On the Structure of Wave Functions in Complex Chern-Simons Theory

TL;DR

The paper develops a structural understanding of wave functions in complex Chern–Simons theory on hyperbolic knot complements, unveiling an integrality structure in holomorphic blocks that enforces cancellation of singularities at rational values of the coupling $\hbar$. It proposes a decomposition of the exact wave function into holomorphic blocks and their $S$-dual partners, and provides three independent methods to compute the rational-point wave function, including a matrix $q$-difference approach and direct state-integral evaluation. The authors validate the framework on the figure-eight knot and the three-twist knot, demonstrating explicit rational-wave data, admissibility via KS theory, and consistency across the state-integral and holomorphic-block formalisms. The work links complex CS wave functions to TS/ST-type structures, with implications for AJ quantization, ground-state interpretations in quantum mirror curves, and potential enumerative/BPS interpretations in the dual theory.

Abstract

We study the structure of wave functions in complex Chern-Simons theory on the complement of a hyperbolic knot, emphasizing the similarities with the topological string/spectral theory correspondence. We first conjecture a hidden integrality structure in the holomorphic blocks and show that this structure guarantees the cancellation of potential singularities in the full non-perturbative wave function at rational values of the coupling constant. We then develop various techniques to determine the wave function at such rational points. Finally, we illustrate our conjectures and obtain explicit results in the examples of the figure-eight and three-twist knots. In the case of the figure-eight knot, we also perform a direct evaluation of the state integral in the rational case and observe that the resulting wave function has the features of the ground state for a quantum mirror curve.

Figures (3)

• Figure 1: The hyperbolic knots known as the figure-eight knot ($\bm{4}_1$), on the left side, and the three-twist knot ($\bm{5}_2$), on the right side.
• Figure 2: The state integral $\chi_{\bm{4}_1}(u;\hbar)$ in the closed form of equation \ref{['state-fact']} as a function of $x \in {\mathbb R}$, at the bottom, and its components $\chi_{\bm{4}_1}^{(\pm)}(u;\hbar)$ in equation \ref{['state-pieces']}, in the top left and top right, respectively, for $P=Q=1$. The real and imaginary parts are displayed in blue and red.
• Figure 3: The real part of the state integral $\chi_{\bm{4}_1}(u;\hbar)$ in the closed form of equation \ref{['state-fact']} as a function of $x \in {\mathbb R}$ for rational values of ${\mathsf{b}}^2$. We show ${\mathsf{b}}^2 = 1/3$ (in red), $1/2$ (in green), $2/3$ (in blue), and $1$ (in black), on the left, and ${\mathsf{b}}^2 = 3$ (in red), $2$ (in green), $3/2$ (in blue), and $1$ (in black), on the right. The imaginary part is identically zero and not displayed explicitly in the plot.

Theorems & Definitions (2)

• Example 2.1
• Example 2.2