On the quantum SU(2) invariant at q=exp(4πi/N) and the twisted Reidemeister torsion for some closed 3-manifolds
On the quantum SU(2) invariant at q=exp(4πi/N) and the twisted Reidemeister torsion for some closed 3-manifolds
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关于 q=exp(4πi/N) 处的量子 SU(2) 不变量和某些闭 3 流形的扭转 Reidemeister 挠率
DOI：
10.1007/s00220-019-03489-2
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发表时间：
2019
影响因子：
2.4
通讯作者：
大槻知忠,高田敏恵
中科院分区：
文献类型：
--
作者：
Kanenobu Taizo;Sumi Toshio;大槻知忠,高田敏恵
Chern-Simons路径积分的微扰展开预示了三维流形的量子不变量的渐近展开公式。当，已经有一些研究，其中量子不变量的渐近展开由平坦连接的贡献之和表示，其系数是Reidemeister挠率的平方根。最近证明了闭双曲三维流形M的量子不变量是N的指数级，其增长由M的复体积决定。第一作者在以前的工作中表明，这一猜想适用于通过手术沿着8字形结得到的双曲3流形。从物理的观点来看，我们使用（形式）鞍点方法时，而我们已经使用的稳定相方法时，这两种方法从数学的观点给出了相当不同的结果公式。在本文中，我们表明，一个平方根的Reidemeister扭转出现作为一个系数的半经典近似的量子不变量的渐近展开的at。进一步证明了当M是Seifert三维流形M时，M的量子不变量的渐近展开式的半经典近似是由M上的一些非线性连接的贡献之和表示的，而Reidemeister挠率的平方根则是这种贡献的系数.
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, there have been some researches where the asymptotic expansion of the quantuminvariant is presented by a sum of contributions fromflat connections whose coefficients are square roots of the Reidemeister torsions. When, it is conjectured recently that the quantuminvariant of a closed hyperbolic 3-manifoldMis of exponential order ofNwhose growth is given by the complex volume ofM. The first author showed in the previous work that this conjecture holds for the hyperbolic 3-manifoldobtained frombypsurgery along the figure-eight knot. From the physical viewpoint, we use the (formal) saddle point method when, while we have used the stationary phase method when, 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 quantuminvariant ofat. Further, when, we show that the semi-classical approximation of the asymptotic expansion of the quantuminvariant of some Seifert 3-manifoldsMis presented by a sum of contributions from some offlat connections onM, and square roots of the Reidemeister torsions appear as coefficients of such contributions.