  Searching for space-time variations of the constants of Nature is a promising
way to search for new physics beyond General Relativity and the standard model
motivated by unification theories and models of dark matter and dark energy. We
propose a new way to search for a variation of the fine-structure constant
using measurements of late-type evolved giant stars from the S-star cluster
orbiting the supermassive black hole in our Galactic Center. A measurement of
the difference between distinct absorption lines (with different sensitivity to
the fine structure constant) from a star leads to a direct estimate of a
variation of the fine structure constant between the star's location and Earth.
Using spectroscopic measurements of 5 stars, we obtain a constraint on the
relative variation of the fine structure constant below $10^{-5}$. This is the
first time a varying constant of Nature is searched for around a black hole and
in a high gravitational potential. This analysis shows new ways the monitoring
of stars in the Galactic Center can be used to probe fundamental physics.
