Quantum dilogarithm identities arising from the product formula for universal R-matrix of quantum affine algebras

TL;DR

This paper proves quantum dilogarithm identities related to quantum affine algebras using the universal R-matrix formula, connecting algebraic structures with wall-crossing phenomena in BPS invariants.

Contribution

It provides an algebraic proof of existing quantum dilogarithm identities via the universal R-matrix approach, revealing new identities from convex orderings.

Findings

Algebraic proof of quantum dilogarithm identities

Construction of identities from convex orders

Connection to wall-crossing formulas in BPS invariants

Abstract

In arXiv:0912.1346, four quantum dilogarithm identities containing infinitely many factors are proposed as wall-crossing formula for refined BPS invariant. We give algebraic proof of these identities using the formula for universal R-matrix of quantum affine algebra developed by K. Ito, which yields various product presentation of universal R-matrix by choosing various convex orders on affine root system. By the uniqueness of universal R-matrix and appropriate degeneration, we can construct various quantum dilogarithm identities including the ones proposed in arXiv:0912.1346, which turn out to correspond to convex orders of multiple row type.