Abstract
We use the Chern-Simons formulation of higher spin theories in three dimensions to study aspects of holographic W-gravity. Concepts which were useful in studies of pure bulk gravity theories, such as the Fefferman-Graham gauge and the residual gauge transformations, which induce Weyl transformations in the boundary theory and their higher spin generalizations, are reformulated in the Chern-Simons language. Flat connections that correspond to conformal and lightcone gauges in the boundary theory are considered.
Highlights
Which of the symmetries one wants to maintain dictates the choice of the counterterms
In the same way as a CFT can be coupled to an external metric, which sources the energy-momentum tensor of the CFT, a CFT with higher-spin W-symmetries can be coupled to higher-spin gauge fields, which source conserved higher-spin currents
Two-dimensional theories with W-algebras as symmetry algebra were intensively studied about 25 years ago, in the context of string theory, but it is fair to say that the implications of the higher-spin symmetries are much less understood than those of the conformal symmetry
Summary
The goal of the previous chapter was to establish relations between the metric formulation of three-dimensional gravity and its connection formulation as a Chern-Simons theory. Generalizing the discussion of the previous chapter we expect that ζ0 parametrizes Weyl and W-Weyl transformations of the boundary fields and that ζ+ can be used to transform the connections back to FG gauge.. The residual gauge transformations parametrized by σ2 and σ3 act in a simple way on the leading terms of the FG expansion of the metric-like fields: g(−2). In this case the boundary metric has Weyl weight 2 whereas the spin-3 field has Weyl weight 4. In this chapter we identified the W-Weyl transformations as the diagonal gauge transformations generated by the Cartan directions in sl(N ) This leads us to conjecture the change of the effective action, which is a non-local functional of the metric and the higher-spin boundary fields δσs W k 2π σs tr W0(s)(dA + dA). We will come back to this when we discuss specific gauges, which is what we will do
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