Twisted superalgebras and cohomologies of the image superconformal quantum mechanics Original Research Article

We prove that the invariance of the image superconformal quantum mechanics is controlled by subalgebras of a given twisted superalgebra made of 6 fermionic (nilpotent) generators and 6 bosonic generators (including a central charge). The superconformal quantum mechanics actions are invariant under this quite large twisted superalgebra. On the other hand, they are fully determined by a subalgebra with only 2 fermionic and 2 bosonic (the central charge and the ghost number) generators. The invariant actions are image-exact (image), with a image-exact (image) antecedent for all 6 fermionic generators. It follows that the superconformal quantum mechanics actions with Calogero potentials are uniquely determined even if, in its bosonic sector, the twisted superalgebra does not contain the one-dimensional conformal algebra image, but only its Borel subalgebra. The general coordinate covariance of the non-linear sigma-model for the image supersymmetric quantum mechanics in a curved target-space is fully implied only by its worldline invariance under a pair of the 6 twisted supersymmetries. The transformation connecting the ordinary and twisted formulations of the image superconformal quantum mechanics is explicitly presented.