Deformation quantisation of the conic and symplectic reduction of wavefunctions

TL;DR

This paper reviews deformation quantisation and explores how wavefunctions in quantum curve theory emerge from it, illustrating with examples involving conic and symplectic reduction.

Contribution

It demonstrates the construction of wavefunctions via deformation quantisation and applies symplectic reduction techniques following Kontsevich and Soibelman's approach.

Findings

Constructed wavefunction for a specific planar conic

Illustrated symplectic reduction of wavefunctions

Connected deformation quantisation with quantum curve objects

Abstract

We give a short review of the algebraic procedure known as deformation quantisation, which replaces a commutative algebra with a non-commutative algebra. We use this framework to examine how the objects known as wavefunctions, as known in the quantum curve literature, arise from deformation quantisation. We give an example in terms of the planar conic $-y+x^2+2 xy + y^2$, and construct an associated wavefunction. We also give an example of the symplectic reduction of a wavefunction, following a procedure from Kontsevich and Soibelman arXiv:1701.09137.