SIGMA 22 (2026), 033, 34 pages      arXiv:2507.19447      https://doi.org/10.3842/SIGMA.2026.033

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

Daniil Klyuev ^a and Joseph Vulakh ^b
a) Department of Mathematics, Northwestern University, USA
^b) Department of Mathematics, Massachusetts Institute of Technology, USA

Received July 28, 2025, in final form March 15, 2026; Published online April 10, 2026

Abstract
For a noncommutative algebra $\mathcal{A}$ and an antilinear automorphism $\rho$ 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 $\mathcal{A}=K^{{\rm SL}(2,\mathbb{C}[[t]])\rtimes \mathbb{C}_q^{\times}}({\rm Gr}_{{\rm PGL}_2})$ is an algebra containing $K$-theoretic Coulomb branches of pure ${\rm SL}(2)$ and ${\rm PGL}(2)$ gauge theories.

Key words: Coulomb branches; twisted traces; gauge theory; sphere quantization; Schur quantization.

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