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

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 $\mathcal{H}{\vec{\lambda}}$ 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 $\mathbf{q}=\exp[\frac{2\pi i}{k}(1+b^2)]$ and $\widetilde{\mathbf{q}}=\exp[\frac{2\pi i}{k}(1+b^{-2})]$ 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 $\Sigma_{0,m}$, 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 $\mathcal{H}{\vec{\lambda}}$. 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 $\Sigma{0,m}$ are diagonalized.