Gravity as the square of Yang-Mills?

TL;DR

The work argues that gravity, and its rich symmetry structure, can be understood as the square of Yang-Mills theories via a covariant field dictionary built from a star-product construction with a bi-adjoint spectator field. By systematically tensoring left and right Yang-Mills multiplets and organizing the resulting fields with a division-algebra framework, the authors reproduce gravity's covariant fields and linearized symmetries, and extend this to extended supersymmetry and U-duality. The analysis reveals that the resulting global symmetry algebras align with Freudenthal's magic square and its dimensional extensions, producing a unified picture encoded in the magic pyramid. This approach provides both conceptual clarity and practical computational leverage for exploring gravity through the double-copy paradigm and division-algebra structures.

Abstract

In these lectures we review how the symmetries of gravitational theories may be regarded as originating from those of "Yang-Mills squared". We begin by motivating the idea that certain aspects of gravitational theories can be captured by the product, in some sense, of two distinct Yang-Mills theories, particularly in the context of scattering amplitudes. We then introduce a concrete dictionary for the covariant fields of (super)gravity in terms of the product of two (super) Yang-Mills theories. The dictionary implies that the symmetries of each (super) Yang-Mills factor generate the symmetries of the corresponding (super)gravity theory: general covariance, $p$-form gauge invariance, local Lorentz invariance, local supersymmetry, R-symmetry and U-duality.

Figures (6)

• Figure 1: The double-copy procedure. Assuming the gauge theory amplitude on the left has been arranged to display colour-kinematic duality then the gravity amplitude on the right is straight-forwardly obtained by replacing the colour factors with a second copy of the kinematic factors. Note, the second factor does not have correspond to the same Yang-Mills theory. The (supressed) Yang-Mills coupling constants must be replaced by the gravitational coupling constant $g\rightarrow \kappa/2$, where $\kappa^2=16\pi G_N$.
• Figure 2: The Fano plane. The structure constants are determined by the Fano plane, $C_{ijk}=1$ if $ijk$ lies on a line and is ordered according as its orientation. Each oriented line follows the rules of quaternionic multiplication. For example, $e_2e_3=e_5$ and cyclic permutations; odd permutations go against the direction of the arrows on the Fano plane and we pick up a minus sign, e.g. $e_3e_2=-e_5$.
• Figure 3: The $\mathcal{H}$ algebra in terms of the left/right super Yang-Mills theories in a division algebraic language.
• Figure 4: All roads lead to the magic square.
• Figure 5: A magic pyramid of supergravities. The vertical axis labels the spacetime division algebra $\mathds{A}_n$, while the horizontal axes label the algebras associated with the number of supersymmetries $\mathds{A}_{n \mathcal{N}}$ and $\mathds{A}_{n\tilde{\mathcal{N}}}$.