The quantum dilogarithm and representations of quantum cluster varieties

DOI10.1007/s00222-008-0149-3zbMath1183.14037arXivmath/0702397OpenAlexW2084769325MaRDI QIDQ1006321

Vladimir Fock, Alexander B. Goncharov

Publication date: 20 March 2009

Published in: Inventiones Mathematicae (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0702397

zbMATH Keywords

quantization quantum dilogarithm cluster varieties Teichmüller theory Langlands modular double

Mathematics Subject Classification ID

Quantum groups (quantized enveloping algebras) and related deformations (17B37) Poisson manifolds; Poisson groupoids and algebroids (53D17) Deformation quantization, star products (53D55) Arithmetic varieties and schemes; Arakelov theory; heights (14G40) Higher logarithm functions (33B30)

Related Items

Analyticity and resurgence in wall-crossing formulas, Quantum generalized cluster algebras and quantum dilogarithms of higher degrees, Ratio coordinates for higher Teichmüller spaces, On tensor product decomposition of positive representations of $\mathcal{U}_{q\widetilde{q}}(\mathfrak{sl}(2,\mathbb{R}))$, Chern-Simons theory and S-duality, Symplectic double for moduli spaces of $G$-local systems on surfaces, Positive Casimir and central characters of split real quantum groups, Aspects of defects in 3d-3d correspondence, K-decompositions and 3d gauge theories, Note on character varieties and cluster algebras, Theta series, wall-crossing and quantum dilogarithm identities, Quantum traces and embeddings of stated skein algebras into quantum tori, Quantum deformation of planar amplitudes, Central extension of mapping class group via Chekhov-Fock quantization, Exponential complexes, period morphisms, and characteristic classes, Cluster structure on generalized Weyl algebras, Gauge theory loop operators and Liouville theory, Spherical and Whittaker functions via DAHA. II., Irreducible self-adjoint representations of quantum Teichmüller space and the phase constants, Toda conformal blocks, quantum groups, and flat connections, Factorization rules in quantum Teichmüller theory, Quantum groups, quantum tori, and the Grothendieck-Springer resolution, Spectral networks, The quantum trace as a quantum non-abelianization map, Representation of the quantum plane, its quantum double, and harmonic analysis on $GL_q^+(2,\mathbb{R })$, Symplectic groupoids for cluster manifolds, A cluster realization of $U_q(\mathfrak {sl}_{\mathfrak {n}})$ from quantum character varieties, The cluster symplectic double and moduli spaces of local systems, Geometry of canonical bases and mirror symmetry, Wilson lines and their Laurent positivity, Quantum Teichmüller space from the quantum plane, DILOGARITHM IDENTITIES IN CLUSTER SCATTERING DIAGRAMS, Three-dimensional quantum gravity from the quantum pseudo-Kähler plane, Action-angle coordinates on coadjoint orbits and multiplicity free spaces from partial tropicalization, Tetrahedron equation and quantum cluster algebras, Skein and cluster algebras of unpunctured surfaces for ${\mathfrak{sl}}_3$, Extended Goldman symplectic structure in Fock-Goncharov coordinates, Cluster algebras, HYPERBOLIC 3-MANIFOLDS AND CLUSTER ALGEBRAS, Twist automorphisms and Poisson structures, Positive representations with zero Casimirs, Quantum unipotent subgroup and dual canonical basis, Donaldson-Thomas transformations of moduli spaces of G-local systems, Quantum trilogy: discrete Toda, Y-system and chaos, On tensor products of positive representations of split real quantum Borel subalgebra $\mathcal {U}_{q\widetilde {q}}(\mathfrak {b}_\mathbb {R})$, Ideal Webs, Moduli Spaces of Local Systems, and 3d Calabi–Yau Categories, Mathematical structures of non-perturbative topological string theory: from GW to DT invariants, $\mathrm{SL}_2$ quantum trace in quantum Teichmüller theory via writhe, Rogers dilogarithms of higher degree and generalized cluster algebras, Centrally extended mapping class groups from quantum Teichmüller theory, Phase constants in the Fock-Goncharov quantum cluster varieties, Stability conditions, cluster varieties, and Riemann-Hilbert problems from surfaces, On the cluster nature and quantization of geometric $R$-matrices, On Arnold's 14 `exceptional' $ \mathcal{N} = 2 $ superconformal gauge theories, Laminations from the symplectic double, A quantum analog of generalized cluster algebras, Toric degenerations of cluster varieties and cluster duality, Gauss-Lusztig decomposition for positive quantum groups and representation by $q$-tori, Cluster Structures on Double Bott–Samelson Cells, Presentations of cluster modular groups and generation by cluster Dehn twists, Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths, Spectral analysis of stable processes on the positive half-line, The omega deformation, branes, integrability and Liouville theory, Positive representations of non-simply-laced split real quantum groups, The pentagon relation and incidence geometry, More Arnold's $ \mathcal{N} = 2 $ superconformal gauge theories, Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence, CLUSTER STRUCTURES ON HIGHER TEICHMULLER SPACES FOR CLASSICAL GROUPS, A duality map for quantum cluster varieties from surfaces, Supersymmetric Gauge Theories, Quantization of $${\mathcal M}_\mathrm{flat}$$ , and Conformal Field Theory, Cluster realizations of Weyl groups and higher Teichmüller theory, Integrability on the master space, Colored BPS pyramid partition functions, quivers and cluster transformations, Laurent positivity of quantized canonical bases for quantum cluster varieties from surfaces, Stated skein modules of marked 3-manifolds/surfaces, a survey, General Argyres-Douglas theory, Leonhard Euler and a 𝑞-analogue of the logarithm, Langlands duality and Poisson-Lie duality via cluster theory and tropicalization, Noncommutative Homological Mirror Functor, Monodromy representations of meromorphic projective structures, Synchronicity phenomenon in cluster patterns, Riemann-Hilbert problems for the resolved conifold and non-perturbative partition functions, Positive representations of split real simply-laced quantum groups, Wall-Crossing Structures in Donaldson–Thomas Invariants, Integrable Systems and Mirror Symmetry, Quantization of canonical bases and the quantum symplectic double, The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization

Cites Work