Dualities of $K$-theoretic Coulomb branches from a once-punctured torus

We consider the quantized ${\mathrm{S}\mathrm{L}}_2$-character variety of a once-punctured torus. We show that this quantized algebra has three ${\mathbb{Z}}_2$-invariant subalgebras that are isomorphic to quantized $K$-theoretic Coulomb branches in the sense of Braverman, Finkelberg, and Nakajima. These subalgebras are permuted by the ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$ mapping class group action. Our results confirm various predictions from the physics literature about 4d $\mathcal{N}=2^{\ast }$ theories and their dualities.