THE GRADUATE STUDENT SECTION

WHAT IS…

the Amplituhedron?Jacob Bourjaily and Hugh Thomas

Communicated by Cesar E. Silva

The amplituhedron is a space whose geometric struc-ture conjecturally determines scattering amplitudes in acertain class of quantum field theories. When it was intro-duced in 2013 by Arkani-Hamed and Trnka [1], QuantaMagazine1 reported that “Physicists have discovered ajewel-like geometric object that dramatically simplifiescalculations of particle interactions and challenges the

Scatteringamplitudes arethe bread and

butter ofquantum field

theory.

notion that space and timeare fundamental compo-nents of reality…” In thisshort article, we will not getas far as challenging thenotions of space and time,but we will give some ex-planation as to the natureand significance of the am-plituhedron both in physicsand mathematics—where itarises as a natural gener-alization of the subject oftotal positivity for Grassmannians.

Let us begin with the origin of these ideas in the physicsof scattering amplitudes. For any particular quantumfieldtheory, the probability for some number of particles to‘scatter’ and produce some number of others is encodedby a function called the (scattering) amplitude for thatprocess. These functions depend on all the observablenumbers labeling the states involved in the process—inparticular, momenta and helicity. Amplitudes can (often)be determined perturbatively according to the Feynman‘loop’ expansion. At leading order, they are rational

Jacob Bourjaily is assistant professor of physics, Niels Bohr Inter-national Academy, University of Copenhagen. His email addressis bourjaily@nbi.ku.dk.Hugh Thomas is professor of mathematics, Université du Québecà Montreal. His email address is thomas.hugh_r@uqam.ca.1Natalie Wolchover, A Jewel in the Heart of Quantum Physics,Quanta Magazine, September 2013.

For permission to reprint this article, please contact:reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1630

functions of the momenta; and at the ‘ℓ loop’ order ofperturbation, they are integrals over rational forms onthe space of ℓ internal loop momenta.

Scattering amplitudes are the bread and butter ofquantum field theory; but they are notoriously difficultto compute using the Feynman expansion. Moreover,individual Feynman diagrams depend on many unob-servable parameters which can greatly obscure the(often incredible) simplicity of amplitudes’ mathematicalstructure.

A major breakthrough in physicists’ understanding of(and ability to compute) scattering amplitudes came in2004 with the discovery of recursion relations for tree-level (ℓ = 0) amplitudes by Britto, Cachazo, Feng, andWitten [3]. They were first described graphically as inFigure 1:

Figure 1. Recursion relations for amplitudes, encodedin terms of bi-colored, trivalent graphs.

The precise meaning of these graphs will not beimportant to us, but a few aspects are worth mentioning.In Figure1, amplitudes are representedbygrey circles, andany two legs may be chosen for the recursion (indicatedby arrows). Thus, the BCFW recursion relations do notprovide any particular representation of an amplitude interms of graphs, but lead to a wide variety of inequivalentrepresentations according to which two legs are chosenat every subsequent order of the recursion. The variety ofpossible recursive formulae obtained was reminiscent ofdifferent ‘triangulations’ of a polytope—an analogy thatwould be made sharp by the amplituhedron.

For several years after their discovery, the primaryimportance of the recursion relations followed from

February 2018 Notices of the AMS 167

THE GRADUATE STUDENT SECTIONthe fact that they yield incredibly more compact—andphysically more transparent—representations for tree-level amplitudes. Processes that would have requiredmore Feynman diagrams than atoms in the universecould now be represented by a single diagram. Anothersurprising feature of these relations was the discoverythat individual terms enjoy much more symmetry thananyone had anticipated. These new symmetries wereshown to hold for amplitudes, and were later identifiedwith a Yangian Lie algebra.

Another key breakthrough came in 2009, when physi-cists discovered that the graphs arising in Figure 1 couldbe related to volume-forms on certain sub-varieties ofGr≥0(𝑘, 𝑛)—the “nonnegative” portion of the Grassman-nian of 𝑘-dimensional subspaces of ℝ𝑛, to be definedmomentarily. Here, 𝑛, 𝑘 are determined by the graph: 𝑛 isthe number of external edges, and 𝑘 = 2𝑛𝐵 +𝑛𝑊 −𝑛𝐼 −2for a graph with 𝑛𝐵, 𝑛𝑊 blue and white vertices, re-spectively, and 𝑛𝐼 internal edges. (The correspondencebetween these graphs and the totally nonnegative Grass-mannian was fully understood by mathematicians, suchas Postnikov, considerably before physicists stumbledupon it independently.) The physical implications of thiscorrespondence, together with its generalization to allorders of perturbation, can be found in [2].

Given the correspondence between the terms gener-ated by BCFW (and its all-orders generalization) and theGrassmannian, it was natural to wonder if a more in-trinsically geometric picture existed. Specifically, can therecursion relations be literally viewed as ‘triangulating’some region in the Grassmannian? And if so, can thisspace be defined intrinsically—without reference to howit should be triangulated? In 2013, Arkani-Hamed andTrnka proposed the amplituhedron as the answer to bothquestions for the case of amplitudes in planar, maximallysupersymmetric Yang-Mills theory [1].

Let us now turn to giving a precise mathematicaldefinition of the amplituhedron, in the 0 loop “tree” case.(The ℓ loop amplituhedron for ℓ > 0 has a definitionwhich is similar but somewhat more complicated; we willnot discuss it further.) Let 𝑀 ∈ Mat(𝑘 × 𝑛) be a 𝑘 × 𝑛matrix of real numbers, with 𝑘 ≤ 𝑛. As we recall fromlinear algebra, the rows of 𝑀 determine a subspace ofℝ𝑛 called its row space. If the row space is 𝑘-dimensional(the maximum possible), we say that 𝑀 has full rank. Twofull-rank matrices 𝑀1 and 𝑀2 have the same row spaceif for some 𝐾 in the space of 𝑘 × 𝑘 invertible matrices,𝐾.𝑀1 = 𝑀2. Clearly, any 𝑘-dimensional subspace of ℝ𝑛

arises as the row space of some full-rank matrix; so thecollection of all 𝑘-dimensional subspaces can be describedasMatfull(𝑘×𝑛)/𝐺𝐿(𝑘). This is theGrassmannianGr(𝑘, 𝑛).

The next ingredient required to define the amplituhe-dron is the notion of positivity in the Grassmannian. Amaximal minor of a matrix 𝑀 ∈ Mat(𝑘 × 𝑛) is the deter-minant of the 𝑘 × 𝑘 submatrix obtained from taking asubset of 𝑘 of the columns, in the order they appear inthe matrix. The totally positive Grassmannian, denotedGr>0(𝑘, 𝑛), is the subspace of the Grassmannian for which

all maximal minors have the same sign. The totally non-negative Grassmannian, Gr≥0(𝑘, 𝑛), is defined in the sameway, except that we also allow some minors to vanish.

The totally nonnegative Grassmannian was studied byPostnikov as a concrete instance of Lusztig’s theory oftotal positivity for algebraic groups. It is topologicallyequivalent to a ball, and has a beautiful cell structure thatcan be indexed by decorated permutations (permutationsof 𝑆𝑛 with one extra bit of information for each fixedpoint). Importantly, cells in this ‘positroid’ stratificationcan be endowed with cluster coordinates and a naturalvolume form.

We now have everything required to define the treeamplituhedron. It is indexed by three integers, 𝑛, 𝑘, and𝑑, and a totally positive 𝑛 × (𝑑 + 𝑘) matrix 𝑃. From thepoint of view of physics, 𝑛 is the number of particles,𝑘, defined above in the context of graphs, encodes thetotal helicity flow, 𝑑 is the space-time dimension, and 𝑃encodes the momenta and helicities of the particles. Let𝑀 ∈ Gr≥0(𝑘, 𝑛) vary. The amplituhedron 𝒜, as definedby Arkani-Hamed and Trnka [1] is simply the total imagelocus,

(1) 𝒜 = 𝑀.𝑃 ,

inside Gr(𝑘, 𝑑+𝑘). Notice that 𝑀.𝑃 is a 𝑘×(𝑑+𝑘) matrix,automatically having full rank, so 𝑀.𝑃 really does definea point in Gr(𝑘, 𝑑 + 𝑘).

To help build intuition, it is useful to consider somespecial cases. If 𝑛 = 𝑑+ 𝑘, then the fact that 𝑃 is totallypositive implies it is invertible, so we may as well take itto be the identity matrix; in this case, the amplituhedronis simply Gr≥0(𝑘, 𝑑+𝑘). Another simple case arises when𝑘 = 1, for which 𝑀 is a totally nonnegative 1×𝑛 matrix—i.e. a vector of nonnegative numbers, not all zero. In thiscase, 𝒜 consists of all non-negative linear combinationsof the rows of 𝑃, and is thus a convex polytope inprojective space. The fact that 𝑃 is totally positive meansthat𝒜 is what is called a cyclic polytope; these have manyinteresting extremal properties. In this case, repeatedlyapplying the BCFW recursion literally triangulates thispolytope. A point 𝑌 in projective space determines ahyperplane 𝑌∗ in the dual projective space, and similarly,the amplituhedron, being in this case a polytope inprojective space, determines a dual polytope 𝒜∗. Thevolume form on 𝒜 at a point 𝑌 is given by the volume of𝒜∗ when 𝑌∗ is the hyperplane at infinity.

In general, the volume form on 𝒜 can conjecturallybe defined by the fact that it has logarithmic singular-ities on the boundaries of 𝒜. It can also be calculatedexplicitly by pushing forward the volume form on BCFWcells. However, it would be desirable to have an explicit,intrinsic description, as given above for 𝑘 = 1. This wouldpresumably require an answer to the question, “What isthe dual amplituhedron?”—something that wemust leavefor the future.

168 Notices of the AMS Volume 65, Number 2

THE GRADUATE STUDENT SECTIONReferences

[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, GrassmannianGeometry of Scattering Amplitudes, Cambridge UniversityPress, 2016. MR 3467729

[3] R. Britto, F. Cachazo, B. Feng, and E. Witten, DirectProof of Tree-Level Recursion Relation in Yang- MillsTheory, dx.doi.org/10.1103/PhysRevLett.94.181602Phys. Rev. Lett. 94 (2005) 181602. MR 2260976

Image CreditsFigure 1 courtesy of Jacob Bourjaily.Photo of Jacob Bourjaily by Ola Joensen, courtesy of the Niels

Bohr Institute.Photo of Hugh Thomas courtesy of Charles Paquette.

ABOUT THE AUTHORS

Jacob Bourjaily enjoys readingpoetry in languages he doesn’tknow.

Jacob Bourjaily

Hugh Thomas enjoys translatingpoetry from languages he doesn’tknow.

Hugh Thomas

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