Journal of High Energy Physics

, 2019:92 | Cite as

invariants at rational τ

  • Piotr KucharskiEmail author
Open Access
Regular Article - Theoretical Physics


invariants of 3-manifolds were introduced as series in q = e2πiτ in order to categorify Witten-Reshetikhin-Turaev invariants corresponding to τ = 1/k. However modularity properties suggest that all roots of unity are on the same footing. The main result of this paper is the expression connecting Reshetikhin-Turaev invariants with invariants for τ. We present the reasoning leading to this conjecture and test it on various 3-manifolds.


Chern-Simons Theories Quantum Groups Supersymmetric Gauge Theory Topological Field Theories 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    S. Gukov, P. Putrov and C. Vafa, Fivebranes and 3-manifold homology, JHEP 07 (2017) 071 [arXiv:1602.05302] [INSPIRE].
  2. [2]
    S. Gukov, M. Mariño and P. Putrov, Resurgence in complex Chern-Simons theory, arXiv:1605.07615 [INSPIRE].
  3. [3]
    S. Gukov, D. Pei, P. Putrov and C. Vafa, BPS spectra and 3-manifold invariants, arXiv:1701.06567 [INSPIRE].
  4. [4]
    M.C.N. Cheng et al., 3d modularity, arXiv:1809.10148 [INSPIRE].
  5. [5]
    E. Witten, Elliptic genera and quantum field theory, Commun. Math. Phys. 109 (1987) 525 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    A. Gadde, S. Gukov and P. Putrov, Walls, lines and spectral dualities in 3d gauge theories, JHEP 05 (2014) 047 [arXiv:1302.0015] [INSPIRE].
  7. [7]
    H.-J. Chung, BPS invariants for Seifert manifolds, arXiv:1811.08863 [INSPIRE].
  8. [8]
    S. Gukov and C. Manolescu, A two-variable series for knot complements, arXiv:1904.06057 [INSPIRE].
  9. [9]
    S. Chun, A resurgence analysis of the SU(2) Chern-Simons partition functions on a Brieskorn homology sphere Σ(2, 5, 7), arXiv:1701.03528 [INSPIRE].
  10. [10]
    R. Lawrence and D. Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J. Math. 3 (1999) 93.Google Scholar
  11. [11]
    K. Hikami, On the quantum invariant for the Brieskorn homology spheres, Int. J. Math. 16 (2005) 661 [math-ph/0405028].
  12. [12]
    K. Hikami, Quantum invariant, modular form, and lattice points, IMRN 3 (2005) 121 [math-ph/0409016].
  13. [13]
    K. Hikami, On the quantum invariant for the spherical Seifert manifold, Commun. Math. Phys. 268 (2006) 285 [math-ph/0504082].
  14. [14]
    K. Hikami, Mock (false) theta functions as quantum invariants, Regul. Chaotic Dyn. 10 (2005) 509 [math-ph/0506073].
  15. [15]
    K. Hikami, Quantum invariants, modular forms, and lattice points II, J. Math. Phys. 47 (2006)102301 [math-ph/0604091].
  16. [16]
    K. Hikami, Decomposition of Witten-Reshetikhin-Turaev invariant: linking pairing and modular forms, in Chern-Simons gauge theory: 20 years after, J.E. Andersen et al. eds., AMS/IP Studies in Advanced Mathematics, American Mathematical Society, U.S.A. (2011).Google Scholar
  17. [17]
    E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989)351.Google Scholar
  18. [18]
    N. Reshetikhin and V.G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547.Google Scholar
  19. [19]
    H.-J. Chung, BPS invariants for 3-manifolds at rational level K, arXiv:1906.12344 [INSPIRE].
  20. [20]
    R. Lawrence and L. Rozansky, Witten-Reshetikhin-Turaev invariants of Seifert manifolds, Commun. Math. Phys. 205 (1999) 287.Google Scholar
  21. [21]
    V.G. Turaev, Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics volume 18, De Gruyter, Berlin, Germany (1994).Google Scholar
  22. [22]
    L.C. Jeffrey, Chern-Simons-Witten invariants of lens spaces and torus bundles and the semiclassical approximation, Commun. Math. Phys. 147 (1992) 563 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    M. Dedushenko, S. Gukov, H. Nakajima, D. Pei and K. Ye, 3d TQFTs from Argyres-Douglas theories, arXiv:1809.04638 [INSPIRE].
  24. [24]
    T.T.Q. Le, Quantum invariants of 3-manifolds: Integrality, splitting, and perturbative expansion, Topol. Its Appl. 127 (2003) 125.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    K. Habiro and T.T.Q. Le, Unified quantum invariants for integral homology spheres associated with simple Lie algebras, Geom. Topol. 20 (2016) 2687 [arXiv:1503.03549] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    K. Hikami, Quantum invariant for torus link and modular forms, Commun. Math. Phys. 246 (2004) 403 [math-ph/0305039].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Walter Burke Institute for Theoretical Physics, California Institute of TechnologyPasadenaU.S.A.
  2. 2.Faculty of PhysicsUniversity of WarsawWarsawPoland

Personalised recommendations