Summary: 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 \(\hslash \to 0\), 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 \(\widehat{A}\left({\widehat{x},\widehat{y}} \right)\) using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing \(\widehat{A}\) that, surprisingly, turns out to be much simpler than any of the existent methods. In particular, as a bonus feature of our approach comes a curious observation that, for all curves that come from knots or topological strings, their non-commutative counterparts can be determined just from the first few steps of the topological recursion. We also propose a K-theory criterion for a curve to be “quantizable,” and then apply our construction to many examples that come from applications to knots, strings, instantons, and random matrices.

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