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

A bstract This paper develops a framework for the Hamiltonian quantization of complex Chern-Simons theory with gauge group $$\text{SL}(2,{\mathbb{C}})$$ 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 $${\mathcal{H}}_{\overrightarrow{\lambda }}$$ and closely connects to the infinite-dimensional *-representation of the quantum deformed Lorentz group $${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$ . The quantum group $${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$ 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 $${\mathcal{H}}_{\text{phys}}$$ is identified by imposing the gauge invariance and the flatness constraint. The states in $${\mathcal{H}}_{\text{phys}}$$ are the $${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$ -invariant linear functionals on a dense domain in $${\mathcal{H}}_{\overrightarrow{\lambda }}$$ . 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 Σ 0, m are diagonalized.