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The quantum dilogarithm and representations quantum cluster varieties

V. V. Fock, A. B. Goncharov

TL;DR

The paper develops a quantum-dilogarithm–driven framework for quantizing cluster ${\cal X}$-varieties by constructing *-representations of the quantized algebras and unitary projective representations of the saturated cluster modular groups. Central to the construction are the symplectic double ${\cal D}$ and the quantum double ${\cal D}_q$, together with intertwiners encoded by the quantum dilogarithm that decompose mutations into automorphism and monomial parts. This yields a canonical representation theory on Schwartz spaces $\mathcal S_{\cal X}$ and principal-series–type decompositions, with applications to higher Teichmüller spaces ${\cal X}_{G,S}$ and their modular groups, and proposes duality and canonical-basis conjectures in the quantum Langlands setting. The results illuminate a Weil-representation–like structure for quantum cluster varieties, provide a pathway to modular-functor–style constructions, and connect cluster quantization to geometric structures on moduli spaces of local systems and quasifuchsian representations. The framework combines cluster algebra mutations, quantum dilogarithm identities, and symplectic/Poisson groupoid geometry to produce infinite-dimensional unitary representations with potential impact on quantum topology and geometric representation theory of surface groups.

Abstract

We construct, using the quantum dilogarithm, a series of *-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the cluster modular groups. The examples of the latter include the classical mapping class groups of punctured surfaces. One of applications is quantization of higher Teichmuller spaces. The constructed unitary representations can be viewed as analogs of the Weil representation. In both cases representations are given by integral operators. Their kernels in our case are the quantum dilogarithms. We introduce the symplectic/quantum double of cluster varieties and related them to the representations.

The quantum dilogarithm and representations quantum cluster varieties

TL;DR

The paper develops a quantum-dilogarithm–driven framework for quantizing cluster -varieties by constructing *-representations of the quantized algebras and unitary projective representations of the saturated cluster modular groups. Central to the construction are the symplectic double and the quantum double , together with intertwiners encoded by the quantum dilogarithm that decompose mutations into automorphism and monomial parts. This yields a canonical representation theory on Schwartz spaces and principal-series–type decompositions, with applications to higher Teichmüller spaces and their modular groups, and proposes duality and canonical-basis conjectures in the quantum Langlands setting. The results illuminate a Weil-representation–like structure for quantum cluster varieties, provide a pathway to modular-functor–style constructions, and connect cluster quantization to geometric structures on moduli spaces of local systems and quasifuchsian representations. The framework combines cluster algebra mutations, quantum dilogarithm identities, and symplectic/Poisson groupoid geometry to produce infinite-dimensional unitary representations with potential impact on quantum topology and geometric representation theory of surface groups.

Abstract

We construct, using the quantum dilogarithm, a series of *-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the cluster modular groups. The examples of the latter include the classical mapping class groups of punctured surfaces. One of applications is quantization of higher Teichmuller spaces. The constructed unitary representations can be viewed as analogs of the Weil representation. In both cases representations are given by integral operators. Their kernels in our case are the quantum dilogarithms. We introduce the symplectic/quantum double of cluster varieties and related them to the representations.

Paper Structure

This paper contains 43 sections, 49 theorems, 241 equations, 2 figures.

Table of Contents

  1. Introduction
  2. An outline
  3. The quantum double and the intertwiner
  4. The symplectic double and its properties
  5. Quantum dilogarithm, quantum double, and quantization
  6. The dilogarithm function and its two quantum versions
  7. The dilogarithm and cluster varieties
  8. Representations of quantum cluster ${\cal X}$-varieties
  9. Quantization of higher Teichmüller spaces
  10. A canonical representation of the quantum double
  11. The symplectic double and higher Teichmüller theory FG5
  12. The symplectic double of a cluster ${\cal X}$-variety
  13. Cluster ${\cal X}$- and ${\cal A}$-varieties
  14. The symplectic double
  15. Cluster linear algebra
  16. ...and 28 more sections

Key Result

Theorem 1.1

The datum $\Bigl(\widehat{\cal G}, {\bf L}, \{{\cal S}_{\lambda, {\bf i}} \subset {\rm V}_{\lambda, {\bf i}}\}, \{{\bf K}_{{\bf c}^o}\}\Bigr)$ provides a $\ast$-representation of quantum cluster ${\cal X}$-variety. In particular operators (12.15.06.2) provide a unitary projective representation of

Figures (2)

  • Figure 1: The composition of five flips is the identity.
  • Figure 2: A feed for a configuration of $5$ affine flags in $SL_3$.

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 49 more