Conformal Blocks Beyond the Semi-Classical Limit

Abstract

Black hole microstates and their approximate thermodynamic properties can be studied using heavy-light correlation functions in AdS/CFT. Universal features of these correlators can be extracted from the Virasoro conformal blocks in CFT2, which encapsulate quantum gravitational effects in AdS3. At infinite central charge c, the Virasoro vacuum block provides an avatar of the black hole information paradox in the form of periodic Euclidean-time singularities that must be resolved at finite c. We compute Virasoro blocks in the heavy-light, large c limit, extending our previous results by determining perturbative 1/c corrections. We obtain explicit closed-form expressions for both the `semi-classical' $h_L^2 / c^2$ and `quantum' $h_L / c^2$ corrections to the vacuum block, and we provide integral formulas for general Virasoro blocks. We comment on the interpretation of our results for thermodynamics, discussing how monodromies in Euclidean time can arise from AdS calculations using `geodesic Witten diagrams'. We expect that only non-perturbative corrections in 1/c can resolve the singularities associated with the information paradox.

Figures (5)

• Figure 1: This figure shows the similarity in the functional dependence of ${\cal V}_{h_L}^{(1)}$ and ${\cal V}_{h_L^2}^{(1)}$ for different values of $\alpha$. Left,top: The ratio $N(\alpha) \frac{{\cal V}^{(1)}_{h_L}(w; \alpha)}{{\cal V}^{(1)}_{h_L}(w ;\alpha=i \infty)}$, where a normalization $N(\alpha)= -\frac{11 \alpha ^4}{(1-\alpha ) (\alpha +1) \left(11 \alpha ^2+1\right)}$ scales them to agree at $z \sim 0$. Right,top: Same as the left, but for the ${\cal O}(h_L^2)$ term $N(\alpha) \frac{{\cal V}^{(1)}_{h_L^2}(w; \alpha)}{{\cal V}^{(1)}_{h_L^2}(w; \alpha=i \infty)}$. The endpoints $\alpha=1$ and $\alpha = i \infty$ are very close, but the difference becomes more significant near $\alpha \sim 0$. Left and right, bottom: Same as the top, but as a function of $z$ for real $z$.
• Figure 2: This figure shows the analytic continuation in $z$ and $\bar{z}$ that are equivalent to rotating ${\cal O}_L(z)$ around the global AdS cylinder. An operator at infinity is not displayed.
• Figure 3: This figure depicts a 'geodesic Witten diagrams' that can be used to compute a conformal block from AdS Hijano:2015zsa. The lines connecting the two light operators to each other and the two heavy operators to each other are both geodesics, while the wavy line designates a propagator whose endpoints have been fixed to these geodesics. The only integrals are over the positions of the bulk-to-bulk propagator along the geodesics.
• Figure 4: This figure shows what happens when we analytically continue the external points of a geodesic Witten diagram. As $z$ moves around the cylinder, the heavy and light geodesics must cross, and as they do, the propagator connecting them passes through its short-distance singularity. Note that in $d>2$ dimensions this crossing is enforced by geometry, not by topology. This is the origin of the non-trivial monodromy of the conformal block. Similar reasoning leads to a monodromy in Euclidean time for non-vacuum heavy-light Virasoro blocks Fitzpatrick:2015zha.
• Figure 5: This figure shows gravitational one-loop diagrams in AdS that could contribute to heavy-light Virasoro blocks at order $1/c$. In the small $h_H / c$ limit, we expect that the two diagrams on the left should correspond with the $1/c$ effects in equation (\ref{['eq:VLinAlphaLinH']}). More generally, the pair of diagrams on the left should be equivalent to the pair on the right with bulk propagators computed in the background gravitational field of the heavy operator.