{"paper":{"title":"Non-Invertible Symmetries and Boundaries for Two-Dimensional Fermions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Two free complex Weyl fermions have anomaly-free Z_k symmetries for each primitive Pythagorean triple, each producing a non-invertible defect that generates all U(1)^2-preserving conformal boundaries.","cross_cats":["cond-mat.str-el"],"primary_cat":"hep-th","authors_text":"Guillermo Arias-Tamargo, Maxwell L. Vel\\'asquez Cotini Hutt, Philip Boyle Smith, Rishi Mouland","submitted_at":"2026-05-13T18:00:00Z","abstract_excerpt":"We study the relation between boundary conditions and categorical symmetries of two-dimensional fermionic conformal field theories. We determine all anomaly-free invertible global symmetries of two free complex Weyl fermions, which take the form $\\mathbb{Z}_k$ for each primitive Pythagorean triple $a^2 + b^2 = k^2$. The theory is self-dual under gauging any of these symmetries, and so to each there is associated a non-invertible topological defect. We study the properties of these lines, and show that any conformal boundary condition of two Dirac fermions that preserves a $U(1)^2$ symmetry can"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We determine all anomaly-free invertible global symmetries of two free complex Weyl fermions, which take the form Z_k for each primitive Pythagorean triple a^2 + b^2 = k^2. ... any conformal boundary condition of two Dirac fermions that preserves a U(1)^2 symmetry can be found by dressing a trivial Dirichlet boundary with one of them.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that gauging any of these Z_k symmetries renders the theory self-dual, allowing the non-invertible defects to be well-defined and to generate all listed boundary conditions without additional anomalies or inconsistencies.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Z_k symmetries from Pythagorean triples in two free Weyl fermions yield non-invertible defects that generate all U(1)^2-preserving boundaries for two Dirac fermions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Two free complex Weyl fermions have anomaly-free Z_k symmetries for each primitive Pythagorean triple, each producing a non-invertible defect that generates all U(1)^2-preserving conformal boundaries.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"94a029c48ab36543f9ab42f57801bf93ccb89ac64135b73770341dcabfdf34d6"},"source":{"id":"2605.13952","kind":"arxiv","version":1},"verdict":{"id":"3d10c23c-1d45-4a87-b1f3-1a14bc362b6e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:42:47.833723Z","strongest_claim":"We determine all anomaly-free invertible global symmetries of two free complex Weyl fermions, which take the form Z_k for each primitive Pythagorean triple a^2 + b^2 = k^2. ... any conformal boundary condition of two Dirac fermions that preserves a U(1)^2 symmetry can be found by dressing a trivial Dirichlet boundary with one of them.","one_line_summary":"Z_k symmetries from Pythagorean triples in two free Weyl fermions yield non-invertible defects that generate all U(1)^2-preserving boundaries for two Dirac fermions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that gauging any of these Z_k symmetries renders the theory self-dual, allowing the non-invertible defects to be well-defined and to generate all listed boundary conditions without additional anomalies or inconsistencies.","pith_extraction_headline":"Two free complex Weyl fermions have anomaly-free Z_k symmetries for each primitive Pythagorean triple, each producing a non-invertible defect that generates all U(1)^2-preserving conformal boundaries."},"references":{"count":58,"sample":[{"doi":"","year":2018,"title":"K. Jensen, E. Shaverin and A. Yarom,’t Hooft anomalies and boundaries,Journal of High Energy Physics2018(2018)","work_id":"43148e46-6935-4062-8736-e444a996ad89","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"R. Thorngren and Y. Wang,Anomalous symmetries end at the boundary,Journal of High Energy Physics2021(2021)","work_id":"248046c4-dff4-4836-93ed-7711a068cc04","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"Boundary conformal field theory and symmetry protected topological phases in $2+1$ dimensions","work_id":"f0a3a51d-1b9c-4493-9308-ed15528f311f","ref_index":3,"cited_arxiv_id":"1704.01193","is_internal_anchor":true},{"doi":"","year":2024,"title":"L. Li, C.-T. Hsieh, Y. Yao and M. Oshikawa,Boundary conditions and anomalies of conformal field theories in 1+1 dimensions,Phys. Rev. B110(2024) 045118 [2205.11190]","work_id":"bbf787b7-5fc1-4e59-8a30-dcac7203080b","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"Boyle Smith and D","work_id":"324954ca-8172-433a-b6aa-88ce753dfbbf","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":58,"snapshot_sha256":"b910e89ab18fd0c1b1a4d142b30257fecb63ca67b6a00aaa66efc282b62e4c08","internal_anchors":15},"formal_canon":{"evidence_count":2,"snapshot_sha256":"d4ac5f20893586d7e69329e1415eabec4504d92dbe7ed3bce20714af8007fee5"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}