Cluster ensembles, quantization and the dilogarithm
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 6, pp. 865-930.

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:𝒜𝒳. 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.

Un ensemble amassé est une paire (𝒳,𝒜) d’espaces positifs (i.e. de variétés munies d’un atlas positif) munis de l’action d’un groupe discret. L’espace 𝒜 est relié au spectre d’une algèbre amassée [12]. Les deux espaces sont liés par un morphisme p:𝒜𝒳. L’espace 𝒜 est muni d’une 2-forme fermée, éventuellement dégénérée, et l’espace 𝒳 est muni d’une structure de Poisson. L’application p est compatible avec ces structures. Le dilogarithme avec ses avatars motiviques et quantiques joue un rôle fondamental dans la structure d’un ensemble amassé. Nous définissons une déformation non-commutative de l’espace 𝒳. Nous montrons que, dans le cas où le paramètre de la déformation q est une racine de l’unité, l’algèbre déformée a un centre qui contient l’algèbre des fonctions sur l’espace 𝒳 originel. Notre exemple principal est celui de l’espace des modules associé dans [7] à une surface topologique S munie d’un nombre fini de points distingués sur le bord et à un groupe algébrique semi-simple G. C’est un avatar algébro-géométrique de la théorie de Teichmüller d’ordre supérieur sur la surface S à valeurs dans G. Nous évoquons l’existence d’une dualité entre les espaces 𝒜 et 𝒳. Une des manifestations de cette dualité est une conjecture de dualité affirmant que les points tropicaux d’un espace paramètrent une base dans l’espace d’une certaine classe de fonctions sur l’espace Langlands-dual. Nous démontrons cette conjecture dans un certain nombre d’exemples.

DOI: 10.24033/asens.2112
Classification: 14T05,  53D17,  53D05,  53D55,  53D30
Keywords: cluster varieties, dilogarithm, quantization, Poisson structure, symplectic structure
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Fock, Vladimir V.; Goncharov, Alexander B. Cluster ensembles, quantization and the dilogarithm. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 6, pp. 865-930. doi : 10.24033/asens.2112. http://archive.numdam.org/articles/10.24033/asens.2112/

[1] A. Berenstein, S. Fomin & A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), 1-52. | MR | Zbl

[2] A. Berenstein & A. Zelevinsky, Quantum cluster algebras, Adv. Math. 195 (2005), 405-455. | MR | Zbl

[3] J. W. Cannon, W. J. Floyd & W. R. Parry, Introductory notes on Richard Thompson's groups, Enseign. Math. 42 (1996), 215-256. | MR | Zbl

[4] F. Chapoton, S. Fomin & A. Zelevinsky, Polytopal realizations of generalized associahedra, Canad. Math. Bull. 45 (2002), 537-566. | MR | Zbl

[5] V. V. Fock & A. B. Goncharov, Cluster 𝒳-varieties, amalgamation, and Poisson-Lie groups, in Algebraic geometry and number theory, Progr. Math. 253, Birkhäuser, 2006, 27-68. | MR | Zbl

[6] V. V. Fock & A. B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1-211. | Numdam | MR | Zbl

[7] V. V. Fock & A. B. Goncharov, Dual Teichmüller and lamination spaces, in Handbook of Teichmüller theory. Vol. I, IRMA Lect. Math. Theor. Phys. 11, Eur. Math. Soc., Zürich, 2007, 647-684. | MR | Zbl

[8] V. V. Fock & A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm II: The intertwiner, Progress in Math. 269 (2009), 513-524. | MR | Zbl

[9] V. V. Fock & A. B. Goncharov, The quantum dilogarithm and representations of quantum cluster varieties, Invent. Math. 175 (2009), 223-286. | MR | Zbl

[10] V. V. Fock & A. B. Goncharov, Completions of cluster varieties, to appear.

[11] V. V. Fok & L. O. Chekhov, Quantum Teichmüller spaces, Teoret. Mat. Fiz. 120 (1999), 511-528. | MR | Zbl

[12] S. Fomin & A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497-529 (electronic). | MR | Zbl

[13] S. Fomin & A. Zelevinsky, The Laurent phenomenon, Adv. in Appl. Math. 28 (2002), 119-144. | MR | Zbl

[14] S. Fomin & A. Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), 63-121. | MR | Zbl

[15] S. Fomin & A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. 158 (2003), 977-1018. | MR | Zbl

[16] M. Gekhtman, M. Shapiro & A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J. 3 (2003), 899-934. | MR | Zbl

[17] M. Gekhtman, M. Shapiro & A. Vainshtein, Cluster algebras and Weil-Petersson forms, Duke Math. J. 127 (2005), 291-311. | MR | Zbl

[18] A. B. Goncharov, Explicit construction of characteristic classes, in I. M. Gel'fand Seminar, Adv. Soviet Math. 16, Amer. Math. Soc., 1993, 169-210. | MR | Zbl

[19] A. B. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995), 197-318. | MR | Zbl

[20] A. B. Goncharov, Pentagon relation for the quantum dilogarithm and quantized 0,5 cyc , in Geometry and dynamics of groups and spaces, Progr. Math. 265, Birkhäuser, 2008, 415-428. | MR | Zbl

[21] M. Imbert, Sur l'isomorphisme du groupe de Richard Thompson avec le groupe de Ptolémée, in Geometric Galois actions, 2, London Math. Soc. Lecture Note Ser. 243, Cambridge Univ. Press, 1997, 313-324. | MR | Zbl

[22] R. M. Kashaev, Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998), 105-115. | MR | Zbl

[23] J. Milnor, Introduction to algebraic K-theory, Annals of Math. Studies 72, Princeton Univ. Press, 1971. | MR | Zbl

[24] R. C. Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987), 299-339. | MR | Zbl

[25] P. Sherman & A. Zelevinsky, Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Mosc. Math. J. 4 (2004), 947-974. | MR | Zbl

[26] A. A. Suslin, K 3 of a field and the Bloch group, Proc. Steklov Inst. Math. N4 (1991), 217-239. | Zbl

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