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A proof of Witten's asymptotic expansion conjecture for WRT invariants of Seifert fibered homology spheres

Jørgen Ellegaard Andersen, Li Han, Yong Li, William Elbæk Mistegård, David Sauzin, Shanzhong Sun

TL;DR

This paper proves Witten's asymptotic expansion conjecture for SU(2) WRT invariants of Seifert fibered integral homology spheres with at least three exceptional fibers by linking WRT values at roots of unity to non-tangential limits of the GPPV invariant via a resurgence- and quantum-modular-form framework. The authors develop a detailed decomposition of the GPPV invariant into Hikami-function components, establish higher-depth quantum modularity, and connect to the Habiro invariant analytically. A new parametrization of the SU(2) moduli space on the orbifold surface is used to identify Chern-Simons invariants and to bound the growth of leading terms, enabling a rigorous Puiseux expansion for WRT invariants. The approach hinges on resurgent analysis, median sums, and a careful analysis of partial theta series, yielding a complete proof for all Seifert fibered integral homology spheres and clarifying the interplay between nonperturbative invariants and semiclassical data. This provides a solid bridge between quantum invariants, modularity phenomena, and gauge-theoretic moduli, with implications for semi-classical approximations and Habiro-type invariants.

Abstract

Let $X$ be a general Seifert fibered integral homology $3$-sphere with $r\ge3$ exceptional fibers. For every root of unity $ζ\not=1$, we show that the SU(2) WRT invariant of $X$ evaluated at $ζ$ is (up to an elementary factor) the non-tangential limit at $ζ$ of the GPPV invariant of $X$, thereby generalizing a result from [Andersen-Mistegard 2022]. Based on this result, we apply the quantum modularity results developed in [Han-Li-Sauzin-Sun 2023] to the GPPV invariant of $X$ to prove Witten's asymptotic expansion conjecture [Witten 1989] for the WRT invariant of $X$. We also prove that the GPPV invariant of $X$ induces a higher depth strong quantum modular form. Moreover, when suitably normalized, the GPPV invariant provides an ``analytic incarnation'' of the Habiro invariant.

A proof of Witten's asymptotic expansion conjecture for WRT invariants of Seifert fibered homology spheres

TL;DR

This paper proves Witten's asymptotic expansion conjecture for SU(2) WRT invariants of Seifert fibered integral homology spheres with at least three exceptional fibers by linking WRT values at roots of unity to non-tangential limits of the GPPV invariant via a resurgence- and quantum-modular-form framework. The authors develop a detailed decomposition of the GPPV invariant into Hikami-function components, establish higher-depth quantum modularity, and connect to the Habiro invariant analytically. A new parametrization of the SU(2) moduli space on the orbifold surface is used to identify Chern-Simons invariants and to bound the growth of leading terms, enabling a rigorous Puiseux expansion for WRT invariants. The approach hinges on resurgent analysis, median sums, and a careful analysis of partial theta series, yielding a complete proof for all Seifert fibered integral homology spheres and clarifying the interplay between nonperturbative invariants and semiclassical data. This provides a solid bridge between quantum invariants, modularity phenomena, and gauge-theoretic moduli, with implications for semi-classical approximations and Habiro-type invariants.

Abstract

Let be a general Seifert fibered integral homology -sphere with exceptional fibers. For every root of unity , we show that the SU(2) WRT invariant of evaluated at is (up to an elementary factor) the non-tangential limit at of the GPPV invariant of , thereby generalizing a result from [Andersen-Mistegard 2022]. Based on this result, we apply the quantum modularity results developed in [Han-Li-Sauzin-Sun 2023] to the GPPV invariant of to prove Witten's asymptotic expansion conjecture [Witten 1989] for the WRT invariant of . We also prove that the GPPV invariant of induces a higher depth strong quantum modular form. Moreover, when suitably normalized, the GPPV invariant provides an ``analytic incarnation'' of the Habiro invariant.

Paper Structure

This paper contains 29 sections, 38 theorems, 283 equations.

Table of Contents

  1. Introduction
  2. Definitions: quantum invariants and quantum modular forms
  3. WRT invariants
  4. Integrality
  5. A formula for WRT invariants in terms of rational surgery presentations
  6. GPPV invariants
  7. Quantum modular forms with higher depth
  8. Reminders on partial theta series
  9. Partial theta series and Laplace transforms
  10. A resurgent asymptotic expansion
  11. Quantum modularity of partial theta series
  12. Quantum modularity of the GPPV invariant of a Seifert fibered integral homology sphere
  13. The WRT invariant of $X$
  14. The modified GPPV invariant of $X$
  15. The normalized GPPV invariant of $X$ and Hikami functions
  16. ...and 14 more sections

Key Result

Theorem 1.1

For the Seifert integral homology sphere $X$, the formal series $W_0(\tau)$ of eqdefWzerolaX is resurgentThroughout this paper, we say that a formal series $\widetilde{\Theta}(\tau)=\sum_{p\ge0} a_p\tau^p$ is resurgent in $\tau$ if its formal Borel transform $\widehat{\Theta}(\xi):=\sum_{p\ge0} a_{p where the $\mathop{\mathrm{SU}}\nolimits(2)$ Chern-Simons actions $S\in\mathop{\mathrm{CS}}\nolimit

Theorems & Definitions (94)

  • Conjecture 1.1: The asymptotic expansion conjecture Andersen02Andersen13Witten98
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.3
  • Definition 2.1: RT91RT90
  • Definition 2.2: GPPV
  • Definition 2.3: adapted from BMM20bZagier10
  • Remark 2.4
  • ...and 84 more