Grassmannian Mandelstam Variables

• Grassmannian Mandelstam variables are geometric realizations of scattering invariants that use Grassmannian minors and Plücker coordinates to reformulate kinematic constraints.
• They generalize traditional Mandelstam invariants into higher-rank tensors and symmetric formulations, naturally encoding on-shell conditions and momentum conservation.
• These variables underpin modern approaches in gravity, SYM, and string theory, linking generalized scattering equations with Mellin transforms and factorization channels.

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 (Bourjaily et al., 2010, Gu, 2012, Stieberger et al., 2013, Cachazo et al., 2019, Cortes et al., 6 May 2025, De et al., 25 Mar 2026, Arundine et al., 20 May 2026).

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 (Gu, 2012, Cachazo et al., 2019, Cortes et al., 6 May 2025, De et al., 25 Mar 2026).

Framework	Grassmannian data	Mandelstam realization
S-variables	$G(k,n)$0-plane $G(k,n)$1 and $G(k,n)$2-plane $G(k,n)$3 in $G(k,n)$4	$G(k,n)$5
Tree amplitudes in $G(k,n)$6	$G(k,n)$7 matrix $G(k,n)$8 modulo $G(k,n)$9	$2$0
Generalized scattering equations	points on $2$1 and $2$2	$2$3 generalizes $2$4
Orthogonal Grassmannian $2$5	$2$6 minors of $2$7	$2$8, $2$9, $4$0
Symplectic Grassmannian $4$1	maximal minors of $4$2	$4$3

A common source of confusion is the status of the symbols $4$4. In some papers they are unnormalized orthogonal-Grassmannian minors obeying a deformed sum rule, while in another they are normalized ratios with $4$5. The underlying objects are related by convention rather than contradiction: one formulation works directly with the minors, and another divides by their sum (De et al., 25 Mar 2026, Arundine et al., 20 May 2026).

2. Grassmannian lifts of ordinary flat-space invariants

A particularly explicit lift is provided by the $4$6-variables for graviton scattering. The kinematic data are specified by a $4$7-plane $4$8, represented by the spinors $4$9, and a $\mathbb C^n$0-plane $\mathbb C^n$1, represented by four-component twistors $\mathbb C^n$2. Their Plücker coordinates are the usual holomorphic brackets $\mathbb C^n$3 and the $\mathbb C^n$4-plane minors $\mathbb C^n$5. The square bracket is then replaced by

$\mathbb C^n$6

so that

$\mathbb C^n$7

Multi-particle invariants are likewise written as $\mathbb C^n$8 (Gu, 2012).

In this formulation, on-shellness and momentum conservation are built in. Each momentum satisfies $\mathbb C^n$9 automatically because $\textsf{s}_{a_1\ldots a_k}$0 and $\textsf{s}_{a_1\ldots a_k}$1 by antisymmetry. Total momentum conservation is equivalent to the Schouten identity

$\textsf{s}_{a_1\ldots a_k}$2

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 $\textsf{s}_{a_1\ldots a_k}$3 and replacing the anti-holomorphic $\textsf{s}_{a_1\ldots a_k}$4-plane $\textsf{s}_{a_1\ldots a_k}$5 by an arbitrary $\textsf{s}_{a_1\ldots a_k}$6-plane $\textsf{s}_{a_1\ldots a_k}$7, with $\textsf{s}_{a_1\ldots a_k}$8 (Gu, 2012).

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 $\textsf{s}_{a_1\ldots a_k}$9 and $\mathbb{CP}^{k-1}$0 is built from the $\mathbb{CP}^{k-1}$1-variables (Gu, 2012).

3. Minors as factorization channels in $\mathbb{CP}^{k-1}$2 and two-copy gravity

In the Grassmannian integral for tree-level $\mathbb{CP}^{k-1}$3 SYM, a $\mathbb{CP}^{k-1}$4 matrix $\mathbb{CP}^{k-1}$5 parametrizes $\mathbb{CP}^{k-1}$6 modulo $\mathbb{CP}^{k-1}$7, and the integrand contains consecutive minors $\mathbb{CP}^{k-1}$8 in the denominator. The constraints $\mathbb{CP}^{k-1}$9 and $s_{ij}=(p_i+p_j)^2$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,

$s_{ij}=(p_i+p_j)^2$1

At six points in the NMHV sector, for example, $s_{ij}=(p_i+p_j)^2$2, and the amplitude is recovered as a sum over residues with rational factors $s_{ij}=(p_i+p_j)^2$3 multiplied by the overall $s_{ij}=(p_i+p_j)^2$4 (Bourjaily et al., 2010).

A gravity analogue appears in the Grassmannian formulation of the superstring/supergravity Mellin correspondence. There one introduces two independent homogeneous coordinates $s_{ij}=(p_i+p_j)^2$5 and $s_{ij}=(p_i+p_j)^2$6, each defining a point on a copy of $s_{ij}=(p_i+p_j)^2$7, with integration measures taken modulo separate $s_{ij}=(p_i+p_j)^2$8 actions. The central object is a generalized Hodges determinant density $s_{ij}=(p_i+p_j)^2$9, built from a matrix $k=2$0 whose entries carry explicit factors of $k=2$1. The details state that $k=2$2 is homogeneous of weight $k=2$3 in each set of link variables, and that the overall factors in $k=2$4 exactly cancel the projective weights, making the integrand $k=2$5-invariant (Stieberger et al., 2013).

Tree-level $k=2$6-graviton amplitudes in the $k=2$7 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 $k=2$8. The same relations render $k=2$9 well defined. The resulting amplitude is symmetric in the external labels and reproduces the usual KLT form (Stieberger et al., 2013).

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,

$\mathbb{CP}^{k-1}$0

After fixing the residual $\mathbb{CP}^{k-1}$1, the open-superstring disk amplitude contains the usual Koba-Nielsen factor $\mathbb{CP}^{k-1}$2. The $\mathbb{CP}^{k-1}$3-integration over the ordered chamber $\mathbb{CP}^{k-1}$4 defines a multidimensional Mellin transform into a space whose variables are the Mandelstam invariants themselves (Stieberger et al., 2013).

The low-point cases make the correspondence explicit. For $\mathbb{CP}^{k-1}$5, the Mellin integral is the Euler Beta-function $\mathbb{CP}^{k-1}$6. For $\mathbb{CP}^{k-1}$7, it becomes an Appell-type double hypergeometric function $\mathbb{CP}^{k-1}$8. 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 (Stieberger et al., 2013).

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

$\mathbb{CP}^{k-1}$9

so that the kinematic factors $G(k,n)$00 entering the gravity-side determinant become the parameters of the hypergeometric kernels on the string side (Stieberger et al., 2013).

5. Higher-$G(k,n)$01 Grassmannian Mandelstams and generalized scattering equations

A broader generalization replaces pairwise invariants by a completely symmetric rank-$G(k,n)$02 tensor $G(k,n)$03. These variables satisfy a massless condition, $G(k,n)$04, and momentum-conservation constraints

$G(k,n)$05

for each label $G(k,n)$06. The geometric setting is $G(k,n)$07 points on $G(k,n)$08, with homogeneous coordinates $G(k,n)$09 and $G(k,n)$10 minors $G(k,n)$11. The potential

$G(k,n)$12

generates the scattering equations as its critical-point equations (Cachazo et al., 2019).

For $G(k,n)$13, the geometry is $G(k,n)$14, and the scattering equations involve the $G(k,n)$15 minors $G(k,n)$16. The associated CHY-like biadjoint amplitudes $G(k,n)$17 are defined by summing over solutions with a reduced determinant $G(k,n)$18 and a $G(k,n)$19 Parke-Taylor factor

$G(k,n)$20

For $G(k,n)$21, the number of solutions is $G(k,n)$22, and the explicit pole structure exhibits three types of poles: $G(k,n)$23-poles $G(k,n)$24, $G(k,n)$25-poles $G(k,n)$26, and the new $G(k,n)$27-poles such as $G(k,n)$28 (Cachazo et al., 2019).

The tropical interpretation is central. $G(k,n)$29 has $G(k,n)$30 rays partitioned into sets corresponding to $G(k,n)$31, $G(k,n)$32, and $G(k,n)$33. Its facets behave as $G(k,n)$34 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,

$G(k,n)$35

with rank-one momentum matrices $G(k,n)$36, making the symmetry and repeated-index vanishing immediate (Cachazo et al., 2019).

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

In cosmological Grassmannian formulations, the relevant space is the orthogonal Grassmannian $G(k,n)$37, represented by a $G(k,n)$38 matrix $G(k,n)$39 satisfying $G(k,n)$40 modulo $G(k,n)$41. The basic Grassmannian Mandelstams are the $G(k,n)$42 minors

$G(k,n)$43

Because of the orthogonality constraint, they obey linear relations; after gauge fixing and solving the kinematic constraints one finds $G(k,n)$44, with $G(k,n)$45, and in the flat-space limit $G(k,n)$46 this reduces to the familiar relation $G(k,n)$47. Geometrically, the vanishing of one minor, such as $G(k,n)$48, is the boundary of the Grassmannian corresponding to $G(k,n)$49-channel factorization (De et al., 25 Mar 2026).

For the minimal Vasiliev scalar four-point function in dS$G(k,n)$50, the ordered $G(k,n)$51 channel is obtained from the simple integrand $G(k,n)$52, and the crossing-symmetric result is

$G(k,n)$53

Its only poles are at $G(k,n)$54, $G(k,n)$55, and $G(k,n)$56; there is no pole at $G(k,n)$57, hence no total-energy singularity. The residue at $G(k,n)$58 is $G(k,n)$59, 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 (De et al., 25 Mar 2026).

A related cosmological convention begins from $G(k,n)$60, identified with the same minors in a convenient chart, and then defines normalized ratios

$G(k,n)$61

so that $G(k,n)$62. In this basis, scalar and spinning four-point exchange solutions are written in closed form: the $G(k,n)$63-channel dependence is a hypergeometric function of $G(k,n)$64, while the spin dependence appears as an overall Legendre polynomial $G(k,n)$65. The integration constants are fixed by the absence of unphysical singularities for $G(k,n)$66 and by matching the collapsed limit $G(k,n)$67 (Arundine et al., 20 May 2026).

The symplectic counterpart is $G(k,n)$68, where $G(k,n)$69 satisfies $G(k,n)$70 modulo $G(k,n)$71. In the Coulomb-branch formulation of $G(k,n)$72 SYM, the familiar two-body invariants are maximal Plücker minors: $G(k,n)$73 At four points, the Grassmannian integral localizes to $G(k,n)$74, the super-delta reduces to $G(k,n)$75, 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 $G(k,n)$76 with maximal minors is exact on the support of the bosonic delta constraints (Cortes et al., 6 May 2025).