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

Abstract

Introducing an extended Lie derivative along the dual of $A$, the $3$-form field of $D=11$ supergravity, the full diffeomorphism algebra of $D=11$ supergravity is presented. This algebra suggests a new formulation of the theory, where the $3$-form field $A$ is replaced by bivector $B^{ab}$, bispinor $B^{\alpha\beta}$, and spinor-vector $\eta^{a\beta}$ $1$-forms. Only the bivector $1$-form $B^{ab}$ is propagating, and carries the same degrees of freedom of the $3$-form in the usual formulation, its curl $\mathcal{D}_{[\mu} B^{ab}_{\nu]}$ being related to the $F_{\mu \nu a b}$ curl of the $3$-form. The other $1$-forms are auxiliary, and the transformation rules on all the fields close on the equations of motion of $D=11$ supergravity.