A-polynomial, B-model, and Quantization

Understand

Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential.

• Typically, the geometry of the curve can be seen most clearly in a suitable semi-classical limit, as $\hbar \to 0$, and becomes non-commutative or "quantum" away from this limit.
• For a classical curve defined by the zero locus of a polynomial $A(x,y)$, we provide a construction of its non-commutative counterpart $\hat{A} (\hat x, \hat y)$ using the technique of the topological recursion.
• This leads to a powerful and systematic algorithm for computing $\hat{A}$ that, surprisingly, turns out to be much simpler than any of the existent methods.