The AdS$_3$ Propagator and the Fate of Locality

TL;DR

The paper uses an exact bulk proto-field $\phi$ constructed from Virasoro symmetry to probe bulk locality in AdS$_3$/CFT$_2$ via the propagator $K=\langle\phi\phi\rangle$. It develops both semiclassical (via a generalized monodromy method) and exact (via Zamolodchikov-type recursions in $c$ and $h$) techniques to compute the holomorphic and full propagators, corroborating perturbative gravity results and exposing nonperturbative locality-violating effects. UV/IR mixing appears at one loop, with a new short-distance scale $\sigma_*\sim (G_N R_{AdS}^3)^{1/4}$, while in the large-$h$ or small-$c$ regimes locality breaks at the AdS length, signaled by branch cuts or imaginary parts. All-orders in $1/c$ for light fields reveal a robust short-distance resummation and a nonperturbative ambiguity captured by a parameter $\kappa$, yielding a finite-$\sigma_*$ scale that governs locality breakdown. These results collectively suggest that bulk locality in quantum gravity is only approximate and that an IR-safe reformulation of bulk observables or a deeper understanding of holographic encoding is essential for a full, local bulk description.

Abstract

We recently used Virasoro symmetry considerations to propose an exact formula for a bulk proto-field $φ$ in AdS$_3$. In this paper we study the propagator $\langle φφ\rangle$. We show that many techniques from the study of conformal blocks can be generalized to compute it, including the semiclassical monodromy method and both forms of the Zamolodchikov recursion relations. When the results from recursion are expanded at large central charge, they match gravitational perturbation theory for a free scalar field coupled to gravity in our chosen gauge. We find that although the propagator is finite and well-defined at long distances, its perturbative expansion in $G_N = \frac{3}{2c}$ exhibits UV/IR mixing effects. If we nevertheless interpret $\langle φφ\rangle$ as a probe of bulk locality, then when $G_N m_φ\ll 1$ locality breaks down at the new short-distance scale $σ_* \sim \sqrt[4]{G_N R_{AdS}^3}$. For $φ$ with very large bulk mass, or at small central charge, bulk locality fails at the AdS length scale. In all cases, locality `breakdown' manifests as singularities or branch cuts at spacelike separation arising from non-perturbative quantum gravitational effects.

Figures (7)

• Figure 1: Left: The trace of the monodromy matrix $M$ computed numerically as a function of $\rho$ and $\rho g'(\rho)$, for three values of $\rho$: $\rho=0.4$(black, solid), $\rho=0.5$, (red, dashed), and $\rho=0.6$, (blue, dotted). Right: The critical value of $\sigma_c$(black, solid) where the semiclassical part of the $\langle \phi \phi\rangle$ correlator first develops an imaginary piece, as a function of $h/c$. For comparison, we show (red, dashed) the analytic small $h/c$ behavior, $\sigma_c \approx (9 \sqrt{3} h/c)^{1/3}$ from (\ref{['eq:SigmacAnalyticSCSmallHC']}), and (blue, dot-dashed) the large $h$ behavior $\sigma_c \approx \log (2+\sqrt{3})$, from (\ref{['eq:SigmacAnalyticSCLargeHC']}).
• Figure 2: This plot compares the ratios of successive coefficients in the holomorphic and mixed terms contributing the the full propagator. We see that at large $c$, the coefficients of $\rho^n$ grow at the same rate, meaning that the holomorphic propagator provides a good estimate for the behavior of the full propagator.
• Figure 3: This figure displays the scalar-graviton one-loop diagram that contributes to $\langle \phi(X_1) \phi(X_2) \rangle$ at order $1/c$. There is also a contact interaction, but the associated diagram vanishes. The computation is performed in appendix \ref{['app:GravitonLoop']}.
• Figure 4: This figure displays fits to logarithms of ratios of successive coefficients in the $\rho$ expansion of equation (\ref{['eq:DefinitionSeriesCoeffs']}) up to the $400$th order. In all cases we have set $h = 0$ identically, and the value of $c$ increases from the top to the bottom of the plot, ranging from $1.5$ to $10^5$. Each line corresponds to one of the points on Fig. \ref{['fig:NumericFitsFinalDataZeroh']}, but for legibility we have only included every fifth point.
• Figure 5: In this plot we used the fits of Fig. \ref{['fig:NumericFitsZeroh']} to extract an approximate asymptotic ratio $\frac{a_{n+1}}{a_n}$, which was then used to identify $\sigma_*$, the scale at which bulk locality appears to break down, for each value of $c$. For very small values of $c$ we find $\sigma_*$ of order the AdS scale, so that at $c \to 1$ we smoothly match the large $h$ results of section \ref{['sec:LargehLimit']}, as indicated by the red line. At large $c$ we enter the flat space regime of small $\sigma_*$, where we extract the fit $\sigma_* \propto c^{-0.27}$. Varying the details of the fitting shifts the exponent, but we consistently find that it lies between $0.25$ and $0.28$.