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Asymptotic Symmetries, Holography and Topological Hair

Rashmish K. Mishra, Raman Sundrum

TL;DR

The work develops a framework in which AdS$_4$ asymptotic symmetries acquire infinite-dimensional structure by coupling the holographic CFT$_3$ to 3D topological sectors (Chern-Simons and gravity) and by restricting attention to boundary subregions with effectively two-dimensional geometry. In the large-level (probe) limit, a residual 2D CFT$_2$ structure emerges on AdS$_3$ boundaries, enabling Virasoro and Kac-Moody algebras to act as 4D hair for AdS$_4$ states, while Wheeler-DeWitt boundary regions reveal memory-like and holographic shadow effects. The maximal spacetime AS are realized by 3D conformal gravity (CGR$_3$), which yields an extended Virasoro-Kac-Moody system compatible with AdS$_4$/2 and AdS$_4$ holography; the Poincaré patch further clarifies the holographic grammar linking AS to the dual CFT$_3$. The results offer a concrete program for understanding Minkowski AS via a 3D holographic perspective and suggest memory/hair-like roles for AS in 4D quantum gravity.

Abstract

Asymptotic symmetries of AdS$_4$ quantum gravity and gauge theory are derived by coupling the dual CFT$_3$ to Chern-Simons gauge theory and 3D gravity in a "probe" large-level limit. The infinite-dimensional symmetries are shown to arise when one is restricted to boundary subspaces with effectively two-dimensional geometry. A canonical example of such a restriction occurs within the 4D subregion described by a Wheeler-DeWitt wavefunctional of AdS$_4$ quantum gravity. An AdS$_4$ analog of Minkowski "super-rotation" asymptotic symmetry is probed by 3D Einstein gravity, yielding CFT$_2$ structure, via AdS$_3$ foliation of AdS$_4$ and the AdS$_3$/CFT$_2$ correspondence. The maximal asymptotic symmetry is however probed by 3D conformal gravity. Both 3D gravities have Chern-Simons formulation, manifesting their topological character. Chern-Simons structure is also shown to be emergent in the Poincare patch of AdS$_4$, as soft/boundary limits of 4D gauge theory, rather than "put in by hand", with a finite effective Chern-Simons level. Several of the considerations of asymptotic symmetry structure are found to be simpler for AdS$_4$ than for Mink$_4$, such as non-zero 4D particle masses, 4D non-perturbative "hard" effects, and consistency with unitarity. The last of these, in particular, is greatly simplified, because in some set-ups the time dimension is explicitly shared by each level of description: Lorentzian AdS$_4$, CFT$_3$ and CFT$_2$. The CFT$_2$ structure clarifies the sense in which the infinite asymptotic charges constitute a useful form of "hair" for black holes and other complex 4D states. An AdS$_4$ (holographic) "shadow" analog of Minkowski "memory" effects is derived. Lessons from AdS$_4$ provide hints for better understanding Minkowski asymptotic symmetries, the 3D structure of its soft limits, and Minkowski holography.

Asymptotic Symmetries, Holography and Topological Hair

TL;DR

The work develops a framework in which AdS asymptotic symmetries acquire infinite-dimensional structure by coupling the holographic CFT to 3D topological sectors (Chern-Simons and gravity) and by restricting attention to boundary subregions with effectively two-dimensional geometry. In the large-level (probe) limit, a residual 2D CFT structure emerges on AdS boundaries, enabling Virasoro and Kac-Moody algebras to act as 4D hair for AdS states, while Wheeler-DeWitt boundary regions reveal memory-like and holographic shadow effects. The maximal spacetime AS are realized by 3D conformal gravity (CGR), which yields an extended Virasoro-Kac-Moody system compatible with AdS/2 and AdS holography; the Poincaré patch further clarifies the holographic grammar linking AS to the dual CFT. The results offer a concrete program for understanding Minkowski AS via a 3D holographic perspective and suggest memory/hair-like roles for AS in 4D quantum gravity.

Abstract

Asymptotic symmetries of AdS quantum gravity and gauge theory are derived by coupling the dual CFT to Chern-Simons gauge theory and 3D gravity in a "probe" large-level limit. The infinite-dimensional symmetries are shown to arise when one is restricted to boundary subspaces with effectively two-dimensional geometry. A canonical example of such a restriction occurs within the 4D subregion described by a Wheeler-DeWitt wavefunctional of AdS quantum gravity. An AdS analog of Minkowski "super-rotation" asymptotic symmetry is probed by 3D Einstein gravity, yielding CFT structure, via AdS foliation of AdS and the AdS/CFT correspondence. The maximal asymptotic symmetry is however probed by 3D conformal gravity. Both 3D gravities have Chern-Simons formulation, manifesting their topological character. Chern-Simons structure is also shown to be emergent in the Poincare patch of AdS, as soft/boundary limits of 4D gauge theory, rather than "put in by hand", with a finite effective Chern-Simons level. Several of the considerations of asymptotic symmetry structure are found to be simpler for AdS than for Mink, such as non-zero 4D particle masses, 4D non-perturbative "hard" effects, and consistency with unitarity. The last of these, in particular, is greatly simplified, because in some set-ups the time dimension is explicitly shared by each level of description: Lorentzian AdS, CFT and CFT. The CFT structure clarifies the sense in which the infinite asymptotic charges constitute a useful form of "hair" for black holes and other complex 4D states. An AdS (holographic) "shadow" analog of Minkowski "memory" effects is derived. Lessons from AdS provide hints for better understanding Minkowski asymptotic symmetries, the 3D structure of its soft limits, and Minkowski holography.

Paper Structure

This paper contains 34 sections, 71 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Lightning Review of CS/GR$_3$ AS and $\text{CFT}_2$ Currents
  3. Non-abelian CS gauge theory
  4. $\text{GR}_3$
  5. Holographic Matter coupled to CS/$\text{GR}_3$ on $\text{AdS}_3$
  6. $\text{CFT}_3$ in isolation on AdS$_3$
  7. CS and $\text{GR}_3$ coupled to $\text{CFT}_3$
  8. The Large-Level "Probe" Limit
  9. Abelian CS
  10. Non-abelian CS and GR$_3$
  11. Non-Standard $\partial \text{AdS}_4/2$ Correlators as CFT$_2$ Correlators
  12. Abelian gauge theory
  13. Non-abelian gauge theory and gravity
  14. Compatibility with 4D quantum loops and masses
  15. Evading the No-Go for Infinite-dimensional AS in $\text{AdS}_4/2$
  16. ...and 19 more sections

Figures (12)

  • Figure 1: $\text{CFT}_3$ state living on $S^2$ at $\tau=0$ on $\partial \text{AdS}_4$ (shown in red), dual to Wheeler-DeWitt wavefunctional describing the subregion of $\text{AdS}_4$ enclosed by the black cones. This subregion is spanned by all spacelike hypersurfaces ending on this boundary $S^2$. An example of such a hypersurface is shown in green. A vertical cross-section is shown on the right.
  • Figure 2: $\text{AdS}_3$ foliation of $\text{AdS}_4$ in global coordinates. A $\text{CFT}_3$ on $\text{AdS}_3$ projects only the upper half of $\text{AdS}_4$.
  • Figure 3: $\text{AdS}_3$ foliation of $\text{AdS}_4$ in "product-space" coordinates.
  • Figure 4: Witten diagrams for $\partial \text{AdS}_3$ correlators of KK modes ($\equiv \text{CFT}_3$ "hadrons"). External lines are $\text{AdS}_3$ bulk-boundary propagators for masses proportional to $\ell$. Blob consists of $\text{AdS}_3$ KK interactions and bulk-bulk lines (Fourier transformed in $\xi$ from $\text{AdS}_4$).
  • Figure 5: Witten diagram for $\partial \text{AdS}_3$ correlators, involving internal and external graviton/gluon lines.
  • ...and 7 more figures