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Cluster ensembles, quantization and the dilogarithm II: The intertwiner

V. V. Fock, A. B. Goncharov

TL;DR

The paper addresses the problem of quantizing cluster ${\cal X}$-varieties and constructing a canonical representation by developing intertwiners built from the quantum dilogarithm. The authors formulate a $\ast$-quantization framework via a Heisenberg algebra ${\cal H}^{\hbar}_{\bf i}$ and a mutation map $\kappa^{\hbar}(\mu_k)$, leveraging the quantum logarithm $\phi^{\hbar}$ and the modular double to connect seed variants and Langlands duals. The main contributions include an explicit intertwiner ${\bf K}_{\mathbf i\to\mathbf i'}$ realizing the mutation between seeds, a bimodule structure on functions on the $\cal A$-space, and a canonical projective representation of the cluster modular groupoid on $L^2({\cal A}^+)$, all grounded in the properties of the quantum dilogarithm $\Phi^{\hbar}$ and quantum logarithm $\phi^{\hbar}$. This framework advances a robust, representation-theoretic approach to quantum cluster varieties with potential applications in quantum geometry and related areas of mathematical physics.

Abstract

The main result is a construction, via the quantum dilogarithm, of certain intertwiner operators, which play the crucial role in the quantization of the cluster X-varieties and construction of the corresponding canonical representation.

Cluster ensembles, quantization and the dilogarithm II: The intertwiner

TL;DR

The paper addresses the problem of quantizing cluster -varieties and constructing a canonical representation by developing intertwiners built from the quantum dilogarithm. The authors formulate a -quantization framework via a Heisenberg algebra and a mutation map , leveraging the quantum logarithm and the modular double to connect seed variants and Langlands duals. The main contributions include an explicit intertwiner realizing the mutation between seeds, a bimodule structure on functions on the -space, and a canonical projective representation of the cluster modular groupoid on , all grounded in the properties of the quantum dilogarithm and quantum logarithm . This framework advances a robust, representation-theoretic approach to quantum cluster varieties with potential applications in quantum geometry and related areas of mathematical physics.

Abstract

The main result is a construction, via the quantum dilogarithm, of certain intertwiner operators, which play the crucial role in the quantization of the cluster X-varieties and construction of the corresponding canonical representation.

Paper Structure

This paper contains 11 sections, 10 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Cluster ensembles
  3. Basic definitions
  4. Connections between quantum ${\cal X}$-varieties
  5. Motivation: $\ast$-quantization of cluster ${\cal X}$-varieties
  6. $\ast$-quantization of the space ${\cal X}^+$ via the quantum logarithm
  7. Modular double of a cluster ${\cal X}$-variety and $\ast$-quantization of the space ${\cal X}^+$
  8. The intertwiner
  9. A bimodule structure on functions on the ${\cal A}$--space
  10. The intertwiner via the quantum dilogarithm
  11. The quantum logarithm and dilogarithm functions

Key Result

Lemma 2.1

a) There is a canonical isomorphism of quantum spaces (Given in on the generators of any cluster coordinate system by $X_i \longmapsto X_i$). b) There is a canonical isomorphism of quantum spaces (Given in any cluster coordinate system by $X^o_i \longmapsto X_i^{-1}$, where $X^o_i$ are the generators of ${\cal X}_{{\bf i}^o, {q^{-1}}}$). c) There is a canonical isomorphism of quantum spaces

Theorems & Definitions (15)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 3.1
  • Claim 3.2
  • Lemma 3.3
  • Definition 3.4
  • Proposition 3.5
  • Claim 3.6
  • Definition 3.7
  • Lemma 4.1
  • ...and 5 more