### 抄録

The perturbative expansion of the Chern–Simons path integral predicts a formula of the asymptotic expansion of the quantum invariant of a 3-manifold. When q=exp(2π-1/N), there have been some researches where the asymptotic expansion of the quantum SU (2) invariant is presented by a sum of contributions from SU (2) flat connections whose coefficients are square roots of the Reidemeister torsions. When q=exp(4π-1/N), it is conjectured recently that the quantum SU (2) invariant of a closed hyperbolic 3-manifold M is of exponential order of N whose growth is given by the complex volume of M. The first author showed in the previous work that this conjecture holds for the hyperbolic 3-manifold M_{p} obtained from S^{3} by p surgery along the figure-eight knot. From the physical viewpoint, we use the (formal) saddle point method when q=exp(4π-1/N), while we have used the stationary phase method when q=exp(2π-1/N), and these two methods give quite different resulting formulas from the mathematical viewpoint. In this paper, we show that a square root of the Reidemeister torsion appears as a coefficient in the semi-classical approximation of the asymptotic expansion of the quantum SU (2) invariant of M_{p} at q=exp(4π-1/N). Further, when q=exp(4π-1/N), we show that the semi-classical approximation of the asymptotic expansion of the quantum SU (2) invariant of some Seifert 3-manifolds M is presented by a sum of contributions from some of SL _{2}C flat connections on M, and square roots of the Reidemeister torsions appear as coefficients of such contributions.

元の言語 | 英語 |
---|---|

ページ（範囲） | 151-204 |

ページ数 | 54 |

ジャーナル | Communications in Mathematical Physics |

巻 | 370 |

発行部数 | 1 |

DOI | |

出版物ステータス | 出版済み - 8 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### これを引用

**On the Quantum SU(2) Invariant at q=exp(4π√-1/N) and the Twisted Reidemeister Torsion for Some Closed 3-Manifolds.** / Ohtsuki, Tomotada; Takata, Toshie.

研究成果: ジャーナルへの寄稿 › 記事

*Communications in Mathematical Physics*, 巻. 370, 番号 1, pp. 151-204. https://doi.org/10.1007/s00220-019-03489-2

}

TY - JOUR

T1 - On the Quantum SU(2) Invariant at q=exp(4π√-1/N) and the Twisted Reidemeister Torsion for Some Closed 3-Manifolds

AU - Ohtsuki, Tomotada

AU - Takata, Toshie

PY - 2019/8/1

Y1 - 2019/8/1

N2 - The perturbative expansion of the Chern–Simons path integral predicts a formula of the asymptotic expansion of the quantum invariant of a 3-manifold. When q=exp(2π-1/N), there have been some researches where the asymptotic expansion of the quantum SU (2) invariant is presented by a sum of contributions from SU (2) flat connections whose coefficients are square roots of the Reidemeister torsions. When q=exp(4π-1/N), it is conjectured recently that the quantum SU (2) invariant of a closed hyperbolic 3-manifold M is of exponential order of N whose growth is given by the complex volume of M. The first author showed in the previous work that this conjecture holds for the hyperbolic 3-manifold Mp obtained from S3 by p surgery along the figure-eight knot. From the physical viewpoint, we use the (formal) saddle point method when q=exp(4π-1/N), while we have used the stationary phase method when q=exp(2π-1/N), and these two methods give quite different resulting formulas from the mathematical viewpoint. In this paper, we show that a square root of the Reidemeister torsion appears as a coefficient in the semi-classical approximation of the asymptotic expansion of the quantum SU (2) invariant of Mp at q=exp(4π-1/N). Further, when q=exp(4π-1/N), we show that the semi-classical approximation of the asymptotic expansion of the quantum SU (2) invariant of some Seifert 3-manifolds M is presented by a sum of contributions from some of SL 2C flat connections on M, and square roots of the Reidemeister torsions appear as coefficients of such contributions.

AB - The perturbative expansion of the Chern–Simons path integral predicts a formula of the asymptotic expansion of the quantum invariant of a 3-manifold. When q=exp(2π-1/N), there have been some researches where the asymptotic expansion of the quantum SU (2) invariant is presented by a sum of contributions from SU (2) flat connections whose coefficients are square roots of the Reidemeister torsions. When q=exp(4π-1/N), it is conjectured recently that the quantum SU (2) invariant of a closed hyperbolic 3-manifold M is of exponential order of N whose growth is given by the complex volume of M. The first author showed in the previous work that this conjecture holds for the hyperbolic 3-manifold Mp obtained from S3 by p surgery along the figure-eight knot. From the physical viewpoint, we use the (formal) saddle point method when q=exp(4π-1/N), while we have used the stationary phase method when q=exp(2π-1/N), and these two methods give quite different resulting formulas from the mathematical viewpoint. In this paper, we show that a square root of the Reidemeister torsion appears as a coefficient in the semi-classical approximation of the asymptotic expansion of the quantum SU (2) invariant of Mp at q=exp(4π-1/N). Further, when q=exp(4π-1/N), we show that the semi-classical approximation of the asymptotic expansion of the quantum SU (2) invariant of some Seifert 3-manifolds M is presented by a sum of contributions from some of SL 2C flat connections on M, and square roots of the Reidemeister torsions appear as coefficients of such contributions.

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UR - http://www.scopus.com/inward/citedby.url?scp=85068772352&partnerID=8YFLogxK

U2 - 10.1007/s00220-019-03489-2

DO - 10.1007/s00220-019-03489-2

M3 - Article

AN - SCOPUS:85068772352

VL - 370

SP - 151

EP - 204

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -