English

Quantum Invariant, Modular Form, and Lattice Points

Mathematical Physics 2007-05-23 v1 math.MP

Abstract

We study the Witten--Reshetikhin--Turaev SU(2) invariant for the Seifert manifold with 4-singular fibers. We define the Eichler integrals of the modular forms with half-integral weight, and we show that the invariant is rewritten as a sum of the Eichler integrals. Using a nearly modular property of the Eichler integral, we give an exact asymptotic expansion of the WRT invariant in NN\to\infty. We reveal that the number of dominating terms, which is the number of the non-vanishing Eichler integrals in a limit τNZ\tau\to N\in\mathbb{Z}, is related to that of lattice points inside 4-dimensional simplex, and we discuss a relationship with the irreducible representations of the fundamental group.

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Cite

@article{arxiv.math-ph/0409016,
  title  = {Quantum Invariant, Modular Form, and Lattice Points},
  author = {Kazuhiro Hikami},
  journal= {arXiv preprint arXiv:math-ph/0409016},
  year   = {2007}
}

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29 pages