Four-point functions, Twistors and Supersymmetry in the Symplectic Bi-Grassmannian for CFT$_4$ and AdS$_5$

Abstract: We initiate the study of four point functions in the symplectic bi-Grassmannian framework for four dimensional conformal field theories. We derive factorization formulae analogous to those for scattering amplitudes and bootstrap several examples of AdS$_5$ exchange correlation functions involving scalars, photons, fermions, gluons and gravitons. These correlators are rational functions of the minors of the Grassmannian matrices in contrast to their momentum space counterparts which are much more complicated. We then develop the $GL(2,\mathbb{R})$ twistor space formalism, deriving the Penrose transform and the twistorial Grassmannian where we find remarkably simple expressions. We also find an elegant and natural extension to $\mathcal{N}=1$ supersymmetry, deriving the supersymmetric Penrose transform and the super-twistor Grassmannian. Performing a half-Fourier transform from twistor space, we derive the supersymmetric symplectic bi-Grassmannian corresponding to spinor-helicity variables. We test our formalism in the context of AdS$_5$ $\mathcal{N}=1$ super-Yang Mills theory where we impose consistent factorization and input the bootstrapped four-fermion correlator to obtain those involving gluons, scalars and fermions. Finally, we discuss the application to supergravity theories.

• The paper establishes a four-point factorization bootstrap in which AdS₅ exchange correlators are rational functions of Grassmannian minors and are fixed by three-point data up to contact terms.
• It derives twistor and super-twistor formulations that make conformal and N=1 superconformal symmetry manifest, including supersymmetric incidence relations and Ward-identity solutions.
• It applies the framework to N=1 super Yang–Mills and supergravity, reproducing component correlators, identifying the universal Klein-quadric pole, and exhibiting an exact graviton/Yang–Mills squaring relation.

Overview and context

This paper by Dhruva K.S. extends the symplectic bi-Grassmannian formalism for four-dimensional conformal field theories, initiated in (Bala et al., 7 May 2026), to four-point functions, twistor space, and $\mathcal{N}=1$ supersymmetry. The construction lives in four-dimensional Klein space $\mathbb{R}^{2,2}$, where the conformal group is $SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R$ and momenta admit an off-shell spinor-helicity parametrization $$p_{\alpha\dot\alpha}=\epsilon^{IJ}\lambda_{I\alpha}\tilde\lambda_{J\dot\alpha}$$ with little group $SL(2,\mathbb{R})\times GL(1,\mathbb{R})$. Correlators of symmetric traceless conserved currents of dimension $\Delta=s+2$ are represented as integrals over two $n$-planes $C,\tilde C$ in a $2n$-dimensional space, constrained to be symplectically orthogonal to each other and to the external data $\Lambda,\tilde\Lambda$.

The paper's central claim is that holographic four-point exchange correlators in AdS$\mathbb{R}^{2,2}$0 — involving scalars, photons, fermions, gluons and gravitons — are rational functions of Grassmannian minors, in contrast to their considerably more complicated momentum-space counterparts. This rationality, together with a derived factorization principle, enables a bootstrap program analogous to that for scattering amplitudes and for the CFT$\mathbb{R}^{2,2}$1/AdS$\mathbb{R}^{2,2}$2 cosmological Grassmannian (Arundine et al., 6 Feb 2026).

The symplectic bi-Grassmannian at four points

The $\mathbb{R}^{2,2}$3-point correlator is written as

$\mathbb{R}^{2,2}$4

with a $\mathbb{R}^{2,2}$5 redundancy requiring $\mathbb{R}^{2,2}$6. After stripping the momentum-conserving delta function, $\mathbb{R}^{2,2}$7-point functions contain $\mathbb{R}^{2,2}$8 non-trivial integrals. The paper introduces a bold spinor notation (following (Arkani-Hamed et al., 2017)) that makes classification of $\mathbb{R}^{2,2}$9 invariants and covariants efficient. At four points, the Mandelstam-like $SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R$0 invariants are

$SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R$1

where, for instance, $SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R$2 and $SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R$3, obeying the relation $SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R$4.

Factorization and the bootstrap principle

The main technical result is a Grassmannian factorization formula. Taking a discontinuity of the four-point function in $SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R$5 factorizes it into a convolution of three-point functions; translating this into the Grassmannian, the discontinuity localizes the two minors $SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R$6 and $SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R$7 (two of the four non-trivial integrals), yielding

$SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R$8

The derivation (in an appendix) assumes $SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R$9 has only simple poles in $$p_{\alpha\dot\alpha}=\epsilon^{IJ}\lambda_{I\alpha}\tilde\lambda_{J\dot\alpha}$$0 and $$p_{\alpha\dot\alpha}=\epsilon^{IJ}\lambda_{I\alpha}\tilde\lambda_{J\dot\alpha}$$1; under this assumption, the most singular part $$p_{\alpha\dot\alpha}=\epsilon^{IJ}\lambda_{I\alpha}\tilde\lambda_{J\dot\alpha}$$2 of the correlator is fixed by three-point data, up to AdS$$p_{\alpha\dot\alpha}=\epsilon^{IJ}\lambda_{I\alpha}\tilde\lambda_{J\dot\alpha}$$3 contact terms whose discontinuity vanishes. This is the bootstrap principle: exchange contributions are determined entirely by three-point data.

Applying this, the paper obtains a family of exchange correlators, all sharing the denominator $$p_{\alpha\dot\alpha}=\epsilon^{IJ}\lambda_{I\alpha}\tilde\lambda_{J\dot\alpha}$$4:

Correlator	Exchange	Result
$$p_{\alpha\dot\alpha}=\epsilon^{IJ}\lambda_{I\alpha}\tilde\lambda_{J\dot\alpha}$$5	scalar	$$p_{\alpha\dot\alpha}=\epsilon^{IJ}\lambda_{I\alpha}\tilde\lambda_{J\dot\alpha}$$6
$$p_{\alpha\dot\alpha}=\epsilon^{IJ}\lambda_{I\alpha}\tilde\lambda_{J\dot\alpha}$$7	spin $$p_{\alpha\dot\alpha}=\epsilon^{IJ}\lambda_{I\alpha}\tilde\lambda_{J\dot\alpha}$$8	$$p_{\alpha\dot\alpha}=\epsilon^{IJ}\lambda_{I\alpha}\tilde\lambda_{J\dot\alpha}$$9
$SL(2,\mathbb{R})\times GL(1,\mathbb{R})$0	scalar	numerator of cubic minors over $SL(2,\mathbb{R})\times GL(1,\mathbb{R})$1
$SL(2,\mathbb{R})\times GL(1,\mathbb{R})$2	gluon	quartic minors over $SL(2,\mathbb{R})\times GL(1,\mathbb{R})$3
$SL(2,\mathbb{R})\times GL(1,\mathbb{R})$4	gluon (colour-ordered)	$SL(2,\mathbb{R})\times GL(1,\mathbb{R})$5
$SL(2,\mathbb{R})\times GL(1,\mathbb{R})$6	graviton	$SL(2,\mathbb{R})\times GL(1,\mathbb{R})$7

Two structural observations follow. First, the spin-$SL(2,\mathbb{R})\times GL(1,\mathbb{R})$8 uplift via Gegenbauer polynomials is unique up to contact diagrams: replacing $SL(2,\mathbb{R})\times GL(1,\mathbb{R})$9 by Legendre polynomials (with matched residues) shifts the answer only by terms regular in $\Delta=s+2$0. Second, the graviton four-point function satisfies the exact squaring relation

$\Delta=s+2$1

a double-copy structure at the level of four-point boundary correlators, mirroring the three-point relation $\Delta=s+2$2.

The universal pole $\Delta=s+2$3 admits a geometric interpretation. Assembling $\Delta=s+2$4 into an antisymmetric $\Delta=s+2$5 matrix $\Delta=s+2$6, its Pfaffian is precisely $\Delta=s+2$7. The six entries are homogeneous coordinates on $\Delta=s+2$8 with the AdS$\Delta=s+2$9 metric, and the Pfaffian constraint defines the Klein quadric — the boundary of AdS$n$0. The paper flags this as suggestive but leaves its physical significance open.

Twistor space formulation

The paper performs a half-Fourier transform $n$1 on the rescaled currents, in which all conformal generators act linearly via $n$2, realizing $n$3. The resulting twistor-space Grassmannian is strikingly simple:

$n$4

which is exactly the general solution to the conformal Ward identities (projective super-delta functions, plus $n$5-symbol invariants relevant to parity-odd correlators, which are not treated here). The paper derives the Penrose transform for currents of $n$6, with incidence relation $n$7 (two copies of the usual incidence relation), and shows that integer-spin currents scale as $n$8 under twistor projective rescaling while half-integer ones scale as $n$9 or $C,\tilde C$0. A dual-twistor version is also given, and a connection to ambitwistor space is established in an appendix: the twistor Fourier transform produces factors $C,\tilde C$1, the defining constraints of ambitwistor space, though for spinning correlators this representation obscures little-group covariance.

Supersymmetry: super-twistors and the super bi-Grassmannian

Extending to $C,\tilde C$2 superspace via $C,\tilde C$3, the paper derives the supersymmetric Penrose transform. The super-incidence relations are fixed uniquely by imposing the shortening (conservation) condition $C,\tilde C$4, giving

$C,\tilde C$5

closely paralleling the on-shell Ferber construction [Nucl. Phys. B 132 (1978) 55]. The super-twistor Grassmannian follows by $C,\tilde C$6 in the delta function, and an inverse half-Fourier transform yields the supersymmetric symplectic bi-Grassmannian in spinor-helicity variables, with an extra fermionic delta function $C,\tilde C$7 and modified covariance $C,\tilde C$8. An appendix provides an independent brute-force verification of superconformal invariance directly in spinor-helicity variables, using the fact that the symplectic orthogonality constraint annihilates both $C,\tilde C$9 and $2n$0 actions on the fermionic delta function. The paper also sketches a natural extension to $2n$1, noting that in Klein signature the $2n$2-symmetry is $2n$3 rather than $2n$4 because spinors are real.

Application to AdS$2n$5 $2n$6 super Yang–Mills

The spectrum of five-dimensional $2n$7 SYM on rigid AdS$2n$8 comprises a massless gluon ($2n$9), a scalar with $\Lambda,\tilde\Lambda$0 ($\Lambda,\tilde\Lambda$1), and gluinos ($\Lambda,\tilde\Lambda$2), all in the adjoint of $\Lambda,\tilde\Lambda$3, packaged into a half-integer-spin conserved supercurrent. The three-point superamplitude ansatz $\Lambda,\tilde\Lambda$4 correctly reproduces all component three-point functions, including the vanishing of the $\Lambda,\tilde\Lambda$5 correlator — consistent with the absence of a cubic scalar coupling in the bulk Lagrangian — and the absence of any $\Lambda,\tilde\Lambda$6 contribution.

At four points, the paper inputs the four-fermion correlator, fixes the exchange coefficients by demanding the correct double residues ($\Lambda,\tilde\Lambda$7, $\Lambda,\tilde\Lambda$8, which also reproduces the $\Lambda,\tilde\Lambda$9-channel residue), and derives the full superamplitude

$\mathbb{R}^{2,2}$00

All component correlators extracted from it — notably the four-gluon correlator, which reproduces the independently bootstrapped colour-ordered result — factorize correctly at $\mathbb{R}^{2,2}$01 and $\mathbb{R}^{2,2}$02. This agreement between the supersymmetry construction and the factorization bootstrap is presented as a non-trivial consistency check on the rigidity of the formalism. The residual ambiguity $\mathbb{R}^{2,2}$03, which has vanishing double residues, is left undetermined; fixing it would require classifying AdS$\mathbb{R}^{2,2}$04 contact diagrams.

Supergravity

For the $\mathbb{R}^{2,2}$05 stress-tensor multiplet, the three-point ansatz reproduces the Einstein-gravity three-point function $\mathbb{R}^{2,2}$06 and yields a vanishing Abelian spin-1 three-point function, consistent with Furry's theorem. The proposed four-point graviton superamplitude,

$\mathbb{R}^{2,2}$07

has the correct double residues in the $\mathbb{R}^{2,2}$08 and $\mathbb{R}^{2,2}$09 channels, but the paper is explicit that this represents the exchange contribution only, up to contact diagrams, and defers a detailed study of AdS$\mathbb{R}^{2,2}$10 supergravity.

Limitations and open questions

Several assumptions bound the results. The bootstrap rests on the assumption that correlators have only simple poles in $\mathbb{R}^{2,2}$11 and $\mathbb{R}^{2,2}$12; contact terms and possible single-pole terms $\mathbb{R}^{2,2}$13 are not determined. All correlators computed are discontinuities (Wightman functions with specified analytic prescriptions) rather than full Euclidean correlators; the paper notes that the Euclidean versions are generally UV divergent and require renormalization, and that the detailed integration contours and their relation to the discontinuity prescriptions are left to future work. Parity-odd correlators are excluded, as are $\mathbb{R}^{2,2}$14 theories beyond the sketch in the appendix. The Klein-quadric interpretation of the universal pole is unproven. The flat-space limit, higher-point generalizations, BCFW-type recursion, and colour-kinematics duality in this framework are all identified as specific open problems rather than resolved.

Conclusion

The paper establishes that the symplectic bi-Grassmannian supports a complete four-point bootstrap for AdS$\mathbb{R}^{2,2}$15 exchange correlators, provides a twistorial reformulation in which conformal and superconformal symmetry are manifest, and demonstrates internal consistency between the factorization bootstrap and supersymmetric Ward identities in $\mathbb{R}^{2,2}$16 SYM. The recurring simplicity of the answers — rational functions of minors organized around the Klein-quadric pole $\mathbb{R}^{2,2}$17, with an exact graviton/Yang-Mills squaring relation — indicates that the Grassmannian framework captures substantial structure of CFT$\mathbb{R}^{2,2}$18 observables, while leaving the determination of contact terms, integration cycles, and the full supergravity correlator as concrete outstanding tasks.