Holographic interpretation of 1-point toroidal block in the semiclassical limit

TL;DR

This paper explores the holographic interpretation of the 1-point conformal block on a torus in the semiclassical limit, linking it to geodesic lengths in thermal AdS space.

Contribution

It introduces a linearized approach to interpret the 1-point toroidal conformal block holographically, using perturbation methods around the zero-point configuration.

Findings

Linearized block corresponds to geodesic length of tadpole graph in thermal AdS

Perturbation method used to compute coefficients of the block

Holographic interpretation connects conformal blocks with geometric quantities in AdS

Abstract

We propose the holographic interpretation of the 1-point conformal block on a torus in the semiclassical regime. To this end we consider the linearized version of the block and find its coefficients by means of the perturbation procedure around natural seed configuration corresponding to the zero-point block. From the AdS/CFT perspective the linearized block is given by the geodesic length of the tadpole graph embedded into the thermal AdS plus the holomorphic part of the thermal AdS action.