[Paper Review] Bulk Cluster Decomposition in AdS/CFT and A No-Go Theorem for Correlators in Microstates of Extremal Black Holes

Byungwoo Kang|arXiv (Cornell University)|May 13, 2013

Black Holes and Theoretical Physics39 references3 citations

TL;DR

This paper demonstrates that in extremal black holes in AdS, bulk cluster decomposition implies all correlation functions are identical across all microstates, due to the infinite throat geometry in the zero-temperature limit. Using the thermo-field double formalism, it proves that no measurement—of boundary or bulk operators—can distinguish microstates, leading to a no-go theorem that rules out certain fuzzball proposals with distinct bulk geometries.

ABSTRACT

Applying the thermo-field double formalism to extremal black holes in AdS with a macroscopic horizon, we show that (1) there exists a natural basis for the degenerate microstates of an extremal black hole, and (2) cluster decomposition in the bulk implies that all correlators are exactly the same for every microstate of the extremal black hole. The latter statement can be interpreted in two ways. First, at the fully non-perturbative level of AdS/CFT at finite N, it means that cluster decomposition does not hold in the bulk. This may be viewed as a sharp manifestation of the bulk non-locality at finite N. Second, at the level of the perturbation theory in 1/N, in which case we expect the bulk cluster decomposition, no measurement of either boundary operators or bulk field operators can distinguish the different microstates. The latter interpretation may exclude some versions of the fuzzball conjecture that assert that different microstates of a black hole are realized in the bulk as different metric and field configurations.

Motivation & Objective

• To understand the physical properties of microstates in extremal black holes within AdS/CFT.
• To investigate whether bulk cluster decomposition holds in the context of finite-N AdS/CFT and its implications for microstate distinguishability.
• To test the viability of fuzzball-type proposals that posit distinct bulk geometries for different microstates.
• To clarify the role of the thermo-field double formalism in describing extremal black hole microstates with macroscopic horizons.
• To determine whether correlation functions can distinguish between degenerate microstates of extremal black holes in the bulk.

Proposed method

• Constructs thermo-field double states for families of extremal black holes in AdS, using canonical or grandcanonical ensembles with fixed charges or chemical potentials.
• Applies the thermo-field double formalism to relate the bulk geometry with the entangled state of two CFTs, particularly in the zero-temperature limit.
• Analyzes the infinite throat geometry in the near-horizon region of extremal black holes, where spatial separation between opposite exterior regions diverges.
• Uses cluster decomposition in the bulk to factorize correlators across separated regions, leading to equality of one-sided correlators across all microstates.
• Derives that the variance of correlators between microstates vanishes exponentially at large entropy, implying no distinguishability.
• Considers both non-perturbative finite-N and perturbative 1/N regimes to assess the breakdown of cluster decomposition and its implications.

Experimental results

Research questions

• RQ1 Does bulk cluster decomposition in the AdS bulk imply that all correlation functions are identical across all microstates of an extremal black hole?
• RQ2 Can any measurement—of boundary or bulk field operators—distinguish between different microstates of an extremal black hole?
• RQ3 To what extent does the breakdown of bulk cluster decomposition signal non-locality in the bulk at finite N?
• RQ4 How does the infinite throat geometry in the zero-temperature limit affect the factorization of correlation functions?
• RQ5 Do these results rule out versions of the fuzzball conjecture that posit distinct bulk geometries for different microstates?

Key findings

• In the zero-temperature limit of extremal black holes with macroscopic horizons, the infinite throat causes spatial separation between opposite exterior regions to diverge.
• If cluster decomposition holds in the bulk, all correlation functions (of arbitrary operators at arbitrary points) are identical across all microstates of the extremal black hole.
• The variance of correlators across microstates is exponentially suppressed as $ e^{-S} $, implying no detectable differences between microstates at large entropy.
• At the non-perturbative finite-$ N $ level, bulk cluster decomposition fails, indicating a sharp manifestation of bulk non-locality.
• In the $ 1/N $ perturbative regime, no measurement can distinguish microstates, which rules out fuzzball models with distinct bulk metric configurations.
• The result holds for both boundary operators and bulk field operators constructed from dual boundary operators, regardless of operator type or insertion point.