{"paper":{"title":"Refined Vafa-Witten invariants for toric surfaces from supersymmetric localization in 5D gauge theory","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG","math.DG"],"primary_cat":"hep-th","authors_text":"Alessandro Tanzini, Boris Pioline, Osama Khlaif","submitted_at":"2026-07-20T18:01:26Z","abstract_excerpt":"We study the partition function of five-dimensional $\\mathcal{N}=1$ $U(N)$ supersymmetric Yang-Mills (SYM) theory with an adjoint hypermultiplet of mass $m_{\\rm adj}$ on a toric K\\\"ahler surface $S$ times a circle of radius $\\boldsymbol{\\beta}$. Extending earlier work in $\\mathcal{N}=2^*$ SYM theory on $S$, and in pure $\\mathcal{N}=1$ SYM on $S\\times \\mathbb{S}^1_{\\boldsymbol{\\beta}}$, we find that the path integral localizes to an integral along the Cartan torus of the product of Nekrasov 5D partition functions for each affine patch. Restricting to the gauge group $U(2)$ for simplicity, the i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.18410","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/2607.18410/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"}