Warm molecular Hydrogen at high redshift with the James Webb Space
  Telescope
  The build-up of galaxies is regulated by a complex interplay between
gravitational collapse, galaxy merging and feedback related to AGN and star
formation. The energy released by these processes has to dissipate for gas to
cool, condense, and form stars. How gas cools is thus a key to understand
galaxy formation. \textit{Spitzer Space Telescope} infrared spectroscopy
revealed a population of galaxies with weak star formation and unusually
powerful H$_2$ line emission. This is a signature of turbulent dissipation,
sustained by large-scale mechanical energy injection. The cooling of the
multiphase interstellar medium is associated with emission in the H$_2$ lines.
These results have profound consequences on our understanding of regulation of
star formation, feedback and energetics of galaxy formation in general. The
fact that H$_2$ lines can be strongly enhanced in high-redshift turbulent
galaxies will be of great importance for the \textit{James Webb Space
Telescope} observations which will unveil the role that H$_2$ plays as a
cooling agent in the era of galaxy assembly.
