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 . We reveal that the number of dominating terms, which is the number of the non-vanishing Eichler integrals in a limit , is related to that of lattice points inside 4-dimensional simplex, and we discuss a relationship with the irreducible representations of the fundamental group.
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}
}
Comments
29 pages