  We present an asymptotic expansion for quaternionic self-adjoint matrix
integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon
graphs and their non-orientable counterparts. The result exhibits a striking
duality between quaternionic self-adjoint and real symmetric matrix integrals.
The asymptotic expansions of these integrals are given in terms of summations
over topologies of compact surfaces, both orientable and non-orientable, for
all genera and an arbitrary positive number of marked points on them. We show
that the Gaussian Orthogonal Ensemble (GOE) and Gaussian Symplectic Ensemble
(GSE) have exactly the same graphical expansion term by term (when
appropriately normalized),except that the contributions from non-orientable
surfaces with odd Euler characteristic carry the opposite sign. As an
application, we give a new topological proof of the known duality for
correlations of characteristic polynomials. Indeed, we show that this duality
is equivalent to Poincare duality of graphs drawn on a compact surface. Another
application of our graphical expansion formula is a simple and simultaneous
(re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary
Ensemble) and GSE: The three cases have exactly the same graphical limiting
formula except for an overall constant that represents the type of the
ensemble.
