The Conformal Grassmannian: A Symplectic Bi-Grassmannian for CFT4 Correlators
Abstract
We introduce a formalism for conformal field theory in four dimensions: a symplectic bi-Grassmannian representation of CFT4 Wightman correlators. Working in Klein space with off-shell spinor-helicity variables, we show that correlators of Δ= 2 scalars and symmetric-traceless conserved currents are encoded by integrals over a pair of n-planes in a 2n-dimensional symplectic vector space. These planes are constrained to be mutually symplectically orthogonal and aligned with the external kinematics. Conformal invariance, momentum conservation, and little-group covariance all follow geometrically from this structure. We derive all two- and three-point functions involving scalars, fermions, conserved currents, and stress tensors. As a non-trivial test, we show that the construction reproduces the full set of independent conformally invariant structures of ⟨ JJJ⟩ and ⟨ TTT⟩ in CFT4. The resulting expressions are considerably more compact than their momentum-space counterparts. They also make manifest the double copy between Yang--Mills ⟨ JJJ ⟩ and Einstein-gravity ⟨ TTT ⟩. We further present a helicity-basis reformulation that makes the GL(1,R) and SL(2,R) weights of individual helicity components explicit. This basis also provides a natural starting point for a twistor-space formulation of the correlators.
Related
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BibTeX
@misc{bala2026conformal,
title = {{The Conformal Grassmannian: A Symplectic Bi-Grassmannian for $CFT_ 4$ Correlators}},
author = {Aswini Bala and Sachin Jain and Dhruva K. S},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.06811},
eprint = {2605.06811},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.06811},
note = {CC0},
}
RIS
TY - UNPB
TI - The Conformal Grassmannian: A Symplectic Bi-Grassmannian for CFT_4 Correlators
AU - Aswini Bala
AU - Sachin Jain
AU - Dhruva K. S
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2605.06811
UR - https://arxiv.org/abs/2605.06811
AB - We introduce a formalism for conformal field theory in four dimensions: a symplectic bi-Grassmannian representation of CFT_4 Wightman correlators. Working in Klein space with off-shell spinor-helicity variables, we show that correlators of Δ= 2 scalars and symmetric-traceless conserved currents are encoded by integrals over a pair of n-planes in a 2n-dimensional symplectic vector space. These planes are constrained to be mutually symplectically orthogonal and aligned with the external kinematics. Conformal invariance, momentum conservation, and little-group covariance all follow geometrically from this structure. We derive all two- and three-point functions involving scalars, fermions, conserved currents, and stress tensors. As a non-trivial test, we show that the construction reproduces the full set of independent conformally invariant structures of ⟨ JJJ⟩ and ⟨ TTT⟩ in CFT_4. The resulting expressions are considerably more compact than their momentum-space counterparts. They also make manifest the double copy between Yang--Mills ⟨ JJJ ⟩ and Einstein-gravity ⟨ TTT ⟩. We further present a helicity-basis reformulation that makes the GL(1,R) and SL(2,R) weights of individual helicity components explicit. This basis also provides a natural starting point for a twistor-space formulation of the correlators.
ER -