![](https://csdnimg.cn/release/download_crawler_static/12266013/bgf.jpg)
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
C
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
δ
4k|4k
(C
αa
(∆)Z
a
), (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
3
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
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