A 2d/1d Holographic Duality

TL;DR

The paper introduces a novel class of AdS$_2$/CFT$_1$ dualities tying tractable 1d topological quantum mechanics with vector or matrix large $N$ limits to weakly coupled 2d gauge theories on AdS$_2$. By embedding the 1d theories into 3d ${ m N}=4$ SCFTs and applying supersymmetric localization, the authors derive corresponding 2d YM theories on AdS$_2$ with explicit boundary terms dictated by holographic renormalization and alternate quantization. They present concrete vector- and matrix-large $N$ examples, including ABJM-based constructions, and compute boundary correlators that match bulk YM$_2$ data, uncovering an emergent SDiff$(S^2)$ symmetry in the matrix case. The framework connects localization on the AdS$_4$/CFT$_3$ side to the AdS$_2$/CFT$_1$ dualities, offering a bottom-up plus localization-based top-down picture with explicit non-linear bulk actions and a route to studying higher-spin-like algebras in this 2d/1d setting. The results open avenues for exploring quantum corrections, Lorentzian continuations, and broader holographic realizations of 1d topological sectors within higher-dimensional AdS/CFT webs.

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.

Figures (4)

• Figure 1: The planar and non-planar diagrams that contribute to $B_2$.
• Figure 2: The planar diagram that contributes to $C_{222}$.
• Figure 3: $AdS_4$ as a $S^1_\tau$ fibration over ${1\over 2}AdS_3$.
• Figure 4: Reconstruction of classical solution for bulk gauge fields from Witten diagrams.