Massless Conformal Fields in Ten Dimensions, Minimal Unitary Representation of $E_{7(-25)}$ and Exceptional Supergravity

Abstract

Minimal unitary representation of $SO(d,2)$ and its deformations describe all the conformally massless fields in $d$ dimensional Minkowskian spacetimes. In critical dimensions these spacetimes admit extensions with twistorial coordinates plus a dilatonic coordinate to causal spacetimes coordinatized by Jordan algebras $J_3^{A}$ of degree three over the four division algebras $A= R , C , H , O $. We study the minimal unitary representation (minrep) of the conformal group $E_{7(-25)}$ of the spacetime coordinatized by the exceptional Jordan algebra $J_3^{O}$. We show that the minrep of $E_{7(-25)}$ decomposes into infinitely many massless representations of the conformal group $SO(10,2)$. Corresponding conformal fields transform as symmetric tensors in spinor indices of $SO(9,1)$ subject to certain constraints. Even and odd tensorial fields describe bosonic and fermionic conformal fields, respectively. Each irrep of $SO(10,2)$ falls into a unitary representation of an $SU(1,1)$ subgroup that commutes with $SO(10,2)$. The noncompact generators in spinor representation $16 $ of $SO(10)$ interpolate between the bosonic and fermionic representations and hence act like "bosonic supersymmetry" generators. We also give the decomposition of the minrep of $E_{7(-25)}$ with respect to the subgroup $SO^*(12)\times SU(2)$ with $SO^*(12) $ acting as the conformal group of the spacetime coordinatized by $J_3^{H}$. Group $E_{7(-25)}$ is also the U-duality group of the exceptional $N=2$ Maxwell-Einstein supergravity in four dimensions. We discuss the relevance of our results to the composite scenario that was proposed for the exceptional supergravity so as to accommodate the families of quarks and leptons of the standard model as well as to the proposal that $E_{7(-25)}$ acts as spectrum generating symmetry group of the $5d$ exceptional supergravity