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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.19447 |
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| _version_ | 1866910120076640256 |
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| author | Klyuev, Daniil Vulakh, Joseph |
| author_facet | Klyuev, Daniil Vulakh, Joseph |
| contents | For a noncommutative algebra $\mathcal{A}$ and an antilinear automorphism $ρ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19447 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Positive Traces on Certain ${\rm SL}(2)$ Coulomb Branches Klyuev, Daniil Vulakh, Joseph High Energy Physics - Theory Mathematical Physics Representation Theory For a noncommutative algebra $\mathcal{A}$ and an antilinear automorphism $ρ$ 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. |
| title | Positive Traces on Certain ${\rm SL}(2)$ Coulomb Branches |
| topic | High Energy Physics - Theory Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2507.19447 |