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

K. B. Alkalaev, V. A. Belavin

TL;DR

This work establishes a holographic interpretation for the linearized classical 1-point toroidal conformal block in the semiclassical limit by mapping it to the total on-shell action of a tadpole geodesic network in thermal AdS, with the block coefficient $f^{lin}$ given by $f^{lin} = -S_{total}$. The authors implement a worldline formalism, derive equilibrium and half-cycle constraints, and perform a perturbative expansion around a seed loop, showing that leading loop and leg contributions reproduce the first linear coefficient $f_1(q)$ of the block. In the low-temperature regime where thermal AdS dominates, the results yield a consistent match between bulk geodesic lengths and CFT data, including a modular shift $\hat{\tau}=\tau+1/2$ connecting bulk and boundary moduli. The paper also outlines several avenues for future work, including higher-point toroidal blocks, subleading $1/c$ corrections, and connections to Wilson lines and geodesic Witten diagrams.

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.

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

TL;DR

This work establishes a holographic interpretation for the linearized classical 1-point toroidal conformal block in the semiclassical limit by mapping it to the total on-shell action of a tadpole geodesic network in thermal AdS, with the block coefficient given by . The authors implement a worldline formalism, derive equilibrium and half-cycle constraints, and perform a perturbative expansion around a seed loop, showing that leading loop and leg contributions reproduce the first linear coefficient of the block. In the low-temperature regime where thermal AdS dominates, the results yield a consistent match between bulk geodesic lengths and CFT data, including a modular shift connecting bulk and boundary moduli. The paper also outlines several avenues for future work, including higher-point toroidal blocks, subleading corrections, and connections to Wilson lines and geodesic Witten diagrams.

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.

Paper Structure

This paper contains 10 sections, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Classical 1-point toroidal block
  3. Worldline approach
  4. Equilibrium condition
  5. Half-cycle condition
  6. Perturbative expansion
  7. Perturbative momentum and radius.
  8. Perturbative lengths.
  9. Conclusion
  10. The linearized block coefficients

Figures (2)

  • Figure 1: One-point conformal block realized as the tadpole graph embedded into the thermal AdS. The loop of the conformal block graph is identified with the non-contractible circle of the thermal AdS. $\Delta$ and $\widetilde{\Delta}$ are external and intermediate conformal dimensions, $\epsilon = k \Delta$ and $\widetilde{\epsilon} = k \widetilde{\Delta}$ are classical conformal dimensions ($k=c/6$).
  • Figure 2: Annulus and tadpole graph. The inner and outer black solid circles represent the conformal boundary. The dashed circle goes along the $r=0$ radius. The blue loop is a deformation of the dashed circle when the external field represented by the solid blue segment is switched on. Vertex and boundary attachment points are at $t=\pi$. Routinely, time flows clockwise.