Cluster ensembles, quantization and the dilogarithm

Vladimir V. Fock; Alexander B. Goncharov

Annales scientifiques de l'École Normale Supérieure (2009)

  • Volume: 42, Issue: 6, page 865-930
  • ISSN: 0012-9593

Abstract

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A cluster ensemble is a pair ( 𝒳 , 𝒜 ) of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group Γ . The space 𝒜 is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism p : 𝒜 → 𝒳 . The space 𝒜 is equipped with a closed 2 -form, possibly degenerate, and the space 𝒳 has a Poisson structure. The map p is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central role in the cluster ensemble structure. We define a non-commutative q -deformation of the 𝒳 -space. When q is a root of unity the algebra of functions on the q -deformed 𝒳 -space has a large center, which includes the algebra of functions on the original 𝒳 -space. The main example is provided by the pair of moduli spaces assigned in [7] to a topological surface S with a finite set of points at the boundary and a split semisimple algebraic group G . It is an algebraic-geometric avatar of higher Teichmüller theory on S related to G . We suggest that there exists a duality between the 𝒜 and 𝒳 spaces. In particular, we conjecture that the tropical points of one of the spaces parametrise a basis in the space of functions on the Langlands dual space. We provide some evidence for the duality conjectures in the finite type case.

How to cite

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Fock, Vladimir V., and Goncharov, Alexander B.. "Cluster ensembles, quantization and the dilogarithm." Annales scientifiques de l'École Normale Supérieure 42.6 (2009): 865-930. <http://eudml.org/doc/272148>.

@article{Fock2009,
abstract = {A cluster ensemble is a pair $(\{\mathcal \{X\}\}, \{\mathcal \{A\}\})$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $\Gamma $. The space $\{\mathcal \{A\}\}$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p: \{\mathcal \{A\}\} \rightarrow \{\mathcal \{X\}\}$. The space $\{\mathcal \{A\}\}$ is equipped with a closed $2$-form, possibly degenerate, and the space $\{\mathcal \{X\}\}$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central role in the cluster ensemble structure. We define a non-commutative $q$-deformation of the $\{\mathcal \{X\}\}$-space. When $q$ is a root of unity the algebra of functions on the $q$-deformed $\{\mathcal \{X\}\}$-space has a large center, which includes the algebra of functions on the original $\{\mathcal \{X\}\}$-space. The main example is provided by the pair of moduli spaces assigned in [7] to a topological surface $S$ with a finite set of points at the boundary and a split semisimple algebraic group $G$. It is an algebraic-geometric avatar of higher Teichmüller theory on $S$ related to $G$. We suggest that there exists a duality between the $\{\mathcal \{A\}\}$ and $\{\mathcal \{X\}\}$ spaces. In particular, we conjecture that the tropical points of one of the spaces parametrise a basis in the space of functions on the Langlands dual space. We provide some evidence for the duality conjectures in the finite type case.},
author = {Fock, Vladimir V., Goncharov, Alexander B.},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {cluster varieties; dilogarithm; quantization; Poisson structure; symplectic structure},
language = {eng},
number = {6},
pages = {865-930},
publisher = {Société mathématique de France},
title = {Cluster ensembles, quantization and the dilogarithm},
url = {http://eudml.org/doc/272148},
volume = {42},
year = {2009},
}

TY - JOUR
AU - Fock, Vladimir V.
AU - Goncharov, Alexander B.
TI - Cluster ensembles, quantization and the dilogarithm
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2009
PB - Société mathématique de France
VL - 42
IS - 6
SP - 865
EP - 930
AB - A cluster ensemble is a pair $({\mathcal {X}}, {\mathcal {A}})$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $\Gamma $. The space ${\mathcal {A}}$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p: {\mathcal {A}} \rightarrow {\mathcal {X}}$. The space ${\mathcal {A}}$ is equipped with a closed $2$-form, possibly degenerate, and the space ${\mathcal {X}}$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central role in the cluster ensemble structure. We define a non-commutative $q$-deformation of the ${\mathcal {X}}$-space. When $q$ is a root of unity the algebra of functions on the $q$-deformed ${\mathcal {X}}$-space has a large center, which includes the algebra of functions on the original ${\mathcal {X}}$-space. The main example is provided by the pair of moduli spaces assigned in [7] to a topological surface $S$ with a finite set of points at the boundary and a split semisimple algebraic group $G$. It is an algebraic-geometric avatar of higher Teichmüller theory on $S$ related to $G$. We suggest that there exists a duality between the ${\mathcal {A}}$ and ${\mathcal {X}}$ spaces. In particular, we conjecture that the tropical points of one of the spaces parametrise a basis in the space of functions on the Langlands dual space. We provide some evidence for the duality conjectures in the finite type case.
LA - eng
KW - cluster varieties; dilogarithm; quantization; Poisson structure; symplectic structure
UR - http://eudml.org/doc/272148
ER -

References

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