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 → 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 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 Ktheory 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.

JT gravity, KdV equations and macroscopic loop operators

“…(C.12) can be thought of as the "quantum spectral curve" for the JT gravity. It would be interesting to study the property of (C.12) along the lines of [61,62]. Let us consider the string equation (C.10) for the JT gravity case.…”

Topological strings, strips and quivers

“…Mirror B-model topological string amplitudes can be computed by means of the topological recursion for the mirror curve. Mirror curves can also be quantized into difference operators A( x, y) that impose difference equations for brane amplitudes [2,38]. In the tropical limit, in which pairs of pants arising from a decomposition of the Riemann surface reduce to trivalent vertices, the mirror curve reduces to the toric diagram of the original toric manifold.…”

“…Once we have derived the brane partition function (2.23), we can also find a q-difference equations it satisfies. Such q-difference equations are interpreted as quantum mirror curves, and in the q → 1 limit they should reduce to (classical) mirror curves [2,38]. For strip geometries we can identify such curves explicitly.…”

Topological strings, D-model, and knot contact homology

“…After the transition, the S 3 has shrunk, Q gets an expectation value Q = 1, the curve generally becomes irreducible 5) and the distinction between L K and M K disappears. Since both p ∼ 0 and x ∼ 0 branches lie on the same Riemann surface, M K and L K are smoothly connected once we include disk instanton corrections, with no phase transition between them.…”