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

This is a long-overdue companion paper to [1]. We study the relation between sl(3|2) Chern-Simons supergravity on AdS₃ and two-dimensional CFT’s with $$ \mathcal{N} $$N = 2 super- $$ {\mathcal{W}}_3 $$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 $$ \mathcal{N} $$N = 2 super- $$ {\mathcal{W}}_3 $$W 3 conformal algebra, including its multiplet structure, OPE’s and spectral flow invariance, correcting some minor typos present in the literature.