Quantum Dilogarithm Identities Arising from the Product Formula for the Universal R-Matrix of Quantum Affine Algebras

Abstract

In Dimofte, Gukov, and Soibelman (Lett. Math. Phys. 95 (2011), 1–25), four quantum dilogarithm identities containing infinitely many factors are proposed as wall-crossing formulas for the refined BPS invariant. We give an algebraic proof of these identities using the formula for the universal R-matrix of the quantum affine algebra developed by Ito (Hiroshima Math. J. 40 (2010), 133–183), which yields various product presentations of the universal R-matrix by choosing various convex orders on an affine root system. By the uniqueness of the universal R-matrix and appropriate degeneration, we can construct various quantum dilogarithm identities, including the ones proposed in Dimofte, Gukov, and Soibelman (Lett. Math. Phys. 95 (2011), 1–25), which turn out to correspond to convex orders of multiple row type.