A magic pyramid of supergravities (1312.6523v2)

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).

• The paper introduces a magic pyramid formula that unifies supergravity theories via the division-algebraic structure of Yang-Mills multiplets.
• It employs N=1,2,4,8 Yang-Mills theories in D=3, combining left/right multiplets to build a higher-dimensional structure ending with Type II supergravity in D=10.
• Numerical findings reveal precise U-duality groups—such as E8(8) in D=3 and SL(2,R) in D=10—highlighting deep symmetry unifications in high-energy physics.

A Magic Pyramid of Supergravities

The paper "A Magic Pyramid of Supergravities," authored by Anastasiou et al., explores a fascinating synthesis of supergravity theories using the lens of division algebras. By leveraging the mathematical structure of $\mathcal{N}=1,2,4,8$ Yang-Mills theories in $D=3$, formulated over division algebras $R, C, H, O$, the authors demonstrate how combining left and right multiplets of these theories results in a construction known as the Freudenthal-Rosenfeld-Tits magic square. This construction is subsequently elevated to form a magic pyramid encompassing dimensions $D=3,4,6,10$, linked to the division algebras over space-time and Yang-Mills multiplets.

The magic pyramid consists of supergravity theories arranged such that the base, found at $D=3$, corresponds to the well-known $4 \times 4$ magic square, with the elevated levels forming progressively smaller squares, concluding with the apex at $D=10$ characterized by Type II supergravity. The conceptual advancement involves defining these constructions via a new algebraic structure termed the "magic pyramid formula". This formula is predicated on a triple of division algebras, one related to space-time and the others to the left/right Yang-Mills multiplets.

Notably, while prior studies have explored connections between gravity and Yang-Mills theories through gauge/gravitational amplitude techniques, this paper offers a novel structural interpretation via division algebras and their associated symmetries. The authors extend these formations to a conformal setting, thereby constructing a conformal magic pyramid through the tensoring of conformal supermultiplets in $D=3,4,6$. This yields prospects for an unconventional theory in $D=10$ characterized by $F_{4(4)}$ as a duality structure, albeit speculative due to lack of conventional classification parameters.

Numerical Results and Claims

The paper presents its strongest claims numerically via the dimension of the U-duality groups encapsulating the magic pyramid structure. Notable entries include:

• In $D=3$, the maximal supergravity U-duality embodies $E_{8(8)}$, forming part of the base's complete magic square.
• In $D=4$, maximal supergravity yields $E_{7(7)}$.
• The $D=6$ layer features $\SO(5,5)$ for maximal supergravity, while a non-gravitational (4,0) multiplet corresponds to $E_{6(6)}$.
• At the pyramid's tip in $D=10$, Type IIB supergravity manifests an $\SL(2,R)$ U-duality.

Implications

The paper seamlessly binds division-algebraic symmetry and supergravity formation, offering a fresh examination of U-duality groups via higher-dimensional field theory formulations. Theoretical implications of these findings suggest enhanced symmetry structures for supergravity theories that could lead to pivotal breakthroughs in understanding uniformities across diverse string theories or M-theory contexts. Practically, the insights on dualities might inform computational advancements in amplitude calculations, promoting more robust formulations of quantum gravity.

Future Speculations

Looking ahead, the paper prompts speculation on further developments in AI as related to theoretical physics. The integration of AI with symmetry-finding algorithms could expedite the discovery of new algebraic or geometric structures vital for theoretical advancements. Additionally, leveraging machine learning to simulate higher-dimensional bosonic structures could refine predictions regarding multidimensional supergravity dynamics.

In conclusion, the formation of a magic pyramid presents an intriguing algebraic approach to unifying supergravity theories across pivotal dimensions, hinting at a deeper understanding of symmetry in high-energy physics.