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We extend this relation to noncommutative crepant resolutions. In this case the threefold develops a singularity and classical smooth geometry is replaced by the algebra of paths of a certain quiver. K-theoretic quantities on the quiver representation moduli space can be computed via toric localization and result in certain rational functions of the toric parameters. We discuss in particular the case of the conifold and certain orbifold singularities.<\/jats:p>","DOI":"10.1007\/s11005-022-01579-2","type":"journal-article","created":{"date-parts":[[2022,9,5]],"date-time":"2022-09-05T12:10:45Z","timestamp":1662379845000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["On the M2\u2013Brane Index on Noncommutative Crepant Resolutions"],"prefix":"10.1007","volume":"112","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9940-3250","authenticated-orcid":false,"given":"Michele","family":"Cirafici","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,9,5]]},"reference":[{"issue":"3","key":"1579_CR1","doi-asserted-by":"publisher","first-page":"1307","DOI":"10.4007\/annals.2009.170.1307","volume":"170","author":"K Behrend","year":"2009","unstructured":"Behrend, K.: Donaldson-Thomas type invariants via microlocal geometry. Ann. 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