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

A cluster variety of Fock and Goncharov is a scheme constructed from the data related to the cluster algebras of Fomin and Zelevinsky. A seed is a combinatorial data which can be encoded as an $n\times n$ matrix with integer entries, or as a quiver in special cases, together with $n$ formal variables. A mutation is a certain rule for transforming a seed into another seed; the new variables are related to the previous variables by some rational expressions. To each seed one attaches an $n$-dimensional torus, and by gluing the tori along the birational maps defined by the mutation formulas, one constructs a cluster variety. Quantization of a cluster variety assigns to each seed a non-commutative ring which deforms the classical ring of functions on the torus attached to the seed, as well as to each mutation an isomorphism of skew fields of fractions of these non-commutative rings. A representation realizes the non-commutative rings as algebras of operators on Hilbert spaces, and the quantum mutation isomorphisms as unitary maps between the Hilbert spaces that intertwine the operators for the rings. These unitary intertwiners are one of the major results of the Fock-Goncharov quantization of cluster varieties, and are given by the special function called the quantum dilogarithm. The classical mutations satisfy certain algebraic relations, which were known to be satisfied also by the corresponding intertwiners up to complex constants of modulus $1$. The present paper shows by computation that these constants are all $1$. One implication is that the mapping class group representations resulting from the application of the Fock-Goncharov quantization to the quantum Teichm\"uller theory are genuine, not projective.