Abstract
The vectorial holographic correspondences between higher-spin theories in AdS$_5$ and free vector models on the boundary are extended to the cases where the latter is described by free massless spin-$j$ field. The dual higher-spin theory in the bulk does not include gravity and can only be defined on rigid AdS$_5$ background with $S^4$ boundary. We discuss various properties of these rather special higher-spin theories and calculate their one-loop free energies. We show that the result is proportional to the same quantity for spin-$j$ doubleton treated as if it is a AdS$_5$ field. Finally, we consider even more special case where the boundary theory itself is given by an infinite tower of massless higher-spin fields.
Highlights
Contents can be identified from the correspondence, and the one-loop partition function was computed in [22]
The vectorial holographic correspondences between higher-spin theories in AdS5 and free vector models on the boundary are extended to the cases where the latter is described by free massless spin-j field
The generalization is based on the fact that the short representations of four-dimensional conformal symmetry so(2, 4) are the spin-j massless particles where j is any half-integer number
Summary
Free massless fields of any spin carry representations of conformal symmetry so(2, 4) and they correspond to the UIR D(j + 1, (j, ±j)) , where D(∆, (l1, l2)) is the UIR having lowest energy ∆ and its eigenvector(or tensor) transforms as (l1, l2) Young diagram representation of so(4). Combining χS[j,0] and χS[0,j] , we obtain the character of parity-invariant representation as χSj (β, α+, α−) = χS[j,0] (β, α+, α−) + χS[0,j] (β, α+, α−). When j is an integer, the boundary operator corresponding to Sj is the curvature tensor, Ra1b1,...,ajbj , having (j, j) Young diagram representation. They are traceless — any contraction of two indices vanish — and subject to the Bianchi identity and the conservation condition,. With gauge symmetry, δ φa1···aj = ∂(a1 ξa2···aj ) This makes link the curvature formulation to the Fronsdal’s one [42] having two-derivative equation, Fa1···aj =. For spin j ≥ 3/2 , it can be defined only on a conformally flat and Einstein background [32, 33]
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