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The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces a remarkable new way for computations in quantum field theory based on volume computations of certain polyhedra. This and related works may also have profound implications for the foundation of particle physics. It is related to various beautiful mathematics, and, in particular, to the combinatorics of certain stratifications of the Grassmanians.(See also the Quanta Magazine article, A Jewel at the Heart of Quantum Physics, by Natalie Wolchover, and Nima Arkani-Hamed’s on-line SUSY 2013 video lecture The Amplituhedron.)

The amplituhedron is remarkable new subsequent geometric object which comes in this study extending the notion of "positive grassmanian." (I did not see it explicitly defined in the above paper, at least not by this name.) It is very briefly described in Arkani-Hamed's lecture. enter image description here

My question is quite simple:

what is the mathematical definition of the amplituhedron?

An older-sister MO question: The amplituhedron minus the physics

Update: The paper The amplituhedron by Nima Arkani-Hamed and Jaroslav Trnka is now on the arxive.

Update: See also this blog post by Trnka http://www.preposterousuniverse.com/blog/2014/03/31/guest-post-jaroslav-trnka-on-the-amplituhedron/

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    $\begingroup$ Thanks Gil for asking this question; I tried to skim the talk, wikipedia site etc. after mathoverflow.net/questions/142841/… --- but am waiitng to see a simplified definition. $\endgroup$
    – Suvrit
    Commented Sep 27, 2013 at 16:08
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    $\begingroup$ Dear Sam, yes this is almost a duplicate of Joe's question that I simply did not see. (A few different links, images, and tags). Regarding your second point: In a few cases I asked questions on MO where I can also get an answer by directly asking a colleague. The advantage of asking on MO is that this way more people are exposed to the information and also it is nice to test the MO platform which is a relatively new thing. Of course, in a few cases I both posted on MO and asked colleagues personally. $\endgroup$
    – Gil Kalai
    Commented Sep 27, 2013 at 17:37
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    $\begingroup$ On second thought, the question here is simpler: What is the mathematical definition of the amplituhedron? It can be regarded as coming one step before Joe's question on geometric/polytopal properties of the amplituhedron. $\endgroup$
    – Gil Kalai
    Commented Sep 27, 2013 at 18:00
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    $\begingroup$ I welcome Gil's more focused and knowledgeable question, as my question ("The amplituhedron minus the physics") has faded in visibility. $\endgroup$
    – Joseph O'Rourke
    Commented Sep 27, 2013 at 19:50
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    $\begingroup$ It is somewhat unsettling that we have already had two questions decorated with several links to papers on an object and we are still asking for its precise definition! :-) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Sep 28, 2013 at 1:17

4 Answers 4

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I would think that this presentation by Jaroslav Trnka, given here in Utrecht, goes at least some way towards a mathematical definition of the amplituhedron.

To skip the physical motivation, start at page 13; jargon abbreviations such as NMHV = "next-to-maximally helicity-violating" can be ignored (they relate only to the physical significance of the construction). The construction of the amplituhedron $P_{n,k,m}$ is summarized on page 23. What follows on later pages is the description how to associate a form $\Omega_{n,k,m}$ to the space $P_{n,k,m}$ and use this to calculate the required physical quantity (a scattering amplitude).

My attempt to parse a definition of the amplituhedron from Trnka's presentation:

For given integers $k,n,m$ (with $n\geq k+m$) take a $k\times n$ real matrix $C\in G_{+}(k,n)$ and a $(k+m)\times n$ real matrix $Z\in G_{+}(k+m,n)$. Here $G_+(k,n)$ is the positive Grassmannian space of $k\times n$ matrices with all $k\times k$ minors $>0$. (The $k\times n$ matrices are identified modulo the simultaneous action of a $k\times k$ matrix on each of the column vectors.)

Associated to these two positive Grassmannians is the $k\times (k+m)$ real matrix $Y$ having matrix elements

$$Y_{\alpha}^{\beta}=\sum_{p=1}^{n}C_{p}^{\alpha}Z_{p}^{\beta}.$$

By varying $C\in G_{+}(k,n)$ at fixed $Z\in G_{+}(k+m,n)$, the matrix $Y$ varies over a space $P_{n,k,m}$. This space is called the amplituhedron.

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    $\begingroup$ Many thanks, Carlo. I still dont see what precisely is $P_{n,k,m}$. $\endgroup$
    – Gil Kalai
    Commented Sep 27, 2013 at 22:46
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    $\begingroup$ Noob question probably, but when you write "all minors >0", are you imposing a canonical ordering on the columns? Because otherwise they can't all be >0, since they change sign when swapping two columns. $\endgroup$
    – Federico Poloni
    Commented Sep 28, 2013 at 13:49
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    $\begingroup$ @FedericoPoloni --- indeed, the minors are ordered in ascending order of the columns: the $(k\times n)$ matrix $C$ is in the positive Grasmannian $G_{+}(k,n)$ if for each set of $k$ integers $1\leq a_1<a_2<\cdots a_k\leq n$ the $k\times k$ submatrix of $C$ consisting of the ordered columns $(a_1,a_2,\ldots a_k)$ has a positive determinant. $\endgroup$
    – Carlo Beenakker
    Commented Sep 28, 2013 at 14:30
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    $\begingroup$ @suvrit --- as far as my very_limited understanding goes, strict positivity of the minors ensures that the matrix $Y$ lies inside the amplituhedron $P_{n,k,m}$; a vanishing minor produces a matrix $Y$ on the boundary. For the physics application one is interested in the volume bounded by the amplituhedron, so it should not make a difference if the positive Grassmannian is replaced by a nonnegative Grasmannian. $\endgroup$
    – Carlo Beenakker
    Commented Sep 28, 2013 at 17:40
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    $\begingroup$ Sorry for making this thread long, but I wanted to alert the readers that as per Logan's answer here: mathoverflow.net/a/143421/8430, the above $Y=CZ^T$ is not the whole amplitudhedron, just the tree-case ... so I'll wait for more updates before I shoot off more comments! $\endgroup$
    – Suvrit
    Commented Sep 28, 2013 at 18:04
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There is now an AMS Notices article whose title is the same as the title of this question (and which therefore may be enlightening to anyone on this page): http://www.ams.org/journals/notices/201802/rnoti-p167.pdf

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In addition to Carlo Beenakker's wonderful answer: it seems Penrose's twistor formalism (twistor diagrams) plays a central role in this amplituhedron business. See http://www.twistordiagrams.org.uk/papers/index.html for the description of historical developments that led to the discovery of amplituhedron. Other useful information can be found at http://ncatlab.org/nlab/show/amplituhedron

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Let me give a brief version of an answer I posted on Joe's sister question.

Alexander Postnikov gave a detailed series of lectures on the positive Grassmanian at the Hebrew University of Jerusalem (see links below). He also briefly referred to the amplituhedron.

Folowing Postinkov, I can briefly explain the situation as follows:

A) The stratification of the positive Grassmanian:

Regard the Grassmanian as represented by equivalence classes of $m$ by $n$ matrices ($m \le n$) under row operators. The non-negative Grassmanian is the set of totally-non negative matrices namely those where all $m$ by $m$ minors are nonnegative. It has an important cell-like structure that can be defined by regarding two matrices in the same cell if whenever an $m$ by $m$ minor of one matrix is zero so is the corresponding matrix of the other.

B) The amplituhedron:

Recall that every polytope is a projection of a simplex and projections with respect to totally positive matrices give precisely the cyclic polytope.

Now, replace the simplex by the positive Grassmanian: The amplituhedron is a projection of the positive Grassmanian based on a totally positive matrix. So the amplituhedron is a common generalization of the positive grassmanian and the cyclic polytope!

Video1, Video2, Video3, Video4

Postnikov's ten minutes explanation of the amplituhedron starts here.

Update: Some more details and links can be found in the blog post The simplex, the cyclic polytope, the positroidron, the amplituhedron and beyond.

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