QM Research Seminar: Quantization via branes and minimal resolution

Speaker: Jian Qiu (Uppsala University)


Abstract:

I want to explain in this talk the framework of quantization using branes, as developed by Gukov-Witten and later refined by Gaiotto-Witten. This approach embeds the symplectic manifold to be quantized into a bigger space (of double the dimension) as a brane, and the open strings ending on the brane play the role of the operators acting on the Hilbert space.
The open strings are expected to be quantized via the A-model, while in practice the open string wave functions are identifiable as holomorphic functions on the larger space and therefore many algebraic approach can be applied for its quantization and bypass the A-model.
I will apply Kontsevich's deformation quantization to deform the algebra of operators. Contrary to what one expects from deformation quantisation, it is very computable, thanks to some recently developed technical results due to Barmeier and Wang.
I will run you through some examples, the simplest being the Kleinian singularity. In the A_1 case, the quantization gives the Verma module of Usu(2) including the integrality condition for hbar, which Is unusual for deformation quantization.