We replace equivariant homology groups by equivariant $K$-group in the mathematical definition of $3d$ $\mathcal{N}=4$ SUSY gauge theories, and define K-theoretic Coulomb branches. Gaiotto conjectures that they are Coulomb branches of $4d$ $\mathcal{N}=2$ SUSY gauge theories over $\mathbb{R}^3 \times \mathbb{S}^1$ with a generic complex structure among $\mathbb{S}^2$ of complex structures of an expected hyperk\"ahler structure. Many properties are in parallel with homology version, but we expect several new features such as cluster algebra structures.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/8493