Semiclassical Virasoro Blocks from AdS$_3$ Gravity

TL;DR

The paper establishes a cohesive holographic framework for semiclassical Virasoro blocks at large $c$ by constructing bulk duals in AdS$_3$ gravity. It shows that geodesic Witten diagrams in conical-defect backgrounds reproduce the heavy-light blocks (FKW result) and presents a complementary holomorphic block computation via higher spin gravity using a Chern-Simons formulation, yielding direct access to the holomorphic Virasoro block. The work unifies bulk interpretations of known semiclassical blocks, clarifies interpolation between regimes, and opens paths to systematic $1/c$ corrections and generalizations to $W$-algebras. This provides a tangible bulk framework linking Zamolodchikov monodromy, geodesic probes, and higher-spin gauge dynamics to the structure of Virasoro blocks in two-dimensional CFTs.

Abstract

We present a unified framework for the holographic computation of Virasoro conformal blocks at large central charge. In particular, we provide bulk constructions that correctly reproduce all semiclassical Virasoro blocks that are known explicitly from conformal field theory computations. The results revolve around the use of geodesic Witten diagrams, recently introduced in arXiv:1508.00501, evaluated in locally AdS$_3$ geometries generated by backreaction of heavy operators. We also provide an alternative computation of the heavy-light semiclassical block -- in which two external operators become parametrically heavy -- as a certain scattering process involving higher spin gauge fields in AdS$_3$; this approach highlights the chiral nature of Virasoro blocks. These techniques may be systematically extended to compute corrections to these blocks and to interpolate amongst the different semiclassical regimes.

Figures (4)

• Figure 1: The spectrum of gravity duals of large $c$ Virasoro blocks. Operator dimensions increase from bottom to top; $h_i$ and $h_p$ denote external and internal holomorphic operator dimensions, respectively. In the limit of fixed dimensions, the Virasoro block becomes the global block, represented by a geodesic Witten diagram. Upon ramping up two external dimensions to enter the heavy-light regime, the bulk dual becomes a geodesic Witten diagram evaluated in a conical defect geometry. Further taking the remaining dimensions to scale with $c$, albeit perturbatively, one minimizes the worldline action of a cubic vertex of geodesics in the presence of the defect. This is equivalent to making a saddle-point approximation to the heavy-light geodesic Witten diagram. Not shown is the fully non-perturbative Virasoro block for all heavy operators, whose form is unknown.
• Figure 2: This is a geodesic Witten diagram in AdS$_{d+1}$, for the exchange of a symmetric traceless spin-$\ell$ tensor with $m^2=\Delta(\Delta-d)-\ell$ in AdS units, introduced in Hijano:2015zsa. The vertices are integrated over the geodesics connecting the two pairs of boundary points, here drawn as dashed orange lines. This computes the conformal partial wave for the exchange of a CFT$_d$ primary operator of spin $\ell$ and dimension $\Delta$. When $d=2$, this yields the product of holomorphic and anti-holomorphic global conformal blocks. To compute the heavy-light Virasoro blocks instead, we allow one geodesic to backreact, creating a conical defect.
• Figure 3: Bulk setup for computing a heavy-light semiclassical Virasoro block. The heavy operators ${\cal O}_{H_{1,2}}$ set up a conical defect geometry centered at the dotted line in the bulk. The conical defect sources a bulk field dual to the exchanged primary operator ${\cal O}_p$. The external light operators ${\cal O}_{L_{1,2}}$ interact with the bulk field along a geodesic; in particular, the interaction vertex is to be integrated over the bulk geodesic (dashed orange line) connecting the light operator insertion points. In the Poincaré figure, the corresponding Virasoro block in the CFT is indicated by the dashed black lines.
• Figure 4: Setup for computing the chiral heavy-light semiclassical Virasoro block using higher spin gauge fields, analogous to our previous construction using of scalar fields. Each gauge field is dual to a higher spin current. The heavy operators in this picture are $J^{(S_1)}$ and $J^{(S_2)}$, whose spins are taken to infinity. Rather then computing the diagram using propagators and vertices, we will obtain it through the use of higher spin gauge transformations, taking advantage of the fact that massless fields of positive integer spin in three-dimensions have no local degrees of freedom.