Holographic duals of large-c torus conformal blocks

TL;DR

The paper investigates semiclassical, large-$c$ torus conformal blocks in CFT$_2$ and provides a holographic realization via geodesic networks in thermal AdS$_3$. It develops and analyzes both $s$- and $t$-channel blocks, introducing perturbative schemes (superlight and double-leg) and showing that exponentiated global torus blocks reproduce classical blocks in appropriate heavy/light regimes. A unified holographic framework is proposed where the classical $s$-channel block equals the sum of the holomorphic thermal AdS action $S_{therm}$ and the dual geodesic-network length $L_{dual}$; the worldline formulation and vertex-momentum conservation are used to construct and solve the dual networks, with explicit results worked out for the 2-point case. The work also demonstrates links between global and classical blocks, supports conjectures relating global blocks to perturbative classical blocks, and provides explicit seed-and-correction schemes that can be extended to higher-point blocks and additional channels. Overall, the results illuminate how semiclassical CFT data on a torus maps to bulk geodesic networks, offering practical computational methods and insight into the structure of torus blocks in AdS$_3$/CFT$_2$.

Abstract

We study CFT2 conformal blocks on a torus and their holographic realization. The classical conformal blocks arising in the regime where conformal dimensions grow linearly with the large central charge are shown to be holographically dual to the geodesic networks stretched in the thermal AdS bulk space. We discuss the n-point conformal blocks and their duals, the 2-point case is elaborated in full detail. We develop various techniques to calculate both quantum and classical conformal block functions. In particular, we show that exponentiated global torus blocks reproduce classical torus blocks in the specific perturbative regimes of the conformal parameter space.

Figures (3)

• Figure 1: Two-point conformal blocks in the $s$-channel (right) and $t$-channel (left). The $s$-channel loop consists of two segments with different conformal dimensions.
• Figure 2: $n$-point $s$-channel conformal blocks with the modular parameter $\tau$ in the $s$-channel holographically realized as the necklace graph on two-dimensional slice of the thermal AdS. Time runs along the non-contractible cycle $t \sim t+ \beta$. Primary operators are inserted at $x_i$ at the boundary. These points are mapped to boundary attachment points $y_i$ in the bulk.
• Figure 3: The necklace graph on the annulus. The inner and outer black solid circles represent the conformal boundary. The blue dashed circle goes along the $r=0$ radius and corresponds to the 0-point block. The blue solid loop is a deformation of the dashed circle by external operators. On the conformal boundary, we choose $y_1=0$ and denote $y_2\equiv y$.