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Witten’s semi-classical approximation conjecture (coefficients of leading terms)

Prove Witten’s semi-classical approximation conjecture, which asserts an explicit gauge-theoretic formula for the coefficients of the leading terms in the asymptotic expansion of SU(2) Witten–Reshetikhin–Turaev invariants, expressed as integrals of gauge-theoretic quantities over components of the moduli space of flat SU(2) connections.

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Background

The semi-classical approximation conjecture (referenced as Conjecture 1.3 in related work) complements the asymptotic expansion by prescribing the leading coefficients via gauge-theoretic invariants.

The authors explain that their geometric parametrization of moduli spaces (Theorem \ref{thmMflat}) and known intersection-pairing formulas enable progress for Seifert fibered homology spheres, and they plan a separate publication to address this case, but the conjecture remains open in general.

References

We highlight that, complementary to Conjecture~\ref{ConjAEC}, there are also the so-called growth rate conjecture [Conjecture~1.2]{Andersen13}, which gives an explicit conjecture for the order of the leading terms of the expansion~eq:AEC, and Witten's semi-classical approximation conjecture (see also Conjecture~1.3 and references in this paper), which gives an explicit formula for the coefficient of the leading terms of the expansion~eq:AEC.

— A proof of Witten's asymptotic expansion conjecture for WRT invariants of Seifert fibered homology spheres (2510.10678 - Andersen et al., 12 Oct 2025) in Introduction, discussion following Conjecture \ref{ConjAEC}