Resurgence analysis of quantum invariants of Seifert fibered homology spheres

Jørgen Ellegaard Andersen; William Elbæk Mistegård

doi:10.1112/jlms.12506

Resurgence analysis of quantum invariants of Seifert fibered homology spheres

Jørgen Ellegaard Andersen, William Elbæk Mistegård^*

^*Corresponding author for this work

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Abstract

For a Seifert fibered homology sphere 𝑋, we show that the 𝑞-series invariant Ẑ 0(𝑋; 𝑞), introduced by Gukov– Pei–Putrov–Vafa, is a resummation of the Ohtsuki series Z0(𝑋). We show that for every even 𝑘 ∈ ℕ there exists a full asymptotic expansion of Ẑ 0(𝑋; 𝑞) for 𝑞 tending to 𝑒2𝜋𝑖∕𝑘, and in particular that the limit Ẑ 0(𝑋; 𝑒2𝜋𝑖∕𝑘) exists and is equal to the Witten–Reshetikhin–Turaev quantum invariant 𝜏𝑘(𝑋). We show that the poles of the Borel transform of Z0(𝑋) coincide with the classical complex Chern–Simons values, which we further show classifies the corresponding components of the moduli space of flat SL(2, ℂ)-connections.

Original language	English
Journal	Journal of the London Mathematical Society
Volume	105
Issue number	2
Pages (from-to)	709-764
ISSN	0024-6107
DOIs	https://doi.org/10.1112/jlms.12506
Publication status	Published - Mar 2022

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@article{d92868da56e84b97aa2bacbff9f77681,

title = "Resurgence analysis of quantum invariants of Seifert fibered homology spheres",

abstract = "For a Seifert fibered homology sphere 푋, we show that the 푞-series invariant {\^Z} 0(푋; 푞), introduced by Gukov– Pei–Putrov–Vafa, is a resummation of the Ohtsuki series Z0(푋). We show that for every even 푘 ∈ ℕ there exists a full asymptotic expansion of {\^Z} 0(푋; 푞) for 푞 tending to 푒2휋푖∕푘, and in particular that the limit {\^Z} 0(푋; 푒2휋푖∕푘) exists and is equal to the Witten–Reshetikhin–Turaev quantum invariant 휏푘(푋). We show that the poles of the Borel transform of Z0(푋) coincide with the classical complex Chern–Simons values, which we further show classifies the corresponding components of the moduli space of flat SL(2, ℂ)-connections.",

author = "Andersen, {J{\o}rgen Ellegaard} and Misteg{\aa}rd, {William Elb{\ae}k}",

note = "Funding Information: We warmly thank S. Gukov for valuable discussions on the GPPV invariant . The first author was supported in part by the center of excellence grant {\textquoteleft}Center for Quantum Geometry of Moduli Spaces{\textquoteright} from the Danish National Research Foundation (DNRF95) and by the ERC‐Synergy grant {\textquoteleft}ReNewQuantum{\textquoteright}. The second author received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sk{\l}odowska‐Curie grant agreement no. 754411. Publisher Copyright: {\textcopyright} 2022 The Authors. Journal of the London Mathematical Society is copyright {\textcopyright} London Mathematical Society.",

year = "2022",

month = mar,

doi = "10.1112/jlms.12506",

language = "English",

volume = "105",

pages = "709--764",

journal = "Journal of the London Mathematical Society",

issn = "0024-6107",

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T1 - Resurgence analysis of quantum invariants of Seifert fibered homology spheres

AU - Andersen, Jørgen Ellegaard

AU - Mistegård, William Elbæk

N1 - Funding Information: We warmly thank S. Gukov for valuable discussions on the GPPV invariant . The first author was supported in part by the center of excellence grant ‘Center for Quantum Geometry of Moduli Spaces’ from the Danish National Research Foundation (DNRF95) and by the ERC‐Synergy grant ‘ReNewQuantum’. The second author received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska‐Curie grant agreement no. 754411. Publisher Copyright: © 2022 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society.

PY - 2022/3

Y1 - 2022/3

N2 - For a Seifert fibered homology sphere 푋, we show that the 푞-series invariant Ẑ 0(푋; 푞), introduced by Gukov– Pei–Putrov–Vafa, is a resummation of the Ohtsuki series Z0(푋). We show that for every even 푘 ∈ ℕ there exists a full asymptotic expansion of Ẑ 0(푋; 푞) for 푞 tending to 푒2휋푖∕푘, and in particular that the limit Ẑ 0(푋; 푒2휋푖∕푘) exists and is equal to the Witten–Reshetikhin–Turaev quantum invariant 휏푘(푋). We show that the poles of the Borel transform of Z0(푋) coincide with the classical complex Chern–Simons values, which we further show classifies the corresponding components of the moduli space of flat SL(2, ℂ)-connections.

AB - For a Seifert fibered homology sphere 푋, we show that the 푞-series invariant Ẑ 0(푋; 푞), introduced by Gukov– Pei–Putrov–Vafa, is a resummation of the Ohtsuki series Z0(푋). We show that for every even 푘 ∈ ℕ there exists a full asymptotic expansion of Ẑ 0(푋; 푞) for 푞 tending to 푒2휋푖∕푘, and in particular that the limit Ẑ 0(푋; 푒2휋푖∕푘) exists and is equal to the Witten–Reshetikhin–Turaev quantum invariant 휏푘(푋). We show that the poles of the Borel transform of Z0(푋) coincide with the classical complex Chern–Simons values, which we further show classifies the corresponding components of the moduli space of flat SL(2, ℂ)-connections.

U2 - 10.1112/jlms.12506

DO - 10.1112/jlms.12506

M3 - Journal article

AN - SCOPUS:85124598813

SN - 0024-6107

VL - 105

SP - 709

EP - 764

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 2

ER -

Resurgence analysis of quantum invariants of Seifert fibered homology spheres

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