Torus Knots and Minimal Models Revisited : Rational VOA characters from non-hyperbolic knots

TL;DR

This paper explores the connection between torus knots and minimal models by linking 3D gauge theories, boundary chiral algebras, and rational VOA characters, extending previous work with new algebraic expressions.

Contribution

It combines 3D--3D and bulk--boundary correspondences to derive new formulas for rational VOA characters from non-hyperbolic knots.

Findings

Boundary chiral algebras reproduce minimal model characters

Flow to TQFT or SCFT depending on knot parameters

Provides a systematic method to construct VOA characters from knot data

Abstract

In 2003, Hikami and Kirillov uncovered an intriguing connection between torus knots $\mathcal{K}_{(P,Q)}$ and Virasoro minimal models $\mathcal{M}(P,Q)$ by relating the Kashaev invariants of the knots to the characters of the corresponding minimal models. In this work, we recover and extend this connection by combining the 3D--3D correspondence with a bulk--boundary correspondence. More concretely, we study the 3D $\mathcal{N}=2$ gauge theories associated with torus-knot complements via the Dimofte--Gaiotto--Gukov construction and show that, in the infrared, these theories either flow to a unitary TQFT (when $|P-Q| = 1$), whose boundary chiral algebra reproduces that of the associated unitary minimal model, or to a 3D $\mathcal{N}=4$ rank-0 SCFT (when $|P-Q| > 1$), which realizes the corresponding non-unitary chiral minimal model at the boundary after an appropriate topological twist. This framework yields new Nahm-sum-like expressions for the characters of Virasoro minimal models and other related rational conformal field theories, providing a systematic algorithm for constructing characters of rational VOAs directly from the combinatorial data of an ideal triangulation of a non-hyperbolic knot complement.