Hamiltonian quantization of complex Chern-Simons theory at level-$k$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908335689695232 |
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| author | Han, Muxin |
| author_facet | Han, Muxin |
| contents | This paper develops a framework for the Hamiltonian quantization of complex Chern-Simons theory with gauge group $\mathrm{SL}(2,\mathbb{C})$ at an even level $k\in\mathbb{Z}_+$. Our approach follows the procedure of combinatorial quantization to construct the operator algebras of quantum holonomies on 2-surfaces and develop the representation theory. The $*$-representation of the operator algebra is carried by the infinite dimensional Hilbert space $\mathcal{H}_{\vecλ}$ and closely connects to the infinite-dimensional $*$-representation of the quantum deformed Lorentz group $\mathscr{U}_{\mathbf{q}}(sl_2)\otimes \mathscr{U}_{\widetilde{\mathbf{q}}}(sl_2)$, where $\mathbf{q}=\exp[\frac{2πi}{k}(1+b^2)]$ and $\widetilde{\mathbf{q}}=\exp[\frac{2πi}{k}(1+b^{-2})]$ with $|b|=1$. The quantum group $\mathscr{U}_{\mathbf{q}}(sl_2)\otimes \mathscr{U}_{\widetilde{\mathbf{q}}}(sl_2)$ also emerges from the quantum gauge transformations of the complex Chern-Simons theory. Focusing on a $m$-holed sphere $Σ_{0,m}$, the physical Hilbert space $\mathcal{H}_{phys}$ is identified by imposing the gauge invariance and the flatness constraint. The states in $\mathcal{H}_{phys}$ are the $\mathscr{U}_{\mathbf{q}}(sl_2)\otimes \mathscr{U}_{\widetilde{\mathbf{q}}}(sl_2)$-invariant linear functionals on a dense domain in $\mathcal{H}_{\vecλ}$. Finally, we demonstrate that the physical Hilbert space carries a Fenchel-Nielsen representation, where a set of Wilson loop operators associated with a pants decomposition of $Σ_{0,m}$ are diagonalized. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16367 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hamiltonian quantization of complex Chern-Simons theory at level-$k$ Han, Muxin High Energy Physics - Theory General Relativity and Quantum Cosmology Geometric Topology Quantum Algebra This paper develops a framework for the Hamiltonian quantization of complex Chern-Simons theory with gauge group $\mathrm{SL}(2,\mathbb{C})$ at an even level $k\in\mathbb{Z}_+$. Our approach follows the procedure of combinatorial quantization to construct the operator algebras of quantum holonomies on 2-surfaces and develop the representation theory. The $*$-representation of the operator algebra is carried by the infinite dimensional Hilbert space $\mathcal{H}_{\vecλ}$ and closely connects to the infinite-dimensional $*$-representation of the quantum deformed Lorentz group $\mathscr{U}_{\mathbf{q}}(sl_2)\otimes \mathscr{U}_{\widetilde{\mathbf{q}}}(sl_2)$, where $\mathbf{q}=\exp[\frac{2πi}{k}(1+b^2)]$ and $\widetilde{\mathbf{q}}=\exp[\frac{2πi}{k}(1+b^{-2})]$ with $|b|=1$. The quantum group $\mathscr{U}_{\mathbf{q}}(sl_2)\otimes \mathscr{U}_{\widetilde{\mathbf{q}}}(sl_2)$ also emerges from the quantum gauge transformations of the complex Chern-Simons theory. Focusing on a $m$-holed sphere $Σ_{0,m}$, the physical Hilbert space $\mathcal{H}_{phys}$ is identified by imposing the gauge invariance and the flatness constraint. The states in $\mathcal{H}_{phys}$ are the $\mathscr{U}_{\mathbf{q}}(sl_2)\otimes \mathscr{U}_{\widetilde{\mathbf{q}}}(sl_2)$-invariant linear functionals on a dense domain in $\mathcal{H}_{\vecλ}$. Finally, we demonstrate that the physical Hilbert space carries a Fenchel-Nielsen representation, where a set of Wilson loop operators associated with a pants decomposition of $Σ_{0,m}$ are diagonalized. |
| title | Hamiltonian quantization of complex Chern-Simons theory at level-$k$ |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology Geometric Topology Quantum Algebra |
| url | https://arxiv.org/abs/2504.16367 |