{"paper":{"title":"The $r^\\sharp$ invariant as a discriminant for the survival of the H-flux under T-duality on product manifolds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The cohomological invariant r^sharp determines whether the H-flux converts to geometric flux or survives under T-duality on product manifolds.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Alexander Pigazzini, Magdalena Toda","submitted_at":"2026-05-13T14:36:28Z","abstract_excerpt":"We show that the cohomological invariant $r^\\sharp$, introduced in [1] as a lower bound for the off-diagonal holonomy dimension of metric connections with totally skew torsion on product manifolds, predicts the behaviour of the torsion $3$-form under both dimensional reduction and Buscher T-duality. On a product $M = \\Sigma_g \\times M_2$ equipped with a product metric, when $r^\\sharp = 0$ the parallel-form strata identify a flat circle factor $S^1_\\beta \\subset M_2$ via the de Rham splitting theorem, and the entire $H$-flux is converted into geometric flux under T-duality along $S^1_\\beta$ (th"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that the cohomological invariant r^sharp, introduced in [1] as a lower bound for the off-diagonal holonomy dimension of metric connections with totally skew torsion on product manifolds, predicts the behaviour of the torsion 3-form under both dimensional reduction and Buscher T-duality.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The manifold is a product M = Σ_g × M_2 equipped with a product metric, and the parallel-form strata are identified via the de Rham splitting theorem when r^sharp = 0.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"r^sharp discriminates between regimes where H-flux survives T-duality or converts to geometric flux on product manifolds with product metrics.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The cohomological invariant r^sharp determines whether the H-flux converts to geometric flux or survives under T-duality on product manifolds.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8ee2da1f9e46ebace32556f3de6cda3d44dc603cd7207e1be7a0267ce2526734"},"source":{"id":"2605.13603","kind":"arxiv","version":1},"verdict":{"id":"085cbb25-571b-4182-ac5f-34bcb72e5648","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:00:49.479081Z","strongest_claim":"We show that the cohomological invariant r^sharp, introduced in [1] as a lower bound for the off-diagonal holonomy dimension of metric connections with totally skew torsion on product manifolds, predicts the behaviour of the torsion 3-form under both dimensional reduction and Buscher T-duality.","one_line_summary":"r^sharp discriminates between regimes where H-flux survives T-duality or converts to geometric flux on product manifolds with product metrics.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The manifold is a product M = Σ_g × M_2 equipped with a product metric, and the parallel-form strata are identified via the de Rham splitting theorem when r^sharp = 0.","pith_extraction_headline":"The cohomological invariant r^sharp determines whether the H-flux converts to geometric flux or survives under T-duality on product manifolds."},"references":{"count":9,"sample":[{"doi":"","year":2025,"title":"Cohomological Calibration and Curvature Constraints on Product Manifolds: A Topological Lower Bound","work_id":"37748c5b-6882-4038-b56e-69cfac6b88a8","ref_index":1,"cited_arxiv_id":"2509.11834","is_internal_anchor":true},{"doi":"","year":2026,"title":"A. Pigazzini, M. Toda,Local topological constraints on Berry curvature in spin–orbit coupled BECs, accepted in JGA (Springer) (2026)","work_id":"d3aa00ec-6755-4285-b38f-db6e070d18d7","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1987,"title":"Buscher,A symmetry of the string background field equations, Phys","work_id":"27e54986-e662-4116-b622-dbac68d7c9ab","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1988,"title":"Buscher,Path integral derivation of quantum duality in nonlinear sigma models, Phys","work_id":"ce551654-277b-4125-b303-9485a9c1ab0a","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"P. Bouwknegt, J. Evslin, V. Mathai,T-duality: topology change fromH-flux, Com- mun. Math. Phys.249(2004), 383–415","work_id":"eb121527-0884-451c-87c7-a156b71debdb","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":9,"snapshot_sha256":"a159d874271ecfb117e3e971e720104df8057251c56182e7740ac85363e9d40c","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"91fae869d864d3a96f044bef8e7d1ae5acded95db7e6cf56f6b2232a159b38a8"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}