ${\rm PD}(A)\cup (-{\cal Z})$ can be deformed into a smooth hypersurface and pushed inside the 5d bulk, with the total action unchanged. This means that the product (\ref{['DZ4-def']}) is a well-defined topological defect in 4d, not requiring a choice of extension of ${\cal Y}$ into the bulk.