Supersymmetric holography on AdS3

TL;DR

This work presents substantial evidence for a holographic duality between Vasiliev's ${ m N}=2$ supersymmetric higher spin theory on AdS$_3$ and the large $N$ limit of ${ m N}=2$ Kazama–Suzuki models. The authors compute and match the 1-loop partition functions on both sides in the 't Hooft limit, developing a first-principles free-field construction to derive coset branching functions and their large $N$ behaviour. They extend the bosonic non-supersymmetric matching to the SUSY case by formulating the higher spin supergravity partition function with ${ m gl}( frac{ ext{infty}}{ ext{infty})}_+$ structures and by expressing the CFT side in terms of ${ m W}_{N,k}$ representations, NS-sector characters, and free-field correlates. The result is a consistent partition-function equality, indicating a robust SUSY higher spin holography and revealing the role of ${ m gl}( ext{infty}| ext{infty})_+$ symmetry and null-state decoupling in the large-$N$ limit.

Abstract

The proposed duality between Vasiliev's supersymmetric higher spin theory on AdS3 and the 't Hooft limit of the 2d superconformal Kazama-Suzuki models is analysed in detail. In particular, we show that the partition functions of the two theories agree in the large N limit.

Figures (4)

• Figure 1: Young diagrams that are finite only in the horizontal direction, and that have a single infinite vertical step label $\mathop{\mathrm{\mathfrak{su}}}\nolimits(N)$ representations generated by the tensor product of finitely many fundamental and dual representations in the limit $N\to\infty$.
• Figure 2: A $\mathop{\mathrm{\mathfrak{u}}}\nolimits(N)$ representation is labelled by a pair of (finite) Young diagrams $\boldsymbol{\Lambda}=(\Lambda_l,\Lambda_r)$ such that the sum of their rows is at most $N$. The corresponding $\mathop{\mathrm{\mathfrak{su}}}\nolimits(N)$ dominant weight is represented by the Young diagram with a bold contour, denoted by $\boldsymbol{\Lambda}_N$.
• Figure 3: Conformal dimensions of the scalar and spinor fields in the two short $\mathcal{N}=2$ complex supermultiplets. Here $Q^\pm$ and $\tilde{Q}^\pm$ are the left- and right-moving $\mathcal{N}=2$ supercharges in the CFT. Since the representation is short, one of the two supercharges of each chirality always acts trivially. The Dirac fermions have multiplicity $2$ since the scalar fields are complex.
• Figure 4: A supertableau of shape $\Lambda$ and type $\mathop{\mathrm{\mathfrak{gl}}}\nolimits(\infty|\infty)_+$ is a filling of the boxes of a Young diagram $\Lambda$ with elements from $\mathbb{N}_0$ such that the entries of the boxes are ordered as indicated in the figure.