Bailey pairs, Eichler integrals and unified Witten-Reshetikhin-Turaev invariants

TL;DR

This paper develops new identities linking $q$-multisums at roots of unity with Eichler integrals of modular forms, unifying and extending previous results and conjectures in quantum invariants of 3-manifolds.

Contribution

It introduces a Bailey pair-based method and a novel relation between quadratic Gauss sums to derive broad families of identities involving quantum invariants.

Findings

Includes all of Hikami's results and conjectures.

Generalizes Lawrence and Zagier's original result.

Establishes infinite families of identities at roots of unity.

Abstract

In 1999, Lawrence and Zagier expressed the Witten-Reshetikhin-Turaev (WRT) invariant of the Poincar\'e homology sphere as the limiting value of the Eichler integral of a weight 3/2 modular form. Habiro's construction of the unified WRT invariant subsequently recast this result as an identity for a $q$-hypergeometric series at roots of unity. This motivated Hikami to prove analogous $q$-series identities involving the unified WRT invariants of certain Brieskorn homology spheres. Hikami also made several conjectures of a similar type for $q$-series with no apparent connection to quantum invariants. In this paper we use the Bailey pair machinery and a novel relation between incomplete quadratic Gauss sums with periodic coefficients to construct infinite families of identities between $q$-multisums at roots of unity and limiting values of Eichler integrals of weight 3/2 modular forms. These identities include all of Hikami's results and conjectures as well as a generalization of the result of Lawrence and Zagier.