Classical and Quantum Dilogarithm Identities

Abstract

Using the quantum cluster algebra formalism of Fock and Goncharov, we present several forms of quantum dilogarithm identities associated with periodicities in quantum cluster algebras, namely, the tropical, universal, and local forms. We then demonstrate how classical dilogarithm identities naturally emerge from quantum dilogarithm identities in local form in the semiclassical limit by applying the saddle point method.

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Cited by in corpus (25)

• Simplex and Polygon Equations
• Functional relations for elliptic polylogarithms
• Line defects and (framed) BPS quivers
• Hyperbolic 3-manifolds and Cluster Algebras
• The Yang-Baxter relation and gauge invariance
• Comments on Exchange Graphs in Cluster Algebras
• Quantum Dilogarithm Identities at Root of Unity
• Exact WKB analysis and cluster algebras
• Representation of the Quantum Plane, its Quantum Double and Harmonic Analysis on $G{L}_q^{+}(2,R)$
• Rogers dilogarithms of higher degree and generalized cluster algebras
• Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm
• Semi-classical quantisation of magnetic solitons in the anisotropic Heisenberg quantum chain
• Cluster transformations, the tetrahedron equation and three-dimensional gauge theories
• Volkov's Pentagon for the Modular Quantum Dilogarithm
• Naturality of ${\mathrm{SL}}_3$ quantum trace maps for surfaces
• The Graphs of Quantum Dilogarithm
• Phase constants in the Fock-Goncharov quantization of cluster varieties: long version
• Note on dilogarithm identities from nilpotent double affine Hecke algebras
• Heisenberg-Euler and the Quantum Dilogarithm
• Braiding Operator via Quantum Cluster Algebra
• Irreducible self-adjoint representations of quantum Teichmüller space and the phase constants
• On the density of the supremum of a stable process
• On webs, polylogarithms and cluster algebras
• Quantum dilogarithm identities and cyclic quivers
• Quantum dilogarithm identities arising from the product formula for universal R-matrix of quantum affine algebras