A note on the $\mathcal{N}=2$ super-$\mathcal{W}_3$ holographic dictionary

TL;DR

This paper establishes a detailed holographic dictionary linking $sl(3|2)$ Chern-Simons supergravity in AdS$_3$ to 2D CFTs with $ =2$ super-$ ext{W}_3$ symmetry, clarifying asymptotic symmetries and boundary correspondences.

Contribution

It provides a complete analysis of the asymptotic symmetries and the explicit holographic dictionary for the $ =2$ super-$ ext{W}_3$ algebra in the context of $sl(3|2)$ supergravity.

Findings

Identified the superalgebra $su(2,1|1,1)$ as the real form in Lorentzian signature.

Established the precise correspondence between bulk currents and boundary operators.

Reviewed and corrected properties of the $ =2$ super-$ ext{W}_3$ algebra.

Abstract

This is a long-overdue companion paper to arXiv:1512.00073. We study the relation between $sl(3|2)$ Chern-Simons supergravity on AdS$_3$ and two-dimensional CFT's with $\mathcal{N}=2$ super-$\mathcal{W}_3$ symmetry. Specifically, we carry out a complete analysis of asymptotic symmetries in a basis that makes the superconformal structure transparent, allowing us to establish the precise dictionary between currents and transformation parameters in the bulk and their boundary counterparts. We also discuss the incorporation of sources and display in full detail the corresponding holographic Ward identities. By imposing suitable hermiticity conditions on the CFT currents, we identify the superalgebra $su(2,1|1,1)$ as the appropriate real form of $sl(3|2)$ in Lorentzian signature. We take the opportunity to review some of the properties of the $\mathcal{N}=2$ super-$\mathcal{W}_3$ conformal algebra, including its multiplet structure, OPE's and spectral flow invariance, correcting some minor typos present in the literature.