Anomaly of 4d Weyl fermions with discrete symmetries

Abstract

We derive explicit anomaly-index formulas for four-dimensional Weyl fermions charged under the finite symmetries Spin$\times {\mathbb{Z}}_n$ and Spin${\times }_{{\mathbb{Z}}_2^{\mathrm{F}}}{\mathbb{Z}}_{2m}^{\mathrm{F}}$. The strategy is to start from the standard perturbative anomaly indices for Spin$\times$U(1) and Spin${\times }_{{\mathbb{Z}}_2^{\mathrm{F}}}$U(1)$=$Spin${}^c$, and then restrict the continuous U(1) symmetry to a finite cyclic subgroup. On the level of invertible field theories this gives natural homomorphisms $${\mathrm{TP}}_5(\mathrm{Spin}\times \mathrm{U}(1))⟶{\mathrm{TP}}_5(\mathrm{Spin}\times {\mathbb{Z}}_n),{\mathrm{TP}}_5({\mathrm{Spin}}^c)⟶{\mathrm{TP}}_5(\mathrm{Spin}{\times }_{{\mathbb{Z}}_2^{\mathrm{F}}}{\mathbb{Z}}_{2m}^{\mathrm{F}}).$$ We compute these maps explicitly by evaluating reduced $\eta$-invariants on geometric representatives of the finite anomaly groups. For Spin$\times {\mathbb{Z}}_n$, the relevant backgrounds are the five-dimensional lens-space bundle $X(n;1,1)$ and the product $L(n;1)\times$K3. For Spin${\times }_{{\mathbb{Z}}_2^{\mathrm{F}}}{\mathbb{Z}}_{2m}^{\mathrm{F}}$, the relevant backgrounds are $L(m;1,1,1)$ and, depending on the parity of $m$, either $L(m;1)\times$Enriques or $L(m;1)\times$K3. The output is a pair of integer-valued anomaly indices for each finite symmetry. These indices are normalized in the cyclic factors of the finite anomaly group, so they can be used directly in anomaly-cancellation checks for fermions with discrete gauge or global symmetries. They also provide the input for symmetry-extension applications to topologically ordered quantum dark matter: in Standard-Model examples, the indices identify the discrete $\mathbf{B}+\mathbf{L}$ mixed gauge--gravitational anomaly left by missing sterile right-handed neutrinos and the finite $K$-gauge topological sector that can match it.

26 pages, PRD version

References in corpus (5)

• (Non-)Abelian discrete anomalies
• Dai-Freed theorem and topological phases of matter
• Categorical Symmetry of the Standard Model from Gravitational Anomaly
• Boundary topological orders of (4+1)d fermionic ${\mathbb{Z}}_{2N}^{\mathrm{F}}$ SPT states
• Topological Leptogenesis