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We discuss the fate of the axial U(1) symmetry in 2-flavor QCD at finitetemperature, where the non-singlet chiral symmetry is restored. We firstsummarize the previous theoretical investigation on the relation between theeigenvalue density of the Dirac operator and the axial U(1) symmetry. We showthat the eigenvalue density near the origin behaves as $\lambda^\gamma$ with$\gamma > 2$ in the chirally symmetric phase, where $\lambda$ is an eigenvalue.This implies that the axial U(1) symmetry is partially restored, so that thelow energy symmetry of the theory becomes SU(2)$\otimes$ SU(2)$\otimes$ Z$_4$.Secondly, we report recent numerical investigations on this issue by latticeQCD simulations with lattice chiral fermions such as Overlap or improveddomain-wall fermions. Our preliminary results indicate that the eigenvaluedensity seems to have a gap at the origin just above $T_c$, the temperature ofthe chiral symmetry restoration, which implies the axial U(1) symmetry iseffectively restored at high temperature. We also point out an importance ofthe exact lattice chiral symmetry to obtain correct results on this issue.

Axial U(1) symmetry in the chiral symmetric phase of 2-flavor QCD at finite temperature