The Conformal Grassmannian: A Symplectic Bi-Grassmannian for $CFT_ 4$ Correlators

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 $\Delta = 2$ 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 $\langle JJJ\rangle$ and $\langle TTT\rangle$ 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 $\langle JJJ \rangle$ and Einstein-gravity $\langle TTT \rangle$. 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.