Exact solution of critical self-dual unitary-matrix models

Models of random surfaces defined by integrals over self-dual unitary quaternion matrices are solved exactly in a double-scaling limit. There are contributions to the specific heat from surfaces with odd Euler characteristic, indicating that these are theories of unoriented strings. For the k=1 model, these contributions are determined from the quantum mechanics of a particle moving in a potential given by the specific heat of the k=1 unitary model, up to a linear term. A by-product of this analysis is a new solution of unitary-matrix models, formulated in terms of matrix differential operators.