On the Quantum SU(2) Invariant at $$q\,{\hbox {=}}\,\exp (4\pi \sqrt{-1}/N)$$ q = exp ( 4 π - 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 $$q=\exp (2 \pi \sqrt{-1}/N)$$ , there have been some researches where the asymptotic expansion of the quantum $$\mathrm{SU}(2)$$ invariant is presented by a sum of contributions from $$\mathrm{SU}(2)$$ flat connections whose coefficients are square roots of the Reidemeister torsions. When $$q=\exp (4 \pi \sqrt{-1}/N)$$ , it is conjectured recently that the quantum $$\mathrm{SU}(2)$$ 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 $$q=\exp (4 \pi \sqrt{-1}/N)$$ , while we have used the stationary phase method when $$q=\exp ({2 \pi \sqrt{-1}}/N)$$ , 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 $$\mathrm{SU}(2)$$ invariant of $$M_p$$ at $$q = \exp (4 \pi \sqrt{-1}/N)$$ . Further, when $$q = \exp (4 \pi \sqrt{-1}/N)$$ , we show that the semi-classical approximation of the asymptotic expansion of the quantum $$\mathrm{SU}(2)$$ invariant of some Seifert 3-manifolds M is presented by a sum of contributions from some of $$\mathrm{SL}_2 {\mathbb C}$$ flat connections on M, and square roots of the Reidemeister torsions appear as coefficients of such contributions.