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

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 ψ. 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 F˜ (containing the F algebra OSp(1|32)), whose gauge fields are B⁽²⁾, B⁽⁶⁾, B⁽¹⁰⁾ and the Weyl projected Majorana gravitino 12(1+Γ13)ψ. Supersymmetry transformations are therefore generated by a Majorana-Weyl supercharge and, being part of a gauge superalgebra, close off-shell. The action is simply ∫STr(R⁶Γ) where R is the OSp(1|64) curvature supermatrix two-form, and Γ is a constant supermatrix involving Γ₁₃ and breaking OSp(1|64) to its F˜ subalgebra. The usual Einstein-Hilbert term is included in the action.