A-Polynomial, B-Model, and Quantization

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 ℏ→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 A^(x^,y^) using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing 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 geometry,” 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. The material contained in this chapter was presented at the conference Mirror Symmetry and Tropical Geometry in Cetraro (July 2011) and is based on the work: Gukov and Sułkowski, “A-polynomial, B-model, and quantization”, JHEP 1202 (2012) 070.