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

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 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 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 ₂C flat connections on M, and square roots of the Reidemeister torsions appear as coefficients of such contributions.