A 2d/1d Holographic Duality

Abstract

We propose $AdS_2$/CFT$_1$ dualities between exactly solvable topological quantum mechanics theories with vector or matrix large $N$ limits (on the boundary) and weakly coupled gauge theories on a fixed $AdS_2$ background (in the bulk). The boundary theories can be embedded as 1d sectors of 3d ${\cal N} = 4$ superconformal field theories with holographic duals, from which they can be obtained using supersymmetric localization. We study a few examples of such 1d theories: theories with vector large $N$ limits that are embedded into 3d theories of many free massless hypermultiplets with $AdS_4$ higher spin duals; and a 1d theory with a matrix large $N$ limit embedded into the 3d ABJM theory at Chern-Simons level $k=1$, which has an $AdS_4$ supergravity dual. We propose that the $U(N)$ singlet sectors of the 1d vector models are dual to 2d gauge theories on $AdS_2$ whose gauge algebras are finite dimensional and whose full non-linear actions we completely determine in some cases. The 1d theory embedded into ABJM theory has a $\mathbb{Z}_2$-invariant sector dual to a 2d gauge theory on $AdS_2$ whose gauge algebra is the infinite dimensional algebra of area preserving diffeomorphisms of a two-sphere. We provide evidence that the 2d gauge theories on $AdS_2$ can be obtained from localizing the $AdS_4$ duals of the 3d SCFTs mentioned above, and thus argue that our 2d/1d dualities can be obtained via supersymmetric localization on both sides of their parent $AdS_4$/CFT$_3$ dualities. We discuss the boundary terms required by holographic renormalization in the 2d gauge theories on $AdS_2$ and show how they arise from supersymmetric localization.