  The weak coupling loop quantum theory with Abelian gauge group provides us a
new perspective to study the weak coupling properties of LQG. In this paper,
the weak coupling theory of all dimensional loop quantum gravity is established
based on a symplectic-morphism between the $SO(D+1)$ holonomy-flux phase space
and the $U(1)^{\frac{D(D+1)}{2}}$ holonomy-flux phase space. More explicitly,
the Gaussian, simplicity, diffeomorphism and scalar constraint operators in
$SO(D+1)$ loop quantum gravity will be generalized to the
$U(1)^{\frac{D(D+1)}{2}}$ loop quantum theory based on the symplectic-morphism,
and the $U(1)^{\frac{D(D+1)}{2}}$ loop quantum theory equipped with these
constraint operators gives the weak coupling $U(1)^{\frac{D(D+1)}{2}}$ loop
quantum gravity, with the corresponding Hilbert space is composed by the
$U(1)^{\frac{D(D+1)}{2}}$ heat-kernel coherent states which are peaked at the
weak coupling region of the $U(1)^{\frac{D(D+1)}{2}}$ holonomy-flux phase
space.
