  We study ring of functions on the (classical and quantized) phase space of
$2d$ BF theory with the gauge group $\mathrm{GL}_N$ coupled to a $1d$ quantum
mechanics with global symmetry $\mathrm{GL}_K$. These functions are
gauge-invariant local observables of the coupled system, which appears as
boundary side of an early example of twisted holography involving stacks of $N$
D2-branes and $K$ D4-branes intersecting along a real line. We first construct
the classical phase space of this system and describe its ring of functions and
their large-$N$ limit. We next compute the Hilbert series of these algebras for
finite-$N$ and also in the large-$N$ limit. We then study the quantization of
this phase space and the deformation quantization of its ring of functions,
elaborate its relation to the Yangian, and construct its co-product. Finally,
we identify these quantized algebras with the quantized Coulomb-branch algebras
of certain $3d$ $\mathcal{N}=4$ quiver gauge theories.
