What about SUGRA?

We want some analogue of (17) for SUGRA. Let us first understand the LHS.

The LHS should give the space of vertex operators in a given background.

Equivalently: this is the same as the space of solutions of the linearized SUGRA equations. (Linearized around a given background solution.)

We will only discuss the case when the background solution is (or flat space as a limit of AdS). (Generalization of these methods to arbitrary backgrounds is work in progress.)

In this case, we can define the BRST cohomology based on the representations of the superconformal algebra.