  Recent studies on lattice QCD have shown the emergence of large symmetries at
high temperature. This includes not only the restoration $SU(n_F)_L \times
SU(n_F)_R$, 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, at
high $T$, 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 the
Dirac spectrum and the presence of $SU(2)_{CS}$.In this paper, we analyze
whether the quark propagator can be invariant under $SU(n_F)_L \times
SU(n_F)_R$ and $SU(2)_{CS}$ transformations, in case of a gap in the Dirac
spectrum, and consequently the invariance of hadron correlators, giving the
condition for a quark propagator to be invariant under $SU(2)_{CS}$.
