The Z2-invariant subalgebras of the quantized SL2 character variety of a once-punctured torus are isomorphic to quantized K-theoretic Coulomb branches for SL2 and PGL2, and are permuted by the SL2(Z) mapping class group.
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Dualities of $K$-theoretic Coulomb branches from a once-punctured torus
The Z2-invariant subalgebras of the quantized SL2 character variety of a once-punctured torus are isomorphic to quantized K-theoretic Coulomb branches for SL2 and PGL2, and are permuted by the SL2(Z) mapping class group.