Deformation quantisation of the conic and symplectic reduction of wavefunctions

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.