月光下的Monster CFT：纠缠与Rényi熵的深度探究

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标题：“可怕的纠缠”：这篇文章探讨的是Monster CFT（Monstrous CFT）在月球魔咒（moonshine）中的关键作用以及它与AdS 3中全息对偶关系的理论探讨，即它被推测为纯重力在AdS 3中的引力对称性描述。全息对偶是一种物理学概念，它将高维空间中的量子引力理论与低维量子场论之间的关系建立起来，这里的重点在于AdS/CFT对应。描述部分深入研究了 Monster CFT与其他极限CFT（极限CFT，指那些具有特殊性质，如卡鲁扎-克莱因（Kaluza-Klein）理论中的CFT）的纠缠现象以及Rényi熵。Rényi熵是量子信息理论中的一个重要量，用于衡量量子系统纯态的多线性信息量，尤其在量子纠缠和量子纠错编码中有重要应用。文章利用了短距离展开技术，即通过对圆环上的单个间隔进行分析，来精确计算Rényi熵的各个阶次。令人惊奇的是，每一阶的表达式都具有封闭形式，这意味着它们在模ularity参数（描述几何形状的参数）中可以被清晰地表述。文中特别关注了q系列的主导项，这些在低温下的Rényi熵中展现出了普遍的修正效应，这表明理论预测与实际观测的数据有很好的一致性。此外，这些计算结果与基于边界上的单环分区函数所得到的Rényi熵进行了对比，进一步验证了理论的准确性和有效性。除了单个间隔的分析，文章还探讨了平面上两个间隔的Rényi熵特性，这是对量子纠缠复杂性的深入理解。通过这些细致的计算和理论研究，文章揭示了Monster CFT在描述极端条件下物理现象，如极低温或强引力环境中的行为时的独特之处。这篇发表于《Journal of High Energy Physics》(JHEP10, 2017)的文章提供了一种新的方法来理解Monster CFT的深层次性质，特别是在量子信息领域中的应用，同时为AdS 3中纯重力和CFT之间复杂的对应关系提供了有力的数学支持。

JHEP10(2017)147

Collecting the leading order terms in the SIE for the n-replicated torus partition function,

we have the following,

= tr

[(ρ

)

] =

2(h

)

2(h

)



1 + b

hT i

+ (b

hAi + b

T T

hT i

)



hBi + b

hDi + b

T A

hT ihAi + b

T T T

hT i



+ ···



φφ

hφi

+ ···





, (2.10)

where contributions from the primaries and their descendants are contained in the φ sum-

mation and the ‘. . . ’ come with higher powers of `/L. Note from (2.9) that a considerable

simpliﬁcation occurs in the n → 1 limit. In particular, in the identity Virasoro module,

only the coeﬃcients of the form, b

...T

survive. As a result, the contribution of the

identity Virasoro module to the entanglement entropy is completely determined by hT i on

the torus and its higher powers. We shall return to this feature in section 6.

3 R´enyi entropies of the Monster CFT on the torus

3.1 The Monster CFT

The Monster CFT (the k = 1 extremal CFT or the moonshine module V

) has been ex-

plicitly constructed by Frenkel, Lepowsky and Meurman [2]; see also [3]. The construction

involves 24 free bosons compactiﬁed on 24-dimensional self-dual the Leech lattice, R

/L.

This is followed by an asymmetric Z

orbifolding, which removes the states at level 1. The

partition function of the theory reads

Z(τ ) =

Leech

2η



(ϑ

)

+ (ϑ

)

− (ϑ

)



= j(τ) − 744 (3.1)

+ 196884q + 21493760q

+ 864299970q

+ 20245856256q

+ ···

In the ﬁrst equality, the ﬁrst term is the contribution from the untwisted sector of the Z

orbifold whilst the other terms are from the twisted sector. As is well-known, the coeﬃcients

of the q-series display ‘moonshine’, i.e., they can be decomposed in terms of dimensions of

irreducible representations of the Fisher-Griess monster, M. More importantly, it has been

shown that automorphism group of the vertex operator algebra of the moonshine module

is indeed M [4].

The 196884 states at level 2 have the following decomposition in terms of irreps of M,

196884 = 196883 + 1, which is the McKay’s equation. This means it decomposes into a

trivial representation and the smallest non-trivial representation of M. In the language of

the conformal ﬁeld theory, these are dimension 2 operators which constitute the lightest

An equivalent depiction of this is that of an Z

asymmetric orbifold of the bosonic string compactiﬁed

on the Leech lattice.

– 6 –

JHEP10(2017)147

states of the theory. The trivial representation corresponds to the stress tensor. In ad-

dition, there are 196883 primary ﬁelds of weight 2. This decomposition of the reducible

representation shall play an important role in section 3.4.

Since we are dealing with a chiral CFT, there is a possible dependence on the conformal

frame in which the R´enyi entropies are being calculated owing to anomalies [57, 58]. This

results in an extra term which is frame-dependent. In what follows, we shall however

ignore these contributions since they are not speciﬁcally important for the theories we are

dealing with.

3.2 Torus 1-point functions

As discussed in section 2, the short interval expansion for the thermal R´enyi entropy

requires the one-point functions of operators on the torus. Let us ﬁrst focus on the identity

Virasoro module. The quasi-primaries contributing in the SIE, till level 6, are T, A, B, D,

as given in equations (2.6), (2.7). We shall now elaborate how to evaluate the one-point

functions of these quasi-primaries. The one-point function of stress tensor can be found

from the partition function itself:

hT i = 2πi∂

log Z . (3.2)

Substituting the partition function (3.1) and upon using the identity for the derivative of

the j-invariant (A.6), we have

hT i =

4π



j − 744



. (3.3)

As expected, the modular weight of the one-point function of a quasi-primary equals its

conformal weight. We shall remark on this further below.

The expectation values of normal ordered products of the stress tensor can be found

as follows. We can ﬁrst use the Ward identity on the torus to ﬁnd the correlation function

of multiple stress-tensors. The Ward identity, derived in [59, 60], is as follows

hT (v)T (v

)T (v

) ···T (v

)i − hT (v)ihT (v

)T (v

) ···T (v



2πi∂

j=1



2(℘(v − v

) + 2η

) + (ζ(v − v

) + 2η

)∂





hT (v

)T (v

) ···T (v

j=1

℘

(v − v

)hT (v

) ···T (v

j−1

)T (v

j+1

) ···T (v

)i. (3.4)

Here, ℘(v) and ζ(v) are the Weierstraß elliptic functions. Once the correlators are evaluated

using the above, we ﬁnd the expectation values of normal ordered powers of the stress tensor

by taking the coincident limit of these operators and subtracting out the OPE singularities.

For instance,

h:T T : (v)i ≡ lim

u→v

[hT (u)T (v)i −singular terms] . (3.5)

– 7 –

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