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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2506.19710 |
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| _version_ | 1866913168536633344 |
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| author | Wan, Zheyan |
| author_facet | Wan, Zheyan |
| contents | We derive explicit anomaly-index formulas for four-dimensional Weyl fermions charged under the finite symmetries $\mathrm{Spin}\times\mathbb Z_n$ and $\mathrm{Spin}\times_{\mathbb Z_2^{\mathrm F}}\mathbb Z_{2m}^{\mathrm F}$. The strategy is to start from the standard perturbative anomaly indices for $\mathrm{Spin}\times\mathrm U(1)$ and $\mathrm{Spin}\times_{\mathbb Z_2^{\mathrm F}}\mathrm U(1)=\mathrm{Spin}^c$, and then restrict the continuous $\mathrm U(1)$ 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 $\mathrm{Spin}\times\mathbb Z_n$, the relevant backgrounds are the five-dimensional lens-space bundle $X(n;1,1)$ and the product $L(n;1)\times\mathrm{K3}$. For $\mathrm{Spin}\times_{\mathbb Z_2^{\mathrm F}}\mathbb Z_{2m}^{\mathrm F}$, the relevant backgrounds are $L(m;1,1,1)$ and, depending on the parity of $m$, either $L(m;1)\times\mathrm{Enriques}$ or $L(m;1)\times\mathrm{K3}$.
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19710 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Anomaly of 4d Weyl fermions with discrete symmetries Wan, Zheyan High Energy Physics - Theory We derive explicit anomaly-index formulas for four-dimensional Weyl fermions charged under the finite symmetries $\mathrm{Spin}\times\mathbb Z_n$ and $\mathrm{Spin}\times_{\mathbb Z_2^{\mathrm F}}\mathbb Z_{2m}^{\mathrm F}$. The strategy is to start from the standard perturbative anomaly indices for $\mathrm{Spin}\times\mathrm U(1)$ and $\mathrm{Spin}\times_{\mathbb Z_2^{\mathrm F}}\mathrm U(1)=\mathrm{Spin}^c$, and then restrict the continuous $\mathrm U(1)$ 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 $\mathrm{Spin}\times\mathbb Z_n$, the relevant backgrounds are the five-dimensional lens-space bundle $X(n;1,1)$ and the product $L(n;1)\times\mathrm{K3}$. For $\mathrm{Spin}\times_{\mathbb Z_2^{\mathrm F}}\mathbb Z_{2m}^{\mathrm F}$, the relevant backgrounds are $L(m;1,1,1)$ and, depending on the parity of $m$, either $L(m;1)\times\mathrm{Enriques}$ or $L(m;1)\times\mathrm{K3}$. 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. |
| title | Anomaly of 4d Weyl fermions with discrete symmetries |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2506.19710 |