Lifting of states in 2-dimensional $N=4$ supersymmetric CFTs

Abstract

We consider states of the D1-D5 CFT where only the left-moving sector is excited. As we deform away from the orbifold point, some of these states will remain BPS while others can `lift'. The lifting can be computed by a path integral containing two twist deformations; however, the relevant 4-point amplitude cannot be computed explicitly in many cases. We analyze an older proposal by Gava and Narain where the lift can be computed in terms of a finite number of 3-point functions. A direct Hamiltonian decomposition of the path integral involves an infinite number of 3-point functions, as well the first order correction to the starting state. We note that these corrections to the state account for the infinite number of 3-point functions arising from higher energy states, and one can indeed express the path-integral result in terms of a finite number of 3-point functions involving only the leading order states that are degenerate. The first order correction to the supercharge $\bar G^{(1)}$ gets replaced by a projection $\bar G^{(P)}$; this projected operator can also be used to group the states into multiplets whose members have the same lifting.