Phase constants in the Fock-Goncharov quantum cluster varieties

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.