This paper is a review of whether a quantum symmetry of the strong force, called axial U(1), can appear to be restored when matter is heated. Axial U(1) is present in the classical equations for massless quarks but is broken by a quantum effect called the axial anomaly. The anomaly always exists, but the review asks whether its visible effects at long distances become small near the temperature where ordinary chiral symmetry is lost.

The authors assemble both theory and first‑principles numerical calculations from lattice quantum chromodynamics (lattice QCD). They trace how the topology of gluon fields — a global count of how the gluon field is twisted, called topological charge — controls the low‑lying modes of the Dirac operator, the mathematical object that governs how quarks propagate. The review explains how those low‑energy “Dirac modes” and their spatial structure determine meson correlation functions and susceptibilities, the observables used to test whether axial U(1) effects remain at long distances.

At a basic level the anomaly links an imbalance of left‑ and right‑handed quarks to topological charge through the index theorem. That imbalance shows up in the spectrum of the Dirac operator and in interactions like the ’t Hooft multi‑fermion vertex. Heating reduces the quark condensate that breaks non‑singlet chiral symmetry and reorganizes meson correlations, topological fluctuations, and the density of small Dirac eigenvalues. “Effective restoration” of axial U(1) means that some infrared observables that would signal the anomaly become equal for partners related by the axial rotation. The review discusses how to test this by comparing meson correlators and by analyzing the Dirac eigenvalues and whether the corresponding modes are localized in space, an effect similar to Anderson localization in condensed matter.