Extended Lie Derivatives and a New Formulation of D=11 Supergravity

Introducing an extended Lie derivative along the dual of A, the $3$-form field $D=11$ supergravity, the full diffeomorphism algebra of $D=11$ supergravity presented. This algebra suggests a new formulation of the theory, where the$3$-form field A is replaced by bivector Bᵃᵇ, bispinor B\α\β, spinor-vector \ηa\β $1$-forms. Only the bivector $1$-form Bᵃᵇ propagating, and carries the same degrees of freedom of the $3$-form in the formulation, its curl [\μ Bᵃᵇ\ν] being related to the\μ \ν a b curl of the $3$-form. The other $1$-forms are auxiliary, and transformation rules on all the fields close on the equations of motion of$D=11$ supergravity.