Exact agreement between quantum Chern–Simons–Higgs theory and its symmetry‑reduced theory

Ascertain whether an exact equivalence holds between the full quantum Chern–Simons–Higgs theory and the one‑dimensional theory obtained via spherically symmetric symmetry reduction, beyond the approximate relationship suggested in the semiclassical regime.

Background

For pure Chern–Simons theory on S^3, the authors argue that the semiclassical evaluation agrees with the corresponding symmetry‑reduced theory. For the interacting Chern–Simons–Higgs theory, the situation is more complicated due to the presence of the Higgs sector and the nontrivial structure of the functional Hessian.

The authors explicitly state uncertainty about whether the full quantum Chern–Simons–Higgs theory is exactly captured by its symmetry‑reduced counterpart, even though they sketch an approximate correspondence in the semiclassical regime. Determining exact agreement, or the precise conditions under which it holds, remains unresolved.