Non-linear geometry of multiple zeta values

TL;DR

This paper explores the non-linear geometric representations of multiple zeta values involving determinants, extending the classical linear integral framework and connecting to diverse mathematical and physical theories.

Contribution

It introduces a geometric framework for multiple zeta values based on non-linear, determinantal integral representations, expanding the understanding beyond traditional linear forms.

Findings

Identifies non-linear, determinantal integral representations of multiple zeta values.

Connects non-linear geometry to tropical geometry, moduli spaces, and quantum field theory.

Proposes open questions for future research in the geometric understanding of multiple zeta values.

Abstract

Since their rediscovery in the 1990s, multiple zeta values have become ubiquitous in many areas of mathematics and physics. Their standard integral and sum representations can usually be traced back to a single source, namely the iterated integrals on the Riemann sphere with three punctures. We refer to such representations as the \emph{linear} geometry of multiple zeta values, since the denominators of the corresponding integrands factor completely into linear terms. However, there also exist equally important and entirely distinct integral representations for multiple zeta values arising in mathematics and physics, in which matrix determinants appear in the denominator of the integrand. We call this the \emph{non-linear} geometry of multiple zeta values. These lectures trace the origins of this non-linear geometry and provide an introductory journey through a range of topics including tropical geometry, the moduli spaces of tropical curves, Feynman integrals in quantum field theory, the general linear group of integer matrices, and the reduction theory of quadratic forms. In doing so, we propose a geometric framework for multiple zeta values based on such non-linear, determinantal representations and set out a number of open questions for future research.