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Is it possible to appreciate the geometric/polytopal properties of the amplituhedron without delving into the physics that gave rise to it? All the descriptions I've so far encountered assume familiarity with quantum field theory, but perhaps there exist more purely geometric explications...? If so, I would appreciate a reference—Thanks!
(Image from Quanta Magazine)

(Added). Here is a snapshot from p.64 of the paper which jc cited, "Scattering Amplitudes and the Positive Grassmannian." (I am guessing that what they called the "positroid" in this paper is either closely related or perhaps identical to what was later named the "amplitudedron"?)

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Wow, interesting timing. We were just discussing this critter at the nForum (the discussion board for the nLab): nforum.mathforge.org/discussion/5278/amplituhedron –  Todd Trimble Sep 22 '13 at 0:59
I don't believe any preprint on the amplituhedron proper has been released yet, so as a partial response to your question, even if such a physics-free way of "appreciating" this exists, certainly no such reference exists at the current time. But you might browse this earlier paper arxiv.org/abs/1212.5605 and these slides staff.science.uu.nl/~tonge105/igst13/Trnka.pdf . –  j.c. Sep 22 '13 at 1:13
Wow. That Wikipedia page is a example of what opaque means! –  Mariano Suárez-Alvarez Sep 22 '13 at 2:09
The positroid story is as follows. Given a $k$-plane $V$, let $M(V) = \{ S : p_S(V) \neq 0\}$ where $p_S$ is the Pl\"ucker coordinate. This $M(V)$ is always a matroid, by definition "realizable". Comparatively few matroids are realizable by $V = rowspan($a real matrix with all $p_S \geq 0)$; those are the "positroids". For each positroid $P$, you can consider the stratum $\{V : M(V) = P\}$, where $V$ is either in the complex Grassmannian, real, or nonnegative real. Postnikov proves that each nonnegative real positroid stratum is a ball, and parametrizes it. –  Allen Knutson Oct 1 '13 at 23:40
...whereas Thomas Lam, David Speyer, and I studied the closures of the corresponding strata in the complex Grassmannian. They're basically every bit as nice as Schubert varieties, and I'm hoping the amplituhedron story can be cast in complex rather than nonnegative real terms. –  Allen Knutson Oct 1 '13 at 23:42

2 Answers 2

up vote 43 down vote accepted

Nima Arkani-Hamed had a series of talks at JHU roughly 6 months ago which I attended related to this topic. He discussed it at Stony Brook a little bit over a week ago (pointed out by Emilio Pisanty in the comments above in which he used the term "amplituhedron", but my understanding of this comes mostly from his earlier talks.

Update: I managed to track Nima down today and get him to explain the details. Surprisingly almost everything was correct, but the loop level description has some changes. I've also made some aesthetic changes, attempting to follow the notation in Trnka's slides as closely as possible while being readable by mathematicians. Also note that Nima claimed that their paper on this would appear "very soon".

Update 2: Their paper has appeared on the arXiv: http://arxiv.org/abs/1312.2007. At first glance everything seems consistent with what I've written below. I think it should be readable to mathematicians simply by skipping the few sections which require knowledge of physics. They do find some combinatorial results therein which may be of interest, but as this answer is already rather large I'll just direct those interested to the paper directly.

Everything here is over $\mathbb R$. We know that the quotient of the subset $\tilde{M}(k,n) \subset M(k,n)$ of matrices with full rank by the $GL(k)$ left action gives the Grassmannian $G(k,n)$, in which each matrix is mapped to its row space. Define $M_+ (k,n)$ to be the subset of $M(k,n)$ in which all $k \times k$ minors are positive (ordering the rows in the process). The image of $M_+(k,n)$ under this quotient is the positive Grassmannian $G_+(k,n)$.

Fix a matrix $Z \in M_+(k+m,n)$ as input data, which comes from physics, but won't really affect the combinatorial structure* at all.

Then there is a map $Y_{n,k,m}: G_+(k,n) \rightarrow G(k,k+m)$ given by $([C],Z) \mapsto [CZ^T]$, where brackets denote equivalence classes under the $GL(k)$ action (the requirement that $CZ^T$ has full rank is automatically satisfied if both are in their respective positive pieces). Its image $\mathcal P_{n,k,m}$ is called the tree-level amplituhedron. The combinatorial structure does depend on $n,k,$ and $m$. This case is apparently fairly well understood thanks to their work with Alexander Postnikov (according to Nima).

(One of the interesting aspects of this positive-real story is that while the map $Y_{n,k,m}$ is only a rational map of algebraic varieties, its base locus doesn't intersect $G_+(k,n)$.)

*I've seen it claimed that $Z$ can be taken to live in $G_+(k+m,n)$. I must admit this doesn't make sense to me because the map described above is not invariant under the right action by $GL(k+m)$. If I'm missing something obvious though than feel free to correct me. At the very least, I'm pretty sure that the above description works, but it might be a bit more redundant than necessary.

As I alluded to above, $\mathcal P_{n,k,m}$ is not the full amplituhedron. Rather, it's just the tree-level case. The full amplituhedron has another nonnegative integer parameter $l$ which determines the loop order. This gives additional coordinates to points in the amplituhedron. The subregion of $M(k+2l, k+m)$ of interest which the $C$ vary through satisfies somewhat more stringent positivity constraints than those of the ordinary positive Grassmannian that we had above in the case $l=0$. I will call this $l$-positivity, though this is my own terminology. $C' = \left( \begin{matrix} C \\ C^{(1)} \\ \vdots \\ C^{(l)} \end{matrix} \right)$ with $C$ as $k \times n$ and each $C^{(i)}$ as $2 \times n$ is $l$-positive iff for any $I = \{i_1 , \ldots, i_r\} \subseteq \{1, \ldots, l\}$ (with $i_1 < i_2 < \cdots < i_r$), the submatrix $\left( \begin{matrix} C \\ C^{(i_1)}\\ \vdots \\ C^{(i_r)} \end{matrix} \right) \in M_+(k+2r,n)$ (including the case $I=\phi$), and each $C^{(i)}$ is only well-defined up to addition of elements of $C$.

For convenience of notation let $\mathcal A_{n,m} = Y_{n,2,m}$. The $l$-loop amplituhedron is then the image of $\{[C'] | C'\text{ is }l\text{-positive}\} \times \{Z\}$ in $G(k,k+m) \times (G(2,k+m))^l$ by just applying $Y_{n,k,m}$ to $([C],Z)$ and $\mathcal A_{n,m}$ to each $([C^{(i)}],Z)$ (all with the same $Z$). This is called $\mathcal P_{n,k,l,m}$ (Trnka drops the $m$, presumably since $m=4$ for physics). The space that this is embedded in has no significance either combinatorially or physically. We could take it to be in the $l$-fold product of the $G(2,m)$ bundle over $G(k,k+m)$ such that the fiber at each point is those $2$-planes orthogonal to that $k$-plane.

It is important to realize that the amplituhedron itself is not so much the object of interest: rather, that is a meromorphic volume form defined on it (and on the Grassmannian, or bundle over Grassmannian, in which the amplituhedron is Zariski-dense). The principal job of the amplituhedron is to help nail down this form: the form is required to be well-defined on the interior of the amplituhedron. It seems very hard to make this statement mean anything without talking about positive real parts of varieties.

The parameters are important for physics, so I'll list them, but of course if you're not interested in the physics you can set them to be whatever you like. $k$ is the order in perturbation theory. $l$, the loop order. $m=4$ is the case for physics, but in principle $m$ can be any even positive integer ($m=2$ makes the loop part trivial, so $m=4$ is in some sense the first interesting case). $n$ is the number of momenta in the scattering process. $Z$ is a positive matrix that represents all of the momenta, but at least for the purpose of combinatorics the structure doesn't really depend on the choice of $Z$. There are of course cases in which the construction does not make sense; these are irrelevant for physics (e.g. $n-k < m$ is unphysical). Also, while I'm talking about physics, to get physical predictions out of the amplituhedron, there is a particular volume form which is simply integrated over the region. This volume gives the amplitude for the process.

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Thank you for investing the time in striving to explain this---It is a considerable service for all of us interested. –  Joseph O'Rourke Sep 29 '13 at 0:54
I found the reference to the map $Y_{n,k,m}$, as taking two inputs, very confusing. $Z$ is fixed and one should take $Y_{n,k,m}$ as a function from $G_+(k,n)$. Also, the amplituhedron as a space is not what's interesting, but as a space plus meromorphic volume form. The positive-real structure is there to help nail down this form; it is supposed to have no poles on the interior of the amplituhedron. –  Allen Knutson Mar 19 '14 at 19:03
Also, there's no orthogonality involved in defining the $G(2,m)$ bundles; rather, they live in the quotient space. (In ordinary algebraic geometry over a field this would be a distinction without a difference. But here it's very difficult to understand positivity when involving an inner product, and we don't have to.) –  Allen Knutson Mar 19 '14 at 19:15
@AllenKnutson You're welcome to edit as you see fit or post your own answer (for reference, I agree with everything you said in your comments). As I said above, the answer is based on what Nima explained to me. I tried to phrase it in a way that would be readable by mathematicians but keep it as close to the way he presented it at that time as possible. In my mind, the answer is already basically obsolete now that they have a couple of papers on the subject available which are quite readable. –  Logan Maingi Mar 19 '14 at 20:00
While quite readable, they do make the two points I'm complaining about above rather difficult! I will edit. –  Allen Knutson Mar 20 '14 at 1:19

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.

Folowwing 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 positive 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 (the open cells are known to be homeomorphic to open balls, their closures are conjectured to be homehomorphic to closed balls) that can be defined in two equivalent ways as follows:

Given an ordering of the $n$ columns the cell of the Shubert stratification indexed by a subset $S$ (of size $m$) are matrices whose lexicographically first non-zero minor (w.r.t. the ordering) is $S$.The matroidal stratification is the common refinement of these Shubert decomposition with respect to all permutations.

The cell structure studied by Postnikov can be seen

a) as the matroidal stratification reduced to this part of the grassmanian;

b) Start with the common refinement of Shubert cells with respect to a cyclic family of permutations (for the whole Grassmanian this lies between the Shubert stratification and the matroid one). Then restrict it to the positive Grassmanian.

This cell structure has beautiful combinatorial description in terms of certain planar graphs and statistics of permutations.

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 amplitutahedron 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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What does it mean to project with respect to totally positive matrices? –  Vít Tuček Jan 28 at 19:47
I meant that the matrix representing the projection is totally positive (all minors are positive). It is enough that all maximal minors are positive. –  Gil Kalai Feb 19 at 21:41

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