正性几何与N=4 SYM散射振幅的解析结构

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本文深入探讨了正性几何在平面N = 4超级杨米尔斯（SYM）散射振幅中的核心作用。作者首先概述了正格拉斯曼几何的基础，这部分内容涵盖了普吕克坐标和简化格拉斯曼几何的表述，这些是理解树级和循环级别的散射过程的关键工具。普吕克坐标是一种特殊坐标系统，它在格拉斯曼子空间中提供了一种直观的几何表示，有助于解析复杂的多粒子相互作用。文章的核心内容分为四个方面： 1. **正成分与BCFW递归关系**：作者提出了“正分量”这一概念，这是一种基础构建块，用于构造正矩阵。通过仅仅依赖于正性原则，作者能够直接推导出树级和1环BCFW（Britto-Cachazo-Feng-Witten）递归关系，这是一种计算高阶散射振幅的有效方法，其在量子场论中扮演着重要角色。 2. **Grassmannian几何与符号表示**：作者利用格拉斯曼几何的性质，特别是普吕克坐标，来解析N2MHV同源性的符号，这些符号揭示了Yangian不变量之间的关联。更重要的是，它们揭示了大多数符号其实隐藏着简单的NMHV（next-to-maximally helicity-violating）多项式身份，这是理论物理中的重要发现。 3. **堆积正关系与矩阵参数化**：文章还介绍了堆积正关系，这是一种强大的工具，用于以d log形式参数化矩阵，使得对给定几何配置的正矩阵的表示更为简洁。这种方法与简化格拉斯曼几何的表示结合，提供了一种独立于BCFW桥组合法的求解路径，对于理解多粒子系统的行为具有重要意义。 4. **优雅的BCFW递归关系**：最后，作者提出了一个优雅且精细的BCFW递归关系版本，特别适用于树级幅。这个版本展示了BCFW轮廓与简化格拉斯曼几何之间深刻的三角形结构，这种几何表示使得对散射振幅的分析更加直观和高效。这篇论文将正性几何与动量旋扭草丛正几何相结合，为平面N = 4 SYM散射振幅的研究提供了新的洞察和计算策略，对于深化我们对量子场论基本原理的理解具有重要的学术价值。

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JHEP12(2017)147

relations. The Jacobian J is ﬁxed such that the integral remains invariant under the

uniform GL(1) rescaling, since it helps compensate the rescaling weight. It is easy to check

that this new integral reduces to the familiar one if columns 1, . . . , k are ﬁxed to be a

unit matrix, without loss of generality. In fact, we have done nothing new but renaming

αa

=(−)

k−α

∆

1 ... bα ... k a

for a=k+1, . . . , n.

In the Pl¨uckerian integral formulation, its gauge ﬁxing is simpliﬁed to choosing a

Pl¨ucker coordinate to be any non-vanishing value (usually, it is unity), rather than ﬁxing

a k × k sub-matrix of C. It is more convenient to switch the gauge in order to suitably

characterize each cell now. We will use this technique extensively in section 4, which proves

to be eﬃcient for determining the signs of homological identities.

One may ask how we express the super delta function in (2.2), in terms of Pl¨ucker

coordinates as

4k|4k

αa

(∆)Z

), (2.23)

the general answer is unknown, but for the canonical gauge above, we can simply follow

the substitution C

αa

=(−)

k−α

∆

1 ... bα ... k a

which is similarly a trivial renaming.

2.3 Reduced Grassmannian geometry

While the Grassmannian geometric conﬁgurations can be explicitly parameterized by

Pl¨ucker coordinates, for plenty of cells it is practical to characterize them in a more com-

pact form, with the price of suppressing their parameterizations. These notations are the

(reduced) Grassmannian geometry representatives, as we have met in the introduction.

Here, a more systematic exposition will be presented.

First, we use “empty slots”, such as [i], to denote removed columns. It is worth

emphasizing that all these symbols only make sense when k and n are speciﬁed. For

example, when k = 1, n = 6, [6] means the 6th entry is absent in the Yangian invariant,

so the resulting quantity is a 5-bracket [12345] (to avoid confusion, we will stress it when

a 5-bracket shows up). They are multiplicative, for instance, [6][7] = [67]. As we have

mentioned, such a product is simply a superposition of all constraints.

Then, for k ≥ 2 we use (ij) to denote that columns i, j are proportional, which is

also multiplicative. For the example (23)(67) in (2.7), it is now nontrivial to read oﬀ the

Yangian invariant, but of course we refrain from doing so. Still we must at least ensure

that it has the correct dimension, which in this case is k(n−k)−2=4k = 8. When k ≥3, the

dimension counting is not transparent in general, and that’s one of the reasons to introduce

the reduced Grassmannian geometry.

The ﬁrst example is the cell (23)(456) of N

MHV n = 8 amplitude in (1.14), where

columns 2, 3 are proportional and columns 4, 5, 6 are “minimally” linearly dependent. We

see that (23) leads to (123) = (234) = 0, but the latter does not necessarily lead to (23).

For both clarity and conciseness, we will use distinct notations for these two diﬀerent

constraints. This is called the reduced Grassmannian geometry, where its term “reduced”

means that we will only extract the essential geometric information and nothing more than

that. For example, when k =4, (34) will lead to (234)= (345)=(1234) =(2345)=(3456) =0,

still we should write (34) only. Note that (34) also denotes the vanishing of an arbitrary

– 13 –

JHEP12(2017)147

Figure 1. Geometric conﬁguration of k = 4, n = 8 labelled by permutation {3,7,6,10,9,8,13,12}

in [11].

2×2 minor of columns 3, 4, and (234) denotes the vanishing of an arbitrary 3×3 minor of

columns 2, 3, 4.

In general, for j ≤ k, (i

. . . i

) denotes the vanishing of an arbitrary j ×j minor of

columns i

, . . . , i

and its implicit vanishing constraints involving more columns are sup-

pressed. All minors here are cyclicly consecutive, which is suﬃcient for planar amplitudes.

It is also convenient to use the abbreviated notation (i

. . . i

j+1

. . . i

j+l−1

)

to denote the

cyclicly adjacent product of consecutive minors

. . . i

)(i

. . . i

j+1

) . . . (i

. . . i

j+l−1

), (2.24)

note the subscript j makes a diﬀerence, for instance, (345)

= (34)(45) is clearly diﬀerent

from (345) when k = 3. Without an explicit subscript, the size of a vanishing minor is its

number of columns by default.

For the example (23)(456), since (23) and (456) have no overlapped column, we can

write it trivially as a product. However, for a “nested” object such as (1.6) in the intro-

duction, one needs a more artistic way to represent it concisely. Now let’s use a second

example for demonstration: in [11], a 9-dimensional geometric conﬁguration of k =4, n =8

labelled by permutation {3, 7, 6, 10, 9, 8, 13, 12} is characterized by the representation (its

pictorial sketch is shown in ﬁgure 1)







(1)(2)(3)(4)(5)(6)(7)(8)

(123)(34)(45)(56)(67)(678)(81)

(3456)(56781)(81234)







, (2.25)

which speciﬁes every layer of linear dependencies, but most of them are trivial or can be

derived from the upstair ones. To encode the geometric information more compactly, we

will instead use

(3 4 5 6) (5

8 1) (8

3 4) (2.26)

which contains nested constraints (123) and (678). Similar to (1.6), each connection be-

tween two column labels stands for one degree of freedom, for instance, column 2 is spanned

by columns 1, 3 as above. This column is diagrammatically divalent, and its two degrees

of freedom can be regarded as additional on top of columns 1, 3, 4, 5, 6, 8.

Now it is trivial to count its dimension from the perspective that separates vanishing

maximal minors and degenerate linear dependencies. For the reduced Grassmannian made

– 14 –

JHEP12(2017)147

of columns 1, 3, 4, 5, 6, 8, there are three constraints, namely (3456)(5681)(8134). In addi-

tion, the two columns attached on it altogether contribute four valencies. In total, we have

4(6 − 4) − 3 + 4 = 9 (2.27)

degrees of freedom, which is harder to be inferred from the un-reduced geometry, let alone

the permutation. This reduced geometry also has a clear pictorial meaning: in ﬁgure 1, we

ﬁrst construct the 3-dimensional (projectively) skeleton made of edges 34, 56, 81 and faces

3456, 5681, 8134, then add points 2 and 7 to the conﬁguration. Each of the latter has two

degrees of freedom, one of which is the translation along a line and the other is suppressed

by the projective sketching.

In general, the dimension of a reduced Grassmannian geometry representative is

given by

d = k(n

−k) − (# vanishing maximal minors) + (# degenerate valencies), (2.28)

where n

is the width of the reduced Grassmannian. This formula also works for former

example (23)(456) since (23) can be regarded as a degenerate valency, then its dimension

is 3(7−3)−1+1 =12. Note that we always list all vanishing constraints in the cyclic order,

and all of them are cyclicly consecutive in the reduced sense. It is also equivalent to choose

any columns of the degenerate constraint to be put upstairs, although for convenience, we

have chosen columns 2, 7 above as in that way we have a minimally reduced Grassmannian

of n

=6.

2.4 Tree BCFW recursion relation in the positive matrix form

Now we will show how the (reduced) Grassmannian geometry representatives of tree am-

plitudes are generated, by the matrix version of BCFW recursion relation [2]. The idea

ﬁrst appeared in [12], and here we will use a more geometric form that manifests positivity,

as presented in the following.

In ﬁgure 2, the ﬁrst row represents a 5-bracket [1 j j+1 n−1 n] and below it, we merge

two positive matrices Y

and Y

in a way such that the entire matrix is positive (we abuse

Y ’s for the corresponding matrices of Yangian invariants). Positivity also demands us to

add sign factors (−)

to c

and c

j+1

, as well as (−)

k−1

to c

n−1

and c

, where k =k

and n = n

−2. When n

= 3, 4, Y

trivially disappears as k

= 0, and so does Y

for

=4. All the rest blank areas are ﬁlled with zeros. Note that although the c

’s are taken

to be positive, it is not the essence of positivity, as will be explained soon.

The deformed sub-columns with subscript ‘D’ are given by

j, D

= L

j+1

, R

j+1, D

= R

j+1

, R

n−1, D

= R

n−1

, (2.29)

where L

, R

j+1

and R

n−1

are the un-deformed columns of Y

and Y

. After all c

’s ﬁnd

their solutions on the support of the bosonic delta function in (2.2), we ﬁnd, for instance:

j, D

+ L

j+1

= L

+ L

j+1

, (2.30)

– 15 –

JHEP12(2017)147

j+1

n-1

L L

j,D

j+1

j+1,D

R R

n-1,D

(-)

k-1

Figure 2. Matrix version of tree BCFW recursion relation.

where

j+1

= ( j j +1) ∩ (n−1 n 1) is the deformed super momentum twistor. It is similar

for R

j+1, D

and R

n−1, D

, so we reach the familiar tree BCFW product of Yangian invariants

[1 j j+1 n−1 n] Y

(1, . . . , j, I) Y

(I, j+1, . . . , n−1, bn) (2.31)

with Z

j+1

=( j j+1) ∩ (n−1 n 1) and

=(n−1 n) ∩ (1 j j+1).

We will derive the matrix in ﬁgure 2 from positivity in section 3, which is exactly

equivalent to the original BCFW recursion relation in momentum twistor space in [2].

It is worth emphasizing that in this article, positivity is only related to the auxiliary

Grassmannian and it has nothing to do with momentum twistors, which is diﬀerent from

the amplituhedron formulation in [19]. It is appealing to set the ordered minors of all n

momentum twistors to be positive, but it contradicts with the principle of this article. To

see why, one can consider a simplest example: NMHV n=5 amplitude in the matrix form.

Figure 2 gives its positive matrix prior to solving the kinematical constraint as a single row

) (2.32)

without signs. If we set up positive kinematical data Z

, Z

so that any hijkli

for i<j <k < l is positive, the bosonic delta function in (2.2) imposes

= h2345i, c

= h3451i, c

= h4512i, c

= h5123i, c

= h1234i, (2.33)

now c

and c

have to be negative. Therefore, we will adopt the perspective that positivity

should be only used for determining the auxiliary Grassmannian. Once we know the explicit

parameterization, positivity is then forgotten in the next step of solving for the positive

parameters. This actually coincides with the idea of manipulating geometric conﬁgurations

instead of momentum twistors as previously mentioned, if we stick to positivity as the

primary guiding principle.

From the mathematical perspective, positivity does not mean literally being positive,

but the essence of subtraction-free expressions. For example, consider a subtraction-free

– 16 –

JHEP12(2017)147

polynomial (x+y +z), there is no way to render it vanish without setting x = y = z = 0,

while for (x+y−z) this is not true. The same property also holds for −(x+y+z), which

is literally negative but subtraction-free as well. Furthermore, neither the Grassmannian

nor momentum twistors should be genuinely taken to be positive, because the former has

contour integrations to be done, while the latter originate from complex massless spinors.

It is intuitive and convenient to pretend that the positive matrix uses genuinely positive

variables, of which the zero limits represent some kinds of singularities. However, we

don’t have positive parameterization or variables for momentum twistors, which are purely

external kinematical data.

The deformed sub-columns in ﬁgure 2 will aﬀect the corresponding Pl¨ucker sub-

coordinates, as those involved are deformed accordingly:

∆

L, D

... j ...

= ∆

... j ...

j+1

∆

... j+1 ...

(2.34)

for the left Pl¨ucker sub-coordinates, and

∆

R, D

... j+1 ...

= ∆

... j+1 ...

j+1

∆

... j ...

, ∆

R, D

... n−1 ...

= ∆

... n−1 ...

n−1

∆

... n ...

(2.35)

for the right counterparts. Pl¨ucker coordinates of the entire matrix are then given by the

schematic form

∆ =

c × ∆

∆

, (2.36)

where ∆

and ∆

may or may not be deformed, depending on whether they contain the

deformed sub-columns. For any ordered minor, the r.h.s. polynomial above is positive (or

subtraction-free precisely), which exactly justiﬁes the sign factors in ﬁgure 2.

Finally, let’s demonstrate ﬁgure 2 with a detailed example. For the N

MHV n = 10

amplitude, there is a set of BCFW cells that belong to the sub-class generated by its

characteristic 5-bracket [1789 10], for which k

=2. They are given by the matrix

C =







0 0 0 0 0 c





0 0







, (2.37)

where L

’s are 2-vectors of the sub-amplitude (Y

)

. One of its sub-cells is (234)

(678)

which will result in the N

MHV n=10 cell

(

2 3

\ /

4 5 6) (6 7 8

9 )

, (2.38)

this can be seen from its matrix form

C =







0 0 0 0 0 c





0 0







, (2.39)

where the positive α

’s help manifest linear dependencies (234)

(678)

of the sub-cell.

– 17 –

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