Quantum Invariant, Modular Form, and Lattice Points

Details Quantum Invariant, Modular Form, and Lattice Points Quantum Invariant, Modular Form, and Lattice Points

2004-09-08

arxiv.org/abs/math-ph/0409016v1

IMRN 2005:3 (2005) 121--154

Physics

Mathematical Physics

29 pages

Scientific paper

10.1155/IMRN.2005.121

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 $N\to\infty$. We reveal that the number of dominating terms, which is the number of the non-vanishing Eichler integrals in a limit $\tau\to N\in\mathbb{Z}$, 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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