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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathfrak {u}(p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">u</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We conjecture that the large N limit algebra acts on the large N limit Hilbert space via the aforementioned degeneration limit, and we prove this conjecture in the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> case by a detailed study of large N limit Hilbert space. This leads to a complete proof of the large N emergence of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\widehat{\mathfrak {u}}(p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi mathvariant="fraktur">u</mi> <mo stretchy="true">^</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> model as proposed by Dorey, Tong and Turner in the case <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. This also suggests a rigorous derivation of edge excitation of a fractional quantum Hall droplet.

Quantum Algebra of Chern-Simons Matrix Model and Large N Limit

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发表日期：

2025年6月6日

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Sen Hu,
Si Li,
Dongheng Ye,
Yehao Zhou

摘要

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 $\mathfrak {u}(p)$ $u (p)$ . We conjecture that the large N limit algebra acts on the large N limit Hilbert space via the aforementioned degeneration limit, and we prove this conjecture in the $p=1$ $p = 1$ case by a detailed study of large N limit Hilbert space. This leads to a complete proof of the large N emergence of the $\widehat{\mathfrak {u}}(p)$ $\hat{u} (p)$ model as proposed by Dorey, Tong and Turner in the case $p=1$ $p = 1$ . This also suggests a rigorous derivation of edge excitation of a fractional quantum Hall droplet.