A Complexity Dichotomy for Quantum Invariants of 3-Manifolds

Abstract

We determine the complexity of exact evaluation of the Reshetikhin--Turaev and Turaev--Viro invariants of closed connected oriented 3-manifolds, with the underlying tensor category fixed. If $\mathcal C$ is a modular category, then the Reshetikhin--Turaev invariant $Z_{\mathcal C}(M)$ can be computed in polynomial time from a framed-link surgery presentation of $M$ precisely when $\mathcal C$ is pointed; otherwise the problem is $\#\mathrm{P}$-hard. If $\mathcal A$ is a spherical fusion category, then the Turaev--Viro invariant $|M|_{\mathcal A}$ can be computed in polynomial time from a triangulation of $M$ precisely when the Drinfeld center $\mathcal Z(\mathcal A)$ is pointed, equivalently when $\mathcal A$ is trivializable pointed; otherwise the problem is $\#\mathrm{P}$-hard. This proves the dichotomy conjectured by Bridges and Samperton and identifies the categorical obstruction to polynomial-time evaluation.