On the Quantum Invariant for the Brieskorn Homology Spheres

Details On the Quantum Invariant for the Brieskorn Homology Spheres On the Quantum Invariant for the Brieskorn Homology Spheres

2004-05-10

arxiv.org/abs/math-ph/0405028v2

Int. J. Math. 16, 661-685 (2005)

Physics

Mathematical Physics

26 pages, 2 figures

Scientific paper

10.1142/S0129167X05003004

We study an exact asymptotic behavior of the Witten-Reshetikhin-Turaev invariant for the Brieskorn homology spheres $\Sigma(p_1,p_2,p_3)$ 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 $\tau\to N \in \mathbb{Z}$. 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.

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