The Microscopic Spectral Density of the Dirac Operator derived from Gaussian Orthogonal and Symplectic Ensembles

The microscopic spectral correlations of the Dirac operator in Yang-Mills theories coupled to fermions in (2+1) dimensions can be related to three universality classes of Random Matrix Theory. In the microscopic limit the Orthogonal Ensemble (OE) corresponds to a theory with 2 colors and fermions in the fundamental representation and the Symplectic Ensemble (SE) corresponds to an arbitrary number of colors and fermions in the adjoint representation. Using a new method of Widom, we derive an expression for the two scalar kernels which through quaternion determinants give all spectral correlation functions in the Gaussian Orthogonal Ensemble (GOE) and in the the Gaussian Symplectic Ensemble (GSE) with all fermion masses equal to zero. The result for the GOE is valid for an arbitrary number of fermions while for the GSE we have results for an even number of fermions.