REVIEW
On limits of superstring in AdS_5 x S^5
High Energy Physics - Theory · Submitted 15 Jan 2002 · 31 references
Pith's verdict: A clean, honest derivation of the λ=0 light-cone action with an explicitly conjectural higher-spin claim; the constraint algebra in the limit is left unexamined.
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Referee report
The paper
The paper claims
Taking the string tension to zero in AdS$_5$ × S$^5$ leaves an action whose zero modes are supergravity, and — the author conjectures — massless higher-spin fields as well.
The reading
What carries the argument
The engine is the light-cone phase-space Hamiltonian of the AdS$_5$ × S$^5$ superstring derived in the earlier light-cone treatment, together with the observation that the tensionless limit is the operation of deleting all $σ$-derivative terms from it. The Hamiltonian density in the 'standard AdS/CFT' normalization reads $H = (1/2p^+)(P_⊥^2 + T^2 e^{4φ} x_⊥'^2 + e^{2φ}[Π^2 + T^2 φ'^2 + P^M P^M + T^2 u'^M u'^M + p^{+2}(η^2)^2 + 2ip^+ η^i (ρ^{MN})_{ij} η^j u^M P^N]) - T e^{2φ} u^M[η^i ρ^M{}_{ij}(θ'^j - i√2 e^φ η^j x') + h.c.]$, with $T = R^2/(2πα') = √λ/(2π)$. Setting $T = 0$ leaves $H_{λ→0} = (1/2p^+)(P_⊥^2 + Y^4 P^M P^M + Y^2[p^{+2}(η^2)^2 + 2ip^+ η ρ^{MN} η Y^M P^N])$, whose zero-mode truncation is the AdS$_5$ × S$^5$ superparticle Hamiltonian. A second piece of machinery is the comparison of two parametrizations of the same ten-dimensional metric: one in which $R$ multiplies the whole metric, making the action depend only on $√λ = R^2/α'$, and one in which $R$ sits only in front of the sphere and radial terms, so that $R → 0$ and $α' → ∞$ are inequivalent limits. In the covariant-gauge version of the second case, integrating over the constant radial mode $φ_0$ converts the limit into an average over an effective tension, $⟨F⟩ ∼ ∫ dT T^{3/2} ∫ Dx Dθ e^{-(i/2)T∫ d^2σ √g (∂_μ x^a - iθσ^a∂_μθ)^2} F[x,θ,T]$.
Extended reading notes
Core claim
The concrete result is the explicit $λ → 0$ form of the light-cone superstring action in AdS$_5$ × S$^5$, Eq. (5.30) of the paper: the $λ → 0$ limit of the phase-space Hamiltonian (5.28) is obtained by omitting the $σ$-derivative terms, leaving $L = (p^+/2)[ẋ_⊥^2 + i(θ^i θ̇^i + h.c.) + (Ż^M - iη^i ρ^{MN}{}_{ij}η^j Z^N Z^{-2})^2 + i(η^i η̇^i + h.c.) - Z^{-2}(η^2)^2]$. Two things follow on the author's own reading. First, the zero-mode part of this action is exactly the superparticle action in AdS$_5$ × S$^5$, so the $λ = 0$ string spectrum necessarily contains the protected type IIB supergravity multiplet, as expected since those masses do not depend on $λ$. Second, and stated as a conjecture rather than a derivation, the same spectrum should contain massless higher-spin states in AdS$_5$, matching the higher-spin conformal currents of the free boundary theory. The paper is explicit that this second step is not established: it names showing that quantizing (5.30) produces those higher-spin states as the important next problem, and it notes that the two transverse 4-d coordinates $x_⊥$ and the four fermions $θ^i$ decouple from the remaining variables in (5.30) while the $S^5$ and $η$ terms generate non-trivial couplings between different Fourier modes in $σ$ — unlike the flat-space tensionless string, where the modes decouple.
Load-bearing premise
The load-bearing premise is that simply deleting the spatial-derivative terms from the light-cone Hamiltonian and then quantizing what remains captures the entire tensionless string spectrum, including the conjectured higher-spin states; the paper proves only that the constant-mode part matches the superparticle and explicitly leaves the interacting-mode quantization as the next problem.
Editorial extensions
If this is right
- The $λ = 0$ string spectrum must contain the full type IIB supergravity multiplet on AdS$_5$ × S$^5$, because the zero-mode truncation of the tensionless action is literally the superparticle action whose quantization gives those states.
- If the conjecture holds, the tensionless AdS$_5$ × S$^5$ string supplies the string-side dual of the conserved traceless higher-spin currents of free $N=4$ super Yang-Mills, giving content to the proposal that free large-$N$ gauge theory is dual to a higher-spin theory in AdS$_5$.
- The higher-spin states should appear as solutions of free light-cone field equations in AdS$_5$ of the form $(∂_x^2 + ∂_Z^2 - Z^{-2}A_{s'})φ_{s'} = 0$ with $A_{s'} = s'^2 - 1/4$, so the spectrum can be checked mode by mode rather than only in principle.
- Because the two parametrizations of the metric give inequivalent $R → 0$ limits, the 'non-standard' correspondence based on the second one predicts a four-dimensional effective description in which the radial AdS direction is frozen and the effective tension is integrated over with measure $T^{3/2}$.
- The $λ → 0$ limit is a candidate solvable corner of the AdS/CFT correspondence: the action loses its $σ$-derivatives, so the surviving dynamics is a quantum-mechanical system with non-linearly coupled Fourier modes rather than a full two-dimensional field theory.
Reading between the lines
- If the higher-spin conjecture is confirmed, the resulting spectrum would be a concrete testing ground for whether the tensionless string reorganizes into an infinite tower of massless fields with integer spin, one for each conserved current of the free boundary theory, which is a sharper statement than 'the dual has higher spins'.
- The paper's remark that $x_⊥$ and $θ^i$ decouple in (5.30) suggests the tensionless model factorizes into a free four-dimensional sector times an interacting $(Z^M, η^i)$ sector; checking whether the interacting sector alone reproduces the AdS$_5$ mass-shell conditions would be a cheaper route to the conjecture than quantizing the whole action.
- The inequivalence of the two $R → 0$ limits is a warning that 'the $λ → 0$ limit' is not a property of the theory alone but of the theory plus a choice of which coordinates and momenta are held fixed; any claimed holographic dual of the tensionless string implicitly commits to one such choice.
- The averaging formula over effective tension resembles a superposition over string tensions, which invites the question of whether the frozen-radial-direction limit defines a consistent world-sheet theory at all, or only a set of boundary correlation functions with restricted operator insertions.
Original abstract
The superstring action in AdS_5 x S^5 depends on two parameters: the inverse string tension a' and the radius R. The standard AdS/CFT correspondence requires that the string coordinates are rescaled so that the action depends only on one combination of the two: (\lambda)^{1/2} = R^2/a'. Then \lambda \to 0 limit is equivalent to R \to 0 for fixed $a'$ or to the zero-tension limit in AdS_5 x S^5: a' \to \infty for fixed R. After reviewing previous work hep-th/0009171 on the light cone superstring we explicitly obtain the \lambda= 0 form of its action. Its zero-mode part is the same as the superparticle action in AdS_5 x S^5, and thus the \lambda=0 string spectrum must include, as expected, the ``protected'' type IIB supergravity states. Following recent suggestions, it is conjectured that the spectrum of this tensionless string should as well contain higher spin massless states in AdS_5. We also discuss the case of another parametrization of the string action which has straightforward R\to\infty flat space limit but where R \to 0 and a' \to \infty limits are not equivalent. There R \to 0 corresponds to shrinking S^5 to zero the size and ``freezing'' the fluctuations of the radial coordinate of AdS_5. This case is the basis of the ``non-standard'' AdS/CFT correspondence suggested in hep-th/0010106. Parts of this work were presented in the talk at ``Supergravity at 25'' conference, Stony Brook, December 1-2, 2001.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The superstring action in AdS_5 x S^5 depends on two parameters: the inverse string tension a' and the radius R. The standard AdS/CFT correspondence requires that the string coordinates are rescaled so that the action depends only on one combination of the two: (\lambda)^{1/2} = R^2/a'. Then \lambda \to 0 limit is equivalent to R \to 0 for fixed $a'$ or to the zero-tension limit in AdS_5 x S^5: a' \to \infty for fixed R. After reviewing previous work hep-th/0009171 on the light cone superstring we explicitly obtain the \lambda= 0 form of its action. Its zero-mode part is the same as the superparticle action in AdS_5 x S^5, and thus the \lambda=0 string spectrum must include, as expected, the ``protected'' type IIB supergravity states. Following recent suggestions, it is conjectured that the spectrum of this tensionless string should as well contain higher spin massless states in AdS_5. We also discuss the case of another parametrization of the string action which has straightforward R\to\infty flat space limit but where R \to 0 and a' \to \infty limits are not equivalent. There R \to 0 corresponds to shrinking S^5 to zero the size and ``freezing'' the fluctuations of the radial coordinate of AdS_5. This case is the basis of the ``non-standard'' AdS/CFT correspondence suggested in hep-th/0010106. Parts of this work were presented in the talk at ``Supergravity at 25'' conference, Stony Brook, December 1-2, 2001.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The superstring action in AdS_5 x S^5 depends on two parameters: the inverse string tension a' and the radius R. | background 1.00 | 0.12 | 0.03 | asserted 1.00 |
| s1 | The standard AdS/CFT correspondence requires that the string coordinates are rescaled so that the action depends only on one combination of the two: (\lambda)^{1/2} = R^2/a'. | background 1.00 | 0.06 | 0.04 | asserted 1.00 |
| s2 | Then \lambda \to 0 limit is equivalent to R \to 0 for fixed $a'$ or to the zero-tension limit in AdS_5 x S^5: a' \to \infty for fixed R. | background 0.95 | 0.14 | 0.08 | asserted 1.00 |
| s3 | After reviewing previous work hep-th/0009171 on the light cone superstring we explicitly obtain the \lambda= 0 form of its action. | contribution 0.81 | 0.93 | 0.13 | asserted 1.00 |
| s4 | Its zero-mode part is the same as the superparticle action in AdS_5 x S^5, and thus the \lambda=0 string spectrum must include, as expected, the ``protected'' type IIB supergravity states. | result 0.61 | 0.77 | 0.21 | asserted 0.97 |
| s5 | Following recent suggestions, it is conjectured that the spectrum of this tensionless string should as well contain higher spin massless states in AdS_5. | interpretation 0.71 | 0.07 | 0.21 | speculative 0.99 |
| s6 | We also discuss the case of another parametrization of the string action which has straightforward R\to\infty flat space limit but where R \to 0 and a' \to \infty limits are not equivalent. | contribution 0.96 | 0.44 | 0.05 | asserted 0.98 |
| s7 | There R \to 0 corresponds to shrinking S^5 to zero the size and ``freezing'' the fluctuations of the radial coordinate of AdS_5. | interpretation 0.81 | 0.23 | 0.05 | asserted 1.00 |
| s8 | This case is the basis of the ``non-standard'' AdS/CFT correspondence suggested in hep-th/0010106. | background 0.63 | 0.09 | 0.05 | asserted 0.92 |
| s9 | Parts of this work were presented in the talk at ``Supergravity at 25'' conference, Stony Brook, December 1-2, 2001. | background 0.95 | 0.44 | 0.01 | asserted 0.96 |
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References
How the literature uses it
67 arXiv papers cite this in 88 sentences
Papers citing it · 55 sentences without a confident label
Compares with it 1 paper
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Massless spinning fields on the Light-Front: quartic vertices and amplitudes · Introduction · 0.83
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Conformal higher spins in 4d [ref] have higher derivative kinetic terms $h_s ^2sh_s$ ($h_s$ are totally symmetric rank $s$ tensors) and thus their generalization to a curved metric background is a non-trivial question (cf. Tseytlin:2013jya,Nutma:2014pua,Grigoriev:2016bzl,Beccaria:2017nco).
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Discussion
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