Deformations of AdS and exact sequences

TL;DR

This work develops a cohomological framework for deformations of $AdS_5\times S^5$ in the pure spinor formalism, framing background deformations as BRST cohomology classes and organizing their extensions via Ext groups. A bicomplex with BRST and Lie algebra differentials yields two spectral sequences that illuminate how BRST-exact products resolve into higher cohomology obstructions, including a concrete toy example that produces a nontrivial $\mathrm{Ext}^1$ class. The paper identifies finite-dimensional AdS/CFT representations connected to deformations, described by super-Young diagrams, and lays out a program to resolve both wedge products and $\mathfrak{g}$-invariant vertices through exact sequences and higher Ext-classes. Open questions include proving conjectures about the irreducibility/semisimplicity of these representations, extending the framework to infinite-dimensional cases, and performing explicit cocycle computations to enable concrete correlation-function analyses within AdS/CFT.

Abstract

We give examples of cohomologies of the superconformal algebra, relevant to computations in the AdS supergravity. Our main examples are deformations of $AdS_5\times S^5$ transforming in finite-dimensional representations of the superconformal algebra at the linearized level. In the study of correlation functions, it is important to compute the resolution of BRST-exact products of vertex operators. The resolution is typically non-covariant, because of cohomological obstacles. Using the pure spinor formalism, we develop a framework to describe these obstacles, and formulate conjectures about their structure.

Figures (4)

• Figure 1: Young diagramms corresponding to $\int d^4x \;\hbox{tr}Z^7$
• Figure 2: Some other Young diagramms
• Figure 3: Super-trace map, lowering the rank of the tensor
• Figure 4: Eq. (\ref{['SDeltaIsDeltaS']})