Quantisation via Branes and Minimal Resolution

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 \({\mathbb {P}}^2\) ${\mathbb{P}}^2$ . 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).