Abstract
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} = 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.
Highlights
One appeal of these higher spin dualities is that they serve as “simple” examples of holography: the abundance of symmetry opens an opportunity to build solvable and tractable instances of the AdS/CFT correspondence
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
In addition to the dictionary for the currents, we establish the holographic relation for the infinitesimal transformation parameters as well as for the sources, which is a key ingredient in the thermodynamic analysis of black hole solutions [1]
Summary
According to [21], the W-algebras that are relevant for the N = 2 version [22] of minimal model holography [23] contain, in addition to the super-Virasoro generators W (1), an infinite number of higher spin multiplets W (s), one for every integer spin s ≥ 2. The same is true for the structure constants appearing in the OPE’s of all the other higher spin multiplets Quite interestingly, it possible to consistently decouple the multiplets W (s) with s ≥ 3 and truncate the full higher spin algebra to one containing just the super-Virasoro currents W (1) and the spin-2 multiplet W (2).
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