Virasoro Blocks and Trouble at the Euclidean Horizon

TL;DR

This paper provides a bulk geometric interpretation of Virasoro blocks in 2D CFTs, clarifying the role of the horizon and associated timescales through geodesic Witten diagram calculations in BTZ black hole coordinates.

Contribution

It introduces a bulk geodesic Witten diagram approach to understand semi-classical Virasoro blocks and their relation to the Euclidean horizon without relying on a periodic thermal circle.

Findings

Half the thermal period corresponds to the boundary timescale where geodesics cross the Euclidean horizon.

Departure from semi-classical blocks begins at half the period, indicating horizon-related effects.

The approach clarifies the geometric role of the horizon in boundary correlators.

Abstract

In the semi-classical ($c \rightarrow \infty$) limit, 4-point HLLH correlators in 2D CFTs exhibit periodic Euclidean singularities. Periodic singularities in Euclidean time are a general feature of thermal correlators, even at weak coupling. Therefore, the bulk significance of this observation (in particular, the role of the horizon) is somewhat obscure. Explicit numerical computations of finite-$c$ Virasoro blocks furthermore suggest that their departure from semi-classical blocks may begin already at half the period. In this paper, we provide a bulk understanding of these facts and clarify the role of the horizon. We present a bulk geodesic Witten diagram calculation of semi-classical Virasoro blocks in coordinates that are naturally adapted to the BTZ black hole. This allows a bulk geometric interpretation for boundary time separation. In this language, half of a thermal time period is the boundary timescale at which the light operator geodesic straddles the Euclidean horizon, capturing both the role of the horizon and the associated timescale. This timescale arises in a calculation that does not involve a periodic thermal circle on the bulk or the boundary.