Supersymmetric Localization in AdS$_5$ and the Protected Chiral Algebra

TL;DR

This paper explores how supersymmetric localization in AdS$_5$ can be used to derive a dual two-dimensional chiral algebra, illustrating the process with a simplified model and proposing a connection to Chern-Simons theory and higher-spin algebras.

Contribution

It demonstrates the application of supersymmetric localization in AdS$_5$ to obtain a dual chiral algebra, using a simplified vector multiplet model and discussing extensions to full supergravity.

Findings

Localization yields Chern-Simons theory on AdS$_3$ with affine Lie algebra boundary dual.

The simplified model captures key features of the full AdS$_5 imes S^5$ supergravity.

Proposes duality between large $N$ chiral algebra and higher-spin Chern-Simons theory.

Abstract

${\cal N} =4$ super Yang-Mills theory admits \cite{Beem:2013sza} a protected subsector isomorphic to a two-dimensional chiral algebra, obtained by passing to the cohomology of a certain supercharge. In the large $N$ limit, we expect this chiral algebra to have a dual description as a subsector of IIB supergravity on $AdS_5 \times S^5$. This subsector can be carved out by a version of supersymmetric localization, using the bulk analog of the boundary supercharge. We illustrate this procedure in a simple model, the theory of an ${\cal N}=4$ vector multiplet in $AdS_5$, for which a convenient off-shell description is available. This model can be viewed as the simplest truncation of the full $AdS_5 \times S^5$ supergravity, in which case the vector multiplet should be taken in the adjoint representation of ${\mathfrak g}_F = \mathfrak {su}(2)_F$. Localization yields Chern-Simons theory on $AdS_3$ with gauge algebra ${\mathfrak g}_F$, whose boundary dual is the affine Lie algebra $\widehat {\mathfrak g}_F$. We comment on the generalization to the full bulk theory. We propose that the large $N$ limit of the chiral algebra of ${\cal N}=4$ SYM is again dual to Chern-Simons theory, with gauge algebra a suitable higher-spin superalgebra.