The quantum dilogarithm and representations quantum cluster varieties

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.

Dedicated to David Kazhdan for his 60th birthday. The final version. To appear in Inventiones Math. The last Section of the previous versions was removed, and will become a separate paper

References in corpus (6)

• Discrete Heisenberg-Weyl Group and Modular Group
• Quantum Dilogarithm
• Strongly coupled quantum discrete Liouville theory. I: Algebraic approach and duality
• An analog of a modular functor from quantized Teichm"uller theory
• Cluster ensembles, quantization and the dilogarithm
• The pentagon relation for the quantum dilogarithm and quantized M_{0,5}

Cited by in corpus (103)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• The Omega Deformation, Branes, Integrability, and Liouville Theory
• R-Twisting and 4d/2d Correspondences
• General Argyres-Douglas Theory
• Spectral networks
• Gauge Theory Loop Operators and Liouville Theory
• Cluster algebras and triangulated surfaces. Part II: Lambda lengths
• Integrating quantum groups over surfaces
• Braids, Walls, and Mirrors
• Discrete integrable systems and Poisson algebras from cluster maps
• On Arnold's 14 `exceptional' N=2 superconformal gauge theories
• Classical and Quantum Dilogarithm Identities
• Opers, surface defects, and Yang-Yang functional
• Quantum Unipotent Subgroup and dual canonical basis
• Aspects of Defects in 3d-3d Correspondence
• A duality map for quantum cluster varieties from surfaces
• Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence
• Periodicities in cluster algebras and dilogarithm identities
• Colored BPS Pyramid Partition Functions, Quivers and Cluster Transformations
• More Arnold's N = 2 superconformal gauge theories
• Mathematical structures of non-perturbative topological string theory: from GW to DT invariants
• Cluster Structures on Higher Teichmuller Spaces for Classical Groups
• The pentagon relation for the quantum dilogarithm and quantized M_{0,5}
• Quantum Riemann Surfaces in Chern-Simons Theory
• Quiver mutation and combinatorial DT-invariants
• Positive Representations of Split Real Simply-laced Quantum Groups
• Chern-Simons Theory and S-duality
• Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I
• 3d N=2 Theories from Cluster Algebras
• Stability conditions, cluster varieties, and Riemann-Hilbert problems from surfaces
• The pentagon relation and incidence geometry
• Leonhard Euler and a q-analogue of the logarithm
• Quantum hypermultiplet moduli spaces in N=2 string vacua: a review
• Hyperbolic 3-manifolds and Cluster Algebras
• Donaldson-Thomas trasnsformations of moduli spaces of G-local systems
• Central extensions of the Ptolemy-Thompson group and quantized Teichmuller theory
• Toric degenerations of cluster varieties and cluster duality
• Quantum Teichmüller space from quantum plane
• Synchronicity phenomenon in cluster patterns
• Cluster realizations of Weyl groups and higher Teichmüller theory
• Motivic correlators, cluster varieties and Zagier's conjecture on zeta(F,4)
• The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization
• Riemann-Hilbert problems for the resolved conifold
• Cluster Algebras: Network Science and Machine Learning
• Integrability on the Master Space
• Supersymmetric gauge theories, quantisation of moduli spaces of flat connections, and Liouville theory
• Skein and cluster algebras of unpunctured surfaces for ${\mathfrak{sl}}_3$
• Quantum Dilogarithm Identities at Root of Unity
• A new take on spherical, Whittaker and Bessel functions (Spherical and Whittaker functions via DAHA I,II)
• Laurent positivity of quantized canonical bases for quantum cluster varieties from surfaces
• Cluster Structures on Double Bott-Samelson Cells
• $Y$ systems, $Q$ systems, and 4D $\mathcal{N}=2$ supersymmetric QFT
• Rogers dilogarithms of higher degree and generalized cluster algebras
• Ratio coordinates for higher Teichmüller spaces
• Laminations from the symplectic double
• Factorization Rules in Quantum Teichmüller Theory
• Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems
• Cluster transformations, the tetrahedron equation and three-dimensional gauge theories
• Note on character varieties and cluster algebras
• Geometry of canonical bases and mirror symmetry
• Dilogarithm identities in cluster scattering diagrams
• Hamiltonian and Lagrangian formalisms of mutations in cluster algebras and application to dilogarithm identities
• Quantum holonomic link invariants derived from stated skein algebras
• Phase constants in the Fock-Goncharov quantum cluster varieties
• Naturality of ${\mathrm{SL}}_3$ quantum trace maps for surfaces
• ${\mathrm{SL}}_2$ quantum trace in quantum Teichmüller theory via writhe
• Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists
• A cluster realization of $U_q(\mathfrak{s}{\mathfrak{l}}_{\mathfrak{n}})$ from quantum character varieties
• On the cluster nature and quantization of geometric $R$-matrices
• A characterization of pseudo-Anosov mapping classes via tropical cluster transformations
• Quantum generalized cluster algebras and quantum dilogarithms of higher degrees
• Quantum decorated character stacks
• Toric Poisson Ideals in Cluster Algebras
• Triangular bases in quantum cluster algebras
• Representations of a quantum-deformed Lorentz algebra, Clebsch-Gordan map, and Fenchel-Nielsen representation of complex Chern-Simons theory at level-$N$
• Symplectic groupoids for cluster manifolds
• Three-dimensional quantum gravity from the quantum pseudo-Kähler plane
• Cluster X-varieties for dual Poisson-Lie groups II
• Central Extension of Mapping Class Group via Chekhov-Fock Quantization
• Generalized double affine Hecke algebras, their representations, and higher Teichmüller theory
• Quantum deformation of planar amplitudes
• Quantization of canonical bases and the quantum symplectic double
• Classical and Quantum Dynamics on Orbifolds
• A quantization of moduli spaces of 3-dimensional gravity
• The geometry of cluster varieties from surfaces
• Quantum duality maps, skein algebras and their ensemble compatibility
• Quantum Trilogy: Discrete Toda, Y-System and Chaos
• Heisenberg-Euler and the Quantum Dilogarithm
• Cluster Structure on Generalized Weyl Algebras
• Parabolic Positive Representations of ${\mathcal{U}}_q({\mathfrak{g}}_{\mathbb{R}})$
• A quantum analogue of generalized cluster algebras
• Braiding Operator via Quantum Cluster Algebra
• Irreducible self-adjoint representations of quantum Teichmüller space and the phase constants
• Positive Casimir and Central Characters of Split Real Quantum Groups
• $K$-theoretic Coulomb branches of quiver gauge theories and cluster varieties
• Wilson lines and their Laurent positivity
• Morita equivalences, moduli spaces and flag varieties
• Twist automorphisms and Poisson structures
• Cluster algebras and their bases
• Poisson Ideals in Cluster Algebras and the Spectra of Quantized Coordinate Rings
• On webs, polylogarithms and cluster algebras
• Quantum dilogarithm identities and cyclic quivers
• Langlands Duality and Poisson-Lie Duality via Cluster Theory and Tropicalization