Virasoro blocks and trouble at the Euclidean horizon

In the semiclassical (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>c</a:mi><a:mo stretchy="false">→</a:mo><a:mi>∞</a:mi></a:math>) limit, 4-point heavy-light-light-heavy correlators in 2D conformal field theory 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-<d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi>c</d:mi></d:math> Virasoro blocks furthermore suggest that their departure from semiclassical 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 semiclassical Virasoro blocks in coordinates that are naturally adapted to the Bañados-Teitelboim-Zanelli 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 involve a periodic thermal circle on the bulk or the boundary.