Supersymmetric localization in AdS5 and the protected chiral algebra

Abstract N=4 $$ \mathcal{N}=4 $$ super Yang-Mills theory admits [1] 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 × 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 N=4 $$ \mathcal{N}=4 $$ 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 × S 5 supergravity, in which case the vector multiplet should be taken in the adjoint representation of gF=su2F $$ {\mathfrak{g}}_F=\mathfrak{s}\mathfrak{u}{(2)}_F $$. Localization yields Chern-Simons theory on AdS 3 with gauge algebra gF $$ {\mathfrak{g}}_F $$, whose boundary dual is the affine Lie algebra g^F $$ {\widehat{\mathfrak{g}}}_F $$. We comment on the generalization to the full bulk theory. We propose that the large N limit of the chiral algebra of N=4 $$ \mathcal{N}=4 $$ SYM is again dual to Chern-Simons theory, with gauge algebra a suitable higher-spin superalgebra