Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 002, 45 pages      arXiv:2312.00624      https://doi.org/10.3842/SIGMA.2026.002

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

Marcos Mariño a and Claudia Rella b
a) 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

Received February 10, 2025, in final form December 19, 2025; Published online January 06, 2026

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.

Key words: complex Chern-Simons theory; hyperbolic knots; $A$-polynomial; wave functions; holomorphic blocks; integrality structure.

pdf (1948 kb)   tex (1179 kb)  

References

  1. Aganagic M., Shakirov S., Knot homology and refined Chern-Simons index, Comm. Math. Phys. 333 (2015), 187-228, arXiv:1105.5117.
  2. Andersen J., Kashaev R., A TQFT from quantum Teichmüller theory, Comm. Math. Phys. 330 (2014), 887-934, arXiv:1109.6295.
  3. Andersen J., Kashaev R., The Teichmüller TQFT, in Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited Lectures, World Scientific, Hackensack, NJ, 2018, 2541-2565, arXiv:1811.06853.
  4. Andersen J., Malusà A., A geometric quantisation view on the AJ-conjecture for the Teichmüller TQFT, Res. Math. Sci. 13 (2026), 41-95, arXiv:1711.11522.
  5. Beem C., Dimofte T., Pasquetti S., Holomorphic blocks in three dimensions, J. High Energy Phys. 2014 (2014), no. 12, 177, 118 pages, arXiv:1211.1986.
  6. Codesido S., Grassi A., Mariño M., Spectral theory and mirror curves of higher genus, Ann. Henri Poincaré 18 (2017), 559-622, arXiv:1507.02096.
  7. Cooper D., Culler M., Gillet H., Long D.D., Shalen P.B., Plane curves associated to character varieties of $3$-manifolds, Invent. Math. 118 (1994), 47-84.
  8. Dimofte T., Complex Chern-Simons theory at level $k$ via the 3d-3d correspondence, Comm. Math. Phys. 339 (2015), 619-662, arXiv:1409.0857.
  9. Dimofte T., Perturbative and nonperturbative aspects of complex Chern-Simons theory, J. Phys. A 50 (2017), 443009, 25 pages, arXiv:1608.02961.
  10. Dimofte T., Gaiotto D., Gukov S., Gauge theories labelled by three-manifolds, Comm. Math. Phys. 325 (2014), 367-419, arXiv:1108.4389.
  11. Dimofte T., Gukov S., Lenells J., Zagier D., Exact results for perturbative Chern-Simons theory with complex gauge group, Commun. Number Theory Phys. 3 (2009), 363-443, arXiv:0903.2472.
  12. Ekholm T., Gruen A., Gukov S., Kucharski P., Park S., Stošić M., Sułkowski P., Branches, quivers, and ideals for knot complements, J. Geom. Phys. 177 (2022), 104520, 75 pages, arXiv:2110.13768.
  13. Faddeev L.D., Discrete Heisenberg-Weyl group and modular group, Lett. Math. Phys. 34 (1995), 249-254, arXiv:hep-th/9504111.
  14. Faddeev L.D., Kashaev R.M., Quantum dilogarithm, Modern Phys. Lett. A 9 (1994), 427-434, arXiv:hep-th/9310070.
  15. François M., Grassi A., Painlevé kernels and surface defects at strong coupling, Ann. Henri Poincaré 26 (2025), 2117-2172, arXiv:2310.09262.
  16. Garoufalidis S., On the characteristic and deformation varieties of a knot, in Proceedings of the Casson Fest, Geom. Topol. Monogr., Vol. 7, Geometry & Topology Publications, Coventry, 2004, 291-309.
  17. Garoufalidis S., Gu J., Mariño M., Peacock patterns and resurgence in complex Chern-Simons theory, Res. Math. Sci. 10 (2023), 29, 67 pages, arXiv:2012.00062.
  18. Garoufalidis S., Kashaev R., Evaluation of state integrals at rational points, Commun. Number Theory Phys. 9 (2015), 549-582, arXiv:1411.6062.
  19. Garoufalidis S., Kashaev R., From state integrals to $q$-series, Math. Res. Lett. 24 (2017), 781-801, arXiv:1304.2705.
  20. Gopakumar R., Vafa C., M-theory and topological strings-II, arXiv:hep-th/9812127.
  21. Grassi A., Hatsuda Y., Mariño M., Topological strings from quantum mechanics, Ann. Henri Poincaré 17 (2016), 3177-3235, arXiv:1410.3382.
  22. Gukov S., Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial, Comm. Math. Phys. 255 (2005), 577-627, arXiv:hep-th/0306165.
  23. Hatsuda Y., Katsura H., Tachikawa Y., Hofstadter's butterfly in quantum geometry, New J. Phys. 18 (2016), 103023, 16 pages, arXiv:1606.01894.
  24. Hatsuda Y., Moriyama S., Okuyama K., Instanton effects in ABJM theory from Fermi gas approach, J. High Energy Phys. 2013 (2013), no. 1, 158, 39 pages, arXiv:1211.1251.
  25. Hatsuda Y., Sugimoto Y., Xu Z., Calabi-Yau geometry and electrons on 2d lattices, Phys. Rev. D 95 (2017), 086004, 24 pages, arXiv:1701.01561.
  26. Hikami K., Hyperbolic structure arising from a knot invariant, Internat. J. Modern Phys. A 16 (2001), 3309-3333, arXiv:math-ph/0105039.
  27. Hikami K., Generalized volume conjecture and the $A$-polynomials: The Neumann-Zagier potential function as a classical limit of the partition function, J. Geom. Phys. 57 (2007), 1895-1940, arXiv:math.QA/0604094.
  28. Hollowood T., Iqbal A., Vafa C., Matrix models, geometric engineering and elliptic genera, J. High Energy Phys. 2008 (2008), no. 3, 069, 81 pages, arXiv:hep-th/0310272.
  29. Källén J., Mariño M., Instanton effects and quantum spectral curves, Ann. Henri Poincaré 17 (2016), 1037-1074, arXiv:1308.6485.
  30. Kameyama M., Nawata S., Refined large $N$ duality for knots, J. Knot Theory Ramifications 32 (2023), 2041001, 49 pages, arXiv:1703.05408.
  31. Kashani-Poor A.-K., Quantization condition from exact WKB for difference equations, J. High Energy Phys. 2016 (2016), no. 6, 180, 33 pages, arXiv:1604.01690.
  32. Kirillov A.N., Dilogarithm identities, Progr. Theoret. Phys. Suppl. 118 (1995), 61-142, arXiv:hep-th/9408113.
  33. Koelink H.T., Swarttouw R.F., On the zeros of the Hahn-Exton $q$-Bessel function and associated $q$-Lommel polynomials, J. Math. Anal. Appl. 186 (1994), 690-710, arXiv:1994.1327.
  34. Kontsevich M., Soibelman Y., Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Number Theory Phys. 5 (2011), 231-352, arXiv:1006.2706.
  35. Labastida J.M.F., Mariño M., A new point of view in the theory of knot and link invariants, J. Knot Theory Ramifications 11 (2002), 173-197, arXiv:math.QA/0104180.
  36. Labastida J.M.F., Mariño M., Vafa C., Knots, links and branes at large $N$, J. High Energy Phys. 2000 (2000), no. 11, 007, 42 pages, arXiv:hep-th/0010102.
  37. Mariño M., Spectral theory and mirror symmetry, in String-Math 2016, Proc. Sympos. Pure Math., Vol. 98, American Mathematical Society, Providence, RI, 2018, 259-294, arXiv:1506.07757.
  38. Mariño M., Zakany S., Exact eigenfunctions and the open topological string, J. Phys. A 50 (2017), 325401, 50 pages, arXiv:1606.05297.
  39. Mariño M., Zakany S., Wavefunctions, integrability, and open strings, J. High Energy Phys. 2019 (2019), no. 5, 014, 36 pages, arXiv:1706.07402.
  40. Ooguri H., Vafa C., Knot invariants and topological strings, Nuclear Phys. B 577 (2000), 419-438, arXiv:hep-th/9912123.
  41. Swarttouw R.F., The Hahn-Exton $q$-Bessel function, Ph.D. Thesis, Technische Universiteit Delft, 2016.
  42. Wang X., Zhang G., Huang M.-X., New exact quantization condition for toric Calabi-Yau geometries, Phys. Rev. Lett. 115 (2015), 121601, 5 pages, arXiv:1505.05360.
  43. Witten E., Quantization of Chern-Simons gauge theory with complex gauge group, Comm. Math. Phys. 137 (1991), 29-66.
  44. Zakany S., Quantized mirror curves and resummed WKB, J. High Energy Phys. 2019 (2019), no. 5, 114, 41 pages, arXiv:1711.01099.

Previous article  Next article  Contents of Volume 22 (2026)