On the quantum invariant for the brieskorn homology spheres

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Abstract

We study an exact asymptotic behavior of the Witten-Reshetikhin-Turaev SU(2) invariant for the Brieskorn homology spheres ∑(p1, p 2, 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.

Original languageEnglish
Pages (from-to)661-685
Number of pages25
JournalInternational Journal of Mathematics
Volume16
Issue number6
DOIs
Publication statusPublished - Jul 2005

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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