  Super-cooled liquids are characterized by their fragility: the slowing down
of the dynamics under cooling is more sudden and the jump of specific heat at
the glass transition is generally larger in fragile liquids than in strong
ones. Despite the importance of this quantity in classifying liquids,
explaining what aspects of the microscopic structure controls fragility remains
a challenge. Surprisingly, experiments indicate that the linear elasticity of
the glass -- a purely local property of the free energy landscape -- is a good
predictor of fragility. In particular, materials presenting a large excess of
soft elastic modes, the so-called boson peak, are strong. This is also the case
for network liquids near the rigidity percolation, known to affect elasticity.
Here we introduce a model of the glass transition based on the assumption that
particles can organize locally into distinct configurations, which are coupled
spatially via elasticity. The model captures the mentioned observations
connecting elasticity and fragility. We find that materials presenting an
abundance of soft elastic modes have little elastic frustration: energy is
insensitive to most directions in phase space, leading to a small jump of
specific heat. In this framework strong liquids turn out to lie the closest to
a critical point associated with a rigidity or jamming transition, and their
thermodynamic properties are related to the problem of number partitioning and
to Hopfield nets in the limit of small memory.
