Quantum polylogarithms

TL;DR

This paper introduces quantum polylogarithms, deformations of multiple polylogarithms depending on a complex parameter, which connect to mixed motives, satisfy modular difference equations, and exhibit unique properties when the parameter is rational or irrational.

Contribution

It defines quantum polylogarithms as a new class of functions with connections to motives, difference equations, and exponential integrals, expanding the understanding of polylogarithm deformations.

Findings

Quantum polylogarithms recover multiple polylogarithms as h approaches 0.

They satisfy holonomic modular difference equations with motivic coefficients.

When h is rational, they can be expressed via multiple polylogarithms; otherwise, they are rational exponential integrals.

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.