Phase constants in the Fock-Goncharov quantum cluster varieties

TL;DR

This paper proves that the constants in quantum mutation relations for Fock-Goncharov cluster varieties are exactly 1, ensuring the representations are genuine and not projective, and derives key operator identities.

Contribution

It establishes that the algebraic relations among quantum mutations hold exactly with constants equal to 1, confirming the genuineness of the quantum mapping class group representations.

Findings

Constants in quantum mutation relations are exactly 1.

Derived hexagon and octagon identities for the quantum dilogarithm.

Confirmed the non-projective nature of the quantum Teichmüller representations.

Abstract

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.