Quantum Algebra of Chern-Simons Matrix Model and Large $N$ Limit
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913882024443904 |
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| author | Hu, Sen Li, Si Ye, Dongheng Zhou, Yehao |
| author_facet | Hu, Sen Li, Si Ye, Dongheng Zhou, Yehao |
| contents | In this paper we study the algebra of quantum observables of the Chern-Simons matrix model which was originally proposed by Susskind and Polychronakos to describe electrons in fractional quantum Hall effects. We establish the commutation relations for its generators and study the large $N$ limit of its representation. We show that the large $N$ limit algebra is isomorphic to the uniform in $N$ algebra studied by Costello, which is isomorphic to the deformed double current algebra studied by Guay. Under appropriate scaling limit, we show that the large $N$ limit algebra degenerates to a Lie algebra which admits a surjective map to the affine Lie algebra of $\mathfrak{u}(p)$. This leads to a complete proof of the large $N$ emergence of the $\mathfrak{u}(p)$ current algebra as proposed by Dorey, Tong and Turner. This also suggests a rigorous derivation of edge excitation of a fractional quantum Hall droplet. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_14046 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantum Algebra of Chern-Simons Matrix Model and Large $N$ Limit Hu, Sen Li, Si Ye, Dongheng Zhou, Yehao Quantum Algebra Strongly Correlated Electrons High Energy Physics - Theory Mathematical Physics Representation Theory 81T32, 81S08, 81V70, 81R60 In this paper we study the algebra of quantum observables of the Chern-Simons matrix model which was originally proposed by Susskind and Polychronakos to describe electrons in fractional quantum Hall effects. We establish the commutation relations for its generators and study the large $N$ limit of its representation. We show that the large $N$ limit algebra is isomorphic to the uniform in $N$ algebra studied by Costello, which is isomorphic to the deformed double current algebra studied by Guay. Under appropriate scaling limit, we show that the large $N$ limit algebra degenerates to a Lie algebra which admits a surjective map to the affine Lie algebra of $\mathfrak{u}(p)$. This leads to a complete proof of the large $N$ emergence of the $\mathfrak{u}(p)$ current algebra as proposed by Dorey, Tong and Turner. This also suggests a rigorous derivation of edge excitation of a fractional quantum Hall droplet. |
| title | Quantum Algebra of Chern-Simons Matrix Model and Large $N$ Limit |
| topic | Quantum Algebra Strongly Correlated Electrons High Energy Physics - Theory Mathematical Physics Representation Theory 81T32, 81S08, 81V70, 81R60 |
| url | https://arxiv.org/abs/2308.14046 |