ON THE QUANTUM INVARIANT FOR THE BRIESKORN HOMOLOGY SPHERES
ON THE QUANTUM INVARIANT FOR THE BRIESKORN HOMOLOGY SPHERES
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关于 Brieskon 同调球的量子不变量
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
K. Hikami
中科院分区：
文献类型：
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作者：
K. Hikami
我们按照 Lawrence 和 Zagier 提出的方法，利用模形式的性质，研究了 Brieskorn 同调球 Σ(p1, p2, p3) 的 Witten-Reshetikhin-Turaev SU(2) 不变量的精确渐近行为。关键观察结果是，该不变量与权重为 3/2 的模形式的艾希勒积分的极限值一致。我们证明卡森不变量与在极限 τ → N ε ℤ 内不消失的艾希勒积分的数量有关。相应地，非零艾希勒积分与基本群的不可约表示之间存在一一对应关系，并且陈-西蒙斯不变量是由艾希勒积分在此极限下给出的。还表明，Ohtsuki 不变量源自艾希勒积分的近模性质，并且我们给出了 L 函数的显式形式。
We study an exact asymptotic behavior of the Witten–Reshetikhin–Turaev SU(2) invariant for the Brieskorn homology spheres Σ(p1, p2, p3) by use of properties of the modular form following a method proposed by Lawrence and Zagier. Key observation is that the invariant coincides with a limiting value of the Eichler integral of the modular form with weight 3/2. We show that the Casson invariant is related to the number of the Eichler integrals which do not vanish in a limit τ → N ∈ ℤ. Correspondingly there is a one-to-one correspondence between the non-vanishing Eichler integrals and the irreducible representation of the fundamental group, and the Chern–Simons invariant is given from the Eichler integral in this limit. It is also shown that the Ohtsuki invariant follows from a nearly modular property of the Eichler integral, and we give an explicit form in terms of the L-function.