{"paper":{"title":"The Sum Over Topological Sectors and $\\theta$ in the 2+1-Dimensional $\\mathbb{C}\\mathbb{P}^1$ $\\sigma$-Model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.AT"],"primary_cat":"cond-mat.str-el","authors_text":"Daniel S. Freed, Nathan Seiberg, Zohar Komargodski","submitted_at":"2017-07-18T03:14:38Z","abstract_excerpt":"We discuss the three spacetime dimensional $\\mathbb{C}\\mathbb{P}^N$ model and specialize to the $\\mathbb{C}\\mathbb{P}^1$ model. Because of the Hopf map $\\pi_3(\\mathbb{C}\\mathbb{P}^1)=\\mathbb{Z}$ one might try to couple the model to a periodic $\\theta$ parameter. However, we argue that only the values $\\theta=0$ and $\\theta=\\pi$ are consistent. For these values the Skyrmions in the model are bosons and fermions respectively, rather than being anyons. We also extend the model by coupling it to a topological quantum field theory, such that the Skyrmions are anyons. We use techniques from geometry"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.05448","kind":"arxiv","version":3},"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"}