AdS box graphs, unitarity and operator product expansions

140

We develop a systematic framework to compute the conformal partial wave expansions (CPWEs) of tree-level four-point Witten diagrams with totally symmetric external fields of arbitrary mass and integer spin in AdS d+1 . As an intermediate step, we identify convenient bases of three-point bulk and boundary structures to invert linear map between spinning three-point conformal structures and spinning cubic couplings in AdS. Given a CFT d , this provides the complete holographic reconstruction of all cubic couplings involving totally symmetric fields in the putative dual theory on AdS d+1 . Employing this framework, we determine the CPWE of a generic four-point exchange Witten diagram with spinning exchanged field. As a concrete application, we compute all four-point exchange Witten diagrams in the type A higher-spin gauge theory on AdS d+1 , which is conjectured to be dual to the free scalar O (N ) model.

“…There are two types of contributions, in accord with the standard lore on CPWEs of Witten diagrams [28,37,75,76,102,103,[126][127][128][129][130][131][132][133]:…”

Section: Generic Spinning Exchange Diagrammentioning

confidence: 99%

Spinning Witten diagrams

Sleight¹,

Taronna²

2017

140

“…Together with ζ a ≡ ∂ a (e it cos ρ) −1 , they form a K µ -invariant basis for the tangent space at each point in AdS d+1 . 11 A general primary tensor is therefore just a product of ξ µ 's and ζ's, times a function f (e it cos ρ). Note further that…”

Section: Primary Wavefunctionsmentioning

confidence: 99%

“…(3.6) 11 We could have chosen ζ a to be a derivative of any function of e it cos ρ, since the K µ 's would annihilate it. The choice (e it cos ρ) −1 is convenient since then ζ a and ξ a µ have the same scaling dimension.…”

Section: Primary Wavefunctionsmentioning

confidence: 99%

See 1 more Smart Citation

Effective conformal theory and the flat-space limit of AdS

Fitzpatrick

Katz

Poland

et al. 2011

173

297

We develop the idea of an effective conformal theory describing the low-lying spectrum of the dilatation operator in a CFT. Such an effective theory is useful when the spectrum contains a hierarchy in the dimension of operators, and a small parameter whose role is similar to that of 1/N in a large N gauge theory. These criteria insure that there is a regime where the dilatation operator is modified perturbatively. Global AdS is the natural framework for perturbations of the dilatation operator respecting conformal invariance, much as Minkowski space naturally describes Lorentz invariant perturbations of the Hamiltonian. Assuming that the lowest-dimension single-trace operator is a scalar, O, we consider the anomalous dimensions, γ(n, l), of the double-trace operators of the form O(∂ 2 ) n (∂) l O. Purely from the CFT we find that perturbative unitarity places a bound on these dimensions of |γ(n, l)| < 4. Non-renormalizable AdS interactions lead to violations of the bound at large values of n. We also consider the case that these interactions are generated by integrating out a heavy scalar field in AdS. We show that the presence of the heavy field "unitarizes" the growth in the anomalous dimensions, and leads to a resonancelike behavior in γ(n, l) when n is close to the dimension of the CFT operator dual to the heavy field. Finally, we demonstrate that bulk flat-space S-matrix elements can be extracted from the large n behavior of the anomalous dimensions. This leads to a direct connection between the spectrum of anomalous dimensions in d-dimensional CFTs and flatspace S-matrix elements in d + 1 dimensions. We comment on the emergence of flat-space locality from the CFT perspective.

“…It is interesting both for its own sake and in connection to the AdS d+1 /CFT d correspondence to ask how the spectrum of a weakly coupled quantum field theory in AdS behaves as a function of its coupling. On the AdS side, this amounts to computing binding energies JHEP10(2020)149 of multi-particle states [1][2][3][4][5][6][7][8][9][10][11], while on the CFT side it corresponds to computing anomalous dimensions of multi-trace operators. Much effort has gone into such computations in the context of the bootstrap program following [12], in which the anomalous dimensions, along with the OPE coefficients, comprise the CFT data.…”

Section: Introductionmentioning

confidence: 99%

Anomalous dimensions from thermal AdS partition functions

Kraus

Megas

Sivaramakrishnan

2020

We develop an efficient method for computing thermal partition functions of weakly coupled scalar fields in AdS. We consider quartic contact interactions and show how to evaluate the relevant two-loop vacuum diagrams without performing any explicit AdS integration, the key step being the use of Källén-Lehmann type identities. This leads to a simple method for extracting double-trace anomalous dimensions in any spacetime dimension, recovering known first-order results in a streamlined fashion.