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Eigenvalues and Random Matrix Theory

An important property of massless QCD is the spontaneous breaking of chiral symmetry. The associated Goldstone pions dominate the low-energy, finite-volume scaling behavior of the Dirac operator spectrum in the microscopic regime, 1/ΛQCD << L << 1/mπ, with L the length of the system [1]. This behavior can be characterized by chiral random matrix theory (RMT). The RMT description of the lowenergy, finite-volume scaling behavior is specified by symmetry properties of the Dirac operator and the topological charge sector being considered [2,3]. The RMT predictions are universal in the sense that the symmetry properties, but not the form of the potential matters [4]. Furthermore, the properties can be derived directly from the effective, finite-volume partition functions of QCD of Leutwyler and Smilga, without the detour through RMT [3], though RMT nicely and succinctly describes and classifies all these properties. The topological charge enters the RMT prediction via the number of fermionic zero modes, related to the topological charge through the index theorem. The symmetry properties of the Dirac operator fall into three classes, corresponding to the chiral orthogonal, unitary, and symplectic ensembles [3]. Examples are, respectively, fermions in the fundamental representation of gauge group SU(2), fermions in the fundamental representation of gauge group SU(Nc) with Nc ≥ 3, and fermions in the adjoint representa-