The Hidden Geometry of Numbers: When Feynman Diagrams Meet Tropical Curves
26 Apr 2026, Yanjiang
Feynman diagrams and tropical curves reveal the hidden geometric foundations of multiple zeta values.
Numbers have secrets. Not the kind that yield to a moment’s reflection, but the kind that require entire new branches of mathematics to uncover. Consider the humble zeta function — that infinite sum of reciprocals that Euler first studied in the 18th century. Its values at even integers, like π²/6, have been known for centuries. But its values at odd integers? Those remain stubbornly mysterious. And then there are the multiple zeta values — sums over several indices — which have quietly become ubiquitous across modern mathematics and theoretical physics.
A new preprint (arXiv:2604.22735) from Francis Brown at All Souls College, Oxford, offers a fresh perspective on these enigmatic numbers. Brown proposes nothing less than a new geometric foundation for understanding them — one that draws unexpected connections between quantum field theory, tropical geometry, and the reduction theory of quadratic forms.
Think of it like an investment portfolio. The standard approach to multiple zeta values — the safe, well-understood bets — represents them as iterated integrals on the Riemann sphere with three punctures. These are what Brown calls the “linear geometry” of multiple zeta values: the denominators of the integrands factor completely into linear terms. It is clean, elegant, and has served mathematicians well for decades.
But there is another, far more interesting portfolio: the high-risk, high-reward “big swings.” These are the integral representations where matrix determinants appear in the denominator of the integrand. Brown calls this the “non-linear geometry” of multiple zeta values, and it is the true subject of his lectures.
The Shape of a Number
Why should anyone care about the geometry of a number? The answer lies in what geometry reveals. Just as a successful career can rest on finding colleagues with the right skills who can be trusted, a successful mathematical framework rests on finding the right structure to reveal hidden relationships.
Multiple zeta values appear in contexts as diverse as the computation of Feynman integrals in quantum electrodynamics — the theory of light and matter — and the study of moduli spaces of algebraic curves. They are the mathematical glue that binds particle physics to pure mathematics. But the standard “linear” representation, while powerful, obscures certain deep connections.
Consider a Feynman diagram representing two electrons exchanging a photon. At lowest order, the calculation is straightforward. But add a second loop — a higher-order quantum correction — and the integrals become vastly more complex. The denominators of these integrals are not simple linear factors; they are determinants of matrices whose entries encode the topology of the diagram.
This is where Brown’s non-linear geometry enters the picture. He proposes that these determinantal representations are not merely computational conveniences but reflect a deeper geometric truth about the numbers themselves.
Tropical Geometry: The Scaffolding
To make this concrete, Brown introduces tropical geometry — a field that might sound exotic but is remarkably intuitive at its core. Imagine trying to understand the shape of a mountain range by studying only its contour lines. Tropical geometry does something similar for algebraic varieties: it replaces complex algebraic equations with piecewise-linear combinatorial structures, much like a topographical map simplifies a three-dimensional landscape into two-dimensional contours.
The moduli spaces of tropical curves — the “shape spaces” of these simplified geometric objects — turn out to be intimately connected to the non-linear geometry of multiple zeta values. It is as if the numbers themselves are casting shadows onto a simpler, combinatorial world, and those shadows reveal structures invisible in the original.
The general linear group of integer matrices — the set of all invertible matrices with integer entries — provides the language for describing how these structures transform. And the reduction theory of quadratic forms — a classical subject dating back to Gauss — tells us how to find the “simplest” representatives of each equivalence class.
Brown weaves these threads together into a single, coherent framework. The result is not a finished theory but a research program — a set of open questions and conjectures that point toward a unified geometric understanding of multiple zeta values.
What This Means for Physics
The implications for physics are significant. Feynman integrals — the computational backbone of quantum field theory — are notoriously difficult to evaluate. Each new loop order requires new mathematical techniques. The non-linear geometry of multiple zeta values may provide a systematic way to understand the structure of these integrals, revealing hidden relationships between seemingly unrelated Feynman diagrams.
In quantum electrodynamics, for example, the graph representing two electrons exchanging a photon with two loops (a process that contributes to the repulsive force between electrons) yields integrals whose denominators contain matrix determinants. Brown’s framework suggests that these determinants are not accidental complications but essential features that encode the geometry of the underlying physical process.
The philosophical implications are profound. For decades, physicists have treated Feynman diagrams as computational tools — bookkeeping devices for perturbation theory. Brown’s work suggests they are something more: geometric objects in their own right, whose structure is deeply connected to the numbers that describe our universe.
Open Questions
Brown’s lectures set out a number of open questions for future research. Can the non-linear geometry be extended to all multiple zeta values, or only a subset? What is the precise relationship between tropical moduli spaces and the reduction theory of quadratic forms? And most tantalizingly: can this geometric framework lead to new computational techniques for Feynman integrals?
This is not a metaphor. It is a precise mathematical statement about the structure of numbers that appear in our most fundamental theories of physics. The geometry of the quantum world is not a metaphor either — it is a mathematical reality that we are only beginning to understand.
Perhaps one day, when physicists want to understand the behavior of particles at the highest energies, they will turn not to brute-force computation but to the elegant geometry of tropical curves and matrix determinants. That day is not yet here. But Brown’s work has opened a door — and the view on the other side is breathtaking.
Yanjiang is an online editor of Loom Science
References
- Francis Brown, Non-linear geometry of multiple zeta values, arXiv:2604.22735