Effective Microstructure

TL;DR

The work demonstrates that intricate microstate geometries in AdS$_3$/CFT$_2$ can be captured by simple, effective three-center descriptions in five dimensions, where momentum waves migrate away from the original brane loci and localize at a sphere-like center governed by bubble equations. By averaging over internal moduli and exploiting both supergravity and worldsheet formalisms, the authors connect five-dimensional effective geometries with six-dimensional superstrata, showing delta-function-like localization in the large mode-number limit and matching these results to exact worldsheet analyses via spectral flow and WZW methods. The key contributions include a concrete averaging procedure that yields tractable effective geometries, explicit localization formulas for momentum centers (e.g., $\cos^2\theta_*=m/k$, $r_*^2/a^2=n/k$), and a unified picture linking probe analyses, bubble equations, and worldsheet spectra. The findings support the broader program of effective microstructure, suggesting that many singular multi-center solutions can encode smooth horizonless microstructure when viewed through appropriate averaged descriptions, with implications for throat-depth bounds and holographic entropy considerations.

Abstract

In AdS$_3$/CFT$_2$ duality, there are large families of smooth, horizonless microstate geometries that correspond to heavy pure states of the dual CFT. The metric and fluxes are complicated functions of up to five coordinates. There are also many duals of heavy pure states that cannot be described in supergravity, but only admit a worldsheet description. Extracting the physical properties of these solutions is technically challenging. In this paper, we show that there are much simpler effective descriptions of these solutions that capture many of their stringy and geometrical features, at the price of sacrificing supergravity smoothness. In particular, the effective description of some families of superstrata, and of certain worldsheet solutions, is given by easy-to-construct three-center solutions. For example, the effective description of a superstratum with a long AdS$_2$ throat is a scaling, three-center solution in which the momentum wave is collapsed to a singular source at one of the three centers. This also highlights how momentum migrates away from the supertube locus in the back-reacted geometry. Our results suggest that effective descriptions can be extended to more general microstates, and that many singular multi-center solutions can in fact correspond to effective descriptions of smooth horizonless microstructure.

Figures (4)

• Figure 1: Single-mode superstratum features. Here $r$ is a radial coordinate in the ${\mathbb R}^4$ transverse to the brane sources, while $a$ is the radius of the underlying two-charge (D1-D5) supertube, upon which the superstratum is built by adding a momentum wave. The latter is supported at the scale $\boldsymbol{\sqrt{n/k}\,a}$ and has charge radius $\boldsymbol{\sqrt{n/k}\,b}$. The quantum numbers $n$ and $k$ are explained in sections \ref{['sec:SingleModeSS']}, \ref{['sec:deep_AdS2']} and \ref{['sec:BHregime']}.
• Figure 2: Example of the $\theta$-dependent part of a Wigner function, $d_{j'{\mathsf m}' \bar{{\mathsf m}}'}(\theta)$, for ${SU(2)}$. For $|\bar{{\mathsf m}}'|=j'$ or $|{\mathsf m}'|=j'$, the wavefunction is peaked at a particular polar angle $\theta_*$, and has a width of order $1/\sqrt{j'}$.
• Figure 3: There are three centers corresponding to the center of space,"(1)", a supertube center, "(2)", and the momentum center, "(3)".
• Figure 4: The 3-center configuration with GLMT centers and a D1-P center. Here "(1)", "(2)", and "(3)" label the centers; the quantities $s+1$ and $-s$ associated to the first two centers refer to their $V$ charges.