  We investigate a system of two-dimensional interacting Brownian particles at
finite density. In the continuum limit, we uncover a hidden symmetry under
area-preserving diffeomorphism transformations. This symmetry leads to the
conservation of local vorticity. By calculating the generating functional
within the saddle-point plus Gaussian fluctuations approximation, we reveal the
emergence of a $U(1)$ gauge symmetry. This emergent symmetry enables us to
describe the dynamics of density fluctuations as a gauge theory. We solve the
corresponding equations of motion for various local and non-local two-body
potentials, demonstrating the presence of multiple dynamical regimes and
associated dynamical phase transitions.
