Large $N$ Invariants of Torus Links in Lens Spaces

TL;DR

This paper computes large N invariants of torus links in lens spaces within Chern-Simons theory, revealing phase transitions and providing a method to derive invariants of complex links from simpler ones.

Contribution

It introduces a collective field theory approach to compute large N invariants of torus links in lens spaces, connecting saddle point analysis with link invariants and phase structure.

Findings

Invariants for Hopf link and unknot expressed via collective field theory.

Method to derive invariants of other torus knots and links.

Identification of potential Douglas-Kazakov phase transition.

Abstract

We compute the invariants for a class of knots and links in arbitrary representations in $S^3/\mathbb{Z}_p$ in the large $k$ (level), large $N$ (rank) limit, keeping $N/(k+N)=\lambda$ fixed, in $U(N)$ and $Sp(N)$ Chern-Simons theories. Using the relation between the saddle point description and collective field theory, we first find that the invariants for the Hopf link and unknot are given by the on shell collective field theory action. We next show that the results of these two invariants can be used to compute the invariants of other torus knots and links. We also discuss the large $N$ phase structure of the Hopf link invariant and observe that the same may admit a Douglas-Kazakov type phase transition depending on the choice of representations and $\lambda$.