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A cluster variety of Fock and Goncharov is a scheme constructed from the datarelated to the cluster algebras of Fomin and Zelevinsky. A seed is acombinatorial data which can be encoded as an $n\times n$ matrix with integerentries, or as a quiver in special cases, together with $n$ formal variables. Amutation is a certain rule for transforming a seed into another seed; the newvariables are related to the previous variables by some rational expressions.To each seed one attaches an $n$-dimensional torus, and by gluing the torialong the birational maps defined by the mutation formulas, one constructs acluster variety. Quantization of a cluster variety assigns to each seed anon-commutative ring which deforms the classical ring of functions on the torusattached to the seed, as well as to each mutation an isomorphism of skew fieldsof fractions of these non-commutative rings. A representation realizes thenon-commutative rings as algebras of operators on Hilbert spaces, and thequantum mutation isomorphisms as unitary maps between the Hilbert spaces thatintertwine the operators for the rings. These unitary intertwiners are one ofthe major results of the Fock-Goncharov quantization of cluster varieties, andare given by the special function called the quantum dilogarithm. The classicalmutations satisfy certain algebraic relations, which were known to be satisfiedalso by the corresponding intertwiners up to complex constants of modulus $1$.The present paper shows by computation that these constants are all $1$. Oneimplication is that the mapping class group representations resulting from theapplication of the Fock-Goncharov quantization to the quantum Teichm\"ullertheory are genuine, not projective.

Phase constants in the Fock-Goncharov quantization of cluster varieties: long version