Anomaly of 4d Weyl fermions with discrete symmetries

[2506.19710] Anomaly of 4d Weyl fermions with discrete symmetries Skip to main content Search Submit Donate Log in High Energy Physics - Theory arXiv:2506.19710 (hep-th) [Submitted on 24 Jun 2025 ( v1 ), last revised 28 Jul 2026 (this version, v5)] Title: Anomaly of 4d Weyl fermions with discrete symmetries Authors: Zheyan Wan View PDF HTML (experimental) 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 $\eta$-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 ${\bf B}+{\bf L}$ mixed gauge--gravitational anomaly left by missing sterile right-handed neutrinos and the finite $K$-gauge topological sector that can match it. Comments: 26 pages, PRD version Subjects: High Energy Physics - Theory (hep-th) Cite as: arXiv:2506.19710 [hep-th] (or arXiv:2506.19710v5 [hep-th] for this version) https://doi.org/10.48550/arXiv.2506.19710 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Phys. Rev. D 114, 036032 (2026) Related DOI : https://doi.org/10.1103/sq16-k99s Focus to learn more DOI(s) linking to related resources Submission history From: Zheyan Wan [ view email ] [v1] Tue, 24 Jun 2025 15:17:44 UTC (12 KB) [v2] Mon, 4 Aug 2025 17:28:22 UTC (14 KB) [v3] Wed, 7 Jan 2026 16:44:28 UTC (20 KB) [v4] Wed, 27 May 2026 18:46:24 UTC (24 KB) [v5] Tue, 28 Jul 2026 15:08:46 UTC (28 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: hep-th < prev | next > new | recent | 2025-06 References & Citations INSPIRE HEP SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. 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