On the Quantum Invariant for the Brieskorn Homology Spheres

TL;DR

This paper investigates the asymptotic behavior of the Witten-Reshetikhin-Turaev invariant for Brieskorn homology spheres using modular form properties, revealing connections to the Casson invariant, irreducible representations, and the Chern-Simons invariant.

Contribution

It introduces a novel approach linking Eichler integrals of modular forms to quantum invariants of Brieskorn spheres, providing explicit formulas and new insights into their topological invariants.

Findings

Invariant coincides with a limit of Eichler integrals.

Casson invariant relates to non-vanishing Eichler integrals.

Explicit form of Ohtsuki invariant via L-function.

Abstract

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.