Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet (1711.09102v2)

Abstract: The search for a theory of the S-Matrix has revealed surprising geometric structures underlying amplitudes ranging from the worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to kinematic space where amplitudes live. In this paper, we propose a novel geometric understanding of amplitudes for a large class of theories. The key is to think of amplitudes as differential forms directly on kinematic space. We explore this picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint cubic scalar, we establish a direct connection between its "scattering form" and a classic polytope--the associahedron--known to mathematicians since the 1960's. We find an associahedron living naturally in kinematic space, and the tree amplitude is simply the "canonical form" associated with this "positive geometry". Basic physical properties such as locality, unitarity and novel "soft" limits are fully determined by the geometry. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between this old "worldsheet associahedron" and the new "kinematic associahedron", providing a geometric interpretation and novel derivation of the bi-adjoint CHY formula. We also find "scattering forms" on kinematic space for Yang-Mills and the Non-linear Sigma Model, which are dual to the color-dressed amplitudes despite having no explicit color factors. This is possible due to a remarkable fact--"Color is Kinematics"--whereby kinematic wedge products in the scattering forms satisfy the same Jacobi relations as color factors. Finally, our scattering forms are well-defined on the projectivized kinematic space, a property that provides a geometric origin for color-kinematics duality.

• The paper introduces a geometric interpretation of scattering amplitudes as differential forms in kinematic space, challenging conventional views.
• It demonstrates that the associahedron naturally underpins tree-level amplitudes in bi-adjoint φ³ theory, elucidating properties like locality and unitarity.
• It extends the framework to Yang-Mills and the Non-linear Sigma Model, highlighting a 'Color is Kinematics' duality and providing a geometric basis for the CHY formula.

Essay on Scattering Forms and the Positive Geometry of Kinematics, Color, and the Worldsheet

The paper "Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet" provides a detailed exploration of the novel geometric understanding of scattering amplitudes, positioning them not as functions, but as differential forms in kinematic space. This perspective not only enriches the theoretical framework of scattering amplitudes but also establishes profound connections with geometric structures such as the associahedron and the amplituhedron, particularly for bi-adjoint scalar theories.

The foundational premise of the paper is the recognition of surprising geometric structures that underpin scattering amplitudes. These geometries, traditionally understood in auxiliary spaces like the string worldsheet or the amplituhedron, are now explicated directly within the kinematic space where the amplitudes reside. The authors navigate through a myriad of theories, including massless theories in arbitrary spacetime dimensions, to elucidate the interplay between these amplitudes and their associated positive geometries.

A focal point of this paper is the associahedron, which draws connections between the classic polytope and tree-level amplitudes in bi-adjoint $\phi^3$ scalar theory. Here, the associahedron emerges naturally in kinematic space, with the canonical form of the polytope determining the amplitude. Crucially, this geometric approach renders certain physical properties, such as locality, unitarity, and soft limits, as outcomes of the combinatorial geometry of the associahedron.

Furthermore, the authors make a striking observation that the moduli space for the open string worldsheet can also be realized as an associahedron. The scattering equations act as a diffeomorphism, linking the interior of this worldsheet associahedron to the kinematic associahedron. This relationship furnishes a geometric interpretation and a simple conceptual derivation of the bi-adjoint CHY formula.

Moving beyond the bi-adjoint $\phi^3$ theory, the paper extends its analysis to Yang-Mills theory and the Non-linear Sigma Model (NLSM). It reveals that these theories possess a dual interpretation in terms of scattering forms. This is remarkable as it illustrates the "Color is Kinematics" duality, where kinematic wedge products in the scattering forms align with the algebraic identities of color factors. By abandoning explicit color factors and focusing on permutation invariant differential forms, the research underscores the universality and flexibility of these geometric structures.

The implications of this work are far-reaching, offering a fresh perspective on both theoretical and practical levels. Theoretically, it challenges conventional methodologies by re-envisioning amplitudes through a geometric lens. Practically, the insights gathered from this framework could potentially streamline calculations in computational physics, especially concerning amplitude computations.

Future developments in AI could greatly benefit from this geometric approach. AI systems, designed to simulate complex physical processes, could leverage such geometric insights to enhance the scalability and efficiency of simulations. By integrating these geometric principles, AI could play a pivotal role in unraveling the complexities inherent in high-dimensional positive geometries and their associated physical theories.

In conclusion, the paper carves a promising pathway towards a geometric unification of scattering amplitudes, offering a deepened understanding of both the mathematical structures and physical phenomena governing these essential constructs in theoretical physics.