  Below the onset temperature $T_\text{o}$, the equilibrium relaxation time of
most glass-forming liquids exhibits glassy dynamics characterized by
super-Arrhenius temperature dependence. In this supercooled regime, the
relaxation dynamics also proceeds through localized elastic excitations
corresponding to hopping events between inherent states, i.e., potential-energy
minimizing configurations of the liquid. Despite its importance in
distinguishing the supercooled regime from the high-temperature regime, the
microscopic origin of $T_\text{o}$ is not yet known. Here, we construct a
theory for the onset temperature in two dimensions and find that inherent-state
melting transition, described by the binding-unbinding transition of dipolar
elastic excitations, delineates the supercooled regime from the
high-temperature regime. The corresponding melting transition temperature is in
good agreement with the onset temperature found in various two-dimensional
atomistic models of glass formers. We discuss the predictions of our theory on
the displacement and density correlations of two-dimensional supercooled
liquids, which are consistent with observations of the Mermin-Wagner
fluctuations in recent experiments and molecular simulations.
