---
title: Grassmannian Mandelstam Variables
url: https://www.emergentmind.com/topics/grassmannian-mandelstam-variables
type: topic
---

# Grassmannian Mandelstam Variables

Grassmannian Mandelstam variables are realizations of flat-space Mandelstam invariants, or of their higher-rank analogues, directly in the geometry of Grassmannians. Depending on context, they appear as Plücker coordinates of a \(k\)-plane in \(G(k,n)\), as bilinears built from a \(2\)-plane and a \(4\)-plane in \(\mathbb C^n\), as completely symmetric tensors \(\textsf{s}_{a_1\ldots a_k}\) governing generalized scattering equations on \(\mathbb{CP}^{k-1}\), or as distinguished minors on orthogonal and symplectic Grassmannians. Across these settings, the common role of the variables is to encode kinematic constraints, factorization channels, and amplitude kernels in coordinates adapted to permutation symmetry, projective geometry, or worldsheet localization [1006.1899], [1205.0226], [1306.1844], [1903.08904], [2505.03705], [2603.24656], [2605.21581].

## 1. Scope of the notion

The literature uses the phrase in several adjacent senses rather than as a single universal definition. In one direction, ordinary invariants \(s_{ij}=(p_i+p_j)^2\) are rewritten in terms of Grassmannian minors or Plücker coordinates. In another, the usual \(k=2\) Mandelstam data are generalized to higher-rank symmetric tensors on \(\mathbb{CP}^{k-1}\). In cosmological and Coulomb-branch settings, distinguished Grassmannian minors themselves are designated as Mandelstam variables because they organize the full analytic structure of correlators or amplitudes [1205.0226], [1903.08904], [2505.03705], [2603.24656].

| Framework | Grassmannian data | Mandelstam realization |
|---|---|---|
| S-variables | \(2\)-plane \(\lambda\) and \(4\)-plane \(Z\) in \(\mathbb C^n\) | \(s_{ij}=\langle ij\rangle \sum_{k,l}\langle kl\rangle \langle ijkl\rangle\) |
| Tree amplitudes in \(G(k,n)\) | \(k\times n\) matrix \(C\) modulo \(\mathrm{GL}(k)\) | \((i_1\cdots i_k)=0 \leftrightarrow s_{i_1\cdots i_k}=0\) |
| Generalized scattering equations | points on \(\mathbb{CP}^{k-1}\) and \(\textsf{s}_{a_1\ldots a_k}\) | \(\textsf{s}_{a_1\ldots a_k}\) generalizes \(s_{ab}\) |
| Orthogonal Grassmannian \(OGr(4,8)\) | \(4\times4\) minors of \(C\) | \(S=(\bar1\,\bar2\,1\,2)\), \(T=(\bar1\,\bar4\,1\,4)\), \(U=(\bar1\,\bar3\,1\,3)\) |
| Symplectic Grassmannian \(SpGr(n,2n)\) | maximal minors of \(C\) | \((i^1\,i^2\,j^1\,j^2)=-2\,p_i\!\cdot\!p_j-2\,m_i m_j\equiv s_{ij}\) |

A common source of confusion is the status of the symbols \(S,T,U\). In some papers they are unnormalized orthogonal-Grassmannian minors obeying a deformed sum rule, while in another they are normalized ratios with \(S+T+U=1\). The underlying objects are related by convention rather than contradiction: one formulation works directly with the minors, and another divides by their sum [2603.24656], [2605.21581].

## 2. Grassmannian lifts of ordinary flat-space invariants

A particularly explicit lift is provided by the \(S\)-variables for graviton scattering. The kinematic data are specified by a \(2\)-plane \(\lambda\subset\mathbb C^n\), represented by the spinors \(\lambda_i^a\), and a \(4\)-plane \(Z\subset\mathbb C^n\), represented by four-component twistors \(Z_i^A\). Their Plücker coordinates are the usual holomorphic brackets \(\langle ij\rangle=\epsilon_{ab}\lambda_i^a\lambda_j^b\) and the \(4\)-plane minors \(\langle ijkl\rangle=\epsilon_{ABCD}Z_i^A Z_j^B Z_k^C Z_l^D\). The square bracket is then replaced by
\[
[i\,j]_S=\sum_{k,l=1}^n \langle k\,l\rangle\,\langle i\,j\,k\,l\rangle,
\]
so that
\[
s_{ij}=\langle i\,j\rangle_S[i\,j]_S
      =\langle i\,j\rangle \sum_{k,l}\langle k\,l\rangle \langle i\,j\,k\,l\rangle.
\]
Multi-particle invariants are likewise written as \(s_I=\sum_{i<j\in I}\langle ij\rangle[i\,j]_S\) [1205.0226].

In this formulation, on-shellness and momentum conservation are built in. Each momentum satisfies \(p_i^2=0\) automatically because \(\langle i\,i\rangle=0\) and \([i\,i]_S=0\) by antisymmetry. Total momentum conservation is equivalent to the Schouten identity
\[
\sum_{k=1}^n \langle i\,k\rangle [k\,j]_S=0.
\]
The construction is permutation invariant and does not require color ordering or a reference spinor. The paper characterizes the variables as arising by keeping the holomorphic spinor-plane \(\lambda\) and replacing the anti-holomorphic \(2\)-plane \(\tilde\lambda\) by an arbitrary \(4\)-plane \(Z\), with \(\tilde\lambda=(\lambda^\perp\cap Z)\) [1205.0226].

This Grassmannian lift is closely tied to gravity-specific amplitude formulae. Gu uses the variables to present reference-free forms of soft factors and tree-level MHV amplitudes of gravity, including a Hodges-determinant representation in which every contraction \(\langle ab\rangle\) and \(\langle ijkl\rangle\) is built from the \(S\)-variables [1205.0226].

## 3. Minors as factorization channels in \(G(k,n)\) and two-copy gravity

In the Grassmannian integral for tree-level \(\mathcal N=4\) SYM, a \(k\times n\) matrix \(C_{\alpha a}\) parametrizes \(G(k,n)\) modulo \(\mathrm{GL}(k)\), and the integrand contains consecutive minors \((a\,a+1\cdots a+k-1)\) in the denominator. The constraints \(C\!\cdot\!\tilde\lambda=0\) and \(C^\perp\!\cdot\!\lambda=0\) imply overall momentum conservation. Within this setup, the vanishing of a minor means that the corresponding set of columns becomes linearly dependent; in the dual momentum-twistor picture this is exactly the vanishing of a multi-particle invariant,
\[
(i_1\,i_2\,\dots\,i_k)=0 \Longleftrightarrow s_{i_1\cdots i_k}=0.
\]
At six points in the NMHV sector, for example, \((i\,i+1\,i+2)\propto s_{i,i+1,i+2}\), and the amplitude is recovered as a sum over residues with rational factors \(1/(s\,s)\) multiplied by the overall \(\delta^4(P)\delta^8(Q)\) [1006.1899].

A gravity analogue appears in the Grassmannian formulation of the superstring/supergravity Mellin correspondence. There one introduces two independent homogeneous coordinates \(\sigma_k^a\) and \(\tilde\sigma_k^a\), each defining a point on a copy of \(\mathbb{CP}^1\), with integration measures taken modulo separate \(\mathrm{GL}(2)\) actions. The central object is a generalized Hodges determinant density \(H_N(\sigma,\tilde\sigma)\), built from a matrix \(X\) whose entries carry explicit factors of \(s_{ij}\). The details state that \(X_{ij}\) is homogeneous of weight \(-1\) in each set of link variables, and that the overall factors in \(H_N\) exactly cancel the projective weights, making the integrand \(\mathrm{GL}(2)\times \mathrm{GL}(2)\)-invariant [1306.1844].

Tree-level \(N\)-graviton amplitudes in the \(\mathrm N^{k-2}\mathrm{MHV}\) sector are then obtained by gluing two copies of the link, or Grassmannian, integrals. The delta-function constraints impose the Grassmannian embedding of the external supertwistor data and enforce momentum conservation and on-shellness, which imply \(\sum_{j\neq i}s_{ij}=0\). The same relations render \(\det'X\) well defined. The resulting amplitude is symmetric in the external labels and reproduces the usual KLT form [1306.1844].

## 4. Mellin correspondence and the emergence of string Mandelstam kernels

The same two-copy formalism acquires a string-theoretic interpretation when one identifies the second set of link coordinates with ordered disk insertion points,
\[
\tilde\sigma_k^1=1,\qquad \tilde\sigma_k^2=z_k.
\]
After fixing the residual \(\mathrm{SL}(2)\), the open-superstring disk amplitude contains the usual Koba-Nielsen factor \(\prod_{i<j}|z_i-z_j|^{s_{ij}}\). The \(z_k\)-integration over the ordered chamber \(0<z_2<\cdots<z_{N-2}<1\) defines a multidimensional Mellin transform into a space whose variables are the Mandelstam invariants themselves [1306.1844].

The low-point cases make the correspondence explicit. For \(N=4\), the Mellin integral is the Euler Beta-function \(B(1+s_{12},1+s_{23})\). For \(N=5\), it becomes an Appell-type double hypergeometric function \(F_1\). In general, the result is a Gelfand-Aomoto-type hypergeometric integral over the simplex, with integral representations and series expansions expressed in products and ratios of Gamma-functions and multivariable Pochhammer symbols [1306.1844].

The significance is that one copy of the Grassmannian remains a link representation of the supergravity amplitude, while the second copy becomes the string worldsheet boundary. The paper summarizes this by the relation
\[
\mathcal A^{\rm string}_{N,k}=\mathcal M\bigl[\mathcal A^{\rm sugra}_{N,k}\bigr],
\]
so that the kinematic factors \(s_{ij}\) entering the gravity-side determinant become the parameters of the hypergeometric kernels on the string side [1306.1844].

## 5. Higher-\(k\) Grassmannian Mandelstams and generalized scattering equations

A broader generalization replaces pairwise invariants by a completely symmetric rank-\(k\) tensor \(\textsf{s}_{a_1a_2\ldots a_k}\). These variables satisfy a massless condition, \(\textsf{s}_{a_1\ldots a_{k-2}bb}=0\), and momentum-conservation constraints
\[
\sum_{a_2<\cdots<a_k}\textsf{s}_{a_1a_2\cdots a_k}=0
\]
for each label \(a_1\). The geometric setting is \(n\) points on \(\mathbb{CP}^{k-1}\), with homogeneous coordinates \(\sigma_a\in\mathbb C^k\) and \(k\times k\) minors \((a_1a_2\ldots a_k)=\det[\sigma_{a_1},\ldots,\sigma_{a_k}]\). The potential
\[
\mathcal S_k(\sigma)=\sum_{1\le a_1<\cdots<a_k\le n} \textsf{s}_{a_1\ldots a_k}\log(a_1a_2\ldots a_k)
\]
generates the scattering equations as its critical-point equations [1903.08904].

For \(k=3\), the geometry is \(\mathbb{CP}^2\), and the scattering equations involve the \(3\times 3\) minors \(\langle abc\rangle\). The associated CHY-like biadjoint amplitudes \(m_n^{(3)}(\alpha|\beta)\) are defined by summing over solutions with a reduced determinant \(\det'\Phi^{(3)}\) and a \(k=3\) Parke-Taylor factor
\[
PT^{(3)}(a_1\cdots a_n)=\frac{1}{\langle a_1a_2a_3\rangle\langle a_2a_3a_4\rangle\cdots\langle a_na_1a_2\rangle}.
\]
For \((k,n)=(3,6)\), the number of solutions is \(26\), and the explicit pole structure exhibits three types of poles: \(s\)-poles \(s_{abc}\), \(t\)-poles \(t_{abcd}\), and the new \(R\)-poles such as \(R_{12,34,56}\) [1903.08904].

The tropical interpretation is central. \( \mathrm{Trop}\,G(3,6)\) has \(65\) rays partitioned into sets corresponding to \(s_{ijk}\), \(t_{ijkl}\), and \(R_{ab,cd,ef}\). Its facets behave as \(k=3\) Feynman diagrams, and summing the facets compatible with two orderings reproduces the rational expressions obtained by direct residue calculations. The same paper also gives a generalized spinor-helicity realization,
\[
\textsf{s}_{a_1\ldots a_k}:=\det\!\bigl[K^{(a_1)}+\cdots+K^{(a_k)}\bigr],
\]
with rank-one momentum matrices \(K^{(a)}_{\alpha\dot\alpha}=\lambda^{(a)}_\alpha\tilde\lambda^{(a)}_{\dot\alpha}\), making the symmetry and repeated-index vanishing immediate [1903.08904].

## 6. Orthogonal and symplectic Grassmannians: cosmological and Coulomb-branch realizations

In cosmological Grassmannian formulations, the relevant space is the orthogonal Grassmannian \(OGr(4,8)\), represented by a \(4\times8\) matrix \(C\) satisfying \(CQC^T=0\) modulo \(\mathrm{GL}(4)\). The basic Grassmannian Mandelstams are the \(4\times4\) minors
\[
S=(\bar1\,\bar2\,1\,2),\qquad T=(\bar1\,\bar4\,1\,4),\qquad U=(\bar1\,\bar3\,1\,3).
\]
Because of the orthogonality constraint, they obey linear relations; after gauge fixing and solving the kinematic constraints one finds \(S+T+U=-2\tau E\), with \(E=\sum_i k_i\), and in the flat-space limit \(E\to0\) this reduces to the familiar relation \(S+T+U=0\). Geometrically, the vanishing of one minor, such as \(S=0\), is the boundary of the Grassmannian corresponding to \(s\)-channel factorization [2603.24656].

For the minimal Vasiliev scalar four-point function in dS\(_4\), the ordered \((s,t)\) channel is obtained from the simple integrand \(U/(ST)\), and the crossing-symmetric result is
\[
\mathcal A_4^{\rm Vas}(S,T,U)=\frac{S^2+T^2+U^2}{STU}.
\]
Its only poles are at \(S=0\), \(T=0\), and \(U=0\); there is no pole at \(S+T+U=0\), hence no total-energy singularity. The residue at \(S=0\) is \(T/U+U/T\), whose expansion yields the even-spin partial-wave decomposition of the higher-spin tower. The abstract further notes that this correlator has the same form as the field-theory limit of the Veneziano amplitude, despite arising from the opposite, tensionless limit of an infinite massless higher-spin tower [2603.24656].

A related cosmological convention begins from \(\tilde S,\tilde T,\tilde U\), identified with the same minors in a convenient chart, and then defines normalized ratios
\[
S=\tilde S/\tilde\Sigma,\qquad T=\tilde T/\tilde\Sigma,\qquad U=\tilde U/\tilde\Sigma,\qquad \tilde\Sigma=\tilde S+\tilde T+\tilde U,
\]
so that \(S+T+U=1\). In this basis, scalar and spinning four-point exchange solutions are written in closed form: the \(s\)-channel dependence is a hypergeometric function of \(S\), while the spin dependence appears as an overall Legendre polynomial \(P_J((U-T)/S)\). The integration constants are fixed by the absence of unphysical singularities for \(S>\tfrac12\) and by matching the collapsed limit \(k_s\to0\) [2605.21581].

The symplectic counterpart is \(SpGr(n,2n)\), where \(C\) satisfies \(C\Omega C^T=0\) modulo \(\mathrm{GL}(n)\). In the Coulomb-branch formulation of \(\mathcal N=4\) SYM, the familiar two-body invariants are maximal Plücker minors:
\[
(i^1\,i^2\,j^1\,j^2)=\det C_*[i^1,i^2,j^1,j^2]
=-2\,p_i\!\cdot\!p_j-2\,m_i m_j\equiv s_{ij}.
\]
At four points, the Grassmannian integral localizes to \(C_*=(\Lambda\mid\widetilde\Lambda)\), the super-delta reduces to \(\delta^4(Q)\delta^4(\widetilde Q)\), and an appropriate cyclic function of minors reproduces the known tree-level amplitude up to a familiar kinematic prefactor. In this setting, the identification of \(s_{ij}\) with maximal minors is exact on the support of the bosonic delta constraints [2505.03705].

Source: https://www.emergentmind.com/topics/grassmannian-mandelstam-variables