@MISC{Cederwall92e7as, author = {Martin Cederwall}, title = {E7 as D = 10 space-time symmetry — Origin of the twistor transform}, year = {1992} }

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Abstract

Massless particle dynamics in D = 10 Minkowski space is given an E7-covariant formulation, including both space-time and twistor variables. E7 contains the conformal algebra as a subalgebra. Analogous constructions apply to D = 3, 4 and 6. Submitted to Physics Letters BIt is well known that massless particle actions are conformal invariant (with the possible exception of D = 10 superparticles, in which case the matter is not clearly understood). However, there is a still larger symmetry present, as will be demonstrated in this paper. Only bosonic particles will be treated. Explicit calculations apply to D = 10 – completely analogous constructions are valid in D = 3, 4 and 6. Some technical details and conventions are found in the appendix. One has traditionally two options for manifestly conformal formulations of particle dynamics, the space-time picture and, in D = 3, 4, 6 or 10, the twistor picture. The space-time formulation, on one hand, with PmP m ≈ 0 as only constraint, is easily made conformally covariant by enlarging the vectors Xm, P m of SO(1, 9) to become vectors X µ, P µ of SO(2, 10). The constraints are taken to be XµX µ ≈ 0, XµP µ ≈ 0 and PµP µ ≈ 0. With gauge choices X ⊕ = c (=constant) and P ⊕ = 0, the ordinary space-time picture is recovered (this of course applies to any dimensionality). The twistor picture[1-4], on the other hand, is reached via the twistor transform P m = 1 2 ψα γ m αβ ψβ ωα = Xmγ m αβ ψβ (1) which implies that the conformal spinor Z A = [ψ α, ωα] t satisfies seven constraints, generating S7 (a covariantly conformal form of these constraints is given in ref.4). Consider now a set of phase-space variables ΞM ∈ 56 of E7. When E7 → Sp(2) × SO(2, 10) the the branching rule[5] is 56 → (2, 12) + (1, 32) and for the adjoint 133 → (3, 1) + (2, 32 ′) + (1, 66), so that ΞM = (Saµ, ZA) and 133 ∋ T A = (T {ab}, T a A ′, T [µν]). If {ΞM, ΞN} = gMN, E7 is generated by T A = 1 2ΞM ΩA MN ΞN, ΩA MN being Clebsh-Gordan coefficients for 56 × 56 × 133 → 1. Saµ = (Xµ, P µ) t is to be interpreted as the conformal space-time vectors and ZA as the twistor variables. The set of constraints is chosen to be