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 Hλ⃗ and closely connects to the infinite-dimensional $*$-representation of the quantum deformed Lorentz group Uq(sl₂)⊗ U_q(sl₂), where q=exp[2π i/k(1+b²)] and q=exp[2π i/k(1+b⁻²)] with $|b|=1$. The quantum group Uq(sl₂)⊗ U_q(sl₂) also emerges from the quantum gauge transformations of the complex Chern-Simons theory. Focusing on a m-holed sphere Σ0,m, the physical Hilbert space Hphys is identified by imposing the gauge invariance and the flatness constraint. The states in Hphys are the Uq(sl₂)⊗ U_q(sl₂)-invariant linear functionals on a dense domain in Hλ⃗. 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 Σ0,m are diagonalized.