The positive orthogonal Grassmannian and loop amplitudes of ABJM

Grassmannian Gr(k, n) can be regarded as the space of k-dimensional subspaces inside n-dimensional complex space, $\mathbb {C}^n$. Take a basis e₁, ..., e_n of $\mathbb {C}^n$, then each point can be represented by a (k × n) matrix up to a GL(k) transformation. For a canonical ordering of e₁, ..., e_n, there is a natural notion of positivity, namely the determinants of all ordered minors Δ_I are non-negative. This notion of positivity is the same as the one defined by Lusztig on partial flag variety (G/P)_⩾0 [20], as the Grassmannian can be identified with the partial flag manifold $GL(\mathbb {C})/P$, where P is a parabolic subgroup of $GL(\mathbb {C})$.

To define orthogonal Grassmannian OG(k, n), one needs to first define a symmetric bilinear form on $\mathbb {C}^n$:

$$\begin{equation} H[x,y]=\eta _{ij} x_i y_j, \end{equation} \tag{ 2.1 }$$

where x_i, y_i are vectors in $\mathbb {C}^n$. In this paper, the bilinear η_ij will be taken to be diagonal: i.e. η_ij = δ_ij. Notice that in the complex case, different signatures of η are isomorphic. The orthogonal Grassmannian is then defined as the space of k-planes inside $\mathbb {C}^{n}$, such that H[x, y] = 0 on this subspace. Again each point of orthogonal Grassmannian can be represented by a k × n matrix, C_ai with a = 1, ..., k, and i = 1, ..., n. The constraints H[x, y] = 0 can be written as:

$$\begin{equation} \eta ^{ij}C_{ai} C_{bj} =0 . \end{equation} \tag{ 2.2 }$$

Note that there are k × (k + 1)/2 independent constraints.

In this paper, we restrict ourselves to the orthogonal Grassmannian with n = 2k, namely OG_k ≡ OG(k, 2k). The dimension of OG_k is given by

$$\begin{equation} k\times 2k-k^2-\frac{k(k+1)}{2}=\frac{k(k-1)}{2} \end{equation} \tag{ 2.3 }$$

where we have removed k² degrees of freedom representing the GL(k) redundancy of linearly recombining the k n-dimensional vectors that span the k-dimensional planes. Let us order the column of C_ai by (1, 2, ..., 2k) and denote $\mathsf { I}$ an ordered k subset. For example, choosing $\mathsf {I}=(1, 2, 3)$, we can use Gauss elimination algorithm to put the C matrix into an $\mathsf { I}$-echelon form, i.e.

$$\begin{eqnarray} C=\left(\begin{array}{@{}cccccc@{}}1& 0&0&*&*&*\\ 0 & 1 &0&*&*&* \\ 0&0&1&*&*&* \end{array}\right)=(\mathsf { I}_{3\times 3}, c_{3\times 3}). \end{eqnarray} \tag{ 2.4 }$$

In the representation, the orthogonal Grassmannian is given by an (9 − 6) = three-dimensional parametrization of c_{3 × 3} such that, with η^ij = δ^ij, cc^T + 1 = 0. As a consequence, the minors constructed from the set of columns $C_{\mathsf { I}}$, denoted by $M_{\mathsf { I}}$ satisfying ${M_{\mathsf { I}}/M_{\bar{\mathsf { I}}}}=\pm (i)^{k}$, where $\bar{\mathsf {I}}$ is the ordered complement of $\mathsf {I}$. For example for k = 3, if $\mathsf { I}=(1, 3, 4)$, then $\bar{\mathsf {I}}=(2, 5, 6)$.

As discussed in [11], the building blocks for scattering amplitudes of ABJM theory can be associated with the positive part of the orthogonal Grassmannian OG_{k +}. To define positive part of orthogonal Grassmannian, we need to first define a real subspace of OG_k, and in this case, the signature of η_ij is important and different signature defines different subspace. As was discussed in [11], to consistently define OG_{k +} for arbitrary k, it is advantageous to use alternative signature η_ij = ( +, −, +, ⋅⋅⋅, −). Note that in this signature, the orthogonal condition becomes 1 − cc^T = 0, and one instead has ${M_{\mathsf { I}}/M_{\bar{\mathsf { I}}}}=\pm 1$. For OG_{k +}, one always has ${M_{\mathsf { I}}/M_{\bar{\mathsf { I}}}}=1$ for both $\mathsf { I}$ and $\bar{\mathsf {I}}$ being ordered.

Using the above notion of positivity, we can define a cell decomposition of non-negative part of OG_{k +}. We consider all possible linear dependency among consecutive chains of columns. The classification of all such linear dependency for the positive Grassmannian Gr(k, n)₊ is referred to as positroid stratification [21, 22]. Each cell is then an element in the stratification. In terms of ordered minors it is defined as

$$\begin{equation} S_{\mathcal{M}}=\lbrace A|M_\sigma (A)>0\ {\rm for}\ \sigma \in \mathcal{M}, {\rm and}\ M_\sigma (A)=0\ {\rm for}\ \sigma\ {\rm not}\ \in \mathcal{M}\rbrace. \end{equation} \tag{ 2.5 }$$

Let us begin with the first non-trivial example, OG_{2 +}.⁶ Taking $\mathsf { I}=(1,2)$, the C-matrix is given by:

$$\begin{equation} C=\left(\begin{array}{@{}cccc@{}}1& 0 &-\cot \theta &-\csc \theta \\ 0 & 1 &\csc \theta &\cot \theta \end{array}\right) . \end{equation} \tag{ 2.6 }$$

On the other hand for $\mathsf { I}=(1,3)$ we instead have:

$$\begin{equation} C=\left(\begin{array}{@{}cccc@{}}1& \cos \theta &0&-\sin \theta \\ 0 & \sin \theta &1&\cos \theta \end{array}\right) . \end{equation} \tag{ 2.7 }$$

As one can easily verify, the ordered minors of the above C-matrix are always positive for $0<\theta <{\pi \over 2}$. In this chapter, we will constrain ourself to this region, and use the abbreviation s_i ≡ sin θ_i, c_i ≡ cos θ_i.

Cells of positive Grassmannian Gr(k, n)₊ can be represented nicely by bipartite network [21], and one can find the so-called cluster coordinates using the network. The combinatorics of the bipartite network is very rich and is an extremely useful tool to study the geometry of the positive Grassmannian. One important ingredient is the equivalence moves for the network. Using those equivalence moves, Postnikov has established a one-to-one correspondence between the positive cells, and the bipartite networks that are distinct up to equivalence moves.

In the work of [2], it was realized that the bipartite networks can be understood as on-shell diagrams of the scattering amplitudes in $\mathcal {N}=4$ SYM, and the aforementioned equivalence moves are given a physical interpretation as either the equivalence of kinematic constraints, or the equivalence of distinct ways to recursively construct the four-point amplitude. Therefore, we have the following triple:

$$\begin{equation*} \hbox{Cells of ${\rm Gr}(k,n)_{+}$}\rightarrow \hbox{planar bi-partite network} = \hbox{on-shell diagram of $\mathcal {N}=4$ SYM.}\nonumber \end{equation*}$$

It was soon realized that similar story can be established for the positive cell of orthogonal Grassmannian:

$$\begin{equation*} \hbox{Cells of ${\rm OG}_{k+}$}\rightarrow \hbox{planar medial graph} = \hbox{on-shell diagram of ABJM theory.} \nonumber \end{equation*}$$

We have reviewed the definition of cells of OG_{k +} in last subsection, here we are going to explain the rules for constructing medial graphs and their equivalence moves.

The basic element for constructing medial graphs is a four-point vertex, see figure 1(A). One can build complicated medial graphs by simply gluing the four-point vertices together as indicated in figure 1(B). We limit ourselves to planar medial graphs, i.e. graphs defined on the disc, and the number of vertices is always even (2k). We will associate one degree of freedom, θ, to each vertex. Such graphs also appear in the study of electric networks [23, 24], which is where the term 'medial graph' came from, and the variable at each vertex is associated with the conductance of the resistors in the electric network⁷.

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As discussed in [2, 11], distinct medial graphs represent distinct cells of OG_{k +}, and as the bipartite network, distinct diagrams are defined modulo equivalence moves. The equivalence moves associated with the medial graphs are shown in figure 2: move A and B are called bubble reduction, and move C is called triangle move. Note that for the bubble reduction, the dimension of the diagram is reduced by one. An important notion is the reduced graph, which is defined as:

• A medial graph is called reduced if no bubble can be formed by doing any sequence of triangle moves.
• Two reduced medial graphs are equivalent and representing the same cell if they can be related by triangle moves.

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Just to distinguish, we will name the bubble (B) in above picture as removable bubble for the reason becoming clear in section 2.4.3. Here is an example of triangle move and bubble reduction:

where we have applied a triangle move and three bubble reductions in that order. It was proposed in [11] that the connection between cells of OG_{k +} and the medial graph is the following:

Cells of OG_{k +} is in one-to-one correspondence with distinct reduced medial graphs.

In the following we will prove the above statement by embedding the medial graphs of OG_{k +} into bipartite graphs, Gr(k, 2k)₊. Since from the work of [21, 22], we know that the stratification of Gr(k, n)₊ is in one-to-one correspondence with the bipartite graphs, then the above statement follows. This will also allow us to derive canonical positive coordinates for OG_{k +} which is equivalent to the coordinates obtained from the iterative gluing procedure discussed in [11].

A cell of Gr(k, 2k)₊ can be represented by a bipartite network. Since OG_k can be considered as a reduction of Gr(k, 2k), it is natural to try to transform the medial graph defined in last subsection into a bipartite network. The crucial thing is to solve the orthogonal constraints on the coordinates once a bipartite network is found.

2.3.1. Review of bipartite network

The various properties of OG_k can be derived by considering a strata preserving embedding of OG_k within Gr(k, 2k). The positroid stratification of Gr(k, n) can be one-to-one identified with bipartite graphs. These are graphs that consist of trivalent vertices, with each vertex associated with two possible colors, black and white, and any each edge is connected to two vertices of opposite color. As an example one has:

For a given bipartite graph, one can associate each face with a variable f_i satisfying ∏_if_i = 1. Due to this constraint, the dimension of the graph is (n_f − 1), where n_f is the number of faces.

One can extract an explicit representation of the C-matrix by providing a perfect orientation to the graph. A perfect orientation is given by assigning an edge with an arrow, and for each white vertex, one must have one incoming, and two outgoings, whereas for a black vertex one has two incomings and one outgoing:

For example, for the four-dimensional top cell in Gr(2, 4)₊, one has:

A given oriented decoration correspond to a particular gauge choice for the C-matrix, with the incoming arrows of the external legs (the source set) indicating the columns that are set to unity. The boundary measurements are then given by the products of face variables on the right-hand side of the path that connects the source to the sink (the outgoing external legs). If there are closed paths, then one obtains a geometric series weighted by ( − 1)^w, where w is the number of times one encircles the closed circle. In the example above, we have:

$$\begin{equation} M_{12}=1/f_1, \; M_{14}=f_4,\quad M_{32}=f_2,\quad M_{34}=1/f_3. \end{equation} \tag{ 2.9 }$$

Using the boundary measurement, we can define the C-matrix. Denote the source set of the network as I = (i₁, i₂, ..., i_k), then C is defined as

• The sub matrix C_I is an identity matrix.
• The remaining element $C_{rj}=(-1)^sM_{i_{r},j}$, where s is the number of elements of I strictly between i_r and j.

So for the network in equation (2.8), the source set is (1, 3), and the C-matrix is

$$\begin{equation} C=\left(\begin{array}{@{}cccc@{}} 1& 1/f_1 &0&-f_4\\ 0 & f_2 &1&1/f_3 \end{array}\right) . \end{equation} \tag{ 2.10 }$$

Using the boundary measurement and the definition of C-matrix, one find a parameterization of cells of Gr(k, n)₊.

The equivalence moves for the bipartite network are shown in figure 3, where we have also indicated how the face variables transform under the moves (it is actually the cluster transformation on the A-variables). Note that the transformation rules are 'subtraction-free', i.e. there are no minus signs in the transformation rules. An important consequence is that once f_i are restricted to be positive, this property will be preserved. Again a reduced graph is defined as:

• A bipartite graph is called reduced network if no bubble is formed after doing any sequence of square moves.
• Two reduced bipartite graphs are equivalent and representing the same cell if they can be related by square moves.

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It is proven in [21] that there is a one-to-one correspondence between the positive cells and bipartite network up to equivalence. Using the parameterization from reduced bipartite network, one can define a simple volume form on the positive cells:

$$\begin{equation} {{\rm d} f_1\over f_1}\wedge {{\rm d} f_2\over f_2} \wedge \cdots \wedge {{\rm d} f_n\over f_n}={\rm d}\ln f_1\wedge {\rm d}\ln f_2 \wedge \cdots \wedge {\rm d} \ln f_n, \end{equation} \tag{ 2.11 }$$

here the f_i are the face variables in the corresponding bipartite network. The remarkable thing about this volume form is that it is invariant under the cluster transformation, up to an overall factor ( − 1)ⁿ.

2.3.2. From medial graph to bipartite network

Now we are ready to consider the 'image' or our medial graph in a bipartite network. The procedure is straightforward: one simply replaces a internal four vertex with a square, see figure 4. Notice that we can choose any of two orderings of black-white vertices, since these two orderings are related by the square move and therefore would give the same point in Grassmannian. The canonical choice is found by doing the blow-up such that the corresponding network is bipartite. It will be advantageous to consider the blow up such the source legs connected to white vertices.

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The next question is how to find a k-dimensional parameterization of the face variables, such that the resulting C-matrix of Gr(k, 2k) is in fact orthogonal. Let us begin with simplest case, OG_{2 +}. Comparing equations (2.7) and (2.10), one immediately sees that orthogonal constraint can be solved by setting

$$\begin{equation} f_1=\frac{1}{c},\;f_{4}=s,\;f_2=s,\;f_3=\frac{1}{c}, \end{equation} \tag{ 2.12 }$$

or diagrammatically:

If one chooses another perfect orientation such that the source set is $\mathsf {I}=(1, 2)$, one can see that equation (2.6) is reproduced using the same face variables. As we reviewed earlier, the boundary measurement is unchanged by doing cluster transformation, and we have a different parameterization if we have a different canonical blow up for OG_{2 +}:

Note that the cluster transformation is nothing but a change of variable in OG_k:

$$\begin{equation} c \rightarrow {1 \over c} , \quad s \rightarrow { 1 \over s } . \end{equation} \tag{ 2.15 }$$

The canonical coordinates in the bipartite network for a general OG_k image are given by:

• The variable for the k new faces associated with the blow up of each four-point vertex is simply $f_v={c_v^2\over s_v^2}$.
• For the remaining faces that were present in the medial graph, identify all vertices that inclose the face and take a clockwise orientation on each face. The contribution from each vertex is 1/c if in the bipartite network, one first encounters the black vertex under this orientation, otherwise the contribution is s. In other words, the face variables are either f = s₁s₂...s_i or $f={1\over c_1 c_2\ldots c_i}$ depending on the orientation.

As an example consider the following embedding for OG₃

The face variables are given by:

$$\begin{eqnarray} \nonumber &&(f_a,f_b,f_c)=(c^2_1/s^2_1, c^2_2/s^2_2, c^2_3/s^2_3),\quad f_0=\frac{1}{c_1c_2c_3}\\ &&f_1=\frac{1}{c_1},\quad f_2=s_1s_2,\quad f_3=\frac{1}{c_2},\quad f_4=s_2s_3,\quad f_5=\frac{1}{c_3},\quad f_6=s_1s_3. \end{eqnarray} \tag{ 2.17 }$$

The boundary measurement is given by:

$$\begin{eqnarray} \nonumber &&M_{16}=f_6/(1+f_0^{-1}),\quad M_{12}=f_1^{-1}+\frac{1}{f_1f_a(1+f_0)},\quad M_{14}=f_4f_5f_6f_c/(1+f_0^{-1})\\ \nonumber &&M_{32}=f_2/(1+f_0^{-1}),\quad M_{34}=f_3^{-1}+\frac{1}{f_3f_b(1+f_0)},\quad M_{36}=f_1f_2f_6f_a/(1+f_0^{-1})\\ \nonumber &&M_{54}=f_4/(1+f_0^{-1}),\quad M_{56}=f_5^{-1}+\frac{1}{f_5f_c(1+f_0)},\quad M_{52}=f_3f_4f_2f_b/(1+f_0^{-1}) . \end{eqnarray} \tag{ 2.18 }$$

One can straightforwardly verify that the C-matrix obtained from the boundary measurements are indeed orthogonal. Note that this parameterization is also subtraction free, ensuring that the image of OG_{k +} is indeed in Gr(k, 2k)₊.

It is important to show that the equivalence moves of medial graph can be reproduced using the equivalence moves of bipartite graph,

In the above, we have used a notation μ_i to indicate that we perform a square move or a bubble reduction on face i. Since all equivalence and reduction moves for the medial graph are faithfully represented by their images in the bipartite network, this confirms the statement that OG_{k +} is a stratification preserving the reduction of Gr(k, 2k)₊. However, it is obvious that not all cells in Gr(k, 2k)₊ contains the image of OG_{k +}. Using the 'left-right permutation paths' of [2], one can easily identify the cells that do contain such images. These are the ones whose left-right path connects both ways between pairs of external vertices, i.e. if there is a left-right path that begins with i and ends in j, then the path beginning with j will end in i.

2.4.1. Coordinates and boundary measurement

As discussed in [2] the parameterization of cells in the orthogonal Grassmannian can also be read off by looking at the medial graph alone. To define the boundary measurement, we need to choose an orientation of the four-point vertex. Using the blow up picture of a four-point vertex, we can see that there are two kinds of orientation: (a) alternating orientation, (b) adjacent orientation, as shown below

Here (a) correspond to the cyclic gauge, equation (2.7), and (b) correspond to equation (2.6) which we will denote as the canonical gauge. It is easy to see, for a given medial graph, it is always possible to only use orientation (a) for each vertex. On the other hand, such statement is not true for orientation (b). For example, for the following diagram, we see that we must use at least one cyclic gauge vertex (denoted by the black dot):

For simplicity, we choose the alternating gauge for each vertex. As discussed in [11], the cyclic gauge is advantageous in that the corresponding parameterization covers all co-dimension boundaries of a given cell. With this choice a given graph can have two orientations related by flipping all arrows in the graph. After choosing the orientation, the graph has an interesting feature: every simple face is oriented⁸. Two faces have the same orientation if they share a single vertex, and two faces have the opposite orientation if they share an edge.

We now associate an angle variable θ_v and functions c_v = cos θ_v and s_v = sin θ_v with each angle of vertex according to the rule shown in equation (2.20). The definition of the boundary measurement is similar to the bipartite network case we reviewed earlier: for a oriented path passing through a vertex, we assign function c_v or s_v depending on how the path turns around this vertex, see equation (2.20). Using the above rule, we can define a measurement between a sink and source node using the path between them:

$$\begin{equation} M_{ij}=\sum _{P:i\rightarrow j} (-1)^{w(p)} \prod _{v\in P} (c_v {\rm or} s_v) \end{equation} \tag{ 2.21 }$$

where w(p) is the winding number of the path. It is easy to see that if there is a loop in a path, the local contribution of this loop would be

$$\begin{eqnarray} & {1\over 1+c_1c_2c_3\ldots }\quad {\rm counterclockwise} \nonumber \\ & {1\over 1+s_1s_2s_3\ldots }\quad {\rm clockwise}. \end{eqnarray} \tag{ 2.22 }$$

Notice that the boundary measurement defined above is actually equivalent to the one defined using the embedding to Gr(k, 2k)₊.

2.4.2. The invariant volume form

Recall that for Gr(k, n)₊, there is an associated volume form that takes a canonical dlog  form in terms of face variables, and is invariant under equivalence moves. In this section we would like to consider a similar volume form for OG_{k +}. We define a canonical volume form to be that of having only logarithmic singularity on the boundaries of the positive region (θ = 0 and θ = π/2), and is preserved under equivalence moves. In [11], a canonical volume form for OG_{2 +} was given as

$$\begin{equation} d\log (s^2/c^2)=d\log (f_v), \end{equation} \tag{ 2.23 }$$

where in the second equality, we have emphasized the fact that the argument of the dlog  is simply the new face variable when OG_{2 +} is embedded in Gr(2, 4)₊. So the measure behaves as dx/x on the boundary of the positive region. For more complicated graphs, it was understood that the canonical volume form is given by:

$$\begin{equation} \mathcal {J} \times \prod _{i\in n_v}d\log f_i , \end{equation} \tag{ 2.24 }$$

where $\mathcal {J}$ is an additional Jacobian factor that appears when there are closed faces in the graph. Note that this is unlike the bipartite network, where the canonical volume form is a product of dlog  that by itself is invariant. We referred to $\mathcal {J}$ as the Jacobian factor associated with the medial graph, that is because it has its origin as the Jacobian factor that arises from the gluing of on-shell diagrams and solving a set of linear (super) momentum-conservation constraints, see [11] for more details⁹. Here we give a set of simple rules to produce the correct Jacobian for any medial graph.

The Jacobian for a given medial graph can be decomposed into several contributions from distinct closed loops. Note that when two closed-loops share an edge, they are necessary of opposite orientation, whereas if two share 2 vertices, they are of the same orientation and can combine to define a bigger loop. We list the distinct contributions as follows:

• $\mathcal {J}_1$: this contains contributions from all single closed loops J_i, products of disjoint pairs of closed loops J_iJ_j, disjoint triples of closed loops J_iJ_jJ_k and so on. We will denote this part as $\mathcal {J}_1$, and it takes the following form:
  $$\begin{equation} \mathcal {J}_1 = \sum _{\rm single } J_i + \sum _{\rm disjoint\ pairs} J_i J_j + \sum _{\rm disjoint\ triples} J_i J_j J_k + \cdots . \end{equation} \tag{ 2.25 }$$
  For our graph, every face forms a oriented closed loop, according to the choice of fundamental vertex (2.7), if the loop is closed in anti-clock wise, it is a product of the c, otherwise it is a product of the s.
• $\mathcal {J}_2$: this takes into account the contributions for closed loops sharing a single vertex. This contribution is given by products of the c or s of the corresponding loops, except that of the shared vertex.
• $\mathcal {J}_3$: it is for closed loops sharing two vertices without sharing an edge. This is given by the sum of products of the c (or s) of all vertices involved plus the product of all vertices except the two shared vertices.
• $\mathcal {J}_{13}$ and $\mathcal {J}_{23}$: since in the case of $\mathcal {J}_3$, the closed loops form a bigger loop, we need to go back and consider case 1 and 2 from the viewpoint of this bigger loop. They are denoted as $\mathcal {J}_{13}$ (and $\mathcal {J}_{23}$) and given by combining rules $\mathcal {J}_3$ with that of $\mathcal {J}_1$(and $\mathcal {J}_2$).
• $\mathcal {J}$: the final Jacobian $\mathcal {J}$ of a medial graph is then given by 1 plus all above contributions,
  $$\begin{equation} \mathcal {J} = 1 + \mathcal {J}_1 + \mathcal {J}_2 + \mathcal {J}_3 + \mathcal {J}_{13} + \mathcal {J}_{23} . \end{equation} \tag{ 2.26 }$$
• Finally, removable bubbles in the diagram can be fully decoupled from the diagram. One only needs to compute the Jacobian for the rest diagram after decoupling according to the rules described above, and in the end multiply with the Jacobian of each removable bubble.

Let us now consider a few concrete examples to illustrate the rules. For instance, consider following two diagrams,

Note these two diagrams are related to each other by a triangle move. The Jacobian of the diagram (a) in above picture only gets contribution from $\mathcal {J}_1$, which is given by

$$\begin{equation} \mathcal {J}_{a} = 1 + \mathcal {J}_{1 a} = 1 + (c_3c_1 c_2 + s_4 s_5 s_3 s_2) . \end{equation} \tag{ 2.28 }$$

While the diagram (b) gets contributions from $\mathcal {J}_1$ as well as $\mathcal {J}_2$, the result is given by

$$\begin{equation} \mathcal {J}_{b} = 1 + \mathcal {J}_{1 b}+ \mathcal {J}_{2 b} = 1 + (s^{\prime }_{1} s^{\prime }_{2} s^{\prime }_3 + s^{\prime }_3 s_4 s_5) +( s^{\prime }_1 s^{\prime }_2 s_4 s_5 ) . \end{equation} \tag{ 2.29 }$$

As we will discuss in next section, the volume forms of diagrams related by triangle moves should be invariant, namely

$$\begin{equation} {\mathcal {J}_{a}} \prod ^5_{i=1} d \log ({\rm tan}_i) = {\mathcal {J}_{b}} \prod ^5_{i=1} d \log ({\rm tan^{\prime }}_i) \, \end{equation} \tag{ 2.30 }$$

where tan_i = s_i/c_i. Here tan'_i = tan_i for i = 4, 5 whereas other tan'_i are related to tan_i according to the triangle move transformation given later in equation (2.55). From equation (2.56) and above relation (2.30), we find $\mathcal {J}_{a}$ and $\mathcal {J}_{b}$ should satisfy following relation,

$$\begin{equation} {\mathcal {J}_{a} \over 1+ c_1 c_2 c_3} = {\mathcal {J}_{b} \over 1+ s^{\prime }_1 s^{\prime }_2 s^{\prime }_3} . \end{equation} \tag{ 2.31 }$$

It is straightforward to verify the above equivalence.

Now we move on to something more complicated. Consider the following two diagrams that are related by two triangle moves:

According to the rules, it is straightforward to read off all the contributions from the diagram. For the diagram (a), we have

$$\begin{eqnarray} &&{\mathcal {J}_a}_1 = c_1 c_2 c_3 + s_2 s_3 s_4 s_5+ s_4 s_6 s_7 + c_6 c_7 + s_6 s_7 s_8 \nonumber\\ &&+\, c_1 c_2 c_3 (s_4 s_6 s_7 + c_6 c_7 + s_6 s_7 s_8 ) + s_2 s_3 s_4 s_5 (c_6 c_7 + s_6 s_7 s_8) , \nonumber\\ &&{\mathcal {J}_a}_2 = s_2 s_3 s_5 s_6 s_7 , \nonumber\\ &&{\mathcal {J}_a}_3 = s_4 s_8 + s_4 c_6 c_7 s_8 . \end{eqnarray} \tag{ 2.33 }$$

Finally there are contributions from ${\mathcal {J}}_{13}$ and ${\mathcal {J}}_{23}$, which give

$$\begin{eqnarray} &&{\mathcal {J}_a}_{13} = c_1 c_2 c_3(s_4 s_8 + s_4 c_6 c_7 s_8) , \nonumber\\ &&{\mathcal {J}_a}_{23} = s_2 s_3 s_5(s_8 + c_6 c_7 s_8). \end{eqnarray} \tag{ 2.34 }$$

So the Jacobian of the diagram (a) is then

$$\begin{equation} \mathcal {J}_a = 1 + {\mathcal {J}_a}_1 + {\mathcal {J}_a}_2 +{\mathcal {J}_a}_3 + {\mathcal {J}_a}_{13} + {\mathcal {J}_a}_{23}. \end{equation} \tag{ 2.35 }$$

Similarly, we can read off the Jacobian for the diagram (b) in equation (2.32), which contains a removable bubble,

$$\begin{equation} \mathcal {J}_b =(1 + c^{\prime }_{6})( 1 + {\mathcal {J}_b}_1 + {\mathcal {J}_b}_2 ) , \end{equation} \tag{ 2.36 }$$

where ${\mathcal {J}}_3$ vanishes in this case, and ${\mathcal {J}_b}_1, {\mathcal {J}_b}_2$ are given by

$$\begin{eqnarray} &&{\mathcal {J}_b}_1 = s^{\prime }_{1} s^{\prime }_{2}s^{\prime }_{3} + s^{\prime }_{3}s_4 s_5 + s_4 s^{\prime }_{7} s^{\prime }_{8} + c^{\prime }_{7}c^{\prime }_{8}+ s^{\prime }_{3}s_4 s_5 c^{\prime }_{7}c^{\prime }_{8} \nonumber\\ &&+\, s^{\prime }_{1} s^{\prime }_{2}s^{\prime }_{3} ( s_4 s^{\prime }_{7} s^{\prime }_{8} + c^{\prime }_{7}c^{\prime }_{8} ) + s^{\prime }_{1}s^{\prime }_{2}s_4 s_5 c_7 c_8 \nonumber\\ &&{\mathcal {J}_b}_2 = s^{\prime }_{1} s^{\prime }_{2} s_4 s_5 + s^{\prime }_{3}s_5 s^{\prime }_{7} s^{\prime }_{8} + s^{\prime }_{1} s^{\prime }_{2} s_5 s^{\prime }_{7} s^{\prime }_{8}. \end{eqnarray} \tag{ 2.37 }$$

Again the differential forms of these two diagrams in equation (2.32) should be invariant under the moves,

$$\begin{equation} \mathcal {J}_a \prod ^8_{i=1} d \log {\rm tan}_i = \mathcal {J}_b \prod ^8_{i=1} d \log {\rm tan^{\prime }}_i, \end{equation} \tag{ 2.38 }$$

which leads to a very non-trivial relation between $\mathcal {J}_a$ and $\mathcal {J}_b$,

$$\begin{equation} {\mathcal {J}_a \over (1 + c_1 c_2 c_3)(1+ s_6 s_7 s_7) } = { \mathcal {J}_b \over (1 + s^{\prime }_{1} s^{\prime }_{2}s^{\prime }_{3} )(1+ c^{\prime }_{6} c^{\prime }_{7}c^{\prime }_{8}) }, \end{equation} \tag{ 2.39 }$$

where $s^{\prime }_4 =s_4, s^{\prime }_5 = s_5$ and other $c^{\prime }_i$ and $s^{\prime }_i$ are related to c_i and s_i according to equation (2.55). We have checked numerically that the above relation indeed holds.

Before closing this section, let us comment on the fact that the Jacobian as well as the boundary measurement of a big graph may be obtained by considering its subgraphs inside the big graph as effective vertices¹⁰. This will also be proved to very useful in next section. For instance for the graph on the right in (2.27), one may consider, for instance, the triangle on the top effectively as a six-point vertex shown as following

We will call the graph with effective vertex as an effective graph. The content of the effective vertex is its own boundary measurements. When constructing the boundary measurement of the entire graph, if the oriented path passes through the effective vertex, instead of the usual s_i or c_i, one picks up the boundary measurement of that effective vertex. For the case in equation (2.40), denote the measurements as m_ij, we have for example,

$$\begin{equation} m_{ab}= {s_3 + s_1 s_2 \over 1+ s_1 s_2 s_3}. \end{equation} \tag{ 2.41 }$$

To calculate the boundary measurements and Jacobian of the full graph, one can use exactly the same rules described in previous sections. The only change is that when encounter the effective vertex, one needs to use its measurements. So the Jacobian of the full graph is simply the Jacobian of the effective graph multiplied with the Jacobian coming from the effective vertex itself. For the example we are considering, we have

$$\begin{equation} \mathcal {J} = (1 + m_{ab} s_4 s_5)(1+ s_1 s_2 s_3), \end{equation} \tag{ 2.42 }$$

where (1 + m_abs₄s₅) is the Jacobian of the effective graph, and (1 + s₁s₂s₃) is the Jacobian of the effective vertex itself. Substitute the result of m_ab, equation (2.41), we find the result we obtained earlier.

2.4.3. Coordinates change under local moves

Here we demonstrate that the Jacobian factors discussed previously is necessary for the volume form to be preserved under the triangle equivalence move. Furthermore, in the bubble reduction, the Jacobian factor is necessary such that after the reduction, one obtains a simple dlog  form. Let us start with the simplest case, the reduction of a removable bubble

The measurement of the removable bubble is trivial, M₁₂ = 1, and the volume form is simply

$$\begin{equation} d \log (1/s -1) . \end{equation} \tag{ 2.44 }$$

Since the boundary measurement of the removable bubble is trivial, which means that one can always decouple it from the rest of diagram, that is the reason we call such bubble as removable. This fact was used to obtain the Jacobian factor for a given graph in previous section. We now move on to consider the other type of bubble, and the triangle equivalence move, as shown following

The boundary measurement for two graphs shown in equation (2.45(A)) is

$$\begin{eqnarray} {\rm lhs}\; (2.45(A)):\quad & M_{12}=M_{34}={c_1c_2\over 1+s_1 s_2},\quad M_{14}=M_{32}={s_1+s_2\over 1+s_1 s_2} \nonumber \\ {\rm rhs}\; (2.45(A)):\quad & M_{12}=M_{34}=c^{\prime },\quad M_{14}=M_{32}=s^{\prime }. \end{eqnarray} \tag{ 2.46 }$$

To make the boundary measurement to be the same, the coordinates are then related by

$$\begin{equation} c^{\prime }={c_1 c_2 \over 1+s_1 s_2},\qquad s^{\prime }={s_1+s_2 \over 1+s_1 s_2}. \end{equation} \tag{ 2.47 }$$

We now calculate how the integration measure is changed. The initial integration measure is

$$\begin{equation} { (1+s_1 s_2)} {d\log \tan _1}\wedge {d\log \tan _2} . \end{equation} \tag{ 2.48 }$$

In the primed coordinates, we can see that the new integration measure is given as¹¹

$$\begin{equation} {\rm d}\log \tan ^{\prime } \wedge {\rm d}\log p={\rm d}\log (\tan ^{\prime }) \wedge d \log \bigg ({s_2 c_1\over s_1 c_2}\bigg ) . \end{equation} \tag{ 2.49 }$$

Thus we see that equipped with the Jacobian factor in equation (2.48), the bubble reduction directly gives a dlog -form where one separates out the degree of freedom that does not appear in the C-matrix.

We have checked the local change of volume form, now we will prove that the above calculation is also valid for an arbitrary graph. Consider following general graph with a bubble inside

As we discussed at the end of previous section, one can view the bubble in the above graph effectively as a fundamental four-point vertex as shown in (2.50) by the graph on the right. The Jacobian of the graph is given by

$$\begin{equation} \mathcal {J} = \mathcal {J}_{\rm bubble} \times \mathcal {J}_{\rm eff} = (1 + s_1 s_2) \mathcal {J}_{\rm eff}. \end{equation} \tag{ 2.51 }$$

Thus the volume form can be rewritten as

$$\begin{eqnarray} \mathcal {J} \prod _{i} {d\log \tan _i} &=&(1 + s_1 s_2) {d\log \tan _1}\wedge {d\log \tan _2} \mathcal {J}_{\rm eff} \prod _{i>2} {d\log \tan _i} \cr & =& d\log ({s_2 c_1\over s_1 c_2}) \times \mathcal {J}_{\rm eff} d\log \tan ^{\prime } \prod _{i>2} {d\log \tan _i}, \end{eqnarray} \tag{ 2.52 }$$

with transformation given by (2.47). This is nothing but the statement of bubble reduction for a general graph, with ${\rm d}\log \big ({s_2 c_1\over s_1 c_2}\big)$ decoupled from the rest of graph.

Let us move on to consider the transformation rule for the vertex coordinates under the triangle move. For the left graph in equation (2.45(B)), the boundary measurements are given as

$$\begin{eqnarray} & M_{12}={c_1+c_2c_3\over 1+c_1c_2c_3},\quad M_{34}={c_2+c_1c_3\over 1+c_1c_2c_3}, \quad M_{56}={c_3+c_1c_2\over 1+c_1c_2c_3} \nonumber \\ & M_{14}={s_1c_3 s_2\over 1+c_1c_2c_3},\quad M_{36}={s_3c_1s_2\over 1+c_1c_2c_3},\quad M_{52}={s_3c_2s_1\over 1+c_1c_2c_3}\nonumber \\ & M_{16}={s_1s_3 \over 1+c_1c_2c_3},\quad M_{32}={s_2s_1\over 1+c_1c_2c_3},\quad M_{54}={s_2s_3\over 1+c_1c_2c_3}. \end{eqnarray} \tag{ 2.53 }$$

After the triangle move, we have

$$\begin{eqnarray} & M_{12}={c^{\prime }_2 c^{\prime }_3\over 1+s^{\prime }_1 s^{\prime }_2 s^{\prime }_3},\quad M_{34}={c^{\prime }_3 c^{\prime }_1\over 1+s^{\prime }_1 s^{\prime }_2 s^{\prime }_3},\quad M_{56}={c^{\prime }_1 c^{\prime }_2\over 1+s^{\prime }_1 s^{\prime }_2 s^{\prime }_3}\nonumber \\ & M_{14}={c^{\prime }_2 s^{\prime }_3 c^{\prime }_1\over 1+s^{\prime }_1 s^{\prime }_2 s^{\prime }_3},\quad M_{36}={c^{\prime }_3 s^{\prime }_1 c^{\prime }_2\over 1+s^{\prime }_1 s^{\prime }_2s^{\prime }_3} ,\quad M_{52}={c^{\prime }_1s^{\prime }_2 c^{\prime }_3\over 1+s^{\prime }_1s^{\prime }_2s^{\prime }_3}\nonumber \\ & M_{16}={s^{\prime }_2+s^{\prime }_3s^{\prime }_1 \over 1+s^{\prime }_2s^{\prime }_3s^{\prime }_1},\quad M_{32}={s^{\prime }_3+s^{\prime }_2s^{\prime }_1 \over 1+s^{\prime }_1s^{\prime }_2s^{\prime }_3},\quad M_{54}={s^{\prime }_1+s^{\prime }_2s^{\prime }_3 \over 1+s^{\prime }_1s^{\prime }_2s^{\prime }_3}. \end{eqnarray} \tag{ 2.54 }$$

By requiring the measurements to be the same, we find

$$\begin{eqnarray} &s^{\prime }_1={s_3c_1 s_2\over c_1+c_2c_3} ,\quad c_1={c^{\prime }_3s^{\prime }_1 c^{\prime }_2\over s^{\prime }_1+s^{\prime }_2s^{\prime }_3 }\nonumber \\ & s^{\prime }_2={s_3c_2 s_1\over c_2+c_1c_3},\quad c_2={c^{\prime }_1s^{\prime }_2 c^{\prime }_3\over s^{\prime }_2+s^{\prime }_3s^{\prime }_1 }\nonumber \\ &s^{\prime }_3={s_1c_3 s_2\over c_3+c_1c_2},\quad c_3={c^{\prime }_2s^{\prime }_3 c^{\prime }_1\over s^{\prime }_3+s^{\prime }_1s^{\prime }_2 }. \end{eqnarray} \tag{ 2.55 }$$

It is straightforward to show that the volume form is also invariant under the above coordinate transformation:

$$\begin{eqnarray} &&({ 1+c_1 c_2 c_3}){d\log \tan _1}\wedge d\log \tan _2\wedge d\log \tan _3\nonumber\\ &&= ({1+s^{\prime }_1 s^{\prime }_2 s^{\prime }_3}) {d\log \tan ^{\prime }_1}\wedge d\log \tan ^{\prime }_2\wedge d\log \tan ^{\prime }_3. \end{eqnarray} \tag{ 2.56 }$$

Indeed as we have emphasized the Jacobian factor is crucial to ensure the invariance of the volume form. Again, we have only checked the invariance of local piece of integration measure, but it is enough for us. To see this, let us consider the following general graphs related to each other by a triangle move,

As shown in the above picture, two graphs on the top are related by a triangle move, where two diagrams on the bottom are the corresponding graphs by considering the triangles as effective vertices. According to the rules, the Jacobian of two graphs are respectively given by

$$\begin{equation} \mathcal {J}_a = {\mathcal {J}_a}_{\rm triangle} \times {\mathcal {J}_a}_{\rm effective} , \quad \mathcal {J}_b = {\mathcal {J}_b}_{\rm triangle} \times {\mathcal {J}_b}_{\rm effective}, \end{equation} \tag{ 2.58 }$$

where we denote ${\mathcal {J}_i}_{\rm triangle}$ as the Jacobian of the triangle in graph (i), whereas ${\mathcal {J}_i}_{\rm effective}$ is the Jacobian of the effective graph (i), for i = a, b. Since the only difference between effective graph (a) and effective graph (b) is that they have different six-point effect vertices. However we have proved that measurements of two six-point effective vertices are actually the same, which then leads to

$$\begin{equation} {\mathcal {J}_a}_{\rm effective} = {\mathcal {J}_b}_{\rm effective}. \end{equation} \tag{ 2.59 }$$

Using this result and the identity (2.56), we thus deduce

$$\begin{equation} \mathcal {J}_a \prod _{i} d\log \tan _{a i} = \mathcal {J}_b \prod _{i} d\log \tan _{b i}. \end{equation} \tag{ 2.60 }$$

Namely the volume form of a general graph is invariant under the triangle move, as we also have shown explicitly by examples in previous section.

2.5.1. Global characterization of medial graph: decorated permutation

In this section, we would like to consider following questions:

• When a medial graph reduced?
• How to tell two that two graphs are equivalent?

Note that since all equivalence and reduction moves are preserved by the embedding of OG_{k +} in Gr(k, 2k)₊, the reducibility of a given medial can be rephrased as a question of reducibility of its image in the bipartite network. In other words the criteria of reducibility is exactly the same as that of the bipartite network.

Reducibility of bipartite network can be deduced by considering the permutation associated with the 'left-right path' [2, 21]: as one enters the graph take a left turn when one hits a white vertex and a right turn when hits a black vertex. Starting with the external vertex i, through the above rule one reaches another external vertex σ(i). This defines a set of permutation i → σ(i) for all external vertices. Equipped with this definition, a reducible bipartite network can be identified as that where closed paths are formed, or two sets of permutation paths crosses each other along more than one common edge in the form of:

As discussed previously, the bipartite network that contains the image of OG_{k +}, is that whose permutation paths all have the property that if σ(i) = j, then σ(j) = i. From the above analysis one can also deduce the corresponding rule for reducible medial graphs:

• A medial graph is reduced if there is no self-intersecting permutation paths.

As an example consider the following two diagrams:

As one can see while (a) is reduced, diagram (b) is reducible via triangle move. Once we have a set of reduced graphs, we can further classify them by the following rule:

• Two reduced graphs are equivalent if and only if they define the same decorated permutation.

All inequivalent medial graphs now correspond to distinct cells in OG_{k +}.

2.5.2. Going to the boundary: removable vertices and Eulerian poset

For a given cell in OG_{k +} we would like to consider its co-dimension one boundaries. These are associated with one extra linear dependency condition among consecutive columns in the C-matrix. The boundaries can be read off from the medial graph via the opening of the four-point vertices [2]:

If the resulting graph is also reduced they are cells which represent the co-dimension one boundaries. Through this procedure, we obtain a poset, i.e. we have A > B if one can obtain the reduced graph corresponding to B from opening vertices in A.

Remarkably the positroid stratification of the OG_{k +} forms an Eulerian poset, i.e. the number of even-dimensional positroids are one more than the number of odd-dimensional ones. For example consider OG_{2 +}, which as a one-dimensional top cell with two zero-dimensional boundaries, giving 2 − 1 = 1. Let us consider OG_{3 +}, starting with the three-dimensional top-cell, we have:

$$\begin{equation} {-}1+3-6+5=1. \end{equation} \tag{ 2.64 }$$

Since each stratification can be represented by an on-shell diagram, we can illustrate the enumeration of the elements in each dimension as follows:

Note that as the face-lattice of a convex polytope forms a Eulerian poset, the above observation suggests that the cell stratification of OG_{k +} is combinatorially a polytope. Recently a new way of enumeration distinct cells for OG_{k +} was developed in [12], and a closed form of a generating function for the enumeration starting from a top cell was found. Use this result, the authors have explicitly shown that the graded poset of all top cells in OG_{k +} are indeed Eulerian.

However, this is not the end of the story. In fact if we begin in the middle of some stratification, one still obtains an Eulerian poset. For example

As a further non-trivial example, let us consider the stratification that corresponds to the permutation path (16) (25) (38) (47), namely 1 → 6 and 6 → 1, etc. We list the elements of each stratification by their permutation paths

$$\begin{eqnarray} &&{\begin{array}{@{}c|c|c@{}}\\ \hline Dimensions & Cell & Multiplicity \\ 4 & (16)(25)(38)(47) &1 \\ \hline 3 & (23)(47)(58)(16), (45)(16)(27)(38), (18)(25)(36)(47), (67)(38)(14)(25) &\; \\ \;&(28)(16)(47)(35), (24)(38)(16)(57), (17)(38)(25)(46), (13)(25)(47)(68)&8\\ \hline 2 & (23)(48)(57)(16), (23)(68)(47)(15), (23)(17)(58)(46), (45)(26)(38)(17) &\; \\ \; & (45)(13)(27)(68), (45)(16)(28)(37), (18)(26)(47)(35), (18)(57)(36)(24) &\; \\ \;&(18)(46)(25)(37), (67)(25)(48)(13), (67)(35)(14)(28), (67)(24)(38)(15) &\;\\ \;&(12)(35)(47)(68), (28)(17)(35)(46), (34)(28)(16)(57), (13)(24)(57)(68) &\;\\ \;&(56)(24)(38)(17), (78)(46)(25)(13)&18\\ \hline 1 & (35)(18)(46)(57), (23)(14)(57)(68), (23)(56)(17)(48), (23)(67)(48)(15) &\; \\ \; & (45)(23)(17)(68), (45)(36)(28)(17), (45)(78)(26)(13), (45)(18)(26)(37) &\; \\ \;&(18)(67)(24)(35), (18)(27)(35)(46), (18)(34)(28)(57), (67)(58)(13)(24) &\;\\ \;&(67)(45)(28)(13), (67)(12)(35)(48), (23)(78)(46)(15), (45)(12)(37)(68) &\;\\ \;&(18)(56)(24)(37), (12)(78)(35)(46), (12)(34)(68)(57), (34)(56)(17)(28) &\;\\ \;&(34)(67)(15)(28), (56)(78)(13)(24)&22\\ \hline 0 & (23)(18)(45)(67), (23)(18)(56)(47), (23)(14)(56)(78), (23)(14)(67)(58) &\; \\ \; & (45)(23)(78)(16), (45)(36)(78)(12), (45)(36)(18)(27), (18)(67)(34)(25) &\; \\ \;&(18)(27)(34)(56), (67)(58)(12)(34), (67)(45)(12)(38), (12)(34)(56)(78) &12\\ \end{array} }\nonumber\\ \end{eqnarray} \tag{ 2.67 }$$

Thus one indeed finds 1 − 8 + 18 − 22 + 12 = 1. For all the examples we have checked this is always the case, it leads us to conjecture that all the cells in OG_{k +} should be Eulerian. It would be of great interest to generalize the result of [12] to find the generating function starting from a general middle cell.

Finally, we would like to answer the question we raised earlier: what is so special about the parametrization of the top cell in Gr(3, 6) that is given by the embedding of OG₃? To answer this question, let us consider the (2 + 1)-dimensional analogue of the factorizability condition of scattering amplitudes for particles in (1 + 1)-dimensions: the Yang–Baxter equation. A suitable generalization was proposed long ago by Zamolodchikov [14, 15], who considered the scattering of infinite straight strings in (2 + 1)-dimensions. Instead of the fundamental 2 → 2 S-matrix in (1 + 1)-dimensions, the first nontrivial amplitude is the 3 → 3 process for three strings (a, b, c) illustrated in the following diagrams:

Diagram (a) denotes the scattering process. The configuration of the three strings can be characterized by three angles (θ_ab, θ_ac, θ_bc). The angles are defined by considering the strings as intersection of three infinite planes, and the angles are associated with the inner product of the normal vectors n_i, of each plane:

$$\begin{equation} n_a\cdot n_{b}=-\cos \theta _{ab},\quad n_a\cdot n_{c}=-\cos \theta _{ac},\quad n_{c}\cdot n_{b}=-\cos \theta _{cb} . \end{equation} \tag{ 2.69 }$$

The scattering amplitude is then a map between the initial $| \theta ^{\prime }_{ab},\theta ^{\prime }_{ac},\theta ^{\prime }_{bc}\rangle$ and final state |θ_ab, θ_ac, θ_bc〉:

$$\begin{equation} | \theta ^{\prime }_{ab},\theta ^{\prime }_{ac},\theta ^{\prime }_{bc}\rangle =R_{abc}(\theta _{ab},\theta _{ac},\theta _{bc})|\theta _{ab},\theta _{ac},\theta _{bc}\rangle . \end{equation} \tag{ 2.70 }$$

A diagrammatic representation of the scattering process is to assign a positive direction to each of the string as shown in diagram (b) of equation (2.68). Denoting each vertex by the two strings that intersect, before scattering we move from vertex (ab) to (ac) to (bc) along the positive direction of each string, while after the scattering the same path moves along the negative direction of each string.

Factorizability of the three-dimensional scattering process is then encoded in the following tetrahedron equation¹²

$$\begin{equation} R_{acb}R_{abd}R_{acd}R_{bcd}=R_{bcd}R_{acd}R_{abd}R_{acb} . \end{equation} \tag{ 2.71 }$$

This equation represent the equivalence of two distinct sequences of 4 (three-string) scattering process that results in the same final string configuration. Diagrammatically the scattering process is represented as:

and

A well-known solution to equation (2.71) is the functional map between two equivalent networks in electric network [23, 24]. Two electric devices are the same between the 'star' and 'triangle' configuration (the Y-Δ transformation):

if their resistance satisfies:

$$\begin{equation} r^{\prime }_1r^{\prime }_2+r^{\prime }_3r^{\prime }_2+r^{\prime }_1r^{\prime }_3=\frac{r_1r_2r_3}{r_1+r_2+r_3} . \end{equation} \tag{ 2.75 }$$

A nontrivial solution is given as:

$$\begin{equation} R: \quad (r_1,r_2,r_3)\quad \rightarrow \quad \left({r_3 r_2 \over r_1+r_2+r_3},{r_1 r_3 \over r_1+r_2+r_3},{r_3 r_2 \over r_1+r_2+r_3}\right). \end{equation} \tag{ 2.76 }$$

The transformation can be shown to satisfy the standard tetrahedron equation, equation (2.71), by making following identification,

$$\begin{eqnarray} \theta _1 &=& -\!{1 \over r_1} , \quad \theta _2 = r_2 , \quad \theta _3 = -{1 \over r_3} ,\nonumber \\ \theta ^{\prime }_1 &=& r^{\prime }_1 , \quad \theta ^{\prime }_2 = -{ 1 \over r^{\prime }_2} , \quad \theta ^{\prime }_3 = r^{\prime }_3 . \end{eqnarray} \tag{ 2.77 }$$

It then leads to a map from θ_i to $\theta ^{\prime }_i$ given by

$$\begin{equation} R: \quad (\theta _1,\theta _2,\theta _3)\,\,\, \rightarrow \,\,\, \left({\theta _1 \theta _2 \over \theta _1+\theta _3 - \theta _1 \theta _2 \theta _3}, \theta _1+\theta _3 - \theta _1 \theta _2 \theta _3,{\theta _2 \theta _3 \over \theta _1+\theta _3 - \theta _1 \theta _2 \theta _3}\right). \end{equation} \tag{ 2.78 }$$

It is straightforward to check the map given above indeed satisfies equation (2.71).

A general class of solutions to equation (2.71) was found by Kashaev, Korepanov and Sergeev [26] by considering the map that arises from the solution to the following equation:

$$\begin{eqnarray} \left(\begin{array}{@{}c@{\ \ \ }c@{\ \ \ }c@{}}A_{1} & B_{1} & 0 \\ C_{1} & D_{1} & 0 \\ 0 & 0 & 1\end{array}\right)\left(\begin{array}{@{}c@{\ \ \ }c@{\ \ \ }c@{}}A_{2} & 0 & B_{2} \\ 0 & 1 & 0 \\ C_{2} & 0 & D_{2}\end{array}\right)\left(\begin{array}{@{}c@{\ \ \ }c@{\ \ \ }c@{}}1 & 0 & 0 \\ 0 & A_{3} & B_{3} \\ 0 & C_{3} & D_{3}\end{array}\right)=\left(\begin{array}{@{}c@{\ \ \ }c@{\ \ \ }c@{}}1 & 0 & 0 \\ 0 & A^{\prime }_{3} & B^{\prime }_{3} \\ 0 & C^{\prime }_{3} & D^{\prime }_{3}\end{array}\right)\left(\begin{array}{@{}c@{\ \ \ }c@{\ \ \ }c@{}}A^{\prime }_{2} & 0 & B^{\prime }_{2} \\ 0 & 1 & 0 \\ C^{\prime }_{2} & 0 & D^{\prime }_{2}\end{array}\right)\left(\begin{array}{@{}c@{\ \ \ }c@{\ \ \ }c@{}}A^{\prime }_{1} & B^{\prime }_{1} & 0 \\ C^{\prime }_{1} & D^{\prime }_{1} & 0 \\ 0 & 0 & 1\end{array}\right)\nonumber\\ \end{eqnarray} \tag{ 2.79 }$$

where each element in the block 2 × 2 matrix is a function of one parameter, i.e. A_i(θ_i), B_i(θ_i), C_i(θ_i) and D_i(θ_i). It was shown in [26], and references there in that given a solution to equation (2.79), the corresponding map $R:\; \theta _i\rightarrow \theta ^{\prime }_i$ will then satisfy the tetrahedron identity in equation (2.71). It is known that using the canonical gauge in equation (2.6), the triangle diagram for the non-trivial part of OG_{3 +} (OG_{3 +} without the 3 × 3 the unity matrix) can be written as products of 3 × 3 matrices in the form equation (2.79) [11],¹³ with,

$$\begin{equation} \left(\begin{array}{@{}cc@{}}A_i(\theta _i) & B_i(\theta _i) \\ C_i(\theta _i) & D_i(\theta _i)\end{array}\right)=\left(\begin{array}{@{}cc@{}}-\cot (\theta _i) & -\csc (\theta _i) \\ \csc (\theta _i) & \cot (\theta _i)\end{array}\right). \end{equation} \tag{ 2.80 }$$

In other words, the coordinate transformation associated with the triangle move satisfies the tetrahedron identity in equation (2.71)! The corresponding medial graphs related by the triangle move are shown below

Solve equation (2.79) with A_i(θ_i), B_i(θ_i), C_i(θ_i) and D_i(θ_i) given in equation (2.80), we obtain the triangle-move transformation of adjacent gauge

$$\begin{equation} R: \quad (c_1,s_2,c_3) \rightarrow \left(\frac{c_1 c_2 s_3 }{c_1 + s_2c_3}, -\frac{s_1 s_2 s_3}{s_2 + c_1 c_3}, \frac{s_1 c_2 c_3}{c_3 + c_1s_2}\right). \end{equation} \tag{ 2.82 }$$

Indeed the above map induces a map of $\theta _i \rightarrow \theta ^{\prime }_i$,

$$\begin{eqnarray} &&(\theta _1, \theta _2, \theta _3) \rightarrow \left( -{\rm ArcCos} \left(\frac{c_1 c_2 s_3 }{c_1 + s_2c_3} \right), -{\rm ArcSin} \left(\frac{s_1 s_2 s_3}{s_2 + c_1 c_3} \right), -{\rm ArcCos} \left( \frac{s_1 c_2 c_3}{c_3 + c_1s_2} \right) \right) \nonumber\\ \end{eqnarray} \tag{ 2.83 }$$

which can be checked to satisfy the tetrahedron equation in the positive region, namely for θ_i ∈ (0, π/2). As for the cyclic gauge equation (2.7), its corresponding transformation, equation (2.55), can be made to be the same as equation (2.82) by some trivial identifications,

$$\begin{equation} \lbrace s^{\prime }_2 \rightarrow -s^{\prime }_2 ,c_2 \leftrightarrow s_2 , s^{\prime }_1 \leftrightarrow c^{\prime }_1 , s^{\prime }_3 \leftrightarrow c^{\prime }_3 \rbrace , \end{equation} \tag{ 2.84 }$$

and with the rest untouched. So with the above identifications the triangle-move transformation of the cyclic gauge satisfies the tetrahedron equation as well, although, interestingly, the Grassmannian of this gauge cannot be decomposed into a factorization form as equation (2.79).

Thus we see that while a generic one-dimensional subspace of G_{2, 4 +} allows us to define a map R₁₂₃(θ₁, θ₂, θ₃) through the triangle equivalence. What is special about the subspace defined by OG_{2 +} is that the corresponding map also defines a S-matrix in (2 + 1)-dimensional describing the scattering of straight strings that is integrable! Note that the diagrams having this interpretation correspond to ones where no two lines intersect twice which implies all such diagrams are reduced. Since these diagrams correspond to three-string scatterings, there must be a total of k(k − 1)(k − 2)/6 number of triangles within a such diagram.

Here, we give a brief review to how the ABJM amplitudes can be given by the on-shell diagrams. The four-point amplitude of ABJM is given by the an integral over OG₂. The two distinct amplitudes are given as:

$$\begin{eqnarray} &&\nonumber \mathcal {A}_4(\bar{1}2\bar{3}4)=\int \frac{d^{2\times 4}C}{(12)(23)}\delta ^{3}(C\cdot C^T)\delta ^{2\times 2|3}(C\cdot \Lambda )\\ &&\mathcal {A}_4(1\bar{2}3\bar{4})=\int \frac{d^{2\times 4}C}{(23)(34)}\delta ^{3}(C\cdot C^T)\delta ^{2\times 2|3}(C\cdot \Lambda ). \end{eqnarray} \tag{ 3.2 }$$

As discussed in [11], the above integral can be rewritten as a sum or difference of two different branches in OG₂. To see this, recall that due to the orthogonal constraint we have:

$$\begin{equation} {\rm OG}_{2+}:\quad (12)=(34), \quad {\rm OG}_{2-}:\quad (12)=-(34) . \end{equation} \tag{ 3.3 }$$

Thus on the positive branch, $\mathcal {A}_4(\bar{1}2\bar{3}4)=\mathcal {A}_4(1\bar{2}3\bar{4})$ whereas on the negative branch $\mathcal {A}_4(\bar{1}2\bar{3}4)=-\mathcal {A}_4(1\bar{2}3\bar{4})$. Thus if $\mathcal {A}_4(\bar{1}2\bar{3}4)$ is given by the sum of the two branches, $\mathcal {A}_4(1\bar{2}3\bar{4})$ would be given by the difference. Starting from

$$\begin{equation} C=\left(\begin{array}{@{}cccc@{}} 1& a &0&b\\ 0 &c &1&d \end{array}\right) \end{equation} \tag{ 3.4 }$$

explicitly solving the orthogonal constraints in equation (3.2), we find that¹⁵

$$\begin{eqnarray} \nonumber \mathcal {A}_4(\bar{1}2\bar{3}4)&=&\sum _{\alpha =\pm } \int \frac{{\rm d}\theta }{2cs} \delta ^{2\times 2|3}(C\cdot \Lambda )\\ \mathcal {A}_4(1\bar{2}3\bar{4})&=&\sum _{\alpha =\pm } \alpha \int \frac{{\rm d}\theta }{2cs} \delta ^{2\times 2|3}(C\cdot \Lambda ), \end{eqnarray} \tag{ 3.5 }$$

where now the OG₂ matrix is given by

$$\begin{equation} C=\left(\begin{array}{@{}cccc@{}}1& c &0& -\alpha s\\ 0 & s &1&\alpha c\, \end{array}\right). \end{equation} \tag{ 3.6 }$$

The form of the four-point amplitude in equation (3.5) can also be found in [27]. The parameter α takes value in +1, −1 and determines the branch of OG₂. Thus we see that while $\mathcal {A}_4(\bar{1}2\bar{3}4)$ is given by the sum of the two branches, and $\mathcal {A}_4(1\bar{2}3\bar{4})$ is given by the difference.

As discussed in [11], the two branches in the orthogonal Grassmannian is intricately related to a unique feature of three-dimensional kinematics. At four-points, through momentum conservation we can identify

$$\begin{equation} \langle 12\rangle =\pm \langle 34\rangle . \end{equation} \tag{ 3.7 }$$

Note that + and − are inequivalent kinematic configurations. The relative signs for the remaining configurations are given by the following identity as a result of momentum conservation:

$$\begin{equation} \frac{\langle 12\rangle }{\langle 34\rangle }=\frac{\langle 14\rangle }{\langle 23\rangle }=\frac{\langle 42\rangle }{\langle 13\rangle }. \end{equation} \tag{ 3.8 }$$

In fact, the two distinct kinematic configurations in equation (3.7) can be mapped to the two branches in OG₂. Indeed from equation (3.6), we can see that the bosonic delta functions δ²(C · λ) implies that:

$$\begin{equation} \left\lbrace \begin{array}{@{}l@{}} \langle 41\rangle + c\langle 42\rangle =0 \\ \langle 23\rangle +\alpha c\langle 24\rangle =0\end{array}\right.\Rightarrow \quad \frac{\langle 14\rangle }{\langle 23\rangle }=1/\alpha . \end{equation} \tag{ 3.9 }$$

Thus the + and − of OG₂ precisely corresponds to the + and − of equation (3.7). We can denote the two branches graphically as:

where the shaded region indicates the kinematic invariant that is constructed clockwise out of the two legs that bound the region are equivalent up to a − sign, whereas the unshaded region indicates the opposite, i.e

$$\begin{equation} (+):\quad \langle 12\rangle = \langle 34\rangle , \;\;\langle 23\rangle = -\langle 41\rangle ,\quad (-): \quad \langle 12\rangle = -\langle 34\rangle , \;\;\langle 23\rangle = \langle 41\rangle . \end{equation} \tag{ 3.11 }$$

The BCFW recursion construction of higher-point tree-level amplitudes in ABJM theory [10], can be mapped into the gluing of the fundamental OG₂'s into OG_k [2, 11]. This is schematically represented as

In other words, each term in the BCFW expansion can be represented as a sum of on-shell diagrams. Note that since each term in the BCFW expansion is a rational function, this implies that the on-shell diagrams must be of dimension 2k − 3, such that the bosonic delta functions associated with each diagram, i.e. δ^2k(C · λ), completely localizes these degrees of freedom.

However, this cannot be the whole story since we have not indicated which branch each the individual vertex should take value in. Naively, one would just sum over all possible local branches resulting in a sum of $2^{n_v}$ number of terms, where n_v is the number of vertices in a given on-shell diagram. On the other hand, by now we are familiar with the fact that distinct configurations in OG_{k +} can be faithfully represented by on-shell diagrams that are inequivalent under various equivalent moves. Thus it is impossible for a given on-shell diagram to represent $2^{n_v}$ number of distinct terms. In fact, for any on-shell diagram, one can only have two distinct configurations, OG_{k +} and OG_{k −}. Thus half of the $2^{n_v}$ terms that correspond to the same branch must be equivalent to each other via redefinition. Indeed this is the case. Thus summing over all $2^{n_v}$ terms is equivalent to summing over arbitrary two representative of each branch.

Just as the four-point amplitude, the above discussion is also reflected in the branches of kinematics. It is instructive to consider an example beyond four-point, let us merge two vertices and consider all four possible combination of the local OG₂ branches:

Configuration (a) and (c) correspond to two-dimensional cells in OG_{3 +}, whereas (b) and (d) are cells in OG_{3 −}. Instead of showing the equivalence of the two pairs, it is more instructive to show that each pair correspond to the same branch for the external data. Indeed using the prescription in equation (3.10) one can straightforwardly deduce that the four configurations imply:

$$\begin{equation} (a),(c):\quad {\rm i}\langle 3|p_5+p_6|4\rangle =\langle 12\rangle \langle 56\rangle ,\quad (b),(d):\quad {\rm i}\langle 3|p_5+p_6|4\rangle =-\langle 12\rangle \langle 56\rangle . \end{equation} \tag{ 3.12 }$$

The right-hand side are precisely the two distinct six-point kinematic configuration in the factorization limit as can be seen using the following identity:

$$\begin{equation} \langle i|p_j+p_k| l\rangle ^2+(p_j+p_k+p_l+p_i)^2(p_j+p_k)^2=(p_i+p_j+p_k)^2(p_j+p_k+p_l)^2 . \end{equation} \tag{ 3.13 }$$

Next, we consider the diagram with a closed loop:

It is straightforward to show that the two configurations imply:

$$\begin{eqnarray} &&(a): \langle 12\rangle \langle 34\rangle \langle 56\rangle = {\rm i} \langle \ell _3\ell _1\rangle \langle \ell _1\ell _2\rangle \langle \ell _2\ell _3\rangle ,\quad \quad (b): \langle 12\rangle \langle 34\rangle \langle 56\rangle = -{\rm i} \langle \ell _3\ell _1\rangle \langle \ell _1\ell _2\rangle \langle \ell _2\ell _3\rangle\nonumber\\ \end{eqnarray} \tag{ 3.14 }$$

where we have used λ_−I = iλ_I. Again one can straightforwardly check that all other configurations simply correspond to one of these branches. Not surprisingly configurations of OG_{3 −} correspond to (a), while those of OG_{3 +} correspond to (b).

Finally, a question one might pose is whether or not summing over the local branches of a given vertex is valid if the vertex is identified with a BCFW bridge. Let us consider the attachment of a BCFW bridge is to connect two tree-amplitudes:

where the blobs labelled by A_L and A_R are collections of on-shell diagrams whose sums give the tree-amplitudes. The hatted lines indicate that the are expressed in terms of external spinors. It is important to keep in mind that legs $\hat{1}$ sits on a barred leg in A_L, and $\hat{2}$ sits on an un-barred leg in A_R. An immediate consequence of this is that if $\lambda _{\hat{1}}\rightarrow -\lambda _{\hat{1}}$, A_L → −A_L, whereas if $\lambda _{\hat{2}}\rightarrow -\lambda _{\hat{2}}$, A_R →A_R. Now if we sum over the two branches of the upper top vertex, this diagram is given as:

$$\begin{eqnarray} {\rm dia} (3.15) &=& {1 \over 2} \sum _{\alpha }\int d^{2|3}\Lambda _{\hat{1}}d^{2|3}\Lambda _{\hat{2}}d^{2|3}\Lambda _{\rm I} \int {{\rm d}\theta \over cs} \delta ^{4|3}(C(\alpha )\cdot \Lambda )A_LA_R. \end{eqnarray} \tag{ 3.16 }$$

The integral over $\Lambda _{\hat{1}}$ and $\Lambda _{\hat{2}}$ is to be localized by the delta function in the integrand. This localizes

$$\begin{equation} \Lambda _{\hat{1}}=\frac{1}{\alpha s}(\Lambda _1+c\Lambda _2),\quad \Lambda _{\hat{2}}=-\frac{1}{s}(\Lambda _2+c\Lambda _1) . \end{equation} \tag{ 3.17 }$$

This generates a Jacobian factor αs. Now, we see that summing over the two branches, again labeled by α, amount to summing over $\Lambda _{\hat{1}}$ and $-\Lambda _{\hat{1}}$. As discussed previously, under the flip of the sign in front of $\Lambda _{\hat{1}}$, the only $\Lambda _{\hat{1}}$ dependent term in the integrand A_L also flips a sign which precisely compensates for the extra sign that arise from the Jacobian factor αs. Thus the integrand actually takes the same form on the two branches. This proves that in the BCFW recursion, the vertex that is identified as the BCFW bridge can be safely averaged over the two branches of the original the vertex.

Thus in conclusion, the tree-level amplitude of ABJM can be schematically written as:

$$\begin{eqnarray} &&A_{n}(\bar{1}2\cdots n)=\sum _{{\rm dia}}\bigg [\prod _{j\in I_{{\rm dia}}}\int d^{2|3}\Lambda _j\bigg ] \bigg [\prod _{i\in v_{{\rm dia}}}\left(\sum _{\alpha _i}\frac{1}{2}\int {\rm d}\log \tan _i \right)\delta ^{2k|3k}\left(C(\alpha _i)\cdot \Lambda \right)\bigg ].\nonumber\\ \end{eqnarray} \tag{ 3.18 }$$

Let us start with the simplest one-loop amplitude, four-point scattering. Since there is no three-point amplitude, the only the contribution to the loop recursion relation is from the forward limit of the six-point tree-amplitude. Thus the result of the loop-recursion is given by:

where we have illustrated the procedure of obtaining the forward limit of the six-point amplitude, and then attaching a BCFW bridge to obtain the one-loop amplitude.

4.1.1. Forward limit

Before obtaining the one-loop amplitude, as an exercise let us first show that forward-limit on-shell diagram reproduces the correct single cut from the known integrand. Consider the following forward-limit diagram:

where the arrows indicate the gauge choice, and all external momenta are flowing outwards and q, q' are flowing rightward in the diagram. This diagram can be regarded as attaching a BCFW bridge to a bubble diagram, where the latter can be conveniently expressed as¹⁶:

We can now obtain the forward limit diagram by applying a BCFW vertex to the bubble diagram. The explicit form is now given as

$$\begin{eqnarray} F^{(4,1)}_4&=&\sum _{\alpha =\pm }\frac{1}{2}\int {{\rm d}\log \tan \theta }\; \delta ^{2|3}(C\cdot \Lambda )\;B_4 \nonumber\\ &=&-\delta ^3 ( P_{{\rm full}})\delta ^6 ( Q_{{\rm full}})\sum _{\alpha =\pm }\frac{1}{2}\int {{\rm d}\theta \over s}\;\int {\rm d}^2 \lambda _q \; \delta ((q + p_{\hat{3}} + p_4)^2) { \langle \hat{3} 4 \rangle \over \langle 1 q\rangle \langle q 4\rangle } \end{eqnarray} \tag{ 4.5 }$$

where the shifted $\lambda _{\hat{3}}$ is defined as

$$\begin{equation} \lambda _{\hat{3}} = -\sec (\theta ) \lambda _3 + \alpha \tan (\theta ) \lambda _2 . \end{equation} \tag{ 4.6 }$$

Note that since the integrand is little-group even on leg 2, we can simply combine the two branches. As usual we treat the delta-function, $\delta ((q + p_{\hat{3}} + p_4)^2)$, as a contour integral. The only other place where there is θ dependence is in the integration measure, thus we can deform the counter and instead localize on the pole s = 0, i.e. θ = 0, π. Since the integrand contain odd number of $\lambda _{\hat{3}}$, the two residue combine to give:

$$\begin{equation} F^{(4,1)}_4=\delta ^3 ( P _{{\rm full}})\delta ^6 ( Q_{{\rm full}})\int {\rm d}^2 \lambda _q { \langle 3 4 \rangle \over \langle 1 q\rangle \langle q 4\rangle (q + p_{3} + p_4)^2 } , \end{equation} \tag{ 4.7 }$$

which agrees precisely the single-cut of one-loop four-point amplitude [28]:

$$\begin{eqnarray} &&\mathcal {A}_4^{\rm tree}\int {\rm d}^3 \ell { -\langle 3 4 \rangle ^2 \langle 4 |\ell |1 \rangle \langle 14 \rangle + \ell ^2\langle 12\rangle \langle 24\rangle \langle 41\rangle \over \ell ^2 (\ell - p_1 )^2 (\ell + p_4)^2 (\ell - p_1 - p_2)^2 }\;\;\mathop{\longrightarrow}^{\; \ell ^2=0}\int {\rm d}^2 \lambda _{\ell } {\delta ^3 ( P_{{\rm full}})\delta ^6 ( Q_{{\rm full}}) \langle 3 4 \rangle \over \langle 1 \ell \rangle \langle \ell 4\rangle (\ell + p_{3} + p_4)^2 }.\nonumber\\ \end{eqnarray} \tag{ 4.8 }$$

It is instructive to consider the forward limit entirely in terms of on-shell variables. We begin with the explicit integrand associated with the pre-reduced on-shell diagram:

$$\begin{equation} J_{\rm F} {d\tan _1 } \wedge {d\tan _2} \wedge {d\tan _3} \end{equation} \tag{ 4.9 }$$

where the labelling of the vertices are consistent with equation (4.2), and the Jacobian factor J_F may be determined according to the rules given in section 2.4.3

$$\begin{equation} J_{\rm F} = 1 + c_2 c_3 + s_1 s_2 s_3. \end{equation} \tag{ 4.10 }$$

The Grassmannian C_F in δ²(C_F · λ) is given as,

$$\begin{eqnarray} C_{\rm F}=\left(\begin{array}{@{}cccc@{}}1 & \displaystyle\frac{c_4 c_1}{1 + s_1 s_4} & 0 & -\displaystyle\frac{s_1 + s_4 }{1 + s_1 s_4} \\ 0 & \displaystyle\frac{s_1 + s_4 }{1 + s_1 s_4} & 1 & \displaystyle\frac{c_4 c_1}{1 + s_1 s_4}\end{array}\right), \end{eqnarray} \tag{ 4.11 }$$

with s₄(and c₄) given by:

$$\begin{equation} s_4 = {s_2 s_3 \over 1+ c_2 c_3 } , \quad c_4 = {c_2 + c_3 \over 1+ c_2 c_3 }. \end{equation} \tag{ 4.12 }$$

Starting from this integrand, we can explicitly preform two-step bubble reductions, under which the diagram is reduced to a tree-level four-point amplitude,

The result of this reduction, one finds that $F^{(4,1)}_4$ can be written in a nice dlog  form,

$$\begin{equation} F^{(4,1)}_4 = A^{\rm tree}_4 \int d\log \left({\tan _2 \over \tan _3} \right) \wedge d\log \left( {\tan _4 \over \tan _1}\right), \end{equation} \tag{ 4.14 }$$

with s₄(c₄) defined in equation (4.12) and s₅(c₅) given by,

$$\begin{eqnarray} s_5 &=& {s_4 + s_1 \over 1+ s_4 s_1 } , \quad c_5 = {c_4 c_1 \over 1+ s_4 s_1 }. \end{eqnarray} \tag{ 4.15 }$$

For simplicity we only focus on the positive branch, and s₅(c₅) are fixed by external kinematics,

$$\begin{equation} s_5 = - i {\langle 12\rangle \over \langle 42\rangle } = -i {\langle 34 \rangle \over \langle 42\rangle } , \quad c_5 = i {\langle 41 \rangle \over \langle 42\rangle } = -i {\langle 23 \rangle \over \langle 42\rangle }. \end{equation} \tag{ 4.16 }$$

The equivalence between equations (4.7) and (4.14) can be verified by exchanging λ_q in equation (4.7), in terms of on-shell variables,

$$\begin{equation} \lambda _{q} = {1 \over s_3}(\lambda _4 + c_3 \lambda _{\hat{3}} ) = {1 \over s_3}\bigg (\lambda _4 - {c_3 \over c_1} \lambda _3 + {c_3 s_1 \over c_1} \lambda _2 \bigg ) . \end{equation} \tag{ 4.17 }$$

Thus in summary, we see that the forward limit on-shell diagram not only reproduces the correct single cut, but it reveals the simplicity of the result: it is nothing but a product of two dlog 's. Note that this structure only reveals itself after one performs bubble reduction, and the dlog  form crucially depends on the presence of the Jacobian factor.

4.1.2. One-loop amplitude

We will now show that the full one-loop amplitude is also given by three dlog 's. Again, the one-loop on-shell diagram is given as:

The associated integrand is:

$$\begin{equation} J_{\rm loop} \;\;{ d\log \tan _1 } \wedge { d\log \tan _2 } \wedge { d\log \tan _3 } \wedge { d\log \tan _4 } , \end{equation} \tag{ 4.19 }$$

with the Jacobian

$$\begin{equation} J_{\rm loop} = 1 + s_1 s_2 s_3 + s_2 s_3 s_4 + c_2 c_3 + s_1 s_4 + c_2 c_3 s_1 s_4 . \end{equation} \tag{ 4.20 }$$

Just as with the case of forward-limit diagram, one can also perform bubble reductions on four-point one-loop diagram, leading to:

Remarkably, after the reduction the final form reveals itself to be a wedge product of dlog  forms:

$$\begin{eqnarray} &&A^{\rm loop}_4 = \int {\rm d}\log (\tan _7)\prod _{a=1,2}\delta ^{2|3}(C_{a}\cdot \Lambda ) \int {\rm d}\log \left( {\tan _2\over \tan _3}\right) \wedge {\rm d}\log \left( {\tan _4\over \tan _5} \right) \wedge {\rm d}\log \left({\tan _6\over \tan _1} \right) ,\nonumber\\ \end{eqnarray} \tag{ 4.22 }$$

with following identification:

$$\begin{eqnarray} &&s_5 = {s_2 s_3 \over 1+ c_2 c_3 } , \quad c_5 = {c_2 + c_3 \over 1+ c_2 c_3 } \nonumber\\ &&s_6 = {s_4 + s_5 \over 1+ s_4 s_5 } , \quad c_6 = {c_4 c_5 \over 1+ s_4 s_5 } \nonumber\\ &&s_7 = {s_6 + s_1 \over 1+ s_6 s_1 } , \quad c_7 = {c_6 c_1 \over 1+ s_6 s_1 } . \end{eqnarray} \tag{ 4.23 }$$

The parameterization of the Grassmannian is given as:

$$\begin{eqnarray} C_{\rm loop}=\left(\begin{array}{@{}cccc@{}}1 & c_7 & 0 & -s_7 \\ 0 & s_7 & 1 & c_7\end{array}\right) . \end{eqnarray} \tag{ 4.24 }$$

Since only s₇ (c₇) appears in the Grassmannian, the prefactor in equation (4.22) is simply the tree-amplitude as expected from the reduction diagram equation (4.21). Thus we see that the four-point amplitude given by a product of the tree-amplitude and three dlog s.

For the purpose of studying the singularities associated with the on-shell diagram, it is simpler to work with equation (4.19). It is easy to see that at the singularity s₄ = 0, the differential form reduces to the one of forward-limit diagram equation (4.9) with corresponding Jacobian and Grassmannian. Similarly s₁ = 0 corresponds to the forward-limit of opening up BCFW vertex with legs 2 and 3, as one would expect. What is more interesting is the singularities at s₂ = 0, as well as s₃ = 0. For instance at s₂ = 0, the differential form (4.19) reduces to

$$\begin{equation} J_{\rm s_2=0} \;\;{ d\log \tan _1 } \wedge { d\log \tan _3 } \wedge { d\log \tan _4 } , \end{equation} \tag{ 4.25 }$$

with the following Jacobian and Grassmannian

$$\begin{eqnarray} J_{\rm s_2=0} &=& (1 + c_3) (1 + s_1 s_4) , \end{eqnarray} \tag{ 4.26 }$$

$$\begin{eqnarray} C_{\rm s_2=0}=\left(\begin{array}{@{}cccc@{}} 1 & \displaystyle\frac{c_4 c_1}{1 + s_1 s_4} & 0 & -\displaystyle\frac{s_1 + s_4 }{1 + s_1 s_4} \\ 0 & \displaystyle\frac{s_1 + s_4 }{1 + s_1 s_4} & 1 & \displaystyle\frac{c_4 c_1}{1 + s_1 s_4}\end{array}\right). \end{eqnarray} \tag{ 4.27 }$$

Diagrammatically this singularity can be represented as¹⁷

This is nothing but the forward-limit singularity of one-loop four-point amplitude by opening up the BCFW vertex with legs 1 and 2, which can be shown explicitly by performing a triangle move on the diagram 4.3. The same analysis can be applied to the singularity at s₃ = 0. We can summarize our finding in the following diagram,

So all four physical forward-limit singularities are presented, two of them appear trivially whereas the other two appear in a rather non-trivial way by opening up the internal vertices. The fact that half of forward-limit singularities are obtained by opening up internal vertices, turns out to be a general feature of loop amplitudes in ABJM theory.

Note the singularities coming from opening internal vertex corresponds to collinear limit between q and BCFW shifted momentum $p_{\hat{1}}$ (and $p_{\hat{4}}$). To see that the collinear limit $p_{\hat{1}}||q$ indeed arises from the singularity denoted by the vertex θ₁, notice that the residue contains a tadpole attached to the line between $p_{\hat{1}}$ and $p_{\hat{2}}$. The constraint implied by such configuration can be found by explicitly identify the two legs on a four vertex:

In the above we have identified legs q and q'. Note that the resulting kinematic constraint forces $p_{\hat{2}}=p_{\hat{1}}$, and $q||p_{\hat{1}}$, as promised. This fact makes evident by writing down the one-loop integrand as a BCFW shift acting on the forward-limit,

$$\begin{eqnarray} \nonumber A^{1\hbox{-}{\rm loop}} &=&\sum _{\alpha =\pm }\frac{1}{2}\int {{\rm d}\log \tan \theta }\; \delta ^{2|3}(C\cdot \Lambda )\;F_4\\ &=&\delta ^3 ( P_{{\rm full}})\delta ^6 ( Q_{{\rm full}})\sum _{\alpha =\pm }\frac{1}{2}\int {\rm d}^2 \lambda _{q} \int { {\rm d} \theta \over s} { \langle 3 \hat{4} \rangle \over \langle \hat{1} q\rangle \langle q \hat{4} \rangle (q - p_{\hat{1}} - p_2)^2 } , \end{eqnarray} \tag{ 4.31 }$$

with

$$\begin{equation} \lambda _{\hat{4}} = -\alpha _4 (1/c_4)\lambda _4 + (s_4/c_4) \lambda _1 , \quad \lambda _{\hat{1}} = (\alpha _4 s_4/c_4) \lambda _4 -(1/ c_4 )\lambda _1. \end{equation} \tag{ 4.32 }$$

Above analysis also justifies the reason why we did not consider following singularity,

where one opens up internal vertex 3 in a different way. That is because this singularity correspondences to a higher dimension comparing to those in (4.28). In particular for the example given in (4.21), it requires $p_{\hat{1}} + q =0$, instead of the collinear limit $p_{\hat{1}}||q$. So it is a higher-dimensional singularity. Since one-loop four-point diagram is obtained by taking forward-limit of six-point tree-level amplitude, so this singularity can be traced back to six-point tree-level diagram. It turns out that such singularity correspondences to so-called soft singularity discussed in [11], which is also a higher-dimensional singularity there. We will encounter such singularities in higher-point and higher-loop amplitudes as well, but since they are higher-dimensional comparing to usual cut singularities, it's not necessary to consider them.

4.1.3. Comparison to known integrand

Finally, we can establish the equivalence of equation (4.19) with the known four-point one-loop integrand. We first note that the four-point one-loop amplitude in [11, 28] can be rewritten as:

$$\begin{equation} A_4^ {1\hbox{-}{\rm loop}} = A_4^{{\rm Tree}} \int {d}\log { (\ell -p_1)^2\over (\ell -p_1-p_2)^2 } {d}\log { \ell ^2\over (\ell -p_1-p_2)^2} {d}\log {(\ell +p_4)^2\over (\ell -p_1-p_2)^2} . \end{equation} \tag{ 4.34 }$$

Note that in this form, it is clear that the integrand not only is a total derivative, but furthermore since there is no (complex)spurious singularity, the integral integrates to zero [29]. The validity of the above form can be straightforwardly derived by using dual coordinates in five-dimensional embedding space. The integrand is given as:

$$\begin{equation} \int _{X^2_0=0} \frac{\langle X_0{d}X_0{d}X_0{d}X_0{d}X_0\rangle }{X_0^2}\frac{\langle 0,1,2,3,4\rangle }{(0.1)(0.2)(0.3)(0.4)},\quad (i.j)\equiv X_i\cdot X_j \end{equation} \tag{ 4.35 }$$

where X_i lives on the five-dimensional null projective space in which the three-dimensional Minkowski space is embedded, 〈⋅⋅⋅〉 indicate the contraction with the five-dimensional Levi-Cevita tensor, and the integration contour is understood to incircle the pole $X_0^2=0$. The dual positions are defined via x_i − x_{i + 1} = p_i and the labels are with respect to the following diagram:

We can parametrize the loop-region as:

$$\begin{equation} X_0=a_1X_1+a_2X_2+a_3X_3+a_4X_4+a_\epsilon \langle *,1,2,3,4\rangle . \end{equation} \tag{ 4.36 }$$

Using this parameterization, we can use the projective nature of the integrand to fix a₁ = 1, and integrate localizing $a^2_\epsilon$ on the pole, we find

$$\begin{equation} {\rm Equation}\ (4.35) =\int {d}\log a_2 {d}\log a_3 { d}\log a_4 . \end{equation} \tag{ 4.37 }$$

The dlog  form in equation (4.34) can be verified using the following identification¹⁸:

$$\begin{eqnarray} &&a_2=\frac{(0.4)(1.3)}{(0.3)(2.4)}=\frac{s(\ell +p_4)^2}{t(\ell -p_1-p_2)^2},\quad a_3=\frac{\ell ^2}{(\ell -p_1-p_2)^2},\quad a_4=\frac{s(\ell -p_1)^2}{t(\ell -p_1-p_2)^2}. \nonumber\\ \end{eqnarray} \tag{ 4.38 }$$

Armed with equation (4.37), we can verify that it is indeed equivalent to our on-shell dlog  form in equation (4.22). To simplify our task, we can simply go to the boundary of each logarithmic singularity and compare the two-form. For example, consider the singularity a₄ = 0 in equation (4.37), whose residue is simply the two-form:

$$\begin{equation} F^{(1,2)}_4 = {d}\log a_2 {d}\log a_3 ={d}\log \frac{(\ell ^{\prime }+p_4+p_1)^2}{(\ell ^{\prime }-p_2)^2} {d}\log \frac{(\ell ^{\prime }+p_1)^2}{(\ell ^{\prime }-p_2)^2} . \end{equation} \tag{ 4.39 }$$

where we defined ℓ' = ℓ − p₁ with ℓ'² = 0 under this cut. This is single cut corresponds to the on-shell diagram in equation (4.28), which, according to bubble reduction, can be written as,

$$\begin{equation} F^{(1,2)}_{4, {{\rm on}\hbox{-}{\rm shell}}} = {d}\log ({\rm csc}_3 -1 ) \, {d}\log {{\rm tan}_4 \over {\rm tan}_1 } . \end{equation} \tag{ 4.40 }$$

It is straightforward to check that these two dlog  forms match by expressing ℓ' in terms of on-shell variables

$$\begin{equation} \lambda _{\ell ^{\prime }} = {c_3 \over c_1(1 + s_3) } ( s_1 \lambda _3 + \lambda _2 ) . \end{equation} \tag{ 4.41 }$$

Similar analysis can be done for a₂ and a₃, thus verifying that the integrand in equation (4.31) indeed matches with the known answer equation (4.34).

After extensive analysis of four-point amplitude at one loop, we now move on to the six-point case. The contribution from the forward-limit of the eight-point tree-level amplitude is given as:

We also have two contributions that arise from the factorization of the six-point one-loop amplitude into a four-point one-loop amplitude multiplied with a four-point tree-level amplitude, given by:

Thus the six-point one-loop amplitude is given as:

Notably the above representation of six-point amplitude has manifest i → i + 2 cyclic symmetry. With this two-site cyclic symmetry, and the fact that only scattering amplitudes with even number of legs are non-vanishing in ABJM theory, it is guaranteed that all factorization poles are present. Like the case of four-point one-loop amplitude, actually only half of forward-limit singularities those are related by two-site cyclic rotations are manifest. So what we need to verify is the existence of the other half of single cuts that are related to the original BCFW bridge by a one-site shift. Again, like the case of four-point one-loop amplitude, these singularities are obtained by opening up the internal vertices. Following diagrams show the forward-limit singularities according BCFW bridge with legs 1 and 2,

On the other hand, unlike four-point one-loop amplitude, each term in equation (4.44) contains some unphysical poles, which we need to ensure those singularities are spurious, namely they cancel out each other in the sum. Indeed, all the spurious singularities appeared in pairs, as shown in following

Note actually the cancellation between the first and the last diagram can be traced back to the spurious singularities cancellation of tree-level eight-point amplitude.

One can actually generally argue that spurious singularities should always cancel in pair in the loop recursion relations. As they can be easily associated with singularities arising from the forward limit of sub amplitudes in a factorization diagram, and factorization singularities arising from a forward limit diagrams, represented as:

where we have used the red line to indicate the spurious singularity, and the blue line to represent the singularity that is used to construct the recursion. As one can see, the same spurious diagram can always appear in pair from two sources, exactly as discussed for $\mathcal {N}=4$ SYM [2].

Before proceeding to next section, let us pause here to make a few comments on the result of six-point amplitude. First of all equation (4.44) is just one possible way of presenting six-point one-loop amplitude, however it is a special one by making of manifest two-site cyclic symmetry. As we argued previously, in fact that this symmetry of ABJM amplitudes is rather general due to the simplicity of the on-shell diagrams of OG_k.

Apparently there are representations which do not have the manifest i → i + 2 symmetry, and one may need all kinds of equivalence moves to see this symmetry. As already shown in [11], by solving recursion relation in a particular way, all tree-level amplitudes can be presented in a form with manifest two-site cyclic symmetry. We will refer such representation of amplitudes as the 'canonic' representation. In general such representation is obtained by solving the recursion relation, (4.1), as follows:

• Firstly, we compute the factorization diagram as proceed in [11], where the BCFW bridge vertex, as a triangle, is connected to the rest diagrams through points only.
• Secondly, as for the forward-limit diagram, we always identify two legs on different (adjacent) vertices, then the BCFW vertex again forms a triangle.
• Finally lower-point or lower-loop amplitudes plugged in the recursion relations were obtained in the way described above.

We will show explicitly that higher-point/higher-loop amplitudes obtained from recursion relation in such way does enjoy the symmetry manifestly.

Before moving to the general one-loop amplitudes, let us take eight-point as one more example. It gets contributions from factorization diagrams with tree-level/one-loop four-point amplitude multiplied with one-loop/tree-level six-point amplitude, and the forward-limit diagram of ten-point tree-level amplitude. By solving the recursion in the canonic way we described in previous section, the result of eight-point one-loop amplitude can be represented in a form with manifest i → i + 2 symmetry,

where i → i + 2 means summing over all other diagrams related by two-site cyclic symmetry. It is easy to see that forward-limit singularity corresponding to opening vertex with legs 8 and 1 (one-site shifted from the original BCFW bridge, vertex with legs 1 and 2) comes from the diagrams in the second line of equation (4.48), again by opening up internal vertices,

Although we argued generally spurious singularities should cancel in pairs, it is still instructive to see this more explicitly. Actually the cancellation appears in a nice pattern:

• Box diagram contains four spurious singularities, where at each of the singularities the box reduces to a triangle. These singularities are cancelled out by the spurious singularities from the triangle diagrams–there are four triangle diagrams after summing over all i → i + 2 cyclic diagrams. One example is shown as follows
  
• Triangle diagrams contain another type spurious singularities, where at each of the singularities the triangle reduces to a bubble. Each triangle has three such singularities (in total there are 4 × 3 = 12). They are all cancelled out by the spurious poles from the bubble diagrams,
  
• Bubble diagrams have further spurious singularities, which are cancelled out between bubble diagrams themselves. For instance,
  
  which is essentially the spurious pole cancellation of eight-point amplitude at tree-level.

Detailed analysis of six-point and eight-point examples makes the pattern of general one-loop amplitudes quite clear, as we will present shortly in following section.

As found in [11], (2p + 4)-point tree-level amplitude from recursion relation can be represented as a sum of (2p)!/(p!(p + 1)!) diagrams constructed all by triangles, which are connected with each other through points only. Similarly, the solution of one-loop recursion relation for (2p + 4)-point amplitude contains ${2p+2}\choose{p}$ diagrams, which now are built up by m-gon's, with m = 2, ..., (p + 2), and each edge of such a polygon is connected with a triangle through the edge. The rest of such a diagram are triangles connected through points only, just as the case of tree-level amplitudes. Finally we sum over all possible two-site cyclic permutations, which makes the result with manifest i → i + 2 cyclic symmetry. So, again, all the physical factorizations and half of forward-limit singularities are manifest because of this symmetry. Whereas the other half of forward-limit singularities can be seen by opening up the internal vertices of the diagrams with 2-gon's, namely bubbles.

As for the spurious singularities, they are cancelled out in order: spurious singularities from diagrams with m-gon cancel the spurious singularities from diagrams with (m − 1)-gon; and diagrams with 2-gon's contain further singularities, which are cancelled out between themselves. Clearly one can see all these structures from the six- and eight-point examples.

It's not difficult to write down the results for higher-point one-loop amplitudes according to above description, here is another non-trivial example, ten-point one-loop amplitude,

As the case of tree-level amplitudes [11], to simplify the notation we have omitted the external legs in the diagrams: the diagrams should be understood with two external legs attached to each external vertex, for instance the first diagram in equation (4.53) is really

Summing over all i → i + 2 cyclic rotations, we find 56 different diagrams for ten-point amplitude, which is indeed the result of the recursion relation.

We can proceed to solve the recursion relation for two-loop amplitudes. There are $4(p+1){2p+1}\choose{p}$ diagrams for a (2p + 4)-point amplitude at two loops. The simplest four-point amplitude at two loops can be obtained by attaching a BCFW bridge to forward-limit of six-point one-loop amplitude, equation (4.44). The result presented in the canonic way is given by

we have also omitted the external legs to simplify the notation. In total there are four diagrams because the second and the last diagrams are symmetric under i → i + 2. Just as what we have found in the one-loop case, one half of physical forward-limit singularities can be trivially seen by opening up external vertices, while the other half is again hidden in the internal vertices. For instance the forward-limit singularity corresponding to opening up vertex with legs 2 and 3 can be shown as follows

where a triangle move has been applied to obtain the second diagram of forward-limit singularity. Similarly the result of two-loop six-point amplitude can be presented in the canonic form with manifest two-site cyclic symmetry

Again, besides those manifest physical poles, one half of forward-limit singularities can only be seen by opening up internal vertices,

In the above diagram, we circle out the vertices whose boundary would lead to the physical singularities. It is straightforward to obtain all 15 singularities in equation (4.58), and the combination of these singularities is precisely the forward-limit of one-loop eight-point amplitude, equation (4.48).

All those examples clearly show the advantage of the canonic representation, which not only has manifest of i → i + 2 cyclic symmetry, but also makes the hidden forward-limit singularities transparent: they all come from the diagrams with bubbles.

A remarkable property of the four-point, and six-point amplitudes displayed in equations (4.18), (4.44), (4.55) and (4.57), is that through equivalence moves, they can all be reduced to tree-level on-shell diagrams. For example, consider one of the four-point two-loop diagram given in equation (4.55):

where we've demonstrated the reduction procedures by outlining, in red, the triangle subgraphs that undergoes the equivalence move. All four-point bubbles are automatically reduced, appearing as vacuum bubbles. The fact that all four-and six-point diagrams can be reduced to the tree-diagrams can be understood easily from the dimension of the top-cell in OG_k: k(k − 1)/2. Since the number of kinematics constraint for any on-shell diagram is given by 2k − 3, for k = 2, 3 the dimension of the top-cell is exactly the same as the number of kinematic constraint. Thus all higher dimensional on-shell diagrams must be equivalent to the top-cell, which is simply tree diagrams, with the extra degrees of freedom decoupled from the Grassmaniann. Thus we see for k = 2, 3 the integrands only has logarithmic singularities, just as the MHV amplitudes in $\mathcal {N}=4$ SYM.

We have seen that the on-shell diagram solution automatically gives an answer where uniform transcendentally is manifest. Indeed the known answer at of one-loop and two-loop amplitudes satisfy this result [13, 27, 30, 31], as well as the all order conjecture for four-point amplitudes [32]. Our result implies that L-loop four- and six-point amplitudes are given by uniform transcendental L functions, where there is one two-particle cut discontinuity at each loop order.

Finally, since we have verified that the solutions to the loop recursions indeed reproduce all physical singularities, this implies all reduced medial graphs correspond to the leading singularity of ABJM loop amplitudes. Since each medial graph is one-to-one correspondence to the cell in the OG_{k +}, this proves that the residues of the orthogonal Grassmannian integral [9], which are evaluated on the boundaries of the top-cell indeed correspond to the leading singularities.