On the Gopakumar–Ooguri–Vafa correspondence for Clifford–Klein 3-manifolds

Abstract

Gopakumar, Ooguri and Vafa famously proposed the existence of a correspondence between a topological gauge theory on one hand -U(N ) Chern-Simons theory on the three-sphere -and a topological string theory on the other -the topological A-model on the resolved conifold. On the physics side, this duality provides a concrete instance of the large N gauge/string correspondence where exact computations can be performed in detail; mathematically, it puts forward a triangle of striking relations between quantum invariants (Reshetikhin-Turaev-Witten) of knots and 3-manifolds, curve-counting invariants (Gromov-Witten/Donaldson-Thomas) of local Calabi-Yau 3-folds, and the Eynard-Orantin recursion for a specific class of spectral curves. I here survey recent results on the most general frame of validity of this correspondence and discuss some of its implications. 1 arXiv:1711.07826v1 [hep-th] 20 Nov 2017asymptotic expansion of the Reshetikhin-Turaev invariant associated to the quantum group U q (sl N ) at large N and fixed q N equates the formal Gromov-Witten potential of X in the genus expansion. This has a B-model counterpart due to recent developments in higher genus toric mirror symmetry a [19,42], where the same genus expansion can be phrased in terms of the Eynard-Orantin topological recursion [41] on the Hori-Iqbal-Vafa mirror curve of X [58,59].The GOV correspondence has had a profound impact for both communities involved. In Gromov-Witten / Donaldson-Thomas theory, it has laid the foundations of the use of large N dualities to solve the topological string on toric backgrounds [4,6,77] and to obtain all-genus results that went well beyond the existing localisation computations at the time, as well as some striking results for the intersection theory on moduli spaces of curves [75] and, via the relation of Chern-Simons theory to random matrices, an embryo of the remodeling proposal [19,71,73]. In the other direction, the integral structure of BPS invariants leads to non-trivial constraints for the structure of quantum knot invariants [66,68,74].Since the original correspondence of [47,81] was confined to the case where the gauge group is the unitary group U(N ), the base manifold is S 3 , and the knot is the trivial knot, a natural question to ask is whether a similar connection could be generalised to other classical gauge groups [17,18,89], knots other than the unknot [24,65,66] (see also [3] for a significant generalisation, in a rather different setting), as well as categorified/refined invariants of various types [7,51]. A further natural direction would be to seek the broadest generalisation of the correspondence in its original form beyond the basic case of the three-sphere; this would require to provide a description of the string dual of Chern-Simons both in terms of Gromov-Witten theory and of the Eynard-Orantin theory on a specific spectral curve setup. This program was initiated in [5] (see also [25,52,53]) for the case of lens spaces, and what is presumably its widest frame of validity has been recently ...