On the Quantum Invariant for the Spherical Seifert Manifold

TL;DR

This paper explores the Witten--Reshetikhin--Turaev SU(2) invariants for spherical Seifert manifolds, expressing them via Eichler integrals of modular forms and analyzing their asymptotic behavior and algebraic connections.

Contribution

It provides a novel expression of WRT invariants in terms of Eichler integrals and links modular forms to polyhedral groups for spherical Seifert manifolds.

Findings

WRT invariants expressed through Eichler integrals of modular forms.

Exact asymptotic expansion derived using nearly modular properties.

Connection established between modular forms and polyhedral group equations.

Abstract

We study the Witten--Reshetikhin--Turaev SU(2) invariant for the Seifert manifold $S^3/\Gamma$ where $\Gamma$ is a finite subgroup of SU(2). We show that the WRT invariants can be written in terms of the Eichler integral of the modular forms with half-integral weight, and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the Eichler integral. We further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to $\Gamma$.