Octonions and supergravity

Abstract

This thesis makes manifest the roles of the normed division algebras R,C,H and O in various supergravity theories. Of particular importance are the octonions O, which frequently occur in connection with maximal supersymmetry, and hence also in the context of string and M-theory. Studying the symmetries of M-theory is perhaps the most straightforward route towards understanding its nature, and the division algebras provide useful tools for such study via their deep relationship with Lie groups. After reviews of supergravity and the definitions and properties of R,C,H and O, a division-algebraic formulation of pure super Yang-Mills theories is developed. In any spacetime dimension a Yang-Mills theory with Q real supercharge components is written over the division algebra with dimension Q/2. In particular then, maximal Q = 16 super Yang-Mills theories are written over the octonions, since O is eight-dimensional. In such maximally supersymmetric theories, the failure of the supersymmetry algebra to close off-shell (using the conventional auxiliary field formalism) is shown to correspond to the non-associativity of the octonions. Making contact with the idea of ‘gravity as the square of gauge theory’, these division-algebraic Yang-Mills multiplets are then tensored together in each spacetime dimension to produce a pyramid of supergravity theories, with the Type II theories at the apex in ten dimensions. The supergravities at the base of the pyramid have global symmetry groups that fill out the famous Freudenthal-Rosenfeld-Tits magic square. This magic square algebra is generalised to a ‘magic pyramid algebra’, which describes the global symmetries of each Yang-Mills-squared theory in the pyramid. Finally, a formulation of eleven-dimensional supergravity over the octonions is presented. Toroidally compactifying this version of the theory to four or three spacetime dimensions leads to an interpretation of the dilaton vectors (which organise the coupling of the seven or eight dilatons to the other bosonic fields) as the octavian integers – the octonionic analogue of the integers.