The shape of the universe revealed through algebraic geometry

How can the behavior of elementary particles and the structure of the entire universe be described using the same mathematical concepts? This question is at the heart of recent work by the mathematicians Claudia Fevola from Inria Saclay and Anna-Laura Sattelberger from the Max Planck Institute for Mathematics in the Sciences, recently published in the Notices of the American Mathematical Society.

Mathematics and physics share a close, reciprocal relationship. Mathematics offers the language and tools to describe physical phenomena, while physics drives the development of new mathematical ideas. This interplay remains vital in areas such as quantum field theory and cosmology, where advanced mathematical structures and physical theory evolve together.

In their article, the authors explore how algebraic structures and geometric shapes can help us understand phenomena ranging from particle collisions such as happens, for instance, in particle accelerators to the large-scale architecture of the cosmos. Their research is centered around algebraic geometry. Their recent undertakings also connect to a field called positive geometry—an interdisciplinary and novel subject in mathematics driven by new ideas in particle physics and cosmology.

This field was inspired by the geometrical concept of positive geometry, which expands the standard Feynman diagram approach in particle physics by representing interactions as volumes of high-dimensional geometric objects, such as the amplituhedron, as introduced by the theoretical physicists Nima Arkani-Hamed and Jaroslav Trnka in 2013. It carries a rich combinatorial structure and offers an alternative, potentially simpler way to compute scattering amplitudes, from which one can derive probabilities of scattering events.

This approach has far-reaching implications that go beyond particle physics. In cosmology, scientists are using the faint light of the cosmic microwave background and the distribution of galaxies to infer what shaped the early universe. Similar mathematical tools are now being applied. For instance, cosmological polytopes, which are themselves positive geometries, can represent correlations in the universe’s first light and help reconstruct the physical laws that governed the birth of the cosmos.

A geometry for the universe

The article highlights that positive geometry is not a niche mathematical curiosity but a potential unifying language for forming branches of theoretical physics. These geometric frameworks naturally encode the transfer of information between physical systems, for example, by mapping concrete, sensory-based concepts to abstract structures, a process that mirrors how humans metaphorically understand the world.

The mathematics behind this is sophisticated and spans multiple disciplines. The authors draw on algebraic geometry, which defines shapes and spaces through solutions to systems of polynomial equations, algebraic analysis, which studies differential equations through mathematical objects called D-modules, and combinatorics, which describes the arrangements and interactions within these structures.

The formal objects under consideration, such as Feynman integrals, generalized Euler integrals, or canonical forms of positive geometries, are not merely mathematical abstractions. They correspond to observable phenomena in high-energy physics and cosmology, enabling precision computations of particle behavior and cosmic structures alike.

Bridging scales with mathematics

The study presents an approach with broad applicability and scalability. Scattering processes are often illustrated using Feynman diagrams. Feynman’s approach in the study of scattering amplitudes boils down to the study of intricate integrals associated to such diagrams. Algebraic geometry provides a range of tools for systematically investigating these integrals.

The graph polynomial of a Feynman diagram is defined in terms of the spanning trees and forests of the underlying graph. The associated Feynman integral can be expressed as a Mellin transform of a power of this graph polynomial, interpreted as a function of its coefficients. These coefficients, however, are constrained by the underlying physical conditions. Feynman integrals are therefore closely connected to generalized Euler integrals, specifically through restrictions to the relevant geometric subspaces.

One way to study these holonomic functions is via the linear differential equations they satisfy, which are D-module inverse images of hypergeometric D-modules. Constructing these differential equations explicitly, however, remains challenging. In theoretical cosmology, correlation functions in toy models also take the form of such integrals, with integrands arising from hyperplane arrangements.

The complement of the algebraic variety defined by the graph polynomial in an algebraic torus is a very affine variety, and the Feynman integral can be viewed as the pairing of a twisted cycle and cocycle of this variety. Its geometric and (co-)homological properties reflect physical concepts such as the number of master integrals. These master integrals form a basis for the space of integrals when the kinematic parameters vary, and the size of this basis is, at least generically, equal to the signed topological Euler characteristic of the variety.

A field in motion

Fevola and Sattelberger’s work reflects a growing international effort. It brings together mathematics, particle physics, and cosmology, focusing on precisely these connections between algebra, geometry, and theoretical physics.

“Positive geometry is still a young field, but it has the potential to significantly influence fundamental research in both physics and mathematics,” the authors emphasize. “It is now up to the scientific community to work out the details of these emerging mathematical objects and theories and to validate them. Encouragingly, several successful collaborations have already laid important groundwork.”

The recent developments are not only advancing our understanding of the physical world but also pushing the boundaries of mathematics itself. Positive geometry is more than a tool. It is a language. One that might unify our understanding of nature at all scales.