  Swap algorithms can shift the glass transition to lower temperatures, a
recent unexplained observation constraining the nature of this phenomenon. Here
we show that swap dynamic is governed by an effective potential describing both
particle interactions as well as their ability to change size. Requiring its
stability is more demanding than for the potential energy alone. This result
implies that stable configurations appear at lower energies with swap dynamics,
and thus at lower temperatures when the liquid is cooled. \maa{ The magnitude
of this effect is proportional to the width of the radii distribution, and
decreases with compression for finite-range purely repulsive interaction
potentials.} We test these predictions numerically and discuss the implications
of these findings for the glass transition.We extend these results to the case
of hard spheres where swap is argued to destroy meta-stable states of the free
energy coarse-grained on vibrational time scales. Our analysis unravels the
soft elastic modes responsible for the speed up swap induces, and allows us to
predict the structure and the vibrational properties of glass configurations
reachable with swap. In particular for continuously poly-disperse systems we
predict the jamming transition to be dramatically altered, as we confirm
numerically. A surprising practical outcome of our analysis is new algorithm
that generates ultra-stable glasses by simple descent in an appropriate
effective potential.
