Positroid Stratification of Orthogonal Grassmannian and ABJM Amplitudes

A novel understanding of scattering amplitudes in terms of on-shell diagrams and positive Grassmannian has been recently established for four dimensional Yang-Mills theories and three dimensional Chern-Simons theories of ABJM type. We give a detailed construction of the positroid stratification of orthogonal Grassmannian relevant for ABJM amplitudes. On-shell diagrams are classified by pairing of external particles. We introduce a combinatorial aid called `OG tableaux' and map each equivalence class of on-shell diagrams to a unique tableau. The on-shell diagrams related to each other through BCFW bridging are naturally grouped by the OG tableaux. Introducing suitably ordered BCFW bridges and positive coordinates, we construct the complete coordinate charts to cover the entire positive orthogonal Grassmannian for arbitrary number of external particles. The graded counting of OG tableaux suggests that the positive orthogonal Grassmannian constitutes a combinatorial polytope.


Introduction and Conclusion
The study of scattering amplitudes in gauge theories has gone through many stages of exciting developments (see [1] for an up-to-date review). A common theme in many recent breakthroughs is reformulation of the gauge theory in such a way to uncover hidden structures that are hardly visible in the traditional approach.
The Grassmannian integral [2] is a relatively new reformulation notably far removed from the traditional approach. Tree amplitudes and loop integrands of a planar gauge theory are produced from a contour integral over the Grassmannian G(k, n), where n is the total number of external particles and k is the number of external particles with negative helicity. The geometric structure makes the conformal symmetry and dual conformal symmetry completely manifest while relegating locality and unitarity to an emergent property. Although the Grassmannian integral inherits some crucial features from its predecessors such as the twistor string theory [3,4] and the BCFW recursion relation [5,6], its physical origin had remained elusive for years.
In a remarkable paper [7], Arkani-Hamed et al. introduced the notion of 'on-shell diagrams' which in a sense provides a microscopic structure underlying the Grassmannian integral. The vertices of on-shell diagrams are the gauge invariant three-point amplitudes with (+ + −) or (+ − −) helicity assignments. They showed how to rewrite the vertices in terms of integrals over G (1,3) or G (2,3). By integrating out internal lines, they combine small Grassmannian integrals to form a big integral over G(k, n) for arbitrary k and n. The BCFW deformation of momenta appears naturally in on-shell diagrams and 'bridges' two external legs with a new internal line and two vertices. While probing the structure of Grassmannian integral at a deeper level, the authors of [7] also encountered a number of modern topics in mathematics such as affine permutation, positive Grassmannian, positroid stratification, and cluster algebra. They also noted that the on-shell diagram approach can be applied to any gauge theory in four dimensions, although it takes by far the simplest form for the N = 4 super-Yang-Mills theory (SYM 4 ).
In three dimensions, a class of N = 6 superconformal Chern-Simons matter theories [8][9][10][11][12], widely known as ABJM theory, serves as a main testing ground for novel methods on scattering amplitudes. An initial step toward the on-shell diagram approach to ABJM theory was already taken in [7], based on the an integral over the orthogonal Grassmannian OG(k, 2k) [13]. The on-shell diagrams of ABJM theory consist of a unique quartic vertex and an internal line. The permutation governing the planar diagrams is a complete pairing of the 2k external particles. Two diagrams sharing the same permutation are equivalent to each other up to Yang-Baxter equivalence moves. The BCFW bridge connects two external legs by creating an extra vertex between them.
Recently, Huang and Wen [14] further studied the on-shell diagrams for ABJM amplitudes. They refined the on-shell diagrams with some sign factors to account for the two disjoint 'branches' of orthogonal Grassmannian. This is crucial since the full tree amplitudes receive contributions from both branches, as noted earlier in [15]. By rewriting the BCFW recursion relation for ABJM theory [15] in terms of on-shell diagrams, they constructed on-shell diagram representation of all tree-level amplitudes. They also introduced a set of coordinates in which all consecutive minors take a simple form. Finally, they defined the notion of positive orthogonal Grassmannian. In contrast to the ordinary Grassmannian G(k, n), it is slightly non-trivial to define reality conditions on OG(k, 2k).
The aim of this paper is to cover a topic that was notably missing in [7,14]. Given a permutation, it is desirable to select a particular representative of the equivalence class of on-shell diagrams, and assign canonically positive coordinates to it. The construction of canonical coordinates elucidates how the Grassmannian integral is decomposed into a series of BCFW bridging. The canonical coordinates also enjoy several nice properties; for instance, the measure of the Grassmannian integral takes a simple 'd(log)' form, and the boundaries of the positive orthogonal Grassmannian become zero loci of the coordinates. This problem of assigning canonical coordinates was solved completely in four dimensions in [7] using a relation to the mathematics of 'positroid stratification' [16,17]. We will solve the problem in three dimensions by introducing a combinatorial device similar to those of [16], which we call 'OG tableaux'. To our knowledge, mathematical studies of positive orthogonal Grassmannian remain incomplete and not readily accessible to physicists. See, e.g., [18] for a related work.
The rest of this paper is organized as follows. Sec. 2 is a brief review of what is known for on-shell diagrams for ABJM theory from earlier works [7,14]. The elementary 4-particle vertex can be understood both as a BCFW bridge and an integral over OG (2,4). Integration over internal lines 'amalgamates' copies of OG (2,4) and builds up bigger OG(k, 2k). In Sec. 3, we take a closer look at the geometry of orthogonal Grassmannian. We examine real slices of OG(k, 2k) and discuss how to define positivity on a real slice. The real version of the BCFW bridge OG (2,4) and amalgamation naturally suggests how to introduce manifestly real coordinates on the orthogonal Grassmannian. Sec. 4 reports the main results of this paper. We begin with mapping the on-shell diagrams to what we call 'OG tableaux'. The tableau notation serves several purposes. It fixes the ambiguity coming from Yang-Baxter equivalence relation in a canonical way. That helps us to count distinct on-shell diagrams, which are interpreted geometrically as cells of the positive Grassmannian. Moreover, the OG tableaux can be used to assign 'canonical positive coordinates' for all on-shell diagrams. The integration measure in the canonical coordinates is a product of simple d log factors. The OG tableaux also help us to study mathematics of OG + (k, 2k). The positive Grassmannian G + (k, n) is known to form a combinatorial polytope called 'Eulerian poset' [19]. The graded counting of OG tableaux suggests that the positive orthogonal Grassmannian OG + (k, 2k) similarly defines an Eulerian poset at each k. Geometrically, the graded counting hints at the possibility that OG + (k, 2k) may have a topology of a ball. We verify this conjecture for k = 2, 3 and leave the generalization to higher k as an open problem.
This paper focuses on a formal aspect of orthogonal Grassmannian and makes little direct contributions to ABJM amplitudes. However, the results of this paper clearly opens up a few directions of further research. Here, we list three prominent possibilities.
First, the issue of Yangian symmetry, which unites the ordinary and dual superconformal symmetries, could be revisited. While there are strong evidences for the Yangian symmetry of ABJM amplitudes [13,15,[20][21][22], a formulation with manifest dual superconformal symmetry has not been found. For SYM 4 , such a dual formulation was found earlier [23,24] and laid a foundation for further discoveries such as the 'amplituhedron' [25]. In [7], the Yangian symmetry was interpreted geometrically as diffeomorphisms which leave the measure on G + (k, n) invariant. A similar interpretation for OG + (k, 2k), if possible, would shed new light on the Yangian symmetry of ABJM amplitudes.
Second, our work could be related to twistor string models for ABJM amplitudes. For tree-level amplitudes of SYM 4 , the equivalence between the twistor string formula [4] and the Grassmannian formula [2] was established in [26]. Simply put, the derivation consists of three steps: specifying the integration contour of the Grassmannian integral, deforming its integrand with no loss of residues, and integrating out some variables. Along the same line of reasoning, a twistor string formula was proposed in [27] (and recently rederived from different viewpoints in [28,29]), but the derivation was less solid due to limited understanding of the integration contours. The canonical coordinates defined in this paper could be useful in finding a refined derivation comparable to that of [26].
Third, the formal structure of the positive orthogonal Grassmannian could be probed at a deeper level. To define the canonical coordinate system, we suppressed the Yang-Baxter equivalence moves in a particular 'frame'. If we move to another frame, the new coordinates would be related to the old ones in a non-trivial way. For SYM 4 , the coordinate transformation has an intriguing connection to the mathematics of cluster algebra [30,31]. It would be interesting to figure out the ABJM counterpart of the story. See a recent work by Huang, Wen and Xie [32] for a related discussion.
We conclude this introduction with some shorthand notations. We will write (P)OG for (positive) orthogonal Grassmannian, and abbreviate OG(k, 2k) and OG + (k, 2k) by OG k and POG k , respectively.
Note added While an early version of this paper was being revised for publication, we received a preprint by Lam [38] which rigorously prove a mathematical theorem stating that POG k defines an Eulerian poset for all k.

On-shell Diagrams for ABJM Amplitudes
In this section, we review some salient features of the on-shell diagrams for ABJM amplitudes [7,14]. We first recall the kinematics of the ABJM amplitudes and the definition of the OG integral to set the stage for the on-shell diagram. After introducing the fundamental building blocks, a quartic vertex and an internal line, we examine two ways to construct the most general on-shell diagrams. One is BCFW bridging, which enables us to add a vertex one at a time to a given diagram. The other is amalgamation, which merges two diagrams into a larger one by integrating over an internal line. For both methods, we explain how the diagrammatics is reflected in the OG integral, thereby making the microscopic decomposition of OG integral manifest. Finally, we comment briefly on the 'Yang-Baxter' equivalence relation for different diagrams corresponding to the same amplitude as well as the bubble diagrams.
In this representation, the superconformal generators come in three types: (2.1) The superspace notation will guarantee the invariance of the amplitudes under the (Λ∂/∂Λ) generators. Let us decompose the (ΛΛ) generators, In a scattering process, the invariance under p αβ and q αI can be imposed by the supermomentum conserving delta functions 3) The r IJ invariance introduces a coset O(2k − 4)/U(k − 2) for the 2k-point amplitude [20]. The coset structure was a precursor to the OG integral for ABJM theory [13]. Once the invariance under the (ΛΛ) generators is confirmed, the invariance under the remaining (∂ 2 /∂Λ∂Λ) generators will follow from the superconformal inversion, which acts on Λ as Fourier transformation (Λ ↔ ∂/∂Λ).
The spinors λ i are contracted via the invariant tensor αβ ( 12 = − 12 = 1) of SL(2, R) Spin(1, 2) Lorentz group, The ABJM theory contains two matter multiplets with opposite gauge charge. The particle/anti-particle superfields take the form The color-ordered tree-level super-amplitudes, A 2k (Λ 1 , · · · , Λ 2k ), are functions of Λ i . Following the convention of [13,15], we choose to associate Λ odd/even toΦ/Φ multiplet. As noted in [20], the kinematics and the multiplet structure imply the 'Λ-parity', and symmetry under the cyclic shift by two sites, OG integral A central object of interest in this paper is the OG integral [13]: The integration variable C is a (k × 2k) matrix. The dot products denote (C · C T ) mn = C mi C ni , (C · Λ) m = C mi Λ i . The i-th consecutive minor M i of C is defined by In [13], this formula was conjectured to reproduce the 2k-point tree level amplitude upon a suitable choice of integration contour. The conjecture was verified up to k = 4 in [15]. We will review the on-shell diagram approach to the OG integral (2.8), initiated in [7] and elaborated in [14], in a way to facilitate the introduction of the positroid stratification to be presented in Sec. 4.

Building blocks
The on-shell diagrams for ABJM amplitudes are planar diagrams drawn on a disk with 2k boundary points representing cyclically ordered external particles.
The graphical notations in (2.10) and (2.11) are not fully well-defined as they stand. The 4-point amplitude A 4 does not have a Z 4 cyclic symmetry. Instead, it is odd under a cyclic shift by two sites, (2.12) Accounting for this symmetry, when the simpler notation is likely to cause confusion, we will use a refined notation for the vertex.

BCFW bridge
Vertex as BCFW bridge As shown in [7,14], it is possible to build up complicated on-shell diagrams from a simpler one by adding vertices via 'BCFW bridges'. In this subsection, we review how to interpret the elementary vertex (2.13) as a BCFW bridge, and establish our convention for coordinates and sign factors.
We recall from [15] that the BCFW deformation acts as an O(2k, C) rotation on the kinematic variables Λ i , which leaves the total super-momentum invariant. For the fourparticle vertex, the total super-momentum is Moving λ 3 and λ 4 to the right-hand-side in (2.15) and squaring, we find The sign factor σ defines two 'branches' of kinematic configuration. As we will see shortly, the same σ will define two branches of OG 2 .
To reveal the structure of the BCFW bridge, we begin with the vertex Inserting two identities, 1 changes the momentum-conserving delta function into Taking the change of variables and integrating out r 3 and r 4 , we find where the first delta function is localized at t 3 + t 4 = 0 or π. The fermionic delta function can be rearranged as Collecting all the ingredients, we obtain Pictorially, as shown in Fig. 1(a), the result might be summarized as building a BCFW bridge between two 'free propagators' (14) and (23). During the derivation, we chose two adjacent legs {1, 2} as 'sources' and the other legs {3, 4} as 'sinks'.
Other choices of sources and sinks are possible. Up to the cyclic symmetry (2.12), the only other possibility for adjacent source legs is {1, 4}, as depicted in Fig. 1(b). With the branch parameter defined by 14 = σ 23 , A 4 becomes We may also consider taking non-adjacent source legs, {1, 3} or {2, 4}. They will not be used in subsequent sections, so we omit them here. Interested readers are referred to [14].
General BCFW bridging The BCFW bridge can be used to add a vertex to an on-shell diagram at a fixed number of external legs. The idea is sketched in Fig. 2. acting on the kinematic variables Λ i . It is worth noting whether the rotation matrix belongs to the orientation preserving SO(2k, C) subgroup of O(2k, C) or the orientation reversing one. In the case depicted in Fig. 1(a), σ = +1 preserves orientation. In contrast, in the case of Fig. 1(b), σ = −1 is the one that preserves orientation.
Vertex as OG 2 We proceed to relate the BCFW bridge to OG 2 . Consider the integral, (2.28) We will show that L 4 reproduces A 4 in the form (2.24) or (2.26). For the former, depicted in Fig. 1(a), the choice of source legs {1, 2} naturally translates into a gauge fixing of the matrix C with 'pivot' columns {1, 2}, We can proceed in two different ways. First, as explained in [15], we can solve the bosonic part of the kinematic delta function δ 4|6 (C · Λ). Since there are four delta functions for four variables, for generic values of λ i (i = 1, 2, 3, 4), the solution to the delta function constraint is unique. Inserting the solution back to the integral, we arrive at the expression for A 4 in (2.18). Alternatively, we can leave δ 4|6 (C · Λ) aside and insert the gauge-fixed C (2.29) into the orthogonality constraint delta function, which takes a similar form as (2.20). Taking the change of variables, and integrating out (r 1 , r 2 , t 2 ), we obtain (2.32) in agreement with (2.24). Repeating the same analysis with pivot columns {1, 4}, we find that L 4 (Λ 1 , Λ 2 , Λ 3 , Λ 4 ) reduces precisely to A 4 given by (2.26). Thus, the OG integral provides a geometric representation of the 4-point vertex.

Amalgamation and permutation
An on-shell diagram for k > 2 can be constructed by gluing two or more diagrams together. The corresponding OG k>2 is obtained by 'amalgamating' a collection of OG 2 's by integrating out the internal lines. The amalgamation proceeds as follows (see Fig. 3).
1. Put two on-shell diagrams together, preserving all external legs. The resulting Cmatrix is a direct product of the two sub-matrices and thus lives in OG k+k .
Integrating out the internal line explicitly, we have The resulting matrix C is represented by wherem runs over {m = 2 · · · k, n = 1 · · · k } andĩ over {i = 1 · · · 2k − 1, j = 1 · · · 2k − 1}. As shown in [14], it is straightforward to verify that C respects the orthogonality condition C ·C T = 0 if C and C satisfy the same condition, Equivalences move and reducible diagrams Different on-shell diagrams corresponding to the same amplitude can be related to each other through a series of equivalence moves. The elementary move for ABJM amplitudes is the Yang-Baxter-like move depicted in Fig. 4. The lines 1 to 6 in the figure may be internal or external. In view of the BCFW bridging, the equivalence relation simply amounts to two different Euler angle decomposition of the same SO(3) rotation matrix.
As the name 'Yang-Baxter' suggests, the equivalence move leaves the permutation among external particles invariant. If we draw all possible diagrams with the same permutation content, we may encounter bubble diagrams such as those in Fig. 5. It was shown in [14] that the bubbles can be completely factorized from the rest of the on-shell diagram, leaving an integral with a d log measure. In the rest of this paper, we will work exclusively with bubble-free diagrams.

Reality and Positivity of Orthogonal Grassmannian
The orthogonal Grassmannian OG k is a subspace of G(k, 2k) restricted by the orthogonality constraint. In this section, we shall examine the notion of reality and positivity for OG k . The real slice of OG k is determined by the reality condition on external kinematic variables. For discussion on positivity, a particular reality condition, called the 'split signature' [14] condition, turns out to be the most convenient. Performing a 'Wick rotation' on the kinematic and the BCFW variables, we can rewrite the BCFW bridging rule with manifestly real and positive coordinates on OG k . Following [7,14], we spell out the conversion rule which enables us to read off the postive C-matrix directly from an on-shell diagram without going through BCFW bridging one at a time.

Complex OG
Here we review the geometry of OG k on which the integral (2.8) is defined [15]. Recall that the ordinary Grassmannian G(k, n) is the moduli space of k-planes in n dimensions. In the standard matrix representation, G(k, n) is described by a (k × n) matrix C with rank k subject to the 'gauge symmetry' C ∼ g C with g ∈ GL(k). OG k is a subspace of G(k, 2k) subject to an 'orthogonality' constraint (C · C T ) mn = C mi C ni = 0 (m = 1, . . . , k; i = 1, . . . , 2k). The constraint and the GL(k) gauge symmetry determine the dimension of OG k as It is also known that OG k is isomorphic to the coset, OG k = O(2k)/U(k). Since O(2k) contains two disjoint SO(2k) components, OG k is also decomposed into two disjoint subspaces. We will call them two 'branches' of OG k .
Let us discuss how the two branches are defined in terms of coordinates. Recall that the Plücker coordinates of G(k, n) are determinants of (k×k) submatrices of C, regarded as homogeneous coordinates of some projective space. The Plücker coordinates are subject to quadratic algebraic relations originating from linear dependencies among the columns of C. Coming back to OG k , let I = {i 1 , i 2 , . . . , i k } be an ordered set of indices labeling k distinct columns of C, and M I = (i 1 , i 2 , · · · , i k ) := det(C i 1 , C i 2 , · · · , C i k ) be the corresponding Plücker coordinate. As noted in [13], the orthogonality constraint C · C T = 0 imposes linear relations among M I 's in addition to the quadratic relations for G(k, n). Define the complement of I byĪ = {ī 1 ,ī 2 , . . . ,ī k } such that {i 1 , . . . , i k ,ī 1 , . . . ,ī k } is an even permutation of {1, 2, . . . , 2k}. The linear relation can be written as The overall sign factor σ = ±1 defines the two branches of OG k . This linear relation implies a quadratic relation for consecutive minors on both branches [13], The orthogonality constraint can be expressed in terms of Plücker coordinates as where a runs over all columns of the C-matrix. and renaming the coordinates as (X, Y, Z) = (M 14 , M 24 , M 34 ), we find that each of the two branches, OG ± 2 , is described by an algebraic variety, which is topologically a CP 1 . We can compare this with the coset description, Repeating the algebraic analysis for higher k would be possible but quite cumbersome. For k = 3, we can use the coset description to find

Reality and positivity of OG
Reality In the spinor helicity formulation p αβ = λ α λ β with real momentum, the spinor λ α should be real or purely imaginary. Our convention is such that the spinor λ α is real for outgoing particles and purely imaginary for incoming particles. If we want to work with strictly real momenta, we have to assign reality conditions on each of the external legs.
Momentum conservation forces all on-shell diagrams to have the same number of incoming and outgoing particles. First, the elementary 4-vertex (2.13) does not vanish only if two of the particles are incoming and the other two are outgoing. If all four particles are outgoing, the total momentum, λ 1 λ 1 + λ 2 λ 2 + λ 3 λ 3 + λ 4 λ 4 is positive definite or negative definite, respectively, so cannot vanish. If particle 1, 2, 3 are outgoing and 4 incoming, λ 1 λ 1 + λ 2 λ 2 + λ 3 λ 3 generically has rank two while λ 4 λ 4 has rank one, in conflict with momentum conservation. The other two unbalanced cases (4 incoming or 3 incoming + 1 outgoing) can be treated similarly. Next, an internal line (2.14) always connects an incoming particle and an outgoing particle, hence the balance between incoming and outgoing particles continue to hold for arbitrary on-shell diagrams.
The reality conditions for the kinematic variables λ i naturally translate into the reality conditions for the matrix C representing a point on OG k . The linear delta function δ(C ·Λ) in (2.8) requires that (3.10) We may use the GL(k) gauge symmetry to make the k columns corresponding to incoming particles to form a (k × k) identity matrix, and denote the other k columns by a non-trivial (k × k) matrix: where the index i runs over incoming particles andj over outgoing particles. Reality of λ implies that ic ij is a real matrix. The orthogonality constraint C · C T = 0 then implies that ic ij is an element of O(k, R). Thus, the reality condition for λ i defines a real slice of OG k . The two disconnected components of O(k, R) correspond to the two branches of OG k . For instance, the real slices of k = 3 are two copies of SO(3) = RP 3 .
We find it instructive to give a gauge invariant, geometric description of the reality conditions. We will focus on the simplest example for k = 2. In the algebraic description (3.7), in a coordinate patch with Z = 0, we can consider four distinct reality conditions: But, the condition (d) with X 2 + Y 2 + Z 2 = 0 yields an empty set in CP 2 . The three remaining real slices are identified with great circles, S 1 = SO(2, R), embedded in S 2 = CP 1 as depicted in Fig. 6. The intersections among different real slices are denoted as We should emphasize that the reality condition is a gauge invariant notion. Although it is sometimes useful to align the gauge choice with the reality condition as in (3.11), other gauge choices might be more convenient for certain purposes. In Sec. 4, we will use gauge choices different from (3.11). Figure 6. Real slices of a OG 2+ .
Positivity As pointed out in [14], the 'split signature' reality condition, in which all odd-labelled particles are incoming and all even-labelled ones outgoing, deserves a special attention. It is the only reality condition that respects the cyclic symmetry of A 2k (2.7). Moreover, the split signature allows for a simple notion of 'positivity'. Following [14], we do a 'Wick rotation' on the C-matrix and λ 2i−1 simultaneously such that all elements of C and all λ i are real, while the orthogonality constraint takes the form, C · η · C T = 0 , η ij = (−1) i δ ij = diag(−, +, · · · , −, +) . (3.14) In this convention, the positivity defined in [14] asserts that all ordered minors of C are non-negative. 2 This particular definition of positivity picks out one of the two branches of OG k at each k. For instance, we have These results strongly suggest that the positive orthogonal Grassmannian (POG) has the same dimension as the real slice of OG. One of the main goals of this paper is to introduce a complete set of coordinate patches for POG for all k. We give a simplest example (k = 2) here for illustration and discuss the general construction in Sec. 4. Let us choose a gauge such that The complete solution to this problem is As t approaches ∞, we can take a gauge transformation to find Including the 'point' at t = +∞, the full geometry of POG 2 is an interval with two endpoints included. In Fig. 6, the POG 2 is identified with the interval A − C + .

Conversion rule
Recall that the elementary vertex, A 4 , is resolved in two ways, (2.24) and (2.26), according to the choice of pivot columns for OG 2 . The first one (2.24) contains a delta-function, We perform a Wick rotation on the odd-labelled particles as Λ 2i−1 → −iΛ 2i−1 and on the BCFW variables as t → it. After the Wick rotation, up to an overall phase, the delta-function becomes Similarly, in the other case (2.26), the Wick rotation gives We apply the same Wick rotation to the internal line (2.14) as well, where Λ 1 and Λ 2 represent odd and even multiplets, respectively.
As explained in [7,14], we can collect the linear relations imposed by the delta-functions and read off the components of the C-matrix without going through BCFW bridging and amalgamation one at a time. To find the component c ij in the gauge-fixed form, λ i +c ij λj = 0, we trace all possible paths p from the source i to the sinkj in the corresponding on-shell diagram. Each internal line (3.24) contributes a factor of (−1), since δ(λ 1 + λ 2 ) implies λ 1 = −λ 2 . At each vertex, we pick up (−f v ), where f v is one of the matrix elements of the (2 × 2) matrix defined at the vertex, λ i + c (v) ij λj = 0, chosen by how the path traverses the vertex. 3 The vertex factors in the two cases of BCFW bridging considered above are summarized in Fig. 7. The factors in Fig. 7(a) are read off from (3.22) and those in Fig. 7(b) from (3.23). Combining the contributions from internal lines and vertices, we arrive at a simple conversion rule for c ij : (3.25) The matrix elements are manifestly real. The factors of (−1) have cancelled out completely, since a path always traverse n p internal lines and n p + 1 vertices. We will study how positivity restricts the matrix elements in the next section.

Positroid Stratification
The 'positroid stratification' of G + (k, n) was developed in [16,17] and brought to physics in [7]. It relates the combinatorics of on-shell diagrams to the geometry of G + (k, n). At an intermediate step, a notation similar to Young tableaux plays a crucial role, which encodes the linear dependency among the columns of the matrix representative of G + (k, n). In this section, we shall develop a similar story for the positive stratification of OG. Some partial results in this direction were obtained in [14].
As a first step, we introduce an auxilary aid called 'OG tableau' which encodes the combinatorics of on-shell diagrams. It naturally provides a set of canonical gauge choices for the C-matrix such that the restriction imposed by positivity takes a simple form. With the help of the OG tabeaux, we construct the 'canonical' coordinate system, which exhibits positivity by construction for each cell of POG k for all k. We verify that the canonical coordiates obtained from the positive stratification agrees with the ones given by the conversion rule derived in Sec. 3.
In the last subsection, we turn to the mathematics of POG k . It is known that the positive Grassmannian G + (k, n) forms a combinatorial polytope called 'Eulerian poset' for each (k, n) [19]. The graded counting of OG tableaux suggests that POG k may also define an Eulerian poset for each k. Geometrically, it seems plausible that POG k has a topology of a ball. We verify this conjecture for k = 2, 3. Finally, we give a preliminary discussion on the boundary operation on POG k . We expect that a more complete study of the boundary operation will help us better understand the topology and geometry of POG.

OG tableaux
As we discussed earlier, on-shell diagrams are determined by a splitting of {1, 2, . . . , 2k} into k pairs of integers, {(a 1 b 1 ), · · · , (a k b k )}. By convention, we set a m < b m for all m. Barring bubbles and modulo Yang-Baxter equivalence moves, the diagrams are in oneto-one correspondence with the pairings. The total number of inequivalent diagrams are (2k)!/(2 k k!) = 1, 3, 15, 105, · · · . Some subclasses of diagrams are easy to enumerate. For a given k, there is a unique 'top' diagram with the maximal number, k(k − 1)/2, of vertices (see Fig. 8). All external legs in a top diagram are paired diagonally. At the opposite extreme, there are 'bottom' diagrams with no vertex (see Fig. 9). The counting of non-intersecting diagrams connecting 2k cyclically ordered points is an elementary problem in combinatorics. The answer is the k'th Catalan number C k [35], To enumerate the diagrams with intermediate number of vertices, and relate them to subspaces of OG k , we introduce a new notation called 'OG tableaux'. There are two related versions of the tableaux: 'unfolded' and 'folded'.
Unfolded tableaux Fig. 10 illustrates how to map an on-shell diagram to a tableau with an example. We first prepare the off-diagonal upper-left half of a (2k ×2k) chessboard. The diagonal boxes of the chessboard are numbered from 1 to 2k, with 1 placed at the lowerleft corner and 2k at the upper-right corner. The empty tableau contains 2k(2k − 1)/2 boxes, in one-to-one correspondence with a pair chosen from {1, . . . , 2k}. For each pair  a 1 b 1 ), · · · , (a k b k )}, we put a 'hook' on the corresponding box. If we extend the right/lower arm of the hook horizontally/vertically toward the diagonal, we recover precisely a copy of the on-shell diagram. So far, the only novelty of the tableau notation is that it defines a canonical way to fix the Yang-Baxter ambiguity. Folded tableaux It is possible to 'fold' the unfolded tableaux without reducing its information content. We begin with examining each of the (k − 1) columns and (k − 1) rows of the unfolded tableaux. If a column/row contains no hook, all the boxes in the column/row are removed. The surviving boxes can be moved in horizontal or vertical directions and fit into a (k × k) chessboard. See Fig. 11 for an illustration. The content of an on-shell diagram is preserved through the folding procedure, so the map between on-shell diagrams and folded tableaux is still bijective. The labels for source legs {a i } are attached to the boundaries at the bottom edges of the folded tableau, while those for sink legs {b i } are attached to the right edges. Thus, on-shell diagrams with the same set of sink/source legs share the same configuration of boxes for the folded tableaux, but are distinguished by the placement of hooks.
One of the fundamental feature of an on-shell diagram is its number of vertices ('level'). We can classify the on-shell diagrams according to the type of the corresponding (folded) tableau and the level. The full classification for k = 2, 3 is given in Fig. 12. (1) We introduced the unfolded tableaux first and switched to the folded tableaux for a pedagogical reason. But, it is no more difficult to work directly with the folded tableaux. We shall construct the OG k tableaux as follows (see Fig. 13).
2. Remove some boxes among the k(k − 1)/2 boxes in the lower-right off-diagonal half, such that the remaining boxes form a Young tableau of at least k(k + 1)/2 boxes.
3. Assign 1 to 2k to the bottom and right edges of the stack of boxes in order.
4. Mark a box with a hook for each column, under the restriction that each row should contain one and only one hook.
Each marked tableau gives an on-shell diagram. Its level can be determined as follows.
1. Count how many boxes were removed from the (k × k) chessboard.
2. Compute the number of row permutations required to arrange marked boxes diagonally from the bottom left corner to the top right.
3. Add up the two numbers, then subtract it from k(k − 1)/2.
As a special case, a top diagram requires no removal of boxes or rearrangement of marked boxes. So, the prescription above gives the expected level k(k − 1)/2. The readers are invited to test the prescription against less trivial examples in Fig. 12.

Canonically positive coordinates
Positroid stratification relates the combinatorics of on-shell diagrams and OG tableaux to the geometry of POG k . Each tableau is mapped to a subspace of POG k . Importing the terminology from the positroid stratification of G(k, n) [16,17], we will call the subspaces 'cells' of POG k . The number of vertices of an on-shell diagram (level of its tableau) equals the dimension of the cell. In this subsection, we will introduce a canonical coordinate system to the cells. The cells sharing the same unmarked tableau (see Fig. 12) will share a common coordinate patch.
From tableaux to matrices The (folded) OG tableaux reveals a decomposition of OG k similar to the standard Schubert decomposition of G(k, n). The source legs translate into 'pivot' columns. Let {p m } (m = 1, . . . , k) be a monotonically increasing labels for the source legs. We put C into a row-echelon form by setting C m,pm = 1, C m,i<pm = 0 and C n =m,pm = 0. An example is given in Fig. 14. Figure 14. Source legs of tableaux translate into pivot columns of C-matrices Note that using the GL(k) gauge symmetry, we can write every element of OG k in the rowechelon form. The whole OG k can be written as a disjoint union, OG k = λ Ω λ , where λ runs over unmarked tableaux. One can interpret Ω λ as disjoint coordinate patches, covering the whole OG k , then an on-shell diagram always belongs to a particular λ.
Unlike the Schubert decomposition of G(k, n), in the current setup, the orthogonality constraint restricts the allowed set of pivot columns. Since the folded tableaux descend from the unfolded tableaux which in turn are copied from allowed on-shell diagrams, the folded tableaux naturally capture the allowed sets of pivot columns.
The matrix elements of non-pivot columns are determined by adopting the conversion rule introduced in Sec. 3.3 and modifying it slightly to fit into the tableaux notation. Two modifications are needed. First, since the definition of positivity requires a specific branch (3.15), we are forced to select the orientation-preserving BCFW bridges: σ = +1 in Fig. 7(a) and σ = −1 in Fig. 7(b). Second, the labels for external legs are ordered clockwise in on-shell diagrams but counter-clockwise on OG tableaux. Taking these factors into account, we arrive at a remarkably simple final rule, depicted in Fig. 15. Note that the two types of bridges in Fig. 7 has been unified into a single one in Fig. 15. An example of the application of the conversion rule is given in Fig. 16.  To reveal the connection, we will turn to an equivalent, and often more convenient, way to determine the matrix elements. We will begin with bottom cells whose matrix elements are completely fixed by positivty. We will then successively turn on BCFW bridges by multiplying the C-matrix from the right by an SO(k, k) rotation matrix.
Bottom cells Each unmarked tableau hosts a unique bottom cell (see Fig. 12). Given an unmarked tableau with pivot columns {p m }, let {q m } be the labels for the 'sink' columns.
By construction, C m,pm = 1 and C m,i = 0 for i = p m , q m . The orthogonality constraint requires that C m,qm = ±1. Positivity determines the sign of C m,qm uniquely. For bottom cells, a minor is non-vanishing if and only if it contains either a pivot column p m or its sink column q m but not both or neither. We start with (p 1 , p 2 , · · · , p k ) = 1, which is positive by construction. Suppose we replace a pivot column p m with its sink column q m . If the two columns are adjacent (q m = p m + 1), the ordering of the columns in the minor will be preserved, and the minor will remain positive if and only if C m,qm = +1. Suppose now p m and q m are not adjacent. Since we are dealing with bottom diagrams with no intersection among lines, the interval between p m and q m may contain a pair (p n , q n ) for some n, but not p n or q n separately. We can recover the ordering of the minor by shifting the column q m to the right by (q m − p m − 1)/2 steps. The resulting ordered minor will become positive if and only if we set C m,qm = (−1) (qm−pm−1)/2 .  When there are two or more BCFW bridges, the order of the rotation matrices can be determined as follows. Let us define the 'floor' of a pivot column as their vertical distance from the bottom of the OG tableau. For example, in Fig. 11(b), pivot columns 1 and 2 reside on the 0th floor, while column 5 resides on the 2nd floor.
The BCFW bridging begins with those pivot columns on the 0th floor. We bridge the two left-most pivot columns, say 1 and 2, in the sense that the rotation matrix acts on the corresponding sink columns. The hook above pivot 1 is lowered from its original location. If there are more pivot columns on the 0th floor, say column 3, then we bridge pivots 1 and 3. We continue the process until the hook above pivot 1 comes down to the 0th floor. Finally, we decouple column 1 and elevate other columns on the 0th floor to the 1st floor.
We proceed in the same way on the 1st floor including those elevated from the 0th floor and those born on the 1st floor. The second leftmost column from the 0th floor, if exists, becomes the leftmost column of the 1st floor. The final result can be summarized in a simple way. Given an empty OG tableau, to go from level 0 to the highest level, we perform the BCFW bridging such that the order or rotation is read off from left to right on the 0th floor, and then from left to right on the 1st floor, and so on. Two examples are given in Fig. 18 and Fig. 19. Positivity A rotation matrix R for swapping a pair of sink columns c i and c j (i < j) is the tensor product of the non-trivial 2 × 2 block, and a (2k − 2) × (2k − 2) identity matrix. The sign factor in (4.3), which resembles (4.2), has been inserted to preserve positivity for t ≥ 0. The rotation swaps the sink columns c i and c j . It leads to mixing of the Plücker coordinates (· · · i · · · ) and (· · · j · · · ), (· · · i · · · ) → R ii (· · · i · · · ) + R ij (· · · j · · · ) (4.4) (· · · j · · · ) → R ji (· · · i · · · ) + R jj (· · · j · · · ) (4.5) When c i and c j are adjacent (j = i + 1), the two minors (· · · i · · · ) and (· · · j · · · ) share a common ordering. Thus, to ensure positivity of minors after the rotation, we should require that R ij ≥ 0. When c i and c j are not adjacent, the ordering of (· · · i · · · ) and (· · · j · · · ) differ precisely by the sign factor (−1) (j−i−1)/2 , such that the rotation (4.3) with the sign factor preserves positivity. Finally, we note that (· · · i · · · j · · · ) → cosh 2 t (· · · i · · · j · · · ) + sinh 2 t (· · · j · · · i · · · ) = (cosh 2 t − sinh 2 t)(· · · i · · · j · · · ) = (· · · i · · · j · · · ) . (4.6) We give an explicit form of the C-matrix for the two examples shown in Fig. 18. Since the pivot columns form a (k × k) identity matrix by construction, it suffices to present the sink columns. For the k = 3, level 2 cell with pivot columns {1, 2, 4}, we have where we used the notations c i = cosh t i , s i = sinh t i . For the k = 3 top-cell with pivot columns {1, 2, 3}, we have It is straightforward to verify that all ordered minors of the corresponding C-matrices are manifestly non-negative, provided that t i ≥ 0.
We have described two ways to construct the C-matrix. One is to use the conversion rule summarized in Fig. 15 and the other is to perform a sequence of BCFW rotations. The latter exhibits manifest positivity, while the former reveals the connection to on-shell diagrams more clearly. As an astute reader may have expected, with hindsight, we have adjusted the variables in the two approaches such that the they agree without any change of variables. Although we have not been able to find a general proof for this agreement, we have verified it in all examples up to k = 5 and expect that it will hold for all k.
Integration measure We note in passing that the integration measure is factorized into a d log form in a way similar to that of [7]. Taking the measures from the elementary BCFW vertices (2.24), (2.26) and taking account of Wick rotation, we see that the integration measure for the full Grassmannian integral can be written as (4.9)

Polytope
It is known that G + (k, n) defines a combinatorial polytope also known as 'Eulerian poset' [19]. We will verify one of the requirements for OG k to be an Eulerian poset.
Eulerian poset It is straightforward to count the number of cells at each level for arbitrary k. The result is most compactly summarized in terms of a generating function, (4.10) T k,l is the number of on-shell diagrams with 2k external legs and l vertices without any bubble. Equivalently, T k,l is the number of OG k tableaux at level l. A counting algorithm based on the construction of OG tableaux in Sec. 4.1 can be easily implemented on a computer and generate T k (q). The results for T k (q) for small values of k are given by T 2 (q) = 2 + q , T 3 (q) = 5 + 6q + 3q 2 + q 3 , (4.11) T 4 (q) = 14 + 28q + 28q 2 + 20q 3 + 10q 4 + 4q 5 + q 6 , T 5 (q) = 42 + 120q + 180q 2 + 195q 3 + 165q 4 + 117q 5 + 70q 6 + 35q 7 + 15q 8 + 5q 9 + q 10 .
After computing T k (q) up to k = 15 using our own algorithm, we found that a beautiful closed-form expression for T k (q) had been known for decades [36], 4 In special cases, this formula reproduces the simple general features discussed in Sec. 4.1, Another property of T k (q) that can be derived from (4.13) is that, for any k, (4.14) We recognize this as one of the central properties of an Eulerian poset. In Sec. 4.2, we assigned a coordinate patch of POG k to each tableau. In this geometric context, T k (−1) is interpreted as the Euler characteristic of POG k . For k = 2, 3 (see Fig. 20), (4.14) matches with the fact that the POG has the topology of a ball. It remains to be seen whether the POG is a topological ball for all k.
Using the positroid stratification, we can refine the Eulerian poset structure. 5 Consider the k = 2 and k = 3 examples in Fig. 12. We can compute an analogue of T k (q) for each chart containing several cells. They turn out to vanish in all charts, except for the smallest one with pivot columns {1, 3, · · · , 2k − 1} which trivially gives 1. We checked that the same phenomenon continues up to k = 7, but have not attempted a proof for all k. Geometrically, it indicates that the subspaces of POG k are likely to be topological balls. For instance, a level 2 cell for k = 3 shown in Fig. 12 is topologically a square with four edges and four vertices. Boundary Operation Following [7], we define the boundary operation ∂ acting on onshell diagrams such that it resolves each BCFW vertex in two ways shown in Fig. 21. The corresponding OG 2 tableaux (see Fig. 12) shows that the first term remains in the same coordinate chart as the original one, {1, 2}, while the second term belongs to {1, 3}. It is a general property of the canonical coordinate system; there always exist some boundaries that cannot be reached without changing coordinates. One possible approach to put every boundary within reach in a single coordinate patch would be to use the cyclic gauge, where the odd-labelled columns form an identity matrix and the even-labelled columns form an SO(k) rotation matrix. This approach has its own drawbacks. First, the cyclic gauge necessarily introduces a BCFW bridge with non-adjacent source legs {1, 3} or {2, 4}. As explained in [14], this type of BCFW bridge is substantially more complicated than the ones used in this paper. Second, positivity imposes coupled, non-linear relations among the angle variables of SO(k) rotation, in contrast to the simple t i ≥ 0 conditions in this paper. For these reasons, we will stay within the canonical coordinate system and look for an alternative way to reach all boundaries.
It is convenient to separate the boundary operation ∂ into ∂ L and ∂ R , according to the orientation of the resolved diagrams (see Fig. 22). In [7], it was shown that ∂ 2 = 0 (mod 2) holds for G + (k, n) and conjectured that the (mod 2) restriction could be dropped if suitable signs are attached to each on-shell diagram. 6 Here, we will outline a similar argument for POG using an example without attempting a general proof. Applying ∂ to the k = 3 top-cell, we first observe that ∂ L results in bubble configurations, which we will discard by hand. Fig. 23 shows not only that each diagram has two incoming arrows, implying ∂ 2 = 0 (mod 2), but also that ∂ 2 R = 0 (mod 2) = ∂ L ·∂ R . In general, we have 2@ L · @ R 2@ L · @ R 2@ L · @ R 2@ R · @ R 2@ R · @ R 2@ R · @ R Boundaries of a top-cell The top-cell at each k has precisely k boundaries as can be seen in (4.11). We will identify them with the vanishing loci of the k independent consecutive minors that appear in the original OG integral (2.8).
We begin with k = 3 (see Fig. 23). In our coordinate system (4.7), the three independent consecutive minors are (123) = 1 , (234) = sinh t 1 sinh t 2 , (345) = sinh t 2 sinh t 3 . (4.16) We can easily obtain these minors without constructing the whole C-matrix. The only relevant matrix elements for (234) and (345) are c 14 and c 36 , respectively, which can be read off from the paths 1 → 4 and 3 → 6, respectively. These paths do not involve a turn so that each vertex contributes a sinh t i . Finally, the construction of POG guarantees that the overall sign should be positive.
A similar argument can be applied to consecutive minors for higher k. We simply summarize the results. First, (123 · · · k) = 1 = (k + 1, · · · , 2k) by construction. For other consecutive minors, we can always draw a rectangle within the OG tableau whose right/bottom edges correspond to the sink/pivot columns participating in the minor. If we collect all BCFW variables inside the rectangle, the product of sinh t factors produces the correct result for the minor. For example, for a general k, (234 · · · , k + 1) = k−1 i=1 sinh t i , (345 · · · k + 2) = 2k−3 i=2 sinh t i , and so on. (4.17) One may try to reach the boundaries of the top-cell by turning off some t i . However, sometimes it forces two or more consecutive minors to vanish at the same time, leaving a bubble configuration. Among the k(k − 1)/2 BCFW variables, only (k − 1) of them can be safely turned off without generating a bubble. It is easy to see that they correspond to the (k − 1) vertices along the diagonal of the OG k tableau.
By definition of the top-cell, the canonical coordinate does not allow (123 · · · k) to vanish. Recall from Sec. 3.2 that, for k = 2, the boundary with '(12)=0' was reached by taking t → ∞ in the canonical coordinates and performing a gauge transformation. A similar change of variable, which leads to a different coordinate patch, can reach the last boundary of a top-cell. In our prescription for computing the minors using rectangles, it is clear that all but (123 · · · k) contains sinh t k−1 coming from the lower right corner of the OG tableau for the top-cell. Taking t k−1 → ∞ makes all consecutive minors except (123 · · · k) diverge. Since only the ratios between minors are gauge invariant, we may divide all minors by sinh t k−1 so that (123 · · · k) converges to zero and all other minors remain finite. This configuration is equal to the OG k tableau whose the rightmost-bottom box is removed. Going back to the k = 3 example, setting t 2 → ∞ and dividing all minors by sinh t 2 gives (123) = 0, (234) = sinh t 1 , (345) = sinh t 3 . (4.18) This agrees with the result obtained from the (13)(25)(46) tableau in Fig. 18.
In summary, we have observed that POG k is likely to form an Eulerian poset and to have a ball topolgy. As a partial attempt to state and prove these conjectures rigorously, we took an initial step to define the notion of boundary operation satifying ∂ 2 = 0. Such ∂ would be naturally identified with the boundary operation of the usual simplicial homology on the geometric side. We verified that ∂ 2 = 0 (mod 2) works for POG k up to k = 3 and is likely to generalize straightforwardly for higher k. But, dropping the 'modulo 2' restriction seems to be a difficult task. We hope to revisit these problems in a future work.