[Paper Review] Phase constants in the Fock-Goncharov quantization of cluster varieties

Hyun Kyu Kim|arXiv (Cornell University)|Feb 2, 2016

Algebraic structures and combinatorial models22 references3 citations

TL;DR

This paper proves that the phase constants in the Fock-Goncharov quantization of cluster varieties are all equal to 1, resolving a long-standing question about the consistency of quantum mutation isomorphisms. By computing these constants explicitly, the authors establish that the resulting mapping class group representations in quantum Teichmüller theory are genuine, not projective, thereby confirming the unitary structure of the quantum theory.

ABSTRACT

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 $n imes n$ 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.

Motivation & Objective

• To resolve the ambiguity in phase constants associated with quantum mutation isomorphisms in Fock-Goncharov quantization.
• To determine whether the quantum intertwiners arising from the quantum dilogarithm satisfy the classical mutation relations up to phase factors or exactly.
• To establish the consistency of the Fock-Goncharov quantization framework by showing that the phase constants are trivial (equal to 1).
• To confirm that the resulting representations of the mapping class group in quantum Teichmüller theory are genuine, not projective.

Proposed method

• Computing the phase constants arising from the composition of quantum mutation isomorphisms in the Fock-Goncharov quantization framework.
• Using the properties of the quantum dilogarithm function as the key tool to derive the intertwiners between skew fields of fractions of non-commutative rings.
• Analyzing the algebraic relations satisfied by the quantum mutations and comparing them to the classical relations in the cluster variety setting.
• Applying explicit algebraic manipulations and functional identities of the quantum dilogarithm to evaluate the constants of proportionality in the quantum intertwiners.
• Verifying that the composition of successive quantum mutations yields the identity map up to a scalar, and showing that this scalar is 1.

Experimental results

Research questions

• RQ1 Are the phase constants in the quantum mutation isomorphisms of Fock-Goncharov quantization equal to 1, or do they remain nontrivial complex numbers of modulus 1?
• RQ2 Do the quantum intertwiners satisfy the classical braid-like relations from cluster mutations up to a phase, and if so, what is the value of that phase?
• RQ3 Does the vanishing of the phase constants imply that the mapping class group representations constructed via Fock-Goncharov quantization are genuine rather than projective?
• RQ4 Can the consistency of the quantum cluster algebra structure be confirmed by showing that the quantum mutation maps compose to the identity with trivial phase?

Key findings

• The phase constants in the quantum mutation isomorphisms are all equal to 1, not just complex numbers of modulus 1.
• The quantum intertwiners satisfy the classical mutation relations exactly, without any nontrivial phase factors.
• The mapping class group representations arising from Fock-Goncharov quantization are genuine unitary representations, not projective ones.
• The consistency of the Fock-Goncharov quantization framework is confirmed at the level of the quantum mutation composition.
• The use of the quantum dilogarithm as the intertwiner leads to a fully consistent and unitary quantum theory of cluster varieties.