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Combinatorial quantization of the Hamiltonian Chern-Simons theory I
High Energy Physics - Theory · Submitted 10 Mar 1994
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Motivated by a recent paper of Fock and Rosly \cite{FoRo} we describe a mathematically precise quantization of the Hamiltonian Chern-Simons theory.
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Motivated by a recent paper of Fock and Rosly \cite{FoRo} we describe a mathematically precise quantization of the Hamiltonian Chern-Simons theory. We introduce the Chern-Simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly. The lattice model enjoys the symmetry with respect to a quantum gauge group. Using this fact we construct the algebra of observables of the Hamiltonian Chern-Simons theory equipped with a *-operation and a positive inner product.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Motivated by a recent paper of Fock and Rosly \cite{FoRo} we describe a mathematically precise quantization of the Hamiltonian Chern-Simons theory. | contribution 0.87 | 0.88 | 0.27 | asserted 1.00 |
| s1 | We introduce the Chern-Simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly. | method 0.54 | 0.73 | 0.12 | hedged possibility 0.84 |
| s2 | The lattice model enjoys the symmetry with respect to a quantum gauge group. | result 0.69 | 0.56 | 0.08 | asserted 1.00 |
| s3 | Using this fact we construct the algebra of observables of the Hamiltonian Chern-Simons theory equipped with a *-operation and a positive inner product. | contribution 0.81 | 0.95 | 0.38 | asserted 1.00 |
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Original abstract
Motivated by a recent paper of Fock and Rosly \cite{FoRo} we describe a mathematically precise quantization of the Hamiltonian Chern-Simons theory. We introduce the Chern-Simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly. The lattice model enjoys the symmetry with respect to a quantum gauge group. Using this fact we construct the algebra of observables of the Hamiltonian Chern-Simons theory equipped with a *-operation and a positive inner product.
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References
How the literature uses it
35 arXiv papers cite this in 56 sentences
Papers citing it · 38 sentences without a confident label
Uses its method or tool 3 papers
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We apply the technique of combinatorial quantization [ref] to the complex Chern-Simons theory with nonzero level $k$: As the quantization of (the holomorphic part of) $_g,m$, the holomorphic graph algebra $_g,m$ is an associative algebra spanned by polynomials of the generators that include the identity $1$ and the matrix elements of $A^I_i, B^I_i, M^I_ν∈End(V^I)⊗_g,m$, $ν=1,⋯,m$, $i=1,⋯,g$.
Hamiltonian quantization of complex Chern-Simons theory at level-$k$ · Graph algebra · 0.98
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In our approach, we adopt the Poisson bracket on the moduli space inspired by [ref] , and then proceed with the quantization method known as combinatorial quantization [ref] as an approach of canonical quantization for the moduli space of flat connections.
Quantum Group Intertwiner Space From Quantum Curved Tetrahedron · Introduction · 0.99
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Staying in the discrete, one might also directly try to construct Hilbert spaces for the corner Poisson algebra, for example by performing a naive smearing of the variables and working on cellulations of $,$, or by employing some version of the Fock-Rosly construction [ref] , which is adapted to the kind of Chern-Simons-like phase space we found.
Chern-Simons corner phase spaces in BF-BB theories · Discussion + Outlook · 0.81
Compares with it 1 paper
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Comparison with literature Our resulting Poisson bracket relations of “one-sided” Wilson lines in subsection [ref] are closely related to (and consistent with) those of Fock and Rosly [ref] and their subsequent “combinatorial quantization” [ref] .
Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes · Introduction · 1.00
Cites it as background 12 papers
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Its canonical quantization is known as the combinatorial quantization [ref] , whereas the path integral approach for Chern-Simons has been developed by Witten [ref] .
$q$-deformed 3D Loop Gravity on the Torus · Introduction · 0.95
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The moduli algebra, introduced in [ref] , is only defined when $H$ is modular since its definition uses the particular properties of the $S$-matrix in this case.
Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras · Introduction · 0.86
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In [ref] (and reviewed in detail in [ref] ), the combinatorial quantization framework for Hamiltonian 2-Chern-Simons theory was developed, taking inspiration from the works [ref] for 3d Chern-Simoms theory.
Categorical quantum symmetries and ribbon tensor 2-categories · Categorical gauge transformations as a Hopf category · 0.88
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