{"paper":{"title":"Cutting and gluing with running couplings in $\\mathcal{N}=2$ QCD","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG","math.NT"],"primary_cat":"hep-th","authors_text":"Elias Furrer, Jan Manschot, Johannes Aspman","submitted_at":"2021-07-09T18:00:02Z","abstract_excerpt":"We consider the order parameter $u=\\left<{\\rm Tr}\\phi^2\\right>$ as function of the running coupling constant $\\tau \\in \\mathbb{H}$ of asymptotically free $\\mathcal{N}=2$ QCD with gauge group $SU(2)$ and $N_f\\leq 3$ massive hypermultiplets. If the domain for $\\tau$ is restricted to an appropriate fundamental domain $\\mathcal{F}_{N_f}$, the function $u$ is one-to-one. We demonstrate that these domains consist of six or less images of an ${\\rm SL}(2,\\mathbb{Z})$ keyhole fundamental domain, with appropriate identifications of the boundaries. For special choices of the masses, $u$ does not give ris"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.04600","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/2107.04600/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"}