{"paper":{"title":"Towards a mathematical definition of Coulomb branches of $3$-dimensional $\\mathcal N=4$ gauge theories, I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.AG","math.DG","math.MP","math.RT"],"primary_cat":"math-ph","authors_text":"Hiraku Nakajima","submitted_at":"2015-03-12T11:22:48Z","abstract_excerpt":"Consider the $3$-dimensional $\\mathcal N=4$ supersymmetric gauge theory associated with a compact Lie group $G$ and its quaternionic representation $\\mathbf M$. Physicists study its Coulomb branch, which is a noncompact hyper-K\\\"ahler manifold, such as instanton moduli spaces on $\\mathbb R^4$, $SU(2)$-monopole moduli spaces on $\\mathbb R^3$, etc. In this paper and its sequel, we propose a mathematical definition of the coordinate ring of the Coulomb branch, using the vanishing cycle cohomology group of a certain moduli space for a gauged $\\sigma$-model on the $2$-sphere associated with $(G,\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.03676","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}