Shock-Like Defects in AdS3 Gravity

• The paper demonstrates a holographic link between shock-like (thin-shell) defects in AdS3 gravity and smeared nonlocal operators in CFT2.
• It employs junction conditions and matching techniques with rotating BTZ metrics to derive detailed stress-energy, spin contributions, and shell dynamics.
• The analysis connects ETH, vacuum Virasoro blocks, and operator noncommutativity to elucidate chaotic behavior and unified holographic correspondence.

Shock-like (thin-shell) defects in AdS$_3$ gravity represent non-conformal line defects in the bulk, realized as dynamically backreacting codimension-one shells composed of dust particles, possibly carrying intrinsic spin. These objects provide a concrete holographic dual to smeared nonlocal operator insertions in CFT$_2$, yielding a precise correspondence between gravitational solutions, vacuum Virasoro blocks, and statistical properties of matrix elements in high-energy eigenstates. The chaotic nature of the high-energy sector in AdS$_3$/CFT$_2$ is illuminated by matching bulk shell actions, CFT$_2$ defect correlators, and Eigenstate Thermalization Hypothesis (ETH) analysis, with explicit treatment of spinning domain walls and higher-point correlators of multiple defects (Liu et al., 15 Jun 2025).

1. Bulk Setup and Shell Stress–Energy

The gravitational action for AdS$_3$ with a codimension-one shell $\mathcal{W}$, in units with AdS radius $\ell_\mathrm{AdS}=1$, is given by: $$S = -\frac{1}{16\pi G}\int_{\mathcal M}(R+2)\sqrt{g}\,d^3x + \int_{\mathcal W}\sqrt{h}\,\sigma\,d^2y + S_\mathrm{spin}$$ where $G$ is Newton's constant, $_2$0 is the induced metric on the shell, $_2$1 is its surface energy density, and $_2$2 encodes the spin current for shells composed of spinning particles. For shells without spin, the localized bulk stress tensor reads: $_2$3 with $_2$4 tangent to the worldvolume and $_2$5 its spacelike normal. The total ADM mass of the shell is $_2$6, with $_2$7 labeling the angular coordinate on $_2$8. For shells with continuous spin $_2$9, the spin-current $_3$0 supplements the defect data.

2. Junction Conditions: First-Order Formalism

Shells in AdS$_3$1 obey junction (matching) conditions across their worldvolume, relating the intrinsic geometry and extrinsic curvature (in the metric formalism) or frame fields and connection (in the first-order formalism). For spinning defects, metric continuity fails and one must employ the vielbein $_3$2 and spin connection $_3$3, leading to: $_3$4 Integrating over a thin "pillbox" enclosing the shell, one derives the junction conditions: $_3$5 with $_3$6. The requirement $_3$7 allows the induced metric $_3$8 to jump, provided the area element is continuous. In the spinless case ($_3$9), standard Israel conditions are recovered: $_2$0

3. Rotating BTZ Geometry and Matching Conditions

Both interior and exterior of the shell are described by regions of the rotating BTZ metric: $_2$1 where $_2$2, with subscripts $_2$3 denoting exterior ($_2$4) and interior ($_2$5) patches. The shell's worldvolume, at radius $_2$6, is characterized by the trajectory equations: $_2$7 where $_2$8 are conserved energy and angular momentum per shell particle, and $_2$9 are horizon radii. Imposing conservation laws and continuity of the area element $_2$0, the conserved quantities are explicitly fixed. For spinless shells: $_2$1 For spinning shells (total spin $_2$2): $_2$3 The area-form continuity again constrains $_2$4.

4. Bulk On-Shell Action: Static and Spinning Shells

The Euclidean on-shell action for a static shell is: $_2$5 with $_2$6 the radial turning point specified by

$_2$7

and the "gluing" equations

$_2$8

For shells with spin, one includes an additional term: $_2$9 with $_3$0 the left/right-moving turning points. The result factorizes into sectors echoing the Virasoro block decomposition.

5. Boundary CFT$_3$1 Line Operators and Correlators

In the dual CFT$_3$2, the shell corresponds to a "thin-shell operator" constructed by distributing a primary $_3$3 of dimension $_3$4 uniformly along a circle: $_3$5 Inserted into a thermofield double of energy $_3$6, the two-point correlation functions are: $_3$7 Using the monodromy method for the large-$_3$8 Virasoro block, the perturbed Fuchsian equation

$_3$9

and trivial monodromy leads to a transcendental equation for $\mathcal{W}$0: $\mathcal{W}$1 with $\mathcal{W}$2. The vacuum block exponentiates: $\mathcal{W}$3 and one identifies

$\mathcal{W}$4

under the bulk/CFT dictionary $\mathcal{W}$5, $\mathcal{W}$6, $\mathcal{W}$7. ETH-type ansatz for the defect matrix elements matches both gravitational and CFT blocks.

6. Higher-Point Defects and Correlator Order Dependence

The generalization to multi-defect correlators is explicit: higher-point functions factorize into products of two-point subblocks, with matching achieved via the monodromy method in CFT$\mathcal{W}$8 and by gluing trajectories of multiple shells in AdS$\mathcal{W}$9 bulk. Line operators, as codimension-one nonlocal objects, yield correlators that depend nontrivially on the insertion order, in contrast to Euclidean correlators for local operators. Explicit four-defect calculations reveal this noncommutativity, underlining the distinct operator algebra of shell defects.

7. Triangular Correspondence and Significance

A precise triangular equivalence is established among:

• (i) Bulk on-shell shell actions in AdS$\ell_\mathrm{AdS}=1$0 (including spin contributions)
• (ii) CFT$\ell_\mathrm{AdS}=1$1 vacuum Virasoro blocks for non-conformal line defects
• (iii) ETH-inspired statistical ansatz for defect matrix elements

This correspondence fully characterizes shock-like (thin-shell) defects in AdS$\ell_\mathrm{AdS}=1$2 gravity and their dual non-conformal line operators in CFT$\ell_\mathrm{AdS}=1$3, establishing a unified framework for analyzing backreacting, nonlocal defect dynamics, chaos, and operator algebra in holographic AdS$\ell_\mathrm{AdS}=1$4/CFT$\ell_\mathrm{AdS}=1$5 (Liu et al., 15 Jun 2025).