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Algebraic and Positive Geometry of the Universe: from Particles to Galaxies

Claudia Fevola, Anna-Laura Sattelberger

TL;DR

This article surveys the convergence of algebraic geometry, algebraic analysis, and combinatorics with particle physics and cosmology through the lens of positive geometry. It articulates two complementary formalisms for observables: traditional Feynman integrals $\mathcal{I}_G$ and a geometric framework built from canonical forms on positive geometries, such as the amplituhedron and cosmological polytopes. It develops the mathematical toolkit—twisted (co)homology, $D$-modules, and GKZ systems—for studying generalized Euler integrals, master integrals, and the singularity structure via Landau analysis, while showcasing how positive geometries streamline amplitude computations via triangulations and boundary structures. The work highlights the potential of a unified mathematical language to describe phenomena from subatomic interactions to cosmological correlators, and discusses ongoing directions for extending these ideas to broader quantum field theories and cosmological models.

Abstract

In recent years, the intersection of algebra, geometry, and combinatorics with particle physics and cosmology has led to significant advances. Central to this progress is the twofold formulation of the study of particle interactions and observables in the universe: on the one hand, Feynman's approach reduces to the study of intricate integrals; on the other hand, one encounters the study of positive geometries. This article introduces key developments, mathematical tools, and the connections that drive progress at the frontier between algebraic geometry, the theory of $D$-modules, combinatorics, and physics. All these threads contribute to shaping the flourishing field of positive geometry, which aims to establish a unifying mathematical language for describing phenomena in cosmology and particle physics.

Algebraic and Positive Geometry of the Universe: from Particles to Galaxies

TL;DR

This article surveys the convergence of algebraic geometry, algebraic analysis, and combinatorics with particle physics and cosmology through the lens of positive geometry. It articulates two complementary formalisms for observables: traditional Feynman integrals and a geometric framework built from canonical forms on positive geometries, such as the amplituhedron and cosmological polytopes. It develops the mathematical toolkit—twisted (co)homology, -modules, and GKZ systems—for studying generalized Euler integrals, master integrals, and the singularity structure via Landau analysis, while showcasing how positive geometries streamline amplitude computations via triangulations and boundary structures. The work highlights the potential of a unified mathematical language to describe phenomena from subatomic interactions to cosmological correlators, and discusses ongoing directions for extending these ideas to broader quantum field theories and cosmological models.

Abstract

In recent years, the intersection of algebra, geometry, and combinatorics with particle physics and cosmology has led to significant advances. Central to this progress is the twofold formulation of the study of particle interactions and observables in the universe: on the one hand, Feynman's approach reduces to the study of intricate integrals; on the other hand, one encounters the study of positive geometries. This article introduces key developments, mathematical tools, and the connections that drive progress at the frontier between algebraic geometry, the theory of -modules, combinatorics, and physics. All these threads contribute to shaping the flourishing field of positive geometry, which aims to establish a unifying mathematical language for describing phenomena in cosmology and particle physics.

Paper Structure

This paper contains 11 sections, 24 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Timeliness
  3. Outline
  4. Particle physics
  5. Cosmology
  6. Mathematical toolkit
  7. Algebraic geometry
  8. Algebraic analysis
  9. Combinatorial algebraic geometry
  10. Interactions
  11. Funding statement

Figures (5)

  • Figure 1: A graphical representation of a scattering process between $n$ particles, such as happens for instance in a particle accelerator.
  • Figure 2: The CMB as observed by Planck. Credit: https://www.esa.int/ESA_Multimedia/Images/2013/10/The_Cosmic_microwave_background_CMB_as_observed_by_Planck, 2013.
  • Figure 3: Some Feynman diagrams: the "one-loop triangle" (left) and the "parachute" (right) diagram.
  • Figure 4: The 2-site chain graph (left) and its associated cosmological polytope (right).
  • Figure 5: The set of real zeros of the graph polynomial $\mathcal{G}_G$, where $G$ is the massless parachute diagram as in \ref{['ex:FeynmanInt']}, with parameters $(p_1+p_2)^2=25,$$p_3^2=49,$$p_4^2=9$.

Theorems & Definitions (5)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5