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

TL;DR

The paper develops a physical framework uniting the Hikami–Kirillov link between torus knots and Virasoro minimal models with the 3D–3D correspondence and a bulk–boundary dictionary. By analyzing 3D ${ m N}=2$ theories $T[S^3ackslash { m K}_{(P,Q)}]$ via Dimofte–Gaiotto–Gukov data and a bulk-twisted boundary construction, it shows that IR phases realize either unitary TQFTs or rank-0 ${ m N}=4$ SCFTs, whose twisted sectors yield rational VOAs at the boundary. The work provides explicit Nahm-sum–like expressions and half-indices for Virasoro and related rational CFT characters directly from ideal triangulations, along with extensive examples for torus knots $(P,Q)=(2,5),(2,7),(5,7)$, etc. It also discusses decoupling of $U(1)_m$ in the IR, reduction to simpler UV descriptions, and the interpretation of boundary VOAs via A/B twists and parity. This framework offers a systematic route to construct and identify boundary RCFT characters from combinatorial knot data and suggests broad generalizations to higher rank theories and other non-hyperbolic manifolds with potential Haagerup-like RCFTs.

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.

Figures (3)

• Figure 1: Schematic summary of the main proposal of this work. The 3D $\mathcal{N}=2$ supersymmetric gauge theory $T[S^{3}\backslash \mathcal{K}_{(P,Q)}]$, associated with a torus-knot complement via the 3D--3D correspondence, exhibits two distinct infrared behaviors. For $|P-Q|=1$, the theory flows to a unitary TQFT, whereas for $|P-Q|>1$ it flows to a three-dimensional $\mathcal{N}=4$ rank-$0$ superconformal field theory. The subsequent bulk--boundary correspondence then realizes, depending on the choice of boundary conditions, the rational vertex operator algebra of the corresponding Virasoro minimal model $\mathcal{M}(P,Q)$ or other closely related rational conformal field theories at the boundary.
• Figure 2: Schematic summary of the construction of r-VOA characters from the combinatorial data of an ideal triangulation of $S^3\backslash \mathcal{K}_{(P,Q)}$.
• Figure 3: Absolute value of the round $S^3$ partition function for $T[S^3\backslash {\cal K}_{(P,Q)=(2,7)}]$, computed using the Bethe-sum formula in \ref{['Bethe-sum for S3']}. The partition function attains its minimum at $\nu=\nu_{\rm con}=1$, where the value is $\frac{2}{\sqrt{7}}\sin\!(\frac{6\pi}{7}) \approx 0.327985$. At $\nu=0$, the value $\frac{2}{\sqrt{7}}\sin\!(\frac{2\pi}{7}) \approx 0.591009$ matches $|S_{00}|$ of the minimal model ${\cal M}(2,7)$. At $\nu=2$, the value is $\frac{2}{\sqrt{7}}\sin\!(\frac{4\pi}{7}) \approx 0.736976$.