Twisted N=1N=1, d=4d=4 supergravity and its symmetries

We display the construction of a twisted superalgebra for the N=1N=1 Euclidean supergravity on 4-manifolds with an almost complex structure. It acts on a representation of twisted supersymmetry made of forms with odd and even statistics and it is covariant under a U(2)⊂SO(4)U(2)⊂SO(4) Lorentz invariance of the manifoldʼs tangent-space. It contains 4 twisted supersymmetry generators, one nilpotent scalar, one vector and one pseudo-scalar. The superalgebra closes on the twisted fields of supergravity in its new minimal set of auxiliary fields. Its couplings to the twisted Wess and Zumino and vector multiplets are also determined.