  In this paper, by using the regulator map of Beilinson-Deligne on a curve, we
show that the quantization condition posed by Gukov is true for the SL_2(C)
character variety of the hyperbolic knot in S^3. Furthermore, we prove that the
corresponding $\mathbb{C}^{*}$-valued closed 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 $\mathbb{C}^{*}\times \mathbb{C}^{*}$. Based on this result, we
give a reformulation of Gukov's generalized volume conjecture from a motivic
perspective.
