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We present a detailed derivation of a microscopic theory for the glasstransition of a liquid enclosed between two parallel walls relying on amode-coupling approximation. This geometry lacks translational invarianceperpendicular to the walls, which implies that the density profile and thedensity-density correlation function depends explicitly on the distances to thewalls. We discuss the residual symmetry properties in slab geometry andintroduce a symmetry adapted complete set of two-point correlation functions.Since the currents naturally split into components parallel and perpendicularto the walls the mathematical structure of the theory differs from theestablished mode-coupling equations in bulk. We prove that the equations forthe nonergodicity parameters still display a covariance property similar tobulk liquids.

Mode-coupling theory of the glass transition for confined fluids