  A cluster variety of Fock and Goncharov is a scheme constructed by gluing
split algebraic tori, called seed tori, via birational gluing maps called
mutations. In quantum theory, the ring of functions on seed tori are deformed
to non-commutative rings, represented as operators on Hilbert spaces. Mutations
are quantized to unitary maps between the Hilbert spaces intertwining the
representations. These unitary intertwiners are described using the quantum
dilogarithm function $\Phi^\hbar$. Algebraic relations among classical
mutations are satisfied by the intertwiners up to complex constants. The
present paper shows that these constants are $1$. So the mapping class group
representations resulting from the Chekhov-Fock-Goncharov quantum Teichm\"uller
theory are genuine, not projective. During the course, the hexagon and the
octagon operator identities for $\Phi^\hbar$ are derived.
