THEGRADUATESTUDENTSECTION

WHATIS… the Amplituhedron? Jacob Bourjaily and Hugh Thomas Communicated by Cesar E. Silva

The amplituhedron is a space whose geometric struc- functions of the momenta; and at the ‘ℓ loop’ order of ture conjecturally determines scattering amplitudes in a perturbation, they are integrals over rational forms on certain class of quantum field theories. When it was intro- the space of ℓ internal loop momenta. duced in 2013 by Arkani-Hamed and Trnka [1], Quanta Scattering amplitudes are the bread and butter of Magazine1 reported that “Physicists have discovered a quantum field theory; but they are notoriously difficult jewel-like geometric object that dramatically simplifies to compute using the Feynman expansion. Moreover, calculations of particle interactions and challenges the individual Feynman diagrams depend on many unob- notion that space and time servable parameters which can greatly obscure the are fundamental compo- (often incredible) simplicity of amplitudes’ mathematical nents of reality…” In this Scattering structure. short article, we will not get amplitudes are A major breakthrough in physicists’ understanding of as far as challenging the (and ability to compute) scattering amplitudes came in notions of space and time, the bread and 2004 with the discovery of recursion relations for tree- but we will give some ex- level (ℓ = 0) amplitudes by Britto, Cachazo, Feng, and planation as to the nature butter of Witten [3]. They were first described graphically as in and significance of the am- Figure 1: plituhedron both in physics quantum field and mathematics—where it arises as a natural gener- theory. alization of the subject of total positivity for Grassmannians. Let us begin with the origin of these ideas in the physics of scattering amplitudes. For any particular quantum field theory, the probability for some number of particles to ‘scatter’ and produce some number of others is encoded by a function called the (scattering) amplitude for that Figure 1. Recursion relations for amplitudes, encoded process. These functions depend on all the observable in terms of bi-colored, trivalent graphs. numbers labeling the states involved in the process—in particular, momenta and helicity. Amplitudes can (often) The precise meaning of these graphs will not be be determined perturbatively according to the Feynman important to us, but a few aspects are worth mentioning. ‘loop’ expansion. At leading order, they are rational In Figure 1, amplitudes are represented by grey circles, and any two legs may be chosen for the recursion (indicated Jacob Bourjaily is assistant professor of physics, Niels Bohr Inter- by arrows). Thus, the BCFW recursion relations do not national Academy, University of Copenhagen. His email address provide any particular representation of an amplitude in is [email protected]. terms of graphs, but lead to a wide variety of inequivalent Hugh Thomas is professor of mathematics, Université du Québec representations according to which two legs are chosen à Montreal. His email address is [email protected]. at every subsequent order of the recursion. The variety of 1Natalie Wolchover, A Jewel in the Heart of Quantum Physics, possible recursive formulae obtained was reminiscent of Quanta Magazine, September 2013. different ‘triangulations’ of a polytope—an analogy that For permission to reprint this article, please contact: would be made sharp by the amplituhedron. [email protected]. For several years after their discovery, the primary DOI: http://dx.doi.org/10.1090/noti1630 importance of the recursion relations followed from

February 2018 Notices of the AMS 167 THEGRADUATESTUDENTSECTION the fact that they yield incredibly more compact—and all maximal minors have the same sign. The totally non- physically more transparent—representations for tree- negative Grassmannian, Gr≥0(푘, 푛), is defined in the same level amplitudes. Processes that would have required way, except that we also allow some minors to vanish. more Feynman diagrams than atoms in the universe The totally nonnegative Grassmannian was studied by could now be represented by a single diagram. Another Postnikov as a concrete instance of Lusztig’s theory of surprising feature of these relations was the discovery total positivity for algebraic groups. It is topologically that individual terms enjoy much more symmetry than equivalent to a ball, and has a beautiful cell structure that anyone had anticipated. These new symmetries were can be indexed by decorated permutations (permutations shown to hold for amplitudes, and were later identified of 푆푛 with one extra bit of information for each fixed with a Yangian Lie algebra. point). Importantly, cells in this ‘positroid’ stratification Another key breakthrough came in 2009, when physi- can be endowed with cluster coordinates and a natural cists discovered that the graphs arising in Figure 1 could volume form. be related to volume-forms on certain sub-varieties of We now have everything required to define the tree Gr (푘, 푛)—the “nonnegative” portion of the Grassman- ≥0 amplituhedron. It is indexed by three integers, 푛, 푘, and nian of 푘-dimensional subspaces of ℝ푛, to be defined momentarily. Here, 푛, 푘 are determined by the graph: 푛 is 푑, and a totally positive 푛 × (푑 + 푘) matrix 푃. From the point of view of physics, 푛 is the number of particles, the number of external edges, and 푘 = 2푛퐵 + 푛푊 − 푛퐼 − 2 푘, defined above in the context of graphs, encodes the for a graph with 푛퐵, 푛푊 blue and white vertices, re- total helicity flow, 푑 is the space-time dimension, and 푃 spectively, and 푛퐼 internal edges. (The correspondence between these graphs and the totally nonnegative Grass- encodes the momenta and helicities of the particles. Let mannian was fully understood by mathematicians, such 푀 ∈ Gr≥0(푘, 푛) vary. The amplituhedron 풜, as defined as Postnikov, considerably before physicists stumbled by Arkani-Hamed and Trnka [1] is simply the total image upon it independently.) The physical implications of this locus, correspondence, together with its generalization to all orders of perturbation, can be found in [2]. (1) 풜 = 푀.푃, Given the correspondence between the terms gener- ated by BCFW (and its all-orders generalization) and the inside Gr(푘, 푑 + 푘). Notice that 푀.푃 is a 푘 × (푑 + 푘) matrix, Grassmannian, it was natural to wonder if a more in- automatically having full rank, so 푀.푃 really does define trinsically geometric picture existed. Specifically, can the a point in Gr(푘, 푑 + 푘). recursion relations be literally viewed as ‘triangulating’ To help build intuition, it is useful to consider some some region in the Grassmannian? And if so, can this special cases. If 푛 = 푑 + 푘, then the fact that 푃 is totally space be defined intrinsically—without reference to how positive implies it is invertible, so we may as well take it it should be triangulated? In 2013, Arkani-Hamed and to be the identity matrix; in this case, the amplituhedron Trnka proposed the amplituhedron as the answer to both is simply Gr≥0(푘, 푑 + 푘). Another simple case arises when questions for the case of amplitudes in planar, maximally 푘 = 1, for which 푀 is a totally nonnegative 1 × 푛 matrix— supersymmetric Yang-Mills theory [1]. i.e. a vector of nonnegative numbers, not all zero. In this Let us now turn to giving a precise mathematical case, 풜 consists of all non-negative linear combinations definition of the amplituhedron, in the 0 loop “tree” case. of the rows of 푃, and is thus a convex polytope in (The ℓ loop amplituhedron for ℓ > 0 has a definition projective space. The fact that 푃 is totally positive means which is similar but somewhat more complicated; we will that 풜 is what is called a cyclic polytope; these have many not discuss it further.) Let 푀 ∈ Mat(푘 × 푛) be a 푘 × 푛 interesting extremal properties. In this case, repeatedly matrix of real numbers, with 푘 ≤ 푛. As we recall from applying the BCFW recursion literally triangulates this linear algebra, the rows of 푀 determine a subspace of polytope. A point 푌 in projective space determines a ℝ푛 called its row space. If the row space is 푘-dimensional hyperplane 푌∗ in the dual projective space, and similarly, (the maximum possible), we say that 푀 has full rank. Two the amplituhedron, being in this case a polytope in full-rank matrices 푀 and 푀 have the same row space 1 2 projective space, determines a dual polytope 풜∗. The if for some 퐾 in the space of 푘 × 푘 invertible matrices, 푛 volume form on 풜 at a point 푌 is given by the volume of 퐾.푀1 = 푀2. Clearly, any 푘-dimensional subspace of ℝ 풜∗ when 푌∗ is the hyperplane at infinity. arises as the row space of some full-rank matrix; so the collection of all 푘-dimensional subspaces can be described In general, the volume form on 풜 can conjecturally be defined by the fact that it has logarithmic singular- as Matfull(푘×푛)/퐺퐿(푘). This is the Grassmannian Gr(푘, 푛). The next ingredient required to define the amplituhe- ities on the boundaries of 풜. It can also be calculated dron is the notion of positivity in the Grassmannian. A explicitly by pushing forward the volume form on BCFW maximal minor of a matrix 푀 ∈ Mat(푘 × 푛) is the deter- cells. However, it would be desirable to have an explicit, minant of the 푘 × 푘 submatrix obtained from taking a intrinsic description, as given above for 푘 = 1. This would subset of 푘 of the columns, in the order they appear in presumably require an answer to the question, “What is the matrix. The totally positive Grassmannian, denoted the dual amplituhedron?”—something that we must leave Gr>0(푘, 푛), is the subspace of the Grassmannian for which for the future.

168 Notices of the AMS Volume 65, Number 2 THEGRADUATESTUDENTSECTION

References [1] N. Arkani-Hamed and J. Trnka, The Amplituhedron, arxiv.org/abs/1312.2007 arXiv:1312.2007. [2] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov, and J. Trnka, Grassmannian Moving? Geometry of Scattering Amplitudes, Cambridge University Press, 2016. MR 3467729 Please make sure that the AMS Notices ISSN 0002-9920 (print) ISSN 1088-9477 (online) [3] R. Britto, F. Cachazo, B. Feng, and E. Witten, Direct and Bulletin find their new home.

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Common Core State Standards for Geometry: An Theory, dx.doi.org/10.1103/PhysRevLett.94.181602 Alternative Approach page 24

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Lubbock Meeting Image Credits page 111 Tel Aviv Meeting • page 114 or make the change yourself online About the cover: Plain geometry (see page 88) Figure 1 courtesy of Jacob Bourjaily. at: www.ams.org/cml-update Photo of Jacob Bourjaily by Ola Joensen, courtesy of the Niels • or send the information to: Bohr Institute. Photo of Hugh Thomas courtesy of Charles Paquette. Member and Customer Services American Mathematical Society 201 Charles Street Providence, RI 02904-2294 USA ABOUT THE AUTHORS Phone: (800) 321-4267 (US & Canada) (401) 455-4000 (Worldwide) Jacob Bourjaily enjoys reading poetry in languages he doesn’t know.

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