{"paper":{"title":"A Spinorial Hopf Differential for Associative Submanifolds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Gavin Ball, Jesse Madnick","submitted_at":"2023-05-23T20:26:32Z","abstract_excerpt":"Given a CMC surface in $R^3$, its traceless second fundamental form can be viewed as a holomorphic section called the Hopf differential. By analogy, we show that for an associative submanifold of a 7-manifold $M^7$ with $G_2$-structure, its traceless second fundamental form can be viewed as a twisted spinor. Moreover, if $M$ is $R^7$, $T^7$, or $S^7$ with the standard $G_2$-structure, then this twisted spinor is harmonic. Consequently, every non-totally-geodesic associative 3-fold in $R^7$, $T^7$, and $S^7$ admits non-vanishing harmonic twisted spinors. Analogous results hold for special Lagra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.14508","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2305.14508/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}