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 $$ \mathrm{TP}_5(\mathrm{Spin}\times\mathrm U(1)) \longrightarrow \mathrm{TP}_5(\mathrm{Spin}\times\mathbb Z_n),\quad \mathrm{TP}_5(\mathrm{Spin}^c) \longrightarrow \mathrm{TP}_5(\mathrm{Spin}\times_{\mathbb Z_2^{\mathrm F}}\mathbb Z_{2m}^{\mathrm F}). $$ We compute these maps explicitly by evaluating reduced $η$-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 ${\bf B}+{\bf L}$ mixed gauge--gravitational anomaly left by missing sterile right-handed neutrinos and the finite $K$-gauge topological sector that can match it.