Let M be a closed and oriented 3-manifold. For each positive integer k, let Z_k(M)∈C denote the operatornameSU(2) WRT invariant for M at level k. Let operatornameCS_C(M) denote the set of Chern--Simons invariants defined as the image of the Chern--Simons action from the operatornameSL_2(C) character variety of M to C/Z. The asymptotic expansion conjecture. For each Chern--Simons invariant theta∈operatornameCS_C(M), there exists a Puiseux series Z_theta(k)∈C[k^1/p| p∈Z_>0][[k^-1]] such that Z_k(M)sim∑_theta∈operatornameCS_C(M)cup0e^2 piik thetaZ_theta(k)quadas k→∞. This conjecture predicts the asymptotic expansion of WRT invariants in terms of Chern--Simons invariants and is a central problem in quantum topology. The source gives no resolution status.

Witten's asymptotic expansion conjecture for WRT invariants

From papers

Let $M$ be a closed and oriented $3$ -manifold. For each positive integer $k$ , let $Z_k(M)\in\mathbb{C}$ denote the $\operatorname{SU}(2)$ WRT invariant for $M$ at level $k$ . Let $\operatorname{CS}_{\mathbb{C}}(M)$ denote the set of Chern--Simons invariants defined as the image of the Chern--Simons action from the $\operatorname{SL}_2(\mathbb{C})$ character variety of $M$ to $\mathbb{C}/\mathbb{Z}$ . The asymptotic expansion conjecture. For each Chern--Simons invariant $\theta\in\operatorname{CS}_{\mathbb{C}}(M)$ , there exists a Puiseux series $Z_\theta(k)\in\mathbb{C}[k^{1/p}\mid p\in\mathbb{Z}_{>0}][[k^{-1}]]$ such that

Z_k(M)\sim\sum_{\theta\in\operatorname{CS}_{\mathbb{C}}(M)\cup\{0\}}e^{2\pi\mathrm{i}k\theta}Z_\theta(k)\quad\text{as }k\to\infty.

This conjecture predicts the asymptotic expansion of WRT invariants in terms of Chern--Simons invariants and is a central problem in quantum topology. The source gives no resolution status.

Progress summary

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Witten's asymptotic expansion conjecture for WRT invariants
Let $Y$ be a closed oriented $3$ -manifold. For each $\operatorname{SU}(2)$ Chern–Simons action $S\in\operatorname{CS}(Y)$ , there exists a Puiseux series
$W_S(\tau)\in\bigcup_{n=1}^{\infty}\operatorname{\mathbb{C}}((\tau^{\frac{1}{n}}))$
such that the WRT invariant of $Y$ has the following Poincaré asymptotic expansion. Witten's asymptotic expansion conjecture.
$\operatorname{WRT}_k(Y)\sim\sum_{S\in\operatorname{CS}(Y)}e^{2\pi i kS}W_S(k^{-1})\qquad\text{as }k\to\infty.$
This conjecture predicts that the large-level asymptotics of the WRT invariant decompose into contributions from the finitely many $\operatorname{SU}(2)$ Chern–Simons actions. The supplied status is open, although the paper's abstract states that the conjecture is proved for general Seifert fibered integral homology $3$ -spheres.
source: Jørgen Ellegaard Andersen, Li Han, Yong Li, William Elbæk Mistegård, David Sauzin and Shanzhong Sun, “A proof of Witten's asymptotic expansion conjecture for WRT invariants of Seifert fibered homology spheres”, arXiv:2510.10678 (2025).

Sources & referencesView supporting material

Primary source

Yuya Murakami, “A framework for proving quantum modularity: Application to Witten's asymptotic expansion conjecture”, arXiv:2508.21710 (2025).

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