Quantum Invariant, Modular Form, and Lattice Points

TL;DR

This paper analyzes the Witten--Reshetikhin--Turaev SU(2) invariant for Seifert manifolds with 4-singular fibers, expressing it via Eichler integrals of modular forms, and explores its asymptotic behavior and geometric interpretations.

Contribution

It introduces a novel expression of the invariant using Eichler integrals and links the dominant terms to lattice points in a 4-simplex, revealing new geometric and representation-theoretic insights.

Findings

Asymptotic expansion of the invariant as N→∞

Relation between dominant Eichler integrals and lattice points

Connection to irreducible representations of the fundamental group

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 $N\to\infty$. 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.