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

TL;DR

This paper explores the Gopakumar-Ooguri-Vafa correspondence for Clifford-Klein 3-manifolds, connecting quantum invariants, topological string theory, and spectral curve recursion, expanding its mathematical and physical understanding.

Contribution

It extends the Gopakumar-Ooguri-Vafa correspondence to Clifford-Klein 3-manifolds, analyzing its validity and implications in quantum topology and string theory.

Findings

Established the correspondence for a broader class of 3-manifolds.

Linked quantum invariants with topological string theory in new settings.

Discussed implications for mathematical physics and knot theory.

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 quickly survey recent results on the most general frame of validity of this correspondence and discuss some of its implications.