{"title":"Quantisation via Branes and Minimal Resolution","authors":"Jian Qiu","doi":"10.1007/s00220-024-05137-w","DOIUrl":null,"url":null,"abstract":"<div><p>The ‘brane quantisation’ is a quantisation procedure developed by Gukov and Witten (Adv Theor Math Phys 13(5):1445–1518, 2009). We implement this idea by combining it with the tilting theory and the minimal resolutions. This way, we can realistically compute the deformation quantisation on the space of observables acting on the Hilbert space. We apply this procedure to certain quantisation problems in the context of generalised Kähler structure on <span>\\({\\mathbb {P}}^2\\)</span>. Our approach differs from and complements that of Bischoff and Gualtieri (Commun Math Phys 391(2):357–400, 2022). We also benefitted from an important technical tool: a combinatorial criterion for the Maurer–Cartan equation, developed by Barmeier and Wang (Deformations of path algebras of quivers with relations, 2020. arXiv:2002.10001).\n</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"405 12","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00220-024-05137-w.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s00220-024-05137-w","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
The ‘brane quantisation’ is a quantisation procedure developed by Gukov and Witten (Adv Theor Math Phys 13(5):1445–1518, 2009). We implement this idea by combining it with the tilting theory and the minimal resolutions. This way, we can realistically compute the deformation quantisation on the space of observables acting on the Hilbert space. We apply this procedure to certain quantisation problems in the context of generalised Kähler structure on \({\mathbb {P}}^2\). Our approach differs from and complements that of Bischoff and Gualtieri (Commun Math Phys 391(2):357–400, 2022). We also benefitted from an important technical tool: a combinatorial criterion for the Maurer–Cartan equation, developed by Barmeier and Wang (Deformations of path algebras of quivers with relations, 2020. arXiv:2002.10001).
布莱恩量子化 "是古可夫和威滕（Adv Theor Math Phys 13(5):1445-1518, 2009）提出的一种量子化程序。我们将这一想法与倾斜理论和最小分辨率结合起来加以实现。这样，我们就能在作用于希尔伯特空间的可观测空间上真实地计算形变量子化。我们将这一过程应用于 \({\mathbb {P}}^2\) 上广义凯勒结构背景下的某些量子化问题。我们的方法不同于比肖夫和瓜尔蒂耶里（Commun Math Phys 391(2):357-400, 2022）的方法，也是对其方法的补充。我们还得益于一个重要的技术工具：由 Barmeier 和 Wang 开发的毛勒-卡尔坦方程的组合准则（Deformations of path algebras of quivers with relations, 2020. arXiv:2002.10001）。
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The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.
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