  Experimental observations of unexpected shear rigidity in confined liquids,
on very low frequency scales on the order of 0.01-0.1 Hz, call into question
our basic understanding of the elasticity of liquids and have posed a challenge
to theoretical models of the liquid state ever since. Here we combine the
nonaffine theory of lattice dynamics valid for disordered condensed matter
systems with the Frenkel theory of the liquid state. The emerging framework
shows that applying confinement to a liquid can effectively suppress the low
frequency modes that are responsible for nonaffine soft mechanical response,
thus leading to an effective increase of the liquid shear rigidity. The new
theory successfully predicts the scaling law $G'\sim L^{-3}$ for the
low-frequency shear modulus of liquids as a function of the confinement length
$L$, in agreement with experimental results, and provides the basis for a more
general description of the elasticity of liquids across different time and
length scales.
