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.

Questo articolo esplora i giri e le loro implicazioni.

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