Octonionic D=11 Supergravity and 'Octavian Integers' as Dilaton Vectors

TL;DR

This paper reformulates D=11 supergravity using octonions, interpreting dilaton vectors as octavian integers, and explores their geometric and algebraic properties related to the Fano plane, leading to insights into various supergravity truncations.

Contribution

It introduces a novel octonionic formulation of D=11 supergravity and interprets dilaton vectors as octavian integers using Fano plane duality, enabling new truncations.

Findings

Dilaton vectors can be represented as octavian integers.

The Fano plane duality provides a geometric interpretation of these vectors.

The approach suggests consistent truncations to N=8,4,2,1 supergravities.

Abstract

We formulate D=11 supergravity over the octonions by rewriting 32-component Majorana spinors as 4-component octonionic spinors. Dimensional reduction to D=4 and D=3 suggests an interpretation of the so-called 'dilaton vectors', which parameterise the couplings of the dilatons to other fields in the theory, as unit 'octavian integers' - the octonionic analogues of integers. The parameterisation involves a novel use of the duality between points and lines on the Fano plane, and suggests a series of consistent truncations with N=8,4,2,1, giving the 'four curious supergravities' studied by Duff and Ferrara