Volume Conjecture, Regulator and SL₂(C)-Character Variety of a Knot

In this paper, by using the regulator map of Beilinson-Deligne, we show that the quantization condition posed by Gukov is true for the SL_2(\mathbb{C}) character variety of the hyperbolic knot in S^3. Furthermore, we prove that the corresponding \mathbb{C}^{*}-valued 1-form is a secondary characteristic class (Chern-Simons) arising from the vanishing first Chern class of the flat line bundle over the smooth part of the character variety, where the flat line bundle is the pullback of the universal Heisenberg line bundle over \mathbb{C}^{*}\times \mathbb{C}^{*}. The second part of the paper is to define an algebro-geometric invariant of 3-manifolds resulting from the Dehn surgery along a hyperbolic knot complement in $S^3$. We establish a Casson type invariant for these 3-manifolds. In the last section, we explicitly calculate the character variety of the figure-eight knot and discuss some applications.