Periodic Reporting for period 1 - UNIVERSE PLUS (Positive Geometry in Particle Physics and Cosmology)
Periodo di rendicontazione: 2024-06-01 al 2025-11-30
Insights from physics at both infinitesimal and cosmological scales strongly indicate that our present understanding of the laws of Nature is only approximate and must give way to more fundamental underlying principles. The UNIVERSE+ project aims to establish a new foundation for fundamental physics, connecting phenomena from the scattering of elementary particles to the Big Bang, and revealing a framework that transcends conventional quantum mechanics and spacetime. Recently discovered geometric objects in theoretical physics imply the existence of novel mathematical structures suggesting unexpected connections between combinatorics, algebra, geometry, particle physics, and cosmology. Building on these advances, the project seeks to establish positive geometry as a new mathematical foundation for the laws of physics. To this end, the UNIVERSE+ project brings together expertise in particle physics (Arkani-Hamed and Henn), cosmology (Baumann), and mathematics (Sturmfels). This unique collaboration, enabled by a recent convergence of key insights across mathematics and physics, puts the project on a clear and compelling trajectory. The project's objectives are ambitious: to develop a new mathematical language capable of describing physical phenomena across all scales, from the interactions of elementary particles to the large-scale structure of the universe.
A central focus within this program is the study of cosmological correlators, which provide a fundamental bridge between the physics of the early universe and late-time observations by encoding how quantum fields interacted in the primordial universe. Despite their central importance, calculating and interpreting these correlators remains technically challenging and is poorly understood at a conceptual level. The UNIVERSE+ collaboration aims to overcome these obstacles by developing a new theoretical framework rooted in positive geometry. This framework will expose the hidden structure of cosmological correlators and place them on the same conceptual footing as scattering amplitudes in particle physics.
A central focus within this program is the study of cosmological correlators, which provide a fundamental bridge between the physics of the early universe and late-time observations by encoding how quantum fields interacted in the primordial universe. Despite their central importance, calculating and interpreting these correlators remains technically challenging and is poorly understood at a conceptual level. The UNIVERSE+ collaboration aims to overcome these obstacles by developing a new theoretical framework rooted in positive geometry. This framework will expose the hidden structure of cosmological correlators and place them on the same conceptual footing as scattering amplitudes in particle physics.
During the reporting period, the UNIVERSE+ collaboration made substantial progress at the intersection of mathematics, quantum field theory, and cosmology. This advances the collaboration's overarching goal of uncovering the geometric structures underlying fundamental physical concepts. A significant portion of the work involved developing algebraic, combinatorial, and geometric tools to better understand scattering amplitudes, Feynman integrals, and cosmological correlators. These efforts have led to the systematic characterization of algebraic varieties and semialgebraic sets that govern kinematic spaces. They have also provided new insights into convex and Grassmannian geometries that are relevant for amplitudes and have offered novel descriptions of limit objects arising in positive geometries. Simultaneously, the collaboration advanced the algebraic and analytic structures underlying cosmological correlators. This included developing D-module methods and new computational frameworks based on differential equations.
A central theme across these activities was unifying techniques originally developed for flat-space scattering amplitudes with problems in cosmology. New differential-equation approaches revealed hidden structural patterns in cosmological Feynman integrals. Previously unknown geometric objects were also identified as organizing principles for correlators in de Sitter space. This marks the first appearance of certain Grassmannian structures in a cosmological context. Parallel advances in quantum field theory have led to cutting-edge calculations of scattering amplitudes and Feynman integrals relevant to Quantum Chromodynamics and supersymmetric gauge theories. These calculations have been accompanied by foundational studies of positive geometry. These advances included the discovery of new sign patterns in amplitudes, the development of geometric formulations of Wilson loops, and the identification of novel positive geometries that govern both particle scattering and cosmological observables.
These scientific advances were enabled by strong collaborative efforts across the consortium. Regular interaction among researchers, facilitated through collaboration visits, focused workshops, and extended research stays, proved essential in aligning mathematical and physical perspectives, accelerating the transfer of ideas, and fostering joint work across institutions. This sustained interaction has been a key factor in achieving the integrated, conceptually unified results described above.
Building on these developments, the collaboration pursued deeper conceptual questions about the nature of amplitudes and correlators. New formalisms were developed that recast scattering and cosmological processes in geometric terms. Exploratory work opened the door to a fundamentally non-recursive, dual description of amplitudes and correlators. This framework makes it possible to access regimes, most notably the limit of infinitely many particles, that are inaccessible using traditional recursive or diagrammatic approaches. Together, these results demonstrate the strength of the collaboration's integrated approach and highlight the emergence of positive geometry as a unifying language across particle physics and cosmology.
A central theme across these activities was unifying techniques originally developed for flat-space scattering amplitudes with problems in cosmology. New differential-equation approaches revealed hidden structural patterns in cosmological Feynman integrals. Previously unknown geometric objects were also identified as organizing principles for correlators in de Sitter space. This marks the first appearance of certain Grassmannian structures in a cosmological context. Parallel advances in quantum field theory have led to cutting-edge calculations of scattering amplitudes and Feynman integrals relevant to Quantum Chromodynamics and supersymmetric gauge theories. These calculations have been accompanied by foundational studies of positive geometry. These advances included the discovery of new sign patterns in amplitudes, the development of geometric formulations of Wilson loops, and the identification of novel positive geometries that govern both particle scattering and cosmological observables.
These scientific advances were enabled by strong collaborative efforts across the consortium. Regular interaction among researchers, facilitated through collaboration visits, focused workshops, and extended research stays, proved essential in aligning mathematical and physical perspectives, accelerating the transfer of ideas, and fostering joint work across institutions. This sustained interaction has been a key factor in achieving the integrated, conceptually unified results described above.
Building on these developments, the collaboration pursued deeper conceptual questions about the nature of amplitudes and correlators. New formalisms were developed that recast scattering and cosmological processes in geometric terms. Exploratory work opened the door to a fundamentally non-recursive, dual description of amplitudes and correlators. This framework makes it possible to access regimes, most notably the limit of infinitely many particles, that are inaccessible using traditional recursive or diagrammatic approaches. Together, these results demonstrate the strength of the collaboration's integrated approach and highlight the emergence of positive geometry as a unifying language across particle physics and cosmology.
The UNIVERSE+ project established positive geometry as a new unifying framework for particle physics, cosmology, and mathematics. Beyond its foundational role in the geometric description of scattering amplitudes and cosmological correlators, positive geometry has revealed deep connections to applied fields such as polynomial optimization and computational mathematics. Concurrently, the project has sparked a sustained shift toward open, algebraic geometry–based software tools for theoretical physics, facilitating more robust and scalable computations.
A significant project outcome is the discovery that cosmological correlators are governed by Grassmannian geometry, marking its first appearance in a cosmological context. This result establishes cosmological observables as being on the same conceptual footing as scattering amplitudes and provides strong evidence that positive geometry plays a fundamental role in early-universe physics.
Further advances include applying tropical geometry to study Feynman integrals to yield new insights into their asymptotic and infrared behavior and developing novel geometric frameworks to compute scattering amplitudes that move beyond traditional diagrammatic approaches. The project also revealed previously unknown positivity properties of quantum amplitudes, demonstrating that large classes satisfy infinite hierarchies of positivity conditions, known as complete monotonicity.
Finally, a major discovery of the collaboration were new geometric structures underlying cosmological wavefunctions called cosmohedra. This constitutes a significant advance in the quest to extend ideas originally developed for scattering amplitudes to time-dependent settings, opening new avenues for exploring the interplay between geometry, combinatorics, and cosmology.
A significant project outcome is the discovery that cosmological correlators are governed by Grassmannian geometry, marking its first appearance in a cosmological context. This result establishes cosmological observables as being on the same conceptual footing as scattering amplitudes and provides strong evidence that positive geometry plays a fundamental role in early-universe physics.
Further advances include applying tropical geometry to study Feynman integrals to yield new insights into their asymptotic and infrared behavior and developing novel geometric frameworks to compute scattering amplitudes that move beyond traditional diagrammatic approaches. The project also revealed previously unknown positivity properties of quantum amplitudes, demonstrating that large classes satisfy infinite hierarchies of positivity conditions, known as complete monotonicity.
Finally, a major discovery of the collaboration were new geometric structures underlying cosmological wavefunctions called cosmohedra. This constitutes a significant advance in the quest to extend ideas originally developed for scattering amplitudes to time-dependent settings, opening new avenues for exploring the interplay between geometry, combinatorics, and cosmology.