Riemannian symmetric superspaces and their origin in random-matrix theory

Gaussian random-matrix ensembles defined over the tangent spaces of the large families of Cartan’s symmetric spaces are considered. Such ensembles play a central role in mesoscopic physics, as they describe the universal ergodic limit of disordered and chaotic single-particle systems. The generating function for the spectral correlations of each ensemble is reduced to an integral over a Riemannian symmetric superspace in the limit of large matrix dimension. Such a space is defined as a pair (G/H,Mr), where G/H is a complex-analytic graded manifold homogeneous with respect to the action of a complex Lie supergroup G, and Mr is a maximal Riemannian submanifold of the support of G/H.