Loops in AdS: From the Spectral Representation to Position Space

Abstract

We compute a family of scalar loop diagrams in $AdS$. We use the spectral representation to derive various bulk vertex/propagator identities, and these identities enable to reduce certain loop bubble diagrams to lower loop diagrams, and often to tree-level exchange or contact diagrams. An important example is the computation of the finite coupling 4-point function of the large-$N$ conformal $O(N)$ model on $Ad{S}_3$. Remarkably, the re-summation of bubble diagrams is equal to a tree-level contact diagram: the ${\bar{D}}_{1,1,\frac{3}{2},\frac{3}{2}}(z,\bar{z})$ function. Another example is a scalar with ${\varphi }^4$ or ${\varphi }^3$ coupling in $Ad{S}_3$: we compute various 4-point (and higher point) loop bubble diagrams with alternating integer and half-integer scaling dimensions in terms of a finite sum of contact diagrams and tree-level exchange diagrams. The 4-point function with external scaling dimensions differences obeying ${\Delta }_{12}=0$ and ${\Delta }_{34}=1$ enjoys significant simplicity which enables us to compute in quite generality. For integer or half-integer scaling dimensions, we show that the $M$-loop bubble diagram can be written in terms of Lerch transcendent functions of the cross-ratios $z$ and $\bar{z}$. Finally, we compute 2-point bulk bubble diagrams with endpoints in the bulk, and the result can be written in terms of Lerch transcendent functions of the AdS chordal distance. We show that the similarity of the latter two computations is not a coincidence, but arises from a vertex identity between the bulk 2-point function and the double-discontinuity of the boundary 4-point function.

Corrections made, 52 pages, 20 figures

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• Loops in AdS: From the Spectral Representation to Position Space
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• On spinning loop amplitudes in Anti-de Sitter space
• Recursion relations for 5-point conformal blocks
• One-loop string corrections to AdS amplitudes from CFT
• Towards Bootstrapping RG flows: Sine-Gordon in AdS
• Opacity from Loops in AdS
• Analytical evaluation of AdS${}_4$ Witten diagrams as flat space multi-loop Feynman integrals
• On Holography in General Background and the Boundary Effective Action from AdS to dS
• Loops in AdS: From the Spectral Representation to Position Space II
• Effective Field Theory in AdS: Continuum Regime, Soft Bombs, and IR Emergence
• Bulk renormalization and the AdS/CFT correspondence
• Exploring Confinement in Anti-de Sitter Space
• Holographic S-fold theories at one loop
• On interacting quantum fields in various charts of anti de Sitter space-time
• Simplicity of AdS Supergravity at One Loop
• 1d Conformal Field Theory and Dispersion Relations
• An Elliptic One-Loop Amplitude in Anti-de-Sitter Space
• Recursive construction of the operator product expansion in curved space
• Information Geometry and Holographic Correlators
• A calculation of the Weyl anomaly for 6D Conformal Higher Spins
• Four-point correlation modular bootstrap for OPE densities
• How to Succeed at Witten Diagram Recursions without Really Trying
• Constructing CFTs from AdS flows
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