Grassmannian Geometry of Scattering Amplitudes

An essential guide to peturbative quantum field theory and its connection to Grassmannian geometry.

Nima Arkani-Hamed (Author), Jacob Bourjaily (Author), Freddy Cachazo (Author), Alexander Goncharov (Author), Alexander Postnikov (Author), Jaroslav Trnka (Author)

9781107086586, Cambridge University Press

Hardback, published 5 May 2016

201 pages, 192 b/w illus. 8 tables
25.3 x 18 x 1.4 cm, 0.56 kg

'This book is the 'Diagrammar' of the twenty-first century. Just as Gerard 't Hooft and Martinus Veltman laid out in 1973 the then-new calculus of Feynman diagrams for scattering processes in non-abelian gauge theories, so now have Nima Arkani-Hamed and his co-authors set forth a new perspective on scattering amplitudes, which leads away from locality and unitarity toward other principles, and they have provided a detailed and elegantly illustrated how-to manual for the practitioner.' Lance Dixon, Stanford University

Outlining a revolutionary reformulation of the foundations of perturbative quantum field theory, this book is a self-contained and authoritative analysis of the application of this new formulation to the case of planar, maximally supersymmetric Yang–Mills theory. The book begins by deriving connections between scattering amplitudes and Grassmannian geometry from first principles before introducing novel physical and mathematical ideas in a systematic manner accessible to both physicists and mathematicians. The principle players in this process are on-shell functions which are closely related to certain sub-strata of Grassmannian manifolds called positroids - in terms of which the classification of on-shell functions and their relations becomes combinatorially manifest. This is an essential introduction to the geometry and combinatorics of the positroid stratification of the Grassmannian and an ideal text for advanced students and researchers working in the areas of field theory, high energy physics, and the broader fields of mathematical physics.

Acknowledgements
1. Introduction
2. Introduction to on-shell functions and diagrams
3. Permutations and scattering amplitudes
4. From on-shell diagrams to the Grassmannian
5. Configurations of vectors and the positive Grassmannian
6. Body configurations, graphs, and permutations
7. The invariant top-form and the positroid stratification
8. (Super) conformal and dual conformal invariance
9. Positive diffeomorphisms and Yangian invariance
10. The kinematical support of physical on-shell forms
11. Homological identities among Yangian-invariants
12. (Relatively) orienting canonical coordinate charts on positroids
13. Classification of Yangian-invariants and their relations
14. The Yang–Braxter relation and ABJM theories
15. On-shell diagrams for theories with N<4 supersymmetries
16. Dual graphs and cluster algebras
17. On-shell representations of scattering amplitudes
18. Outlook
References.

Subject Areas: Mathematical physics [PHU], Particle & high-energy physics [PHP], Physics [PH], Mathematics & science [P]