Loops in AdS: From the Spectral Representation to Position Space

TL;DR

This work develops a spectral-representation toolkit for scalar loop diagrams in AdS and derives bulk vertex/propagator identities that convert bubbles into simpler tree-level or contact diagrams. In AdS$_3$ it proves several powerful reductions, including a finite-coupling 4-point function for the large-$N$ O($N$) model that collapses to a contact diagram proportional to $\bar{D}_{1,1,\frac{3}{2},\frac{3}{2}}(z,\bar z)$, and it expresses higher-loop bubbles with integer or half-integer scaling dimensions in terms of Lerch transcendent functions. The paper also establishes a link between bulk 2-point diagrams and the double-discontinuity of boundary 4-point functions via a vertex identity, and it derives a broad set of identities (1–10) that enable systematic reductions of loop Witten diagrams to contact and exchange building blocks. Together, these results provide analytic control over AdS loop amplitudes, expose rich special-function structures, and suggest pathways to generalize to higher loops, other AdS dimensions, and spinning fields.

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 $AdS_3$. 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 $φ^4$ or $φ^3$ coupling in $AdS_3$: 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 $Δ_{12}=0$ and $Δ_{34}=1$ 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.

Figures (20)

• Figure 1: Left: The grey blob represents a general bulk two point function $F(x_1,x_2)$, with points $x_1$ and $x_2$. Other than that, the blob is completely general. Right: Attaching 4 legs to the boundary gives a CFT 4-point function $g_4(z, \bar{z})$. This class of diagrams has 2 bulk-to-boundary propagators attached to $x_1$, and 2 attached to $x_2$.
• Figure 2: An $N$-point function with a pair of external lines originating from the same vertex $x_1$, and ending on the boundary points $P_1$ and $P_2$. The grey blob is completely general.
• Figure 3: Identity 1. For every choice of external scaling dimensions $\Delta_i$, the $4$-point function on the left equals the $4$-point function on the right, where we define $\Delta_{123,4}\equiv \frac{\Delta_1+\Delta_2+\Delta_3-\Delta_4}{2}$. These Witten diagrams are defined to be stripped of the factor $A_{\Delta_i}$ of Eq. \ref{['eq:lkjh7']}. The grey blob 2-point function is general.
• Figure 4: a) Identity 2: The derivative operator raises the scaling dimension of the external legs. b) Identity 2': The derivative operator kills a bulk-to-bulk propagator. The derivative operators are defined in Eqs. \ref{['eq:deriv12']} and \ref{['eq:deriv13']}. We are suppressing numerical coefficients in these plots, and dropping the pre-factor $A_{\Delta_i}$ defined in Eq. \ref{['eq:prefactor']}.
• Figure 5: Identity 3'. This identity generalizes the well known relations between tree level exchange 4-point diagrams and 4-point contact diagrams. We are suppressing numerical coefficients in these plots. The grey blob is general, and the diagrams correspond to stripped correlators.