Quantum polylogarithms

TL;DR

Quantum polylogarithms extend multiple polylogarithms by a complex deformation parameter $\hbar$, with $\hbar\to0$ recapturing classical periods of mixed Tate motives and arbitrary $\hbar$-values encoding a broader quantum deformation landscape. The framework builds on integral presentations, holonomic modular difference equations, and a rich network of relations (shuffle, distribution, differential, and analytic continuation), connecting to both $q$-deformations and companion series to achieve stability across all $\hbar$ values. Rational $\hbar$ yields periods of mixed Tate motives, while irrational $\hbar$ pushes the functions outside the motivic realm, interpreted as rational exponential integrals. The work unifies several strands—quantum dilogarithms, $q$-polylogarithms, and iterated-integral representations—and points to applications in cluster algebras and high-energy scattering, suggesting a general principle: period deformations exist across quantum-variations of motives and their associated integrals, with practical computational tools such as generating functions and residue-based decompositions.

Abstract

Multiple polylogarithms are periods of variations of mixed Tate motives. Conjecturally, they deliver all such periods. We introduce deformations of multiple polylogarithms depending on a complex parameter h. We call them quantum polylogarithms. Their asymptotic expansion as h goes to 0 recovers multiple polylogarithms. The quantum dilogarithm was studied by Barnes in the XIX century. Its exponent appears in many areas of Mathematics and Physics. Quantum polylogarithms satisfy a holonomic systems of modular difference equations with coefficients in variations of mixed Hodge-Tate structures of motivic origin. If h is a rational number, the quantum polylogarithms can be expressed via multiple polylogarithms. Otherwise quantum polylogarithms are not periods of variations of mixed motives, i.e. they can not be given by integrals of rational differential forms on algebraic varieties. Instead, quantum polylogarithms are integrals of differential forms built from both rational functions and exponentials of rational functions. We call them rational exponential integrals. We suggest that quantum polylogarithms reflect a very general phenomenon: Periods of variations of mixed motives should have quantum deformations.

Theorems & Definitions (45)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Theorem 3.1
• proof