GT Seminarium med 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 difficult A-model.
I will apply Kontsevich’s deformation quantization to deform the
algebra of operators. This approach would have been impractical without
further tools such as the minimal bi-algebra resolution and some
recently developed techniques for deformation of quiver algebras.
I will run you through some examples, such as the quantization of
Kleinian singularity, in the A_1 case, the quantization gives the Verma
module ofUsu(2) and integrality condition for the parameters, which Is
unusual for deformation quantization. Further examples include T^*CP^2,
which is useful in the context of generalized Kahler structure.
Future applications include the quantization of character varieties,
Higgs moduli space etc. 

See upcoming seminars in our seminar series on Geometry and Topology (GT).