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

N1 - Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

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.

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

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

SN - 0010-3616

VL - 370

SP - 151

EP - 204

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 1

ER -