Abstract
We show that truncating the exact renormalization group equations of free U(N) vector models in the single-trace sector to the linearized level reproduces the Fronsdal equations on AdS d+1 for all higher spin fields, with the correct boundary conditions. More precisely, we establish canonical equivalence between the linearized RG equations and the familiar local, second order differential equations on AdS d+1, namely the higher spin Fronsdal equations. This result is natural because the second-order bulk equations of motion on AdS simply report the value of the quadratic Casimir of the corresponding conformal modules in the CFT. We thus see that the bulk Hamiltonian dynamics given by the boundary exact RG is in a different but equivalent canonical frame than that which is most natural from the bulk point of view.
Highlights
Background symmetriesThe path integration in (2.10) is over the set of all square integrable complex scalar functions over the space-time R1,d−1, namely L2(R1,d−1)
We show that truncating the exact renormalization group equations of free U(N ) vector models in the single-trace sector to the linearized level reproduces the Fronsdal equations on AdSd+1 for all higher spin fields, with the correct boundary conditions
U(L2) as a background symmetry, under which the sources Wμ and B transform like a connection and an adjoint-valued field respectively: Wμ = L−1·Wμ·L + L−1· [Pμ, L]·, B = L−1·B·L. It was argued in [1], that the relevant geometry here is that of infinite jet bundles, i.e., Wμ is a connection 1-form on the infinite jet bundle over R1,d−1, while B is a section of its endomorphism bundle
Summary
In order to be self-contained, we will review some details of the holographic dual to the free bosonic U(N ) vector model constructed in [2]. The most general U(N )-invariant “single-trace” deformations away from the free fixed point can be incorporated by introducing the two bi-local sources B(x, y) and Wμ(x, y) as follows. Given the “matrix” notation we have introduced above, we can define a product and a trace between bi-locals as follows:. The sources B and Wμ that we have introduced above couple, respectively, to the following bi-local operators. There is an important subtlety in defining U(N ) singlet bi-local operators which should be pointed out — since φm(x) is a section of a U(N ) vector bundle, the only natural contraction between φ∗m(y) and φm(x) should involve a U(N ) Wilson line. We may source all the operators of interest, namely φ∗mφm, φ∗m∂μφm, φ∗m∂μ∂ν φm · · ·. Where we have introduced a new source U (for the identity operator) to keep track of the overall normalization
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