A magic pyramid of supergravities

TL;DR

This work extends a division-algebraic program linking super Yang–Mills and supergravity into a structured magic pyramid of U-dualities across D=3,4,6,10, built by tensoring left/right SYM multiplets valued in division algebras. The authors formulate a universal pyramid formula Pyr(A_n,A_{nN_L},A_{nN_R}) governing the U-duality groups and construct a parallel conformal pyramid from conformal multiplets, uncovering a rich hierarchy of coset spaces and exceptional structures (including an exotic F_{4(4)} tip). By analyzing complex and quaternionic structures, anomaly considerations, and explicit Lie-algebra decompositions, they map the D=3 magic square to higher dimensions and reveal geometric interpretations via projective spaces and buildings. The framework clarifies how division algebras organize spacetime and R-symmetries, suggests avenues for extending to spin-factor Jordan algebras, and connects to double-copy ideas and potential UV properties of these highly structured supergravity theories.

Abstract

By formulating N = 1, 2, 4, 8, D = 3, Yang-Mills with a single Lagrangian and single set of transformation rules, but with fields valued respectively in R,C,H,O, it was recently shown that tensoring left and right multiplets yields a Freudenthal-Rosenfeld-Tits magic square of D = 3 supergravities. This was subsequently tied in with the more familiar R,C,H,O description of spacetime to give a unified division-algebraic description of extended super Yang-Mills in D = 3, 4, 6, 10. Here, these constructions are brought together resulting in a magic pyramid of supergravities. The base of the pyramid in D = 3 is the known 4x4 magic square, while the higher levels are comprised of a 3x3 square in D = 4, a 2x2 square in D = 6 and Type II supergravity at the apex in D = 10. The corresponding U-duality groups are given by a new algebraic structure, the magic pyramid formula, which may be regarded as being defined over three division algebras, one for spacetime and each of the left/right Yang-Mills multiplets. We also construct a conformal magic pyramid by tensoring conformal supermultiplets in D = 3, 4, 6. The missing entry in D = 10 is suggestive of an exotic theory with G/H duality structure F4(4)/Sp(3) x Sp(1).

Figures (6)

• Figure 1: 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 $\mathds{A}_{n \mathcal{N}_L}$ and $\mathds{A}_{n\mathcal{N}_R}$.
• Figure 2: The ranks of the scalar cosets $G/H$, where $G$ is the U-duality and $H$ is its maximal compact subgroup (the entries here apply to the original magic pyramid obtained by squaring SYM rather that from squaring conformal theories).
• Figure 3: The conformal magic pyramid. Note, the exterior faces, up to real forms, are given by the magic square (i.e. the $4\times 4$ base) cut across its diagonal.
• Figure 4: 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 5: Magic pyramid of maximal compact subgroups