  The emergence of glassy dynamics and the glass transition in dense disordered
systems is still not fully understood theoretically. Mode-coupling theory (MCT)
has shown to be effective in describing some of the non-trivial features of
glass formation, but it cannot explain the full glassy phenomenology due to the
strong approximations on which it is based. Generalized mode-coupling theory
(GMCT) is a hierarchical extension of the theory, which is able to outclass MCT
by carefully describing the dynamics of higher order correlations in its
generalized framework. Unfortunately, the theory has so far only been developed
for single component systems and as a result works poorly for highly
polydisperse materials. In this paper, we solve this problem by developing GMCT
for multi-component systems. We use it to predict the glassy dynamics of the
binary Kob-Andersen Lennard-Jones mixture, as well as its purely repulsive
Weeks-Chandler-Andersen analogue. Our results show that each additional level
of the GMCT hierarchy gradually improves the predictive power of GMCT beyond
its previous limit. This implies that our theory is able to harvest more
information from the static correlations, thus being able to better understand
the role of attraction in supercooled liquids from a first-principles
perspective.
