Radial expansion for spinning conformal blocks

Self Cite

100

194

We consider the Regge limit of the CFT correlation functions J J OO and T T OO , where J is a vector current, T is the stress tensor and O is some scalar operator. These correlation functions are related by a type of Fourier transform to the AdS phase shift of the dual 2-to-2 scattering process. AdS unitarity was conjectured some time ago to be positivity of the imaginary part of this bulk phase shift. This condition was recently proved using purely CFT arguments. For large N CFTs we further expand on these ideas, by considering the phase shift in the Regge limit, which is dominated by the leading Regge pole with spin j(ν), where ν is a spectral parameter. We compute the phase shift as a function of the bulk impact parameter, and then use AdS unitarity to impose bounds on the analytically continued OPE coefficients C J J j(ν) and C T T j(ν) that describe the coupling to the leading Regge trajectory of the current J and stress tensor T . AdS unitarity implies that the OPE coefficients associated to non-minimal couplings of the bulk theory vanish at the intercept value ν = 0, for any CFT. Focusing on the case of large gap theories, this result can be used to show that the physical OPE coefficients C J J T and C T T T , associated to non-minimal bulk couplings, scale with the gap ∆ g as ∆ −2 g or ∆ −4 g . Also, looking directly at the unitarity condition imposed at the OPE coefficients C J J T and C T T T results precisely in the known conformal collider bounds, giving a new CFT derivation of these bounds. We finish with remarks on finite N theories and show directly in the CFT that the spin function j(ν) is convex, extending this property to the continuation to complex spin.

Section: Jhep10(2017)197mentioning

confidence: 99%

Bounds for OPE coefficients on the Regge trajectory

Costa

Hansen

Penedones

2017

Self Cite

100

194

“…For even dimensions, equation (3.2) simplifies and an analytic expression can be found in terms of hypergeometric functions [4]. For arbitrary ε the solutions of the Casimir equation are unknown, but there exists rapidly converging power series in terms of radial coordinates [49,50], as well as recursion relations by studying their analytic structure [51][52][53].…”

Section: The Superconformal Casimir Equationmentioning

confidence: 99%

Superconformal blocks for mixed 1/2-BPS correlators with SU(2) R-symmetry

Baume

Fuchs

Lawrie

2019

For SCFTs with an SU (2) R-symmetry, we determine the superconformal blocks that contribute to the four-point correlation function of a priori distinct half-BPS superconformal primaries as an expansion in terms of the relevant bosonic conformal blocks. This is achieved by using the superconformal Casimir equation and the superconformal Ward identity to fix the coefficients of the bosonic blocks uniquely in a dimension-independent way. In addition we find that many of the resulting coefficients are related through a web of linear transformations of the conformal data. a very nice playground to study non-perturbative effects and relations to string theory, for instance their connection to the swampland program [38].As six is the largest dimension allowing for a superconformal algebra [39], N = (1, 0)representation theory provides an overarching language encompassing lower dimensions via dimensional reduction 3 . Let us review the possible multiplets allowed in theories with SU (2) R R-symmetry [19, 20]. There can exist states which are annihilated by a subset of the supercharges. These null states must be absent in unitary theories, and lead to what are referred as short multiplets, as opposed to long multiplets, which do not have null states. It is standard to write long multiplets as L[∆, , J R ], where ∆ is the conformal dimension of the superconformal primary, denotes how the superconformal primary transforms as a traceless-symmetric 4 representation of so(1, d − 1) rotations of the Poincaré algebra, and J R is the charge under SU (2) R . The different short multiplets are denoted as the A-, B-, C-, and D-type multiplets. Unitarity gives lower bounds on the allowed conformal dimensions, ∆, of the superconformal primaries of long multiplets, and is moreover strong enough to completely fix the conformal dimension of the short multiplets as a function of the other group theoretical data and the spacetime dimension. The superconformal multiplets can be summarised as follows : L[∆, , J R ] : ∆ > 2ε J R + + µ , A[ , J R ] : ∆ = 2ε J R + + 4ε − 2 , B[ , J R ] : ∆ = 2ε J R + + 2ε , (1.1) C[J R ] : ∆ = 2ε J R + 2 , D[J R ] : ∆ = 2ε J R ,with ε = (d − 2)/2, µ = 2ε for 2 < d ≤ 4 and µ = 4ε − 2 for 4 ≤ d ≤ 6. We stress again that this corresponds to the standard notation for d = 6. Indeed, A-type multiplets correspond to the unitarity bound of long multiplets, which for d ≤ 4 coincides with the bound of type B. Type C is unique to six dimensions and can be traced back to the presence of self-dual two-forms. Moreover, type C and D will appear only with = 0, since in this work we 3 We note that for d ≤ 4, the nomenclature we are using here might not match the one the reader is familiar with. For instance, in four dimensions, the classification of half-BPS multiplets is refined into Higgs and Coulomb type, commonly denoted E r andB R respectively [40]. We refer to [20] for a dictionary between the 6D notation and lower dimensions.4 In this paper we will consider only multiplets that have a superconformal primary in a traceles...

“…Even though there can be additional exchange channels involving mixed tensor fields (see e.g. [24][25][26]), we leave the detailed holographic analysis to the future work.…”

Section: Jhep05(2017)070 3 Spinning Three Point Functions and Conformmentioning

confidence: 99%

“…Let us first consider the contributions from the double trace operators, as encoded within the Γ-functions in the second line of (5.17). In the first term, after the ν-integration, nonvanishing residues arise from the poles at 26) and in the second term, the the poles at…”

Section: Jhep05(2017)070mentioning

confidence: 99%

Anatomy of geodesic Witten diagrams

Chen

Kuo

Kyono

2017

Abstract:We revisit the so-called "Geodesic Witten Diagrams" (GWDs) [1], proposed to be the holographic dual configuration of scalar conformal partial waves, from the perspectives of CFT operator product expansions. To this end, we explicitly consider three point GWDs which are natural building blocks of all possible four point GWDs, discuss their gluing procedure through integration over spectral parameter, and this leads us to a direct identification with the integral representation of CFT conformal partial waves. As a main application of this general construction, we consider the holographic dual of the conformal partial waves for external primary operators with spins. Moreover, we consider the closely related "split representation" for the bulk to bulk spinning propagator, to demonstrate how ordinary scalar Witten diagram with arbitrary spin exchange, can be systematically decomposed into scalar GWDs. We also discuss how to generalize to spinning cases.