A locally supersymmetric $SO(10,2)$ invariant action for $D=12$ supergravity

Abstract

We present an action for $N=1$ supergravity in $10+2$ dimensions, containing the gauge fields of the $OSp(1|64)$ superalgebra, i.e. one-forms $B^{(n)}$ with $n$=1,2,5,6,9,10 antisymmetric D=12 Lorentz indices and a Majorana gravitino $\psi$. The vielbein and spin connection correspond to $B^{(1)}$ and $B^{(2)}$ respectively. The action is not gauge invariant under the full $OSp(1|64)$ superalgebra, but only under a subalgebra $\tilde{F}$ (containing the $F$ algebra $OSp(1|32)$), whose gauge fields are $B^{(2)}$, $B^{(6)}$, $B^{(10)}$ and the Weyl projected Majorana gravitino $\frac{1}{2}(1+{\Gamma }_{13})\psi$. Supersymmetry transformations are therefore generated by a Majorana-Weyl supercharge and, being part of a gauge superalgebra, close off-shell. The action is simply $\int STr({\mathbf{R}}^6\mathbf{\Gamma })$ where $\mathbf{R}$ is the $OSp(1|64)$ curvature supermatrix two-form, and $\mathbf{\Gamma }$ is a constant supermatrix involving ${\Gamma }_{13}$ and breaking $OSp(1|64)$ to its $\tilde{F}$ subalgebra. The action includes the usual Einstein-Hilbert term.

LaTeX, 13 pages. Added a reference, a Table in Appendix A for the gamma commutations in d=12, and corrected eq. (4.14) for the Einstein-Hilbert term; v4: corrected formulas (A.3), (A.4) and (A.10), modified last paragraph of Section 5, added acknowledgements

References in corpus (1)

• The Integral Form of Supergravity

Cited by in corpus (4)

• Supergravity in the group-geometric framework: a primer
• The Geometry of Exceptional Super Yang-Mills Theories
• Supergravities and Branes from Hilbert-Poincaré Series
• Gauge supergravity in $D=2+2$