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

Marcos Mariño^a, Claudia Rella<sup>b

a</sup> Département de Physique Théorique and Section de Mathématiques, Université de Genève, CH-1211 Genève, Switzerland
^b Institut des Hautes Études Scientifiques, 91440 Bures-sur-Yvette, France

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.

Keywords: complex Chern–Simons theory, hyperbolic knots, $A$-polynomial, wave functions, holomorphic blocks, integrality structure.

MSC: 57K31, 81T45, 81Q05

Received: February 10, 2025; in final form December 19, 2025; Published online January 6, 2026

Language: English

DOI: 10.3842/SIGMA.2026.002

ArXiv: 2312.00624