Positive Traces on Certain $\mathrm{SL}(2)$ Coulomb Branches

Abstract

For a noncommutative algebra $\mathcal{A}$ and an antilinear automorphism $\ddot{I}$ of $\mathcal{A}$, there is a notion of a positive trace. When we have a three-dimensional $\mathcal{N}=4$ gauge theory or four-dimensional $\mathcal{N}=2$ gauge theory compactified on a circle, classification of positive traces on its Coulomb branch $\mathcal{A}$ can give a better understanding of this theory. We classify positive traces on $\mathcal{A}$ in two cases. The first case is when $\mathcal{A}$ is a quantization of a Kleinian singularity of type $D$, with certain restriction on the quantization parameter. The second case is when $\mathcal{A}=K^{\mathrm{SL}(2,\mathbb{C}[[t]])⋊{\mathbb{C}}_q^{\times }}({\mathrm{Gr}}_{{\mathrm{PGL}}_2})$ is an algebra containing $K$-theoretic Coulomb branches of pure $\mathrm{SL}(2)$ and $\mathrm{PGL}(2)$ gauge theories.