  We establish a connection between time evolution of free Fermi droplets and
partition function of \emph{generalised} \emph{q}-deformed Yang-Mills theories
on Riemann surfaces. Classical phases of $(0+1)$ dimensional unitary matrix
models can be characterised by free Fermi droplets in two dimensions. We
quantise these droplets and find that the modes satisfy an abelian Kac-Moody
algebra. The Hilbert spaces $\mathcal{H}_+$ and $\mathcal{H}_-$ associated with
the upper and lower free Fermi surfaces of a droplet admit a Young diagram
basis in which the phase space Hamiltonian is diagonal with eigenvalue, in the
large $N$ limit, equal to the quadratic Casimir of $u(N)$. We establish an
exact mapping between states in $\mathcal{H}_\pm$ and geometries of droplets.
In particular, coherent states in $\mathcal{H}_\pm$ correspond to classical
deformation of upper and lower Fermi surfaces. We prove that correlation
between two coherent states in $\mathcal{H}_\pm$ is equal to the chiral and
anti-chiral partition function of $2d$ Yang-Mills theory on a cylinder. Using
the fact that the full Hilbert space $\mathcal{H}_+ \otimes \mathcal{H}_-$
admits a \emph{composite} basis, we show that correlation between two classical
droplet geometries is equal to the full $U(N)$ Yang-Mills partition function on
cylinder. We further establish a connection between higher point correlators in
$\mathcal{H}_\pm$ and higher point correlators in $2d$ Yang-Mills on Riemann
surface. There are special states in $\mathcal{H}_\pm$ whose transition
amplitudes are equal to the partition function of $2d$ \emph{q}-deformed
Yang-Mills and in general character expansion of Villain action. We emphasise
that the \emph{q}-deformation in the Yang-Mills side is related to special
deformation of droplet geometries without deforming the gauge group associated
with the matrix model.

High Energy Physics - Theory