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Recent studies on lattice QCD have shown the emergence of large symmetries athigh temperature. This includes not only the restoration $SU(n_F)_L \timesSU(n_F)_R$, but also the effective emergence of an unexpected symmetry group,namely $SU(2)_{CS}$, which contains $U(1)_A$ as subgroup. At the same time, athigh $T$, a gap in Dirac spectrum appears. As it is argued in several works of\textit{L. Glozman et al.}, there should be a connection between a gap in theDirac spectrum and the presence of $SU(2)_{CS}$.In this paper, we analyzewhether the quark propagator can be invariant under $SU(n_F)_L \timesSU(n_F)_R$ and $SU(2)_{CS}$ transformations, in case of a gap in the Diracspectrum, and consequently the invariance of hadron correlators, giving thecondition for a quark propagator to be invariant under $SU(2)_{CS}$.

Gap in the Dirac spectrum and quark propagator symmetries in lattice QCD