A-polynomial, B-model, and quantization

Abstract

Abstract 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 $ \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.