2d BF theory coupled to 1d quantum mechanics: The phase space and its quantization

We study the ring of functions on the (classical and quantized) phase space of 2-dimensional BF theory with the gauge group $\mathrm{GL}_N$ coupled to 1-dimensional quantum mechanics with global symmetry $\mathrm{GL}_K$. These functions are gauge-invariant local observables of the coupled system. We first construct the classical phase space of this system and describe its ring of functions, as well as their large-$N$ limit. We next compute the Hilbert series of these algebras for finite $N$ and 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 coproduct. Finally, we identify these quantized algebras with the quantized Coulomb branch algebras of certain $3 \mathrm{~d} \mathcal{N}=4$ quiver gauge theories.