Spectrum of the QCD Dirac operator and chiral random matrix theory

We argue that the spectrum of the QCD Dirac operator near zero virtuality can be described by random matrix theory. As in the case of the classical random matrix ensembles of Dyson we have three different cases: the chiral orthogonal ensemble, the chiral unitary ensemble, and the chiral symplectic ensemble. They correspond to gauge groups SU(2) in the fundamental representation, SU(Nc), Nc3 in the fundamental representation, and non-Abelian gauge groups SU(Nc) for all Nc with fermions in the adjoint representation, respectively. The joint probability density reproduces Leutwyler-Smilga sum rules.