Unitarity and the holographic S-Matrix

By A. Fitzpatrick, Jared Kaplan. 2011

299 citations recorded by the source corpus.

A bstractThe bulk S-Matrix can be given a non-perturbative definition in terms of the flat space limit of AdS/CFT. We show that the unitarity of the S-Matrix, ie the optical theorem, can be derived by studying the behavior of the OPE and the conformal block decomposition in the flat space limit. When applied to perturbation theory in AdS, this gives a holographic derivation of the cutting rules for Feynman diagrams. To demonstrate these facts we introduce some new techniques for the analysis of conformal field theories. Chief among these is a method for conglomerating local primary operators $ {{\mathcal{O}}_1} $ and $ {{\mathcal{O}}_2} $ to extract the contribution of an individual primary $ {{\mathcal{O}}_{{\varDelta, \ell }}} $ in their OPE. This provides a method for isolating the contribution of specific conformal blocks which we use to prove an important relation between certain conformal block coefficients and anomalous dimensions. These techniques make essential use of the simplifications that occur when CFT correlators are expressed in terms of a Mellin amplitude.

Subfields: Applied Mathematics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/2da003a03a161707ce143e4944071e37d8737b38

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Citations in the corpus

Solving the 3d Ising Model with the Conformal Bootstrap II. $$c$$c-Minimization and Precise Critical Exponents (2012, background): “The explicit coefficients pMFT , can be found in [6,7].”
Analyticity and the holographic S-matrix (2011, methodology): “The purpose of this work is to prove Penedones’ conjecture [1, 2] and to explore how the locality vis-a-vis analyticity of the S-Matrix emerges from the Mellin amplitude of the CFT; we will discuss unitarity as a consequence of the OPE in a separate work [9].”
The S-matrix bootstrap. Part I: QFT in AdS (2016, background): “More precisely, one can check that the exact formula for free fields in AdS [45]”
Loops in AdS from conformal field theory (2016, methodology): “This implies that there is a one-loop analog of the tree-level derivative relation, a (1) n,` = 1 2 ∂n(a (0) n,`γ (1) n,`), discovered in [51] and proven in [27].”
Bootstrapping SCFTs with four supercharges (2015, unknown): “The decomposition of the correlator 〈φφ̄φφ̄〉 into ordinary conformal blocks was given in [64].”
Convexity and liberation at large spin (2012, background): “They could be easily found using the results of [68]”
Conformal invariance in the long-range Ising model (2015, background): “in [65,67] the conformal block decomposition and the φ ×φ OPE have been worked out explicitly for any .”
4D scattering amplitudes and asymptotic symmetries from 2D CFT (2016, background): “This ambitious goal has a long history [8–16], typically with a focus on AdS/CFT in the limit of infinite AdS radius.”
Causality constraints in conformal field theory (2015, background): “the perturbed OPE coefficients are c(n, `) = c0(n, `) + δc(n, `) with [26, 33]”
Light-ray operators in conformal field theory (2018, background): “ﬁrst principles, since the Casimir operators are not self-adjoint anymore. We will thus 32For general spin operators we should also include contributions from a discrete series of partial waves. 33In [67], the process of forming the Euclidean partial wave P∆,J is called “conglomeration.” 34The reaso”
Bootstrapping inflationary correlators in Mellin space (2019, background): “The Mellin space representation of conformal correlators [5–13] and Harmonic Analysis for the Euclidean Conformal Group [12, 14–17] have proven instrumental in making this connection manifest, which encode bulk physics in a way which shares key similarities with the flat-space scattering amplitudes.”
The lightcone bootstrap and the spectrum of the 3d Ising CFT (2016, unknown): “with coefficients given by [83] C n,` (∆1,∆2) = (∆1 − ν)n(∆2 − ν)n(∆1)`+n(∆2)`+n `!n!(`+ ν + 1)n(∆1 + ∆2 + n− 2ν − 1)n(∆1 + ∆2 + 2n+ `− 1)`(∆1 + ∆2 + n+ `− ν − 1)n .”
Quartic AdS interactions in higher-spin gravity from Conformal Field Theory (2015, background): “J H E P 1 1 ( 2 0 1 5 ) 1 4 9 The only difference of this formula with the one found by [96] is that we do not symmetrise the index carried by Kj with the remaining ones and for this reason we have more independent structures on the r.”
Looking for a bulk point (2015, background): “J H E P 0 1 ( 2 0 1 7 ) 0 1 3 On,l ≡ O∂2n∂μ1 · · · ∂μlO are given by [41] p̄n,l ≡ f(2) OOOn,l |MFT (B.”
$$ \mathcal{N} $$ = 4 Super-Yang-Mills correlators at strong coupling from string theory and localization (2019, unknown): “Using crossing symmetry and the analytic structure of tree-level Witten diagrams in Mellin space [32–35], we can then expand Mp(s, t) as a series in 1/λ: Mp =c−1 ( B 1Mp + λ− 3 2 [ B 4Mp +B 1Mp ] + λ−2 [ B 5Mp +B 4Mp +B 1Mp ] + λ− 5 2 [ B 6,1M p +B 6,2M p +B 5Mp +B 4Mp +B 1Mp ] +O(λ−3) ) +O(c−2) .”
Half-BPS Wilson loop and AdS2/CFT1 (2017, result): “A relation of this type was found empirically in [54] and proved in [55] (see also [63]).”
Witten diagrams revisited: the AdS geometry of conformal blocks (2015, background): “The expansion of a correlator in the s-channel includes the double-trace terms5 〈O1(x1)O2(x2)O3(x3)O4(x4)〉 ⊃ ∑ m,` P (12)(m, `)W∆(12)(m,`),`(xi)+ ∑ n,` P (34)(n, `)W∆(34)(n,`),`(xi) (2.14) Following [26,43], we have defined a notation for squared OPE coefficients, P (ij)(n, `) ≡ C12OCO34 , where O =”