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

This paper develops a framework for the Hamiltonian quantization of complex Chern-Simons theory with gauge group SL (2, C) at an even level k_+. 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 H_ and closely connects to the infinite-dimensional *-representation of the quantum deformed Lorentz group Uₐ (sl₂) Uₐ (sl₂), where q=2 ik (1+b²) and q=2 ik (1+b^-2) with |b|=1. The quantum group Uₐ (sl₂) Uₐ (sl₂) also emerges from the quantum gauge transformations of the complex Chern-Simons theory. Focusing on a m-holed sphere ₀, ₌, the physical Hilbert space H₏₇ₘₒ is identified by imposing the gauge invariance and the flatness constraint. The states in H₏₇ₘₒ are the Uₐ (sl₂) Uₐ (sl₂) -invariant linear functionals on a dense domain in H_. 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 ₀, ₌ are diagonalized.