Scattering Amplitudes and the Positive Grassmannian (1212.5605v2)

Abstract: We establish a direct connection between scattering amplitudes in planar four-dimensional theories and a remarkable mathematical structure known as the positive Grassmannian. The central physical idea is to focus on on-shell diagrams as objects of fundamental importance to scattering amplitudes. We show that the all-loop integrand in N=4 SYM is naturally represented in this way. On-shell diagrams in this theory are intimately tied to a variety of mathematical objects, ranging from a new graphical representation of permutations to a beautiful stratification of the Grassmannian G(k,n) which generalizes the notion of a simplex in projective space. All physically important operations involving on-shell diagrams map to canonical operations on permutations; in particular, BCFW deformations correspond to adjacent transpositions. Each cell of the positive Grassmannian is naturally endowed with positive coordinates and an invariant measure which determines the on-shell function associated with the diagram. This understanding allows us to classify and compute all on-shell diagrams, and give a geometric understanding for all the non-trivial relations among them. Yangian invariance of scattering amplitudes is transparently represented by diffeomorphisms of G(k,n) which preserve the positive structure. Scattering amplitudes in (1+1)-dimensional integrable systems and the ABJM theory in (2+1) dimensions can both be understood as special cases of these ideas. On-shell diagrams in theories with less (or no) supersymmetry are associated with exactly the same structures in the Grassmannian, but with a measure deformed by a factor encoding ultraviolet singularities. The Grassmannian representation of on-shell processes also gives a new understanding of the all-loop integrand for scattering amplitudes, presenting all integrands in a novel dLog form which directly reflects the underlying positive structure.

• The paper connects scattering amplitudes in planar quantum field theories to the positive Grassmannian geometry using on-shell diagrams, offering a novel computational framework.
• It represents amplitudes through contour integrals over Grassmannians, leveraging Yangian symmetry for simplified calculations and deeper theoretical understanding.
• This geometric approach provides a more natural and computationally efficient way to compute amplitudes, with potential applications beyond the studied N=4 SYM theory.

An Overview of "Scattering Amplitudes and the Positive Grassmannian"

The paper "Scattering Amplitudes and the Positive Grassmannian" explores the profound connections between scattering amplitudes in planar four-dimensional quantum field theories, particularly in $\mathcal{N}=4$ super Yang-Mills (SYM), and mathematical structures involving the positive Grassmannian. This paper is a confluence of advances in theoretical physics and combinatorial mathematics, presenting new ways to compute scattering amplitudes based on the geometry of Grassmannians. Below is an overview and analysis of the key concepts and implications of this research.

Core Concepts and Results

1. On-Shell Diagrams and Amplitudes: The authors conceptualize scattering amplitudes through on-shell diagrams formed by gluing together three-particle amplitudes, the most fundamental interaction vertices, without reference to virtual particles. These diagrams naturally align with the mathematical framework of the Grassmannian, suggesting that scattering processes can be understood in terms of configurations in these spaces.
2. Grassmannian Connection: The Grassmannian, denoted as $G(k,n)$, is the space of all $k$-dimensional planes in an $n$-dimensional vector space. The paper exploits this structure to represent scattering amplitudes through contour integrals over Grassmannians, revealing symmetries not visible in traditional Feynman diagrams and making full use of Yangian symmetry.
3. The Positive Grassmannian: A central innovation is the linkage of physical processes to the positive Grassmannian $G_+(k,n)$, which restricts configurations to those involving only positive minors by cyclically ordering columns. This reflects a convex substructure of the mathematical space, offering a natural and insightful decomposition relevant to scattering amplitudes.
4. Yangian Symmetry: In $\mathcal{N}=4$ SYM, amplitudes exhibit Yangian symmetry, an infinite-dimensional extension of conformal symmetry. The paper elegantly maps these symmetries to structures preserved under positive diffeomorphisms, demonstrating how these transformations manifest as invariances in the Grassmannian formulation.
5. Homological Analysis and Identities: Importantly, the paper explores how identities among Yangian-invariant functions describe relationships between amplitudes, which are seen as emergent from the topological properties of Grassmannian boundaries. This enhances understanding of amplitude singularities and residues.

Implications for Field Theory

• Computational Simplification: By translating complex amplitude calculations into questions about the geometry of the Grassmannian, the paper proposes a more natural unification of kinematical and dynamical aspects of scattering amplitudes, reducing computational complexity and exposing symmetries.
• Potential Extensions to Other Theories: While the focus is on $\mathcal{N}=4$ SYM, the methods presented may be adaptable to other gauge theories, possibly including less supersymmetric or even non-supersymmetric theories, through deformations in the Grassmannian representation.
• Theoretical Foundation for Amplitude Structures: The research provides an elegant theoretical foundation that integrates known results on scattering amplitudes with the combinatorial approaches of the positive Grassmannian, offering a coherent picture that aligns with modern physical theories.

Future Directions

The work opens significant avenues for further exploration. In addition to confirming and extending these results in diverse theoretical frameworks, there remains the potential to explore connections with integrable systems beyond existing gauge-theory correspondences. The methodology may also inspire new insights into the dynamics of higher-dimensional theories or condensed matter systems analogously described by Grassmannian structures.

This research constitutes an important link between abstract algebraic geometry and practical computation in quantum field theory, offering fresh perspectives on fundamental interactions through the lens of Grassmannian geometry.