Bailey pairs, Eichler integrals and unified Witten-Reshetikhin-Turaev invariants
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| Format: | Preprint |
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2026
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| _version_ | 1866913139941965824 |
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| author | Lovejoy, Jeremy Osburn, Robert Storzer, Matthias |
| author_facet | Lovejoy, Jeremy Osburn, Robert Storzer, Matthias |
| contents | In 1999, Lawrence and Zagier expressed the Witten-Reshetikhin-Turaev (WRT) invariant of the Poincaré 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15740 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bailey pairs, Eichler integrals and unified Witten-Reshetikhin-Turaev invariants Lovejoy, Jeremy Osburn, Robert Storzer, Matthias Number Theory Geometric Topology Quantum Algebra 11F27, 33D15, 57K16 In 1999, Lawrence and Zagier expressed the Witten-Reshetikhin-Turaev (WRT) invariant of the Poincaré 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. |
| title | Bailey pairs, Eichler integrals and unified Witten-Reshetikhin-Turaev invariants |
| topic | Number Theory Geometric Topology Quantum Algebra 11F27, 33D15, 57K16 |
| url | https://arxiv.org/abs/2605.15740 |