Twisted Superalgebras and Cohomologies of the N=2 Superconformal Quantum Mechanics

Abstract

We prove that the invariance of the N=2 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 Q_i-exact (i=1,2,...,6), with a Q_{i'}-exact (i'\neq i) 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 sl(2), but only its Borel subalgebra. The general coordinate covariance of the non-linear sigma-model for the N=2 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 N=2 superconformal quantum mechanics is explicitly presented.