{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"98c04c88168060839a77911ff2c1f26ad3a9ee713bd45e52f7e4526dd935e266","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"c7eb238ed19e63ee13e1a831968fe2f604665d1cefc6fa7156c5abcd7c3328a9","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.yang-mills.ym-3d","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"In three dimensions the short-distance problems of Yang-Mills theory are milder, yet even this case has not been built rigorously. It is the natural stepping stone to four dimensions.","posed_since":"","precise":"Construct the Euclidean Yang-Mills measure for $G = \\mathrm{SU}(2)$ on the torus $T^3$ and on $R^3$ as a limit of lattice or smoothed measures, as a gauge-invariant probability measure on distributional connections modulo gauge, satisfying reflection positivity and, on $R^3$, Euclidean invariance. Answer: a construction with proof; a mass gap in 3D is a further question.","problem_ref":null,"references":"","settled_by":"A proof that the lattice or stochastically quantized measures converge to a gauge-invariant limit with the Osterwalder-Schrader properties.","status_note":"Local-in-time solutions of the 3D Yang-Mills-Higgs Langevin dynamics were constructed by Chandra, Chevyrev, Hairer and Shen around 2022; the invariant measure itself is not yet constructed.","title":"Construction of three-dimensional Euclidean Yang-Mills theory","topic_ref":"c1a7c5336cf6e262c1ec8a96010ceec0183209142167af7077f763d0eba2c1d0"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"f6e0d80f1850c21d49dd3668db4dd48d3319f9d50cc72ff9d51b32bf33683ca5","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"4b4c87cc90f910d19ac37e8df83bbba37eca9f39d3617a640af59246397e6802","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"CGBe-dVWQ46SAB7wP645dtFnIcHKBtX2I8YsjLcpj0z1FGiDAUKlpwGdM_1ptabMQY2FpMHp-BMp5SK1typwCA"},"schema":"pubphys.envelope/1"},"record_hash":"f6e0d80f1850c21d49dd3668db4dd48d3319f9d50cc72ff9d51b32bf33683ca5","leaf_index":1664}