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We study ring of functions on the (classical and quantized) phase space of2-dimensional BF theory with the gauge group $\mathrm{GL}_N$ coupled to a1-dimensional quantum mechanics with global symmetry $\mathrm{GL}_K$. Thesefunctions are gauge-invariant local observables of the coupled system. We firstconstruct the classical phase space of this system and describe its ring offunctions and their large-$N$ limit. We next compute the Hilbert series ofthese algebras for finite-$N$ and also in the large-$N$ limit. We then studythe quantization of this phase space and the deformation quantization of itsring of functions, elaborate its relation to the Yangian, and construct itscoproduct. Finally, we identify these quantized algebras with the quantizedCoulomb branch algebras of certain 3-dimensional $\mathcal{N}=4$ quiver gaugetheories.

Phase Space and Quantization of 2D BF Theory Coupled to 1D Quantum Mechanics