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

TL;DR

This paper explores the structure of twisted superalgebras controlling N=2 superconformal quantum mechanics, revealing how invariant actions are uniquely determined by specific subalgebras and their implications for models with Calogero potentials.

Contribution

It identifies the subalgebras governing invariance, demonstrates the unique determination of actions, and clarifies the connection between twisted and ordinary formulations of N=2 superconformal quantum mechanics.

Findings

Invariant actions are Q_i-exact with a common antecedent.

Actions with Calogero potentials are uniquely determined by a subalgebra.

The transformation between twisted and ordinary formulations is explicitly constructed.

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.