  To pinpoint the microscopic mechanism for superconductivity has proven to be
one of the most outstanding challenges in the physics of correlated quantum
matter. Thus far, the most direct evidence for an electronic pairing mechanism
is the observation of a new symmetry of the order-parameter, as done in the
cuprate high-temperature superconductors. Like distinctions based on the
symmetry of a locally defined order-parameter, global, topological invariants
allow for a sharp discrimination between states of matter that cannot be
transformed into each other adiabatically. Here we propose an unconventional
pairing state for the electron fluid in two-dimensional oxide interfaces and
establish a direct link to the emergence of nontrivial topological invariants.
Topological superconductivity and Majorana edge states can then be used to
detect the microscopic origin for superconductivity. In addition, we show that
also the density wave states that compete with superconductivity sensitively
depend on the nature of the pairing interaction. Our conclusion is based on the
special role played by the spin-orbit coupling and the shape of the Fermi
surface in SrTiO$_{3}$/LaAlO$_{3}$-interfaces and closely related systems.
