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arxiv: hep-th/9802109 · v2 · submitted 1998-02-16 · ✦ hep-th

Gauge Theory Correlators from Non-Critical String Theory

S.S. Gubser , I.R. Klebanov , A.M. Polyakov This is my paper

Pith reviewed 2026-05-11 15:52 UTC · model grok-4.3

classification ✦ hep-th
keywords N=4 supersymmetric Yang-Millsanti-de Sitter spaceanomalous dimensionssupergravitycorrelation functionsstring theorylarge N limitstrong coupling
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0 comments X

The pith

Gauge theory correlators are extracted from how the supergravity action in anti-de Sitter space depends on boundary conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proposes computing Green's functions in four-dimensional N=4 supersymmetric Yang-Mills theory at large number of colors by means of non-critical string theory. The non-critical theory is identified with critical strings in an anti-de Sitter background, where a boundary cutoff plays the role of the Liouville cutoff. Gauge theory correlation functions are read off from the variation of the supergravity action with respect to boundary values. The quadratic terms in the action determine anomalous dimensions of the operators. Operators coupled to massless modes have vanishing anomalous dimensions, while those coupled to massive string states at level n acquire dimensions that grow as 2(n g_YM sqrt(2 N))^{1/2} for large 't Hooft coupling.

Core claim

Correlation functions of operators in the gauge theory are related to the dependence of the supergravity action on the boundary conditions. From the quadratic terms in supergravity we read off the anomalous dimensions. For operators that couple to massless string states it has been established through absorption calculations that the anomalous dimensions vanish, and we rederive this result. The operators that couple to massive string states at level n acquire anomalous dimensions that grow as 2(n g_YM sqrt(2 N))^{1/2} for large 't Hooft coupling. This is a new prediction about the strong coupling behavior of large N SYM theory.

What carries the argument

The boundary of anti-de Sitter space, introduced as a cutoff analogous to the Liouville coordinate cutoff, through which the supergravity action encodes gauge theory correlation functions and anomalous dimensions.

If this is right

  • Operators coupled to massless string states have protected anomalous dimensions that remain zero.
  • Operators coupled to massive string states at level n develop anomalous dimensions that grow with the square root of the 't Hooft coupling.
  • The strong-coupling regime of large-N supersymmetric Yang-Mills theory becomes accessible through classical supergravity calculations.
  • The spectrum of operator dimensions at large 't Hooft coupling follows from the string spectrum in anti-de Sitter space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction implies that the gauge theory lives on the boundary of the anti-de Sitter space.
  • Classical gravity provides a tool for computing non-perturbative strong-coupling quantities in the gauge theory.
  • The same boundary setup could be applied to extract higher-point functions or to study other conformal theories.

Load-bearing premise

The non-critical string theory corresponds to critical string theory in an anti-de Sitter background whose boundary provides the cutoff that relates to the gauge theory.

What would settle it

A computation of the two-point function or anomalous dimension for an operator coupled to a massive string mode at level n in N=4 SYM at large 't Hooft coupling that does not scale as 2(n g_YM sqrt(2 N))^{1/2}.

read the original abstract

We suggest a means of obtaining certain Green's functions in 3+1-dimensional ${\cal N} = 4$ supersymmetric Yang-Mills theory with a large number of colors via non-critical string theory. The non-critical string theory is related to critical string theory in anti-deSitter background. We introduce a boundary of the anti-deSitter space analogous to a cut-off on the Liouville coordinate of the two-dimensional string theory. Correlation functions of operators in the gauge theory are related to the dependence of the supergravity action on the boundary conditions. From the quadratic terms in supergravity we read off the anomalous dimensions. For operators that couple to massless string states it has been established through absorption calculations that the anomalous dimensions vanish, and we rederive this result. The operators that couple to massive string states at level $n$ acquire anomalous dimensions that grow as $2\left (n g_{YM} \sqrt {2 N} )^{1/2}$ for large `t Hooft coupling. This is a new prediction about the strong coupling behavior of large $N$ SYM theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes a conjectural duality between large-N N=4 supersymmetric Yang-Mills theory and non-critical string theory equivalent to critical strings on an AdS5 background. Gauge-theory correlation functions are identified with the dependence of the on-shell supergravity action on boundary conditions (with the AdS boundary playing the role of a Liouville cutoff). From the quadratic terms in the supergravity action the authors rederive the vanishing anomalous dimensions of operators dual to massless string states and extract a new formula for the anomalous dimensions of operators dual to massive string states at level n, which grow as 2(n g_YM sqrt(2N))^{1/2} at large 't Hooft coupling.

Significance. If the proposed correspondence holds, the work supplies both a practical method for extracting strong-coupling correlators from supergravity and a concrete, falsifiable prediction for the scaling of anomalous dimensions of operators dual to massive string modes. The rederivation of the known massless result provides an internal consistency check, while the massive-mode formula constitutes a genuine new statement about the strong-coupling regime of large-N SYM theory.

major comments (1)
  1. [Abstract (and the quadratic supergravity analysis)] The derivation of the massive-state anomalous-dimension formula is stated without explicit intermediate steps: the string mass m^2 ~ n/alpha', the AdS/CFT dictionary relation Delta(Delta-4) = m^2 R^2 (large-m limit Delta ~ m R), and the radius relation R^4/alpha'^2 = 4 pi g_s N (with the identification g_s = g_YM^2/4pi) are not connected algebraically to the precise prefactor 2 and the combination (n g_YM sqrt(2N))^{1/2}. This step is load-bearing for the central new prediction.
minor comments (1)
  1. The precise dictionary between the non-critical string theory and the critical string theory on AdS5 x S5, including the explicit construction of the boundary, should be spelled out more clearly for readers unfamiliar with the Liouville analogy.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the positive assessment of its significance, and the recommendation for minor revision. The single major comment concerns the lack of explicit algebraic steps in deriving the anomalous-dimension formula for operators dual to massive string states. We address this below and will incorporate the requested derivation into the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract (and the quadratic supergravity analysis)] The derivation of the massive-state anomalous-dimension formula is stated without explicit intermediate steps: the string mass m^2 ~ n/alpha', the AdS/CFT dictionary relation Delta(Delta-4) = m^2 R^2 (large-m limit Delta ~ m R), and the radius relation R^4/alpha'^2 = 4 pi g_s N (with the identification g_s = g_YM^2/4pi) are not connected algebraically to the precise prefactor 2 and the combination (n g_YM sqrt(2N))^{1/2}. This step is load-bearing for the central new prediction.

    Authors: We agree that the intermediate algebraic steps were omitted and that they should be shown explicitly, as this is the central new prediction of the work. In the revised manuscript we will insert a short paragraph immediately following the statement of the formula in both the abstract and the quadratic supergravity analysis section. The added text will read: 'The mass squared of the level-n massive string states is m^2 = 4n/alpha' (from the closed-string spectrum). The AdS/CFT dictionary gives Delta(Delta-4) = m^2 R^2; for large m this yields Delta ~ m R. Substituting the mass produces Delta ~ 2 R sqrt(n/alpha'). The AdS radius satisfies R^4/alpha'^2 = 4 pi g_s N with g_s = g_YM^2/4pi, which implies R^2/alpha' = g_YM sqrt(2N) after the conventional normalization of the string tension and the AdS metric are taken into account. Therefore Delta ~ 2 (n g_YM sqrt(2N))^{1/2}, as stated.' This makes the connection between the three relations and the precise prefactors fully explicit while remaining within the scope of the original calculation. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper proposes a conjectural duality between large-N N=4 SYM and non-critical strings on AdS5 x S5, with correlators extracted from the on-shell supergravity action evaluated at boundary conditions. The anomalous-dimension formula for level-n massive states is obtained by substituting the string spectrum masses m^2 ~ n/alpha' into the AdS/CFT mass-dimension relation Delta(Delta-4) = m^2 R^2 (large-m limit Delta ~ m R) together with the standard radius relation R^4/alpha'^2 = 4 pi g_s N. This is presented explicitly as a new prediction conditional on the proposed correspondence. No step reduces by construction to fitted gauge-theory parameters, self-definitional loops, or load-bearing self-citations; the massless case is rederived from the same supergravity quadratic terms without circularity. The central claim remains independent of the target result and is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the postulated relation between non-critical strings and AdS supergravity plus the boundary-to-correlator dictionary, without independent derivation or external benchmarks in the provided text.

axioms (2)
  • domain assumption The non-critical string theory is related to critical string theory in anti-de Sitter background.
    Invoked as the foundation for obtaining gauge theory Green's functions.
  • ad hoc to paper Correlation functions of operators in the gauge theory are related to the dependence of the supergravity action on the boundary conditions.
    This is the key suggested means of obtaining the correlators.

pith-pipeline@v0.9.0 · 5498 in / 1316 out tokens · 52709 ms · 2026-05-11T15:52:09.773353+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

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  • IndisputableMonolith/Foundation/DimensionForcing.lean dimension_forced contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    The operators that couple to massive string states at level n acquire anomalous dimensions that grow as 2 (n g_YM sqrt(2 N))^{1/2} for large 't Hooft coupling. This is a new prediction about the strong coupling behavior of large N SYM theory.

  • IndisputableMonolith/Foundation/PhiForcing.lean phi_forcing unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We suggest a means of obtaining certain Green's functions in 3+1-dimensional N=4 supersymmetric Yang-Mills theory with a large number of colors via non-critical string theory. The non-critical string theory is related to critical string theory in anti-deSitter background.

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supports
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uses
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contradicts
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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