The goldstone fields of interacting higher spin field theory in AdS(4)
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PROCEEDINGS OF THE XII INTERNATIONAL CONFERENCE ON SYMMETRY METHODS IN PHYSICS JULY 3–8 (2006), YEREVAN, ARMENIA
The Goldstone Fields of Interacting Higher Spin Field Theory in AdS(4)* ¨ ** W. Ruhl Department of Physics, Kaiserslautern University of Technology, Germany Received August 20, 2007
Abstract—A higher spin field theory on AdS(4) possesses a conformal theory on the boundary R(3) which can be identified with the critical O(N ) sigma model of O(N ) invariant fields only. The notions of quasiprimary and secondary fields can be carried over to the AdS theory. If de Donder’s gauge is applied, the traceless part of the higher spin field on AdS(4) is quasiprimary and the Goldstone fields are quasiprimary fields to leading order too. Those fields corresponding to the Goldstone fields in the critical O(N ) sigma model are odd-rank symmetric tensor currents which vanish in the free-field limit. PACS numbers: 75.10.Jm DOI: 10.1134/S1063778808060148
1. INTRODUCTION If we move from the free-field limit N = ∞ of the critical O(N ) sigma model to the interacting conformal field theory CFT(3) by a renormalization group transformation [1, 2], gauge symmetry of the higher spin field theory HS(4) on the AdS(4) side is broken and a Goldstone field arises. Such Goldstone modes are proper dynamical degrees of freedom that must be represented also in the boundary conformal field theory. Its quantum numbers being known, we must try to find such field in the list of fields for CFT(3). To derive such a list, one looks first for all quasiprimary fields in the free-field limit N = ∞. A simple algorithm tells us [3] how to choose the quasiprimary fields from all composites of the derivatives of the free massless fields and select the O(N ) invariant ones from them. Symmetric traceless tensor fields are then always of even rank (odd-rank tensors belong to nontrivial O(N ) representations and are eliminated). Then one perturbs these composites by switching on the interaction. They obtain in this way anomalous dimensions. Several quasiprimary fields in the free theory have degenerate quantum numbers (tensor type and dimension). Certain linear combinations of them form eigenvectors with respect to the anomalous dimensions as eigenvalues. These are the interacting quasiprimary fields. Among these quasiprimary fields in the free-field theory are exceptional elementary representations ∗ **
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which belong to conserved currents, their divergence being zero. If we switch on the interaction, these currents obtain not only a (positive) anomalous dimension, but their divergence is nonzero and gives a new quasiprimary field. Their two-point function is proportional to their anomalous dimension. In this fashion, we can obtain symmetric traceless tensor fields with odd tensor rank. The Goldstone degree of freedom on the boundary of AdS(4) is represented by these field operators (in de Donder’s gauge). Thus, the renormalization group, which looks continuous, produces a discontinuous jump in the field-the
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