A note on commutation relation in conformal field theory

Nagano, Lento; Terashima, Shiro

doi:10.1007/jhep09(2021)187

Cited by 6 publications

(3 citation statements)

References 24 publications

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“…However, the commutator of the operators is not well-defined for a non-free CFT, so we need to consider the commutator of the smeared operators for time and space directions. This formulation has been constructed [NT21,BDBM20] but has yet to apply it to the conformal bootstrap equations fully. It is a desirable research direction.…”

Section: Future Research Directionmentioning

confidence: 99%

Conformal bootstrap in momentum space without large N

Nishikawa¹

2023

Preprint

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In this paper, we Fourier transform the Wightman function concerning energy and angular momentum on the S D−1 spatial slice in radial quantization in D = 2, 3 dimensions. In each case, we use the conformal Ward Identities to solve systematically for the Fourier components. We then use these Fourier components to build conformal blocks for the four-point function in momentum space, giving a finite-volume version of the momentum-space conformal blocks of [Gil18, Gil20, GLLM20]. We check that this construction is consistent with the known result in infinite volume. Our buildings help find bootstrap equations that can give nontrivial constraints that do not appear in analysis in infinite volume. We show some examples of bootstrap equations and their nontriviality.

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Section: Future Research Directionmentioning

confidence: 99%

Conformal bootstrap in momentum space without large N

Nishikawa¹

2023

Preprint

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“…where we have neglected the terms which are sub-leading in 1/(p v a) expansion. 8 For ∆ / ∈ Z, there is a branching point v 1 = v 2 . Here, we can neglect the contribution around the branching point because 1 > ∆ − q.…”

Section: =mentioning

confidence: 99%

Simple bulk reconstruction in anti-de Sitter/conformal field theory correspondence

Terashima

2023

Progress of Theoretical and Experimental Physics

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In this paper, we show that the bulk reconstruction in the AdS/CFT correspondence is rather simple and has an intuitive picture, by showing that the HKLL bulk reconstruction formula can be simplified. We also reconstruct the wave packets in the bulk theory from the CFT primary operators. With these wave packets, we discuss the causality and duality constraints and find our picture is only the consistent one. Our picture of the bulk reconstruction can be applied to the asymptotic AdS spacetime.

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“…Throughout the paper, we have tried to present explicit formulas and also to emphasize ambiguities and open problems. See [31][32][33][34][35][36][37] for some recent papers that are on the topic of bulk reconstruction.…”

Section: Jhep12(2022)075mentioning

confidence: 99%

HKLL for the non-normalizable mode

Bhattacharjee

Krishnan

Sarkar

2022

J. High Energ. Phys.

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We discuss various aspects of HKLL bulk reconstruction for the free scalar field in AdSd+1. First, we consider the spacelike reconstruction kernel for the non-normalizable mode in global coordinates. We construct it as a mode sum. In even bulk dimensions, this can be reproduced using a chordal Green’s function approach that we propose. This puts the global AdS results for the non-normalizable mode on an equal footing with results in the literature for the normalizable mode. In Poincaré AdS, we present explicit mode sum results in general even and odd dimensions for both normalizable and non-normalizable kernels. For generic scaling dimension ∆, these can be re-written in a form that matches with the global AdS results via an antipodal mapping, plus a remainder. We are not aware of a general argument in the literature for dropping these remainder terms, but we note that a slight complexification of a boundary spatial coordinate (which we call an iϵ prescription) allows us to do so in cases where ∆ is (half-) integer. Since the non-normalizable mode turns on a source in the CFT, our primary motivation for considering it is as a step towards understanding linear wave equations in general spacetimes from a holographic perspective. But when the scaling dimension ∆ is in the Breitenlohner-Freedman window, we note that the construction has some interesting features within AdS/CFT.

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A note on commutation relation in conformal field theory

Cited by 6 publications

References 24 publications

Conformal bootstrap in momentum space without large N

Conformal bootstrap in momentum space without large N

Simple bulk reconstruction in anti-de Sitter/conformal field theory correspondence

HKLL for the non-normalizable mode

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