全息CFT的混沌边界：二维限制与AdS引力的理论挑战

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本文主要探讨了混沌在全息CFT (Conformal Field Theory) 和阿贝尔-德西特 (Anti-de Sitter, AdS) 引力理论中的作用，特别是针对二维全息CFT中的热态系统。作者 Eric Perlmutter 通过研究热态下的李雅普诺夫指数 (Lyapunov exponent, λL)，发现其在量子系统中受到一个基本限制：λL ≤ 2π/β，这个界限反映了混沌现象在系统中开始的速度。这是基于量子力学中的基本原理，即高温状态下系统的复杂性增长受到混沌的约束。在分析过程中，作者首先关注了普通二维全息CFT中的λL = 2π/β与强耦合下的 Operator Product Expansion (OPE) 关系。OPE是一种描述量子场论中两个本地场相互作用的重要工具，其性质在混沌边界上的变化可以揭示理论的某些关键特性。通过对OPE的研究，作者揭示了当λL达到混沌限制时，全息CFT的特定性质，如中心电荷和有界自旋的高阶电流，对于理论的可行性产生了影响。接着，作者提出了一个重要的结论：不存在单一、稀疏的二维CFT，它们具有大中心电荷和一组高自旋的有界电流，同时满足λL = 2π/β。这意味着在弱耦合的AdS 3中，如果不存在无限大的规范场塔，比如SL(N)理论，高自旋的引力理论将面临不一致性问题。这与三维Vasiliev理论或更高自旋的hs[λ]理论形成对比，后者由于拥有经典W∞[λ]对称性，即使不表现混沌也不违反混沌界，因为λL = 0，这表明这些理论不会经历因果关系的破坏。这些结果挑战了关于AdS引力与CFT之间的一般理解，并鼓励进一步探索3D Vasiliev理论的无张力弦理论解释。在高维重力结构中，有限的高自旋场塔可能导致不同的因果律行为，而这种结构与标准的无限规范场塔模型如SL(N)相区别。本文通过混沌界限对全息CFT空间进行了严格的约束，揭示了量子系统中的混沌对理论物理的深远影响，尤其是在AdS引力的理论框架内，这对于理解量子引力和量子信息在高维度中的行为具有重要的理论意义。

JHEP10(2016)069

with thermal periodicity t ∼ t + iβ. For Euclidean correlators, z

= z

∗

, and time is

imaginary. In Lorentzian conﬁgurations, z

6= z

∗

, and time is complex.

As explained in [7], the OTO correlator (2.1) can be obtained by a particular analytic

continuation of (2.3) from the Euclidean regime. When z 6= z

∗

, correlators have a branch

cut running from z ∈ (1, ∞), and the analytic continuation requires crossing this cut. This

leads to new singularities. The relevant limit for chaos has been studied before, known as

the Regge limit, and boils down to the following manipulations of A(z, z). First, take z

clockwise around the branch point at z = 1, leaving z alone:

(1 − z) → e

−2πi

(1 − z)

(2.6)

This deﬁnes a Lorentzian amplitude in which z, but not z, lives on the second sheet of the

function A(z, z). Then take z and z to zero, holding their ratio ﬁxed:

z → 0 , z → 0 ,

≡ η ﬁxed (2.7)

This deﬁnes A

Regge

(z, η).

The appearance of the Regge limit is explained as follows. To obtain the OTO corre-

lator of interest from the Euclidean amplitude A(z, z), one assigns complex time t

+ i

each operator, then evolves the t

to the desired kinematic conﬁguration. We take V and

†

to sit at t = x = 0, and W and W

†

to sit at t > x > 0:

= e

2π

i

, z

= e

−

2π

i

= e

2π

i

, z

= e

−

2π

i

= e

2π

(t+i

−x)

, z

= e

2π

(−t−i

−x)

= e

2π

(t+i

−x)

, z

= e

2π

(−t−i

−x)

(2.8)

The cross-ratio can be read oﬀ from (2.4). At t = x, W crosses the lightcone of V , and

z passes through the cut. See ﬁgure 2. Evolving to later times t − x  β puts us in the

Regge regime, with the desired time-ordering hV

†

V W i. In this regime,

z ≈ −e

2π

(x−t)



∗



, z ≈ −e

2π

(−x−t)



∗



(2.9)

and



≡ i



2π

i

− e

2π

i



(2.10)

To preserve the operator ordering hV

†

V W i, the 

obey 

< 

; when all 

are distinct, all operators are separated in imaginary time.

We occasionally will use the following notation: → means we pass to the second sheet, and ∼ means

that we keep the leading behavior near z = 0 on the second sheet, i.e. we take the Regge limit.

In d = 2, one could opt to use Zamolodchikov’s q-variable, q = exp(iK(1 − z)/K(z)), where K(z) =

(

, 1, z) [78]. This maps all sheets of the cut plane C\(1, ∞) to the unit disk |q| ≤ 1. The Regge

limit corresponds to taking q → i. While we will not ﬁnd occasion to use this variable anymore in this

paper, it is undoubtedly useful for studying Lorentzian correlators in generic CFTs [60].

– 8 –

JHEP10(2016)069

for t − x  β. The range of θ is bounded as indicated in order to preserve the ordering

V W V W . Note that Im (z) ≤ 0. When θ = π/2, the operators are spaced equally.

also note the imaginary time parameterization used in [2],

θ =

−

2π

τ (2.16)

This pairwise arrangement of operators leads to a bound on the rate of chaotic time

evolution [2]. The authors prove a general statement about analytic functions bounded on

the half strip, which they then apply to OTO correlators. Consider a function of complex

time, f(t+iτ ), which obeys the following conditions: i) f (t+iτ ) is analytic in the half-strip

|τ| ≤ β/4, i.e. for 0 ≤ θ ≤ π, ii) f (t) is real, and iii) |f(t + iτ)| ≤ 1 throughout the strip.

Then f(t) obeys

1 − f



≤

2π

+ O(e

−

4π

) (2.17)

We are ignoring possible sources of error in this bound that are carefully discussed in [2] and

reviewed in [17]; these are not important for the large c theories we will discuss. Actually,

f(t) need not be real; the generalization is

1 − |f|



≤

2π

+ O(e

−

4π

) (2.18)

This is necessary when f(t + iτ) is an OTO correlator of non-Hermitian operators.

The bound may be applied to functions of the form

f(t) ≈ 1 −  e

+ . . . (2.19)

where 0 <   1 is a small parameter. Its sign ensures that f(t) decays rather than grows

as t increases. (2.18) implies λ

≤ 2π/β. This expression is valid for λ

−1

 t  λ

−1

log 1/.

The upper bound deﬁnes the scrambling time t

∗

, at which the -expansion breaks down;

resummation of higher-order eﬀects in  ensure a smooth descent towards zero. For complex

 = 

+ i

|f| = 1 − 

+ . . . (2.20)

The bound requires 

> 0 and λ

≤ 2π/β, while 

is unconstrained.

In a large c CFT,  ∝ 1/c. A corollary of the chaos bound in large c theories is a

bound on t

∗

≥

2π

log c (2.21)

This is the (updated version of the) fast scrambling conjecture [20].

One-sided correlators have 

= 

= 0, which leads to divergences in A

Regge

(z, η); these can be

regulated, most crudely by just not taking the 

strictly to zero.

– 10 –

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