Constraints on the Dirac spectrum from chiral symmetry restoration and the fate of $\mathrm{U}(1{)}_A$ symmetry

I discuss chiral symmetry restoration in the chiral limit $m\to 0$ of QCD with two light quark flavours of mass $m$, focussing on its consequences for scalar and pseudoscalar susceptibilities, and on the resulting constraints on the Dirac spectrum. I show that $\mathrm{U}(1{)}_A$ symmetry remains broken in the $\mathrm{S}\mathrm{U}(2{)}_A$ symmetric phase if the spectral density $\rho (\lambda ;m)$ develops a singular near-zero peak, tending to $O(m^4)\mathrm{/}\lambda$ in the chiral limit. Moreover, $\mathrm{S}\mathrm{U}(2{)}_A$ restoration requires that the number of modes in the peak be proportional to the topological susceptibility, indicating that such a peak must be of topological origin.