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Conformal Field Theory Correlators from Classical Field Theory on Anti-de Sitter Space II. Vector and Spinor Fields

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W. Mück, K. S. Viswanathan

High Energy Physics - Theory · Submitted 21 May 1998 · 22 references

Pith's verdict: A solid, honest extension of the Dirichlet-problem treatment to massive vectors and Dirac fields, with real cross-checks and only presentation-level gaps.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on October 29, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

  1. T0 AI review An AI read the paper and answered a fixed list of questions.
  2. T1 Audit Automatic checks that the references and files the paper relies on exist and match.
  3. T2 Compute The paper's code or calculations re-run with the same results.
  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

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arXiv IDhep-th/9805145 (version 2)
Pith IDYJ224SGA, Pith's permanent record of this paper and its review
CitationK. S. Viswanathan, W. M\"uck. "Conformal Field Theory Correlators from Classical Field Theory on Anti-de Sitter Space II. Vector and Spinor Fields." arXiv:hep-th/9805145 (1998). https://pith.science/paper/YJ224SGA
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The paper

The paper claims

The paper argues that a Dirichlet cutoff prescription on anti-de Sitter space reproduces the conformal two- and three-point functions of vector and spinor operators in the boundary field theory.

The reading

This paper asks whether the prescription that worked for a scalar field in anti-de Sitter space also works for massive vector fields and Dirac spinors: impose a Dirichlet condition on the slice $x_0=epsilon$, solve the classical bulk equations, and send $epsilon$ to zero. If it does, the boundary theory should see a conserved current of dimension $tildeDelta=sqrt((d-2)^2/4+m^2)+d/2$ and a spinor of dimension $m+d/2$, with two-point functions fixed by conformal invariance. The authors derive exactly those forms, then add a minimal gauge coupling and compute the current-spinor-spinor three-point function, checking that the gauge Ward identity holds. The payoff is a consistency test of the correspondence beyond scalar operators and a concrete statement about which spinor components can live on the boundary.

What carries the argument

The load-bearing device is the Dirichlet boundary-value problem on the cutoff surface $x_0=epsilon$: the bulk solutions are written in terms of modified Bessel functions $K_nu(kx_0)$, the boundary data are extracted at $x_0=epsilon$, and the $epsilon$ to zero limit is taken only after the action is reduced to a surface term. Rescaling the boundary fields by $epsilon^{tildeDelta-d}$ for the vector and $epsilon^{m-d/2}$ for the spinor converts the cutoff-dependent surface action into the finite conformal two-point functions. The same machinery, plus the relation between the $+$ and $-$ spinor components enforced by the first-order Dirac equation, controls the spinor surface term and the interaction calculation.

Extended reading notes

Core claim

The central claim is that the bulk Dirichlet problem, not a naive power-law identification of the boundary field, is what makes the AdS/CFT dictionary produce CFT correlators for vector and spinor fields. For a massive vector field the current two-point function takes the conformal tensor form with dimension $tildeDelta=sqrt((d-2)^2/4+m^2)+d/2$; for a Dirac field the boundary spinor two-point function is proportional to $gamma_i(x^i-y^i)/|x-y|^{d+2m+1}$ and has dimension $m+d/2$. The authors also show that only half the spinor components can be prescribed at the boundary, so for even $d$ the boundary object is chiral and for odd $d$ it is a Dirac spinor, and that the minimal vector-spinor interaction yields a three-point function satisfying the expected Ward identity without any new surface term.

Load-bearing premise

The load-bearing premise is that every term dropped when the modified Bessel functions are expanded for small $epsilon$ is either a contact term or vanishes as $epsilon$ goes to zero; the paper asserts this classification in passing from Eq. (20) to Eq. (22) rather than verifying it term by term.

Editorial extensions

If this is right

  • Massive vector currents in the boundary CFT have dimension $tildeDelta=sqrt((d-2)^2/4+m^2)+d/2$, and their two-point function has the tensor structure fixed by conformal invariance.
  • Boundary spinors have dimension $m+d/2$; only half their components are independent, giving a chiral spinor for even $d$ and a Dirac spinor for odd $d$.
  • Minimal coupling of the gauge field to the spinor produces a vector-spinor-spinor three-point function of the conformal form and satisfies the gauge Ward identity.
  • The $epsilon$ to zero Dirichlet prescription is validated for fields with spin, extending the earlier scalar-field treatment to vector and spinor operators.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The half-component boundary spinor result suggests that the dictionary naturally produces chiral boundary fermions in even dimensions, which could be used to match the chiral matter content of specific candidate dual gauge theories.
  • The method should extend to higher-spin fields and to correlators with stress-tensor insertions, where cutoff-dependent contact terms are likely to be more delicate than in the vector and spinor cases treated here.
  • The non-universal coefficient of the vector-spinor-spinor three-point function could serve as a fingerprint for distinguishing among candidate boundary CFTs that share the same symmetries.
Original abstract

We use the AdS/CFT correspondence to calculate CFT correlation functions of vector and spinor fields. The connection between the AdS and boundary fields is properly treated via a Dirichlet boundary value problem.

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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,221 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.05 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We use the AdS/CFT correspondence to calculate CFT correlation functions of vector and spinor fields. The connection between the AdS and boundary fields is properly treated via a Dirichlet boundary value problem.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We use the AdS/CFT correspondence to calculate CFT correlation functions of vector and spinor fields. contribution 0.97 0.94 0.40 asserted 1.00
s1 The connection between the AdS and boundary fields is properly treated via a Dirichlet boundary value problem. method 0.98 0.60 0.11 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References
How later papers used it

76 later papers cite it

Evidence: substantial. 76 of the citing papers share no author with it, and 23 of those carry a confident label from Pith's citation classifier.

Pith read 19 of the 140 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Builds on it 1 of 2 papers with this label
Uses its method or tool 1 of 12 papers with this label
Compares with it 3 of 9 papers with this label
How the literature uses it

70 arXiv papers cite this in 133 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 82 sentences without a confident label

Contests it 1 paper
  • Namely, the authors of [ref] [See also Ref. [ref] , which appeared after the present manuscript was submitted to the ArXiv.] missed that the vanishing energy flux boundary conditions at the asymptotic boundary of AdS $-$ that they propose to be “novel" $-$ are nothing else but the AdS/CFT correspondence no-source boundary conditions $-$ often denoted as `standard' or, if allowed, `alternative' quantizations (discussed, for $s=1/2$, originally in [ref] and specially in [ref] ; this discussion applies both to bosonic and fermionic fields).

    Hunting for fermionic instabilities in charged AdS black holes · Introduction · contest 0.91

Builds on it 2 papers
Uses its method or tool 11 papers
Compares with it 9 papers
  • This expression can be made equal to the standard formula for the spinor bulk-to-boundary propagator [ref] after performing normalized gamma matrix contractions.

    Twistor methods for AdS$_5$ · Spinors · 0.91

  • The phase agrees with that of [ref] and the CFT tensor structure [ref] is reproduced upon combining the transverse and longitudinal components.

    Vector Bosons in the Randall-Sundrum 2 and Lykken-Randall models and unparticles · Conclusions · 0.93

  • Using the projection operator $P_±= 1/2(1 ±γ)$, where $γ$ is the chirality matrix [ref], we can summarize the results as [equation] This computation confirms that the application of our smearing functions correctly reproduces the known results of [ref] , which have been obtained through different methods.

    Bulk reconstruction for spinor fields in AdS/CFT · AdS$_odd$ · 0.93

Cites it as background 20 papers
  • If in the CFT we have a primary fermion operator $ψ_+$ of dimension $(h,h+1/2)$ , then we can build a free bulk Dirac fermion field with mass $m=Δ-1=2h-1/2$, $Ψ=[ Ψ_-; Ψ_+ ]$ with [ref] (For the Euclidean version see [ref] )

    Dressing bulk spinor fields in AdS${}_3$ · Background · 0.80

  • Let's recapitulate some highlights for spinor fields in AdS/CFT, which will become handy later on (see [ref] , [ref] , [ref] , [ref] for more details).

    Boundary Terms, Spinors and Kerr/CFT · Introduction · 1.00

  • The Euclidean vacuum two-point function for a spinor operator $$ in a CFT, which is dual a fermonic field $ψ$ in a pure AdS, was obtained in closed form before in [ref] (see also [ref] ).

    Real-time response in AdS/CFT with application to spinors · Pure AdS · 0.82

Discussion

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