A 2d/1d Holographic Duality

Abstract

We propose $Ad{S}_2$/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 $Ad{S}_2$ background (in the bulk). The boundary theories can be embedded as 1d sectors of 3d $\mathcal{N}=4$ 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 $Ad{S}_4$ 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 $Ad{S}_4$ supergravity dual. We propose that the $U(N)$ singlet sectors of the 1d vector models are dual to 2d gauge theories on $Ad{S}_2$ 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 $Ad{S}_2$ 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 $Ad{S}_2$ can be obtained from localizing the $Ad{S}_4$ 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 $Ad{S}_4$/CFT${}_3$ dualities. We discuss the boundary terms required by holographic renormalization in the 2d gauge theories on $Ad{S}_2$ and show how they arise from supersymmetric localization.

70 pages plus appendices, 4 figures

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• Evidence for an Algebra of ${\mathbf{G}}_2$ Instantons
• 4d/2d $\to$ 3d/1d: A song of protected operator algebras
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• Bound on Lyapunov exponent in $c=1$ matrix model
• Three-Point Functions in ABJM and Bethe Ansatz
• Cosmology as a CFT${}_1$
• Fermi-gas correlators of ADHM theory and triality symmetry