Abstract
We study resurgence for some 3-manifold invariants when Gℂ = SL(2, ℂ). We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of the torus knot complement in S3. Via resurgent analysis, we see that the contribution from the abelian flat connections to the analytically continued Chern-Simons partition function contains the information of all non-abelian flat connections, so it can be regarded as a full partition function of the analytically continued Chern-Simons theory on 3-manifolds M3. In particular, this directly indicates that the homological block for the torus knot complement in S3 is an analytic continuation of the full G = SU(2) partition function, i.e. the colored Jones polynomial.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Gukov, M. Mariño and P. Putrov, Resurgence in complex Chern-Simons theory, arXiv:1605.07615 [INSPIRE].
S. Gukov, P. Putrov and C. Vafa, Fivebranes and 3-manifold homology, JHEP 07 (2017) 071 [arXiv:1602.05302] [INSPIRE].
S. Gukov, D. Pei, P. Putrov and C. Vafa, BPS spectra and 3-manifold invariants, J. Knot Theor. Ramifications 29 (2020) 2040003 [arXiv:1701.06567] [INSPIRE].
M. C. N. Cheng, S. Chun, F. Ferrari, S. Gukov and S. M. Harrison, 3d modularity, JHEP 10 (2019) 010 [arXiv:1809.10148] [INSPIRE].
H.-J. Chung, BPS invariants for Seifert manifolds, JHEP 03 (2020) 113 [arXiv:1811.08863] [INSPIRE].
R. Lawrence and L. Rozansky, Witten-Reshetikhin-Turaev invariants of Seifert manifolds, Commun. Math. Phys. 205 (1999) 287.
H.-J. Chung, BPS invariants for 3-manifolds at rational level K , JHEP 02 (2021) 083 [arXiv:1906.12344] [INSPIRE].
S. Gukov and C. Manolescu, A two-variable series for knot complements, arXiv:1904.06057 [INSPIRE].
J. E. Andersen and W. E. Mistegård, Resurgence analysis of quantum invariants of Seifert fibered homology spheres, arXiv:1811.05376 [INSPIRE].
R. Lawrence and D. Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J. Math. 3 (1999) 93.
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].
K. Hikami, Quantum invariant for torus link and modular forms, Commun. Math. Phys. 246 (2004) 403.
P. Kucharski, Ẑ invariants at rational τ, JHEP 09 (2019) 092 [arXiv:1906.09768] [INSPIRE].
E. Witten, Analytic continuation of Chern-Simons theory, AMS/IP Stud. Adv. Math. 50 (2011) 347 [arXiv:1001.2933] [INSPIRE].
L. Rozansky, Higher order terms in the Melvin-Morton expansion of the colored Jones polynomial, Commun. Math. Phys. 183 (1997) 291.
C. Beasley, Localization for Wilson loops in Chern-Simons theory, Adv. Theor. Math. Phys. 17 (2013) 1 [arXiv:0911.2687] [INSPIRE].
E. P. Klassen, Representations of knot groups in SU(2), Trans. Amer. Math. Soc. 326 (1991) 795.
H. Fuji, K. Iwaki, H. Murakami and Y. Terashima, Witten-Reshetikhin-Turaev function for a knot in Seifert manifolds, arXiv:2007.15872 [INSPIRE].
S. Park, Large color R-matrix for knot complements and strange identities, J. Knot Theor. Ramifications 29 (2020) 2050097 [arXiv:2004.02087] [INSPIRE].
S. Gukov, P.-S. Hsin, H. Nakajima, S. Park, D. Pei and N. Sopenko, Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants, arXiv:2005.05347 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2008.02786
Rights and permissions
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.
About this article
Cite this article
Chung, HJ. Resurgent analysis for some 3-manifold invariants. J. High Energ. Phys. 2021, 106 (2021). https://doi.org/10.1007/JHEP05(2021)106
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2021)106