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Multi-centered AdS$_3$ solutions from Virasoro conformal blocks

Ondřej Hulík, Tomáš Procházka, Joris Raeymaekers

TL;DR

This work establishes a precise link between fully backreacted multi-centered AdS$_3$ gravity and Virasoro/classical conformal blocks by recasting the gravity problem as a Liouville theory on the disk with ZZ boundary conditions. The core technical advance is showing that the accessory parameters governing the Liouville stress tensor are fixed by an SU(1,1) monodromy constraint, which is equivalent to a particular classical conformal block on the sphere in the vacuum channel. This reduces the construction of multi-centered gravity solutions to integrating a second-order ODE once the corresponding classical block is known, and it clarifies the holographic interpretation as left-right asymmetric operator insertions in the dual CFT. The paper confirms the approach with explicit examples and discusses abelian monodromy and static multicenter solutions, offering a cohesive framework that links bulk geometry, monodromy problems, and CFT data with potential extensions to higher-spin regimes.

Abstract

We revisit the construction of multi-centered solutions in three-dimensional anti-de Sitter gravity in the light of the recently discovered connection between particle worldlines and classical Virasoro conformal blocks. We focus on multi-centered solutions which represent the backreaction of point masses moving on helical geodesics in global AdS$_3$, and argue that their construction reduces to a problem in Liouville theory on the disk with Zamolodchikov-Zamolodchikov boundary condition. In order to construct the solution one needs to solve a certain monodromy problem which we argue is solved by a vacuum classical conformal block on the sphere in a particular channel. In this way we construct multi-centered gravity solutions by using conformal blocks special functions. We show that our solutions represent left-right asymmetric configurations of operator insertions in the dual CFT. We also provide a check of our arguments in an example and comment on other types of solutions.

Multi-centered AdS$_3$ solutions from Virasoro conformal blocks

TL;DR

This work establishes a precise link between fully backreacted multi-centered AdS gravity and Virasoro/classical conformal blocks by recasting the gravity problem as a Liouville theory on the disk with ZZ boundary conditions. The core technical advance is showing that the accessory parameters governing the Liouville stress tensor are fixed by an SU(1,1) monodromy constraint, which is equivalent to a particular classical conformal block on the sphere in the vacuum channel. This reduces the construction of multi-centered gravity solutions to integrating a second-order ODE once the corresponding classical block is known, and it clarifies the holographic interpretation as left-right asymmetric operator insertions in the dual CFT. The paper confirms the approach with explicit examples and discusses abelian monodromy and static multicenter solutions, offering a cohesive framework that links bulk geometry, monodromy problems, and CFT data with potential extensions to higher-spin regimes.

Abstract

We revisit the construction of multi-centered solutions in three-dimensional anti-de Sitter gravity in the light of the recently discovered connection between particle worldlines and classical Virasoro conformal blocks. We focus on multi-centered solutions which represent the backreaction of point masses moving on helical geodesics in global AdS, and argue that their construction reduces to a problem in Liouville theory on the disk with Zamolodchikov-Zamolodchikov boundary condition. In order to construct the solution one needs to solve a certain monodromy problem which we argue is solved by a vacuum classical conformal block on the sphere in a particular channel. In this way we construct multi-centered gravity solutions by using conformal blocks special functions. We show that our solutions represent left-right asymmetric configurations of operator insertions in the dual CFT. We also provide a check of our arguments in an example and comment on other types of solutions.

Paper Structure

This paper contains 24 sections, 142 equations, 2 figures.

Table of Contents

  1. Introduction and summary
  2. Ansatz for helical multi-particle solutions
  3. Multi-centered solutions from conformal blocks
  4. Properties of the Liouville stress tensor
  5. Pole structure
  6. Reality condition and and Schwarz reflection principle
  7. Constraints on the accessory parameters
  8. Associated ODE problem
  9. Boundary condition
  10. The accessory parameter problem
  11. Conjugacy classes of monodromy matrices
  12. Parameter counting
  13. Relation to classical conformal blocks
  14. Classical conformal blocks on the sphere
  15. Connecting two monodromy problems
  16. ...and 9 more sections

Figures (2)

  • Figure 1: (a) The particle worldlines considered in this paper are helical geodesics which spiral around at constant radius in global AdS$_3$ (pictured here as a solid cylinder with time running vertically). (b) In the dual CFT defined on the plane, our solutions represent left-right asymmetric configurations: purely holomorphic operators (blue dots) are inserted in the points $z_i$ within the unit disk, as well as in the image points $1/\bar{z}_i$. A purely antiholomorphic operator (red dot) is inserted in the origin and its image point at infinity.
  • Figure 2: A diagram of a $10$-point conformal block on a sphere. The black legs represent the external primaries inserted at $z_i$ and $\tilde{z}_i \equiv \bar{z}_i^{-1}$. The blue lines correspond to the exchange of the identity family. This specific channel where the mirror pairs are fused first is the one which is relevant for the discussion of the monodromy problem on a disk. The green circles illustrate contours along which the monodromy is trivial.