Hamiltonian quantization of complex Chern-Simons theory at level-$k$

TL;DR

This work develops a Hamiltonian, combinatorial quantization framework for complex Chern-Simons theory with gauge group $SL(2,\mathbb{C})$ at even level $k$, revealing a deep link to the infinite-dimensional $*$-representation of the quantum Lorentz group $\mathscr{U}_{\mathbf q}(sl_2)\otimes \mathscr{U}_{\widetilde{\mathbf q}}(sl_2)$. By constructing the graph (holonomy) algebra on surfaces and its $*$-representation on $\mathcal H_{\vec{\lambda}}$, the authors identify the physical Hilbert space for an $m$-holed sphere as gauge-invariant linear functionals on a dense domain, effectively implementing the flatness constraint. They develop a Clebsch–Gordan decomposition and an invariant bilinear form to realize physical states, and show that Wilson loop operators diagonalize in a Fenchel-Nielsen type representation tied to a pants decomposition. This provides a robust framework connecting complex CS quantization to (i) a quantum Lorentz symmetry, (ii) a direct integral structure over intermediate representations, and (iii) a FN-coordinate-like spectral decomposition, with potential implications for 3d gravity and non-compact TQFTs. The work also clarifies crossing symmetry through A-moves, establishing unitary equivalence between different pants decompositions and reinforcing the geometric interpretation of the quantum moduli space.

Abstract

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.

Figures (4)

• Figure 1: The standard graph $\Gamma_{g,m}$: the cyclic order at the vertex $p$ is counterclockwise. The dashed line is the cilium.
• Figure 2: A decomposition of a 5-holed sphere into 3 pairs of pants: The holes are colored by $\lambda_1,\cdots,\lambda_5$. 2 cuts (the dashed circle) are colored by $\chi_3$ and $\chi_4$.
• Figure 3: A pair of pants associates to the "3j-symbol" $\Phi_{\lambda_{1}\lambda_{2}}^{\chi}$.
• Figure 4: A change of pants decomposition from cutting along the S-cycle to cutting along the T-cycle.

Theorems & Definitions (18)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Theorem 3.7