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Cluster ensembles, quantization and the dilogarithm

V. V. Fock, A. B. Goncharov

TL;DR

This work introduces cluster ensembles as a rich geometric framework pairing a positive A-space with a positive X-space via a morphism, unifying cluster algebras, Poisson/symplectic structures, and deep motivic aspects. It develops a noncommutative q-deformation of the X-space and a quantum Frobenius mechanism, linking to the quantum dilogarithm and canonical dualities between A and X through Langlands duality. The authors propose duality conjectures and canonical pairings, provide finite-type evidence via tropical and cluster-theoretic constructions, and connect the theory to higher Teichmüller theory and quantum Teichmüller theory. A central theme is the motivic avatar of the Weil-Petersson form, expressed through K_2/Bloch complexes and the dilogarithm, which governs both classical and quantum facets of cluster ensembles. The work further develops a rich algebraic-geometric infrastructure, including modular complexes, Drinfeld-type centers at roots of unity, and canonical maps linking A- and X-spaces, paving the way for a broader quantization program.

Abstract

Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its group of symmetries - the cluster modular group, and a relation with the motivic dilogarithm. We define a q-deformation of the X-space. Formulate general duality conjectures regarding canonical bases in the cluster ensemble context. We support them by constructing the canonical pairing in the finite type case. Interesting examples of cluster ensembles are provided the higher Teichmuller theory, that is by the pair of moduli spaces corresponding to a split reductive group G and a surface S defined in math.AG/0311149. We suggest that cluster ensembles provide a natural framework for higher quantum Teichmuller theory.

Cluster ensembles, quantization and the dilogarithm

TL;DR

This work introduces cluster ensembles as a rich geometric framework pairing a positive A-space with a positive X-space via a morphism, unifying cluster algebras, Poisson/symplectic structures, and deep motivic aspects. It develops a noncommutative q-deformation of the X-space and a quantum Frobenius mechanism, linking to the quantum dilogarithm and canonical dualities between A and X through Langlands duality. The authors propose duality conjectures and canonical pairings, provide finite-type evidence via tropical and cluster-theoretic constructions, and connect the theory to higher Teichmüller theory and quantum Teichmüller theory. A central theme is the motivic avatar of the Weil-Petersson form, expressed through K_2/Bloch complexes and the dilogarithm, which governs both classical and quantum facets of cluster ensembles. The work further develops a rich algebraic-geometric infrastructure, including modular complexes, Drinfeld-type centers at roots of unity, and canonical maps linking A- and X-spaces, paving the way for a broader quantization program.

Abstract

Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its group of symmetries - the cluster modular group, and a relation with the motivic dilogarithm. We define a q-deformation of the X-space. Formulate general duality conjectures regarding canonical bases in the cluster ensemble context. We support them by constructing the canonical pairing in the finite type case. Interesting examples of cluster ensembles are provided the higher Teichmuller theory, that is by the pair of moduli spaces corresponding to a split reductive group G and a surface S defined in math.AG/0311149. We suggest that cluster ensembles provide a natural framework for higher quantum Teichmuller theory.

Paper Structure

This paper contains 72 sections, 65 theorems, 254 equations, 8 figures.

Table of Contents

  1. Introduction and main definitions with simplest examples
  2. Positive schemes and positive spaces
  3. Cluster ensembles: definitions
  4. Seeds and seed tori.
  5. Seed mutations.
  6. Coordinate description.
  7. Cluster transformations.
  8. Cluster modular groupoids.
  9. The ${\cal A}$- and ${\cal X}$- positive spaces.
  10. Examples of trivial cluster transformations.
  11. Special cluster modular groupoid and modular groups.
  12. The cluster ensemble.
  13. The chiral and Langlands duality for seeds.
  14. An example: cluster ${\cal X}$-variety structure of the moduli space ${\cal M}_{0,n+3}$
  15. Cluster ensemble structures
  16. ...and 57 more sections

Key Result

Lemma 1.5

The fibers of the map $p$ are the leaves of the null-foliation of the $2$-form $\Omega$. The subtorus ${\cal U}_{\Lambda}:= {p}({\cal A}_{\Lambda})$ is a symplectic leaf of the Poisson structure on ${\cal X}_{\Lambda}$. The symplectic structure on ${\cal U}_{\Lambda}$ induced by the form $\Omega$ on

Figures (8)

  • Figure 1: Picturing seeds by quivers - we show $d_i$'s only if they differ from $1$.
  • Figure 2: The two triangulations of the hexagon are related by the flip at the edge $E$.
  • Figure 3: A zig-zag triangulations of the $11$-gon.
  • Figure 4: Orbits of mutations.
  • Figure 5: The function $\delta_{E,F}$.
  • ...and 3 more figures

Theorems & Definitions (93)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.5
  • Definition 1.6
  • Lemma 1.7
  • Proposition 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 83 more