扭曲AdS3时空复杂性：手性Liouville作用的洞察

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"这篇研究论文深入探讨了扭曲AdS3 /扭曲CFT2对称情况下的路径积分优化过程，特别是关注手性Liouville作用在复杂性计算中的应用。作者Mahdis Ghodrati详细阐述了再规范化流方程，并对比了与标准CFT（共形场论）的区别。" 在本文中，作者提出了一种新的视角，即手性Liouville作用可以替代传统的Liouville作用作为扭曲AdS3时空复杂性的成本函数。这在理解AdS/CFT对应（反德西特空间/共形场论对应）的扭曲版本中起到了关键作用，因为在这个扭曲的环境中，传统的理论和方法可能不再适用。首先，作者介绍了具体的再规范化流动方程，这是理解量子场论中物理量随能量尺度变化的关键工具。在扭曲AdS3 /扭曲CFT2背景下，这些方程表现出了与标准CFT显著不同的特性。作者通过这种方式揭示了新环境下的基本规则和挑战。接着，作者展示了如何从时空、时向和零扭曲度量出发推导出变形的Liouville动作，同时分析了与扭曲参数相关的边界拓扑项的行为。这些拓扑项是构建扭曲度量的关键部分，它们与扭曲参数的相互作用反映了理论的内在一致性。文章的另一个重要贡献是发展了一系列全息工具，适用于扭曲AdS3的复杂性研究。这些工具包括针对手性扭曲CFT的张量网络结构、纠缠函数、表面/状态对应以及Kac-Moody代数的量子电路和运动空间WAdS/WCFT。这些工具为理解和量化扭曲AdS3时空的复杂性提供了强大的数学框架。此外，作者还讨论了路径积分复杂度的不同计算方法，包括将Polyakov作用、p-adic字符串和Zabrodin作用作为更合适的成本函数的可能性。这些示例进一步拓宽了研究复杂性的视野，强调了不同理论背景下选择合适成本函数的重要性。这篇开放获取的论文在扭曲AdS3 /扭曲CFT2对应的研究领域中开辟了新的道路，对手性Liouville作用的使用深化了我们对量子引力和复杂性的理解。它不仅提供了理论上的进展，还为未来的实证研究提供了实用的工具和技术。

JHEP02(2020)052

Figure 1. The relation between the codimension one surface M

and where it ends on Σ could

be the same as those of [30] shown in the left side and the quantum circuit modeled by the path-

integration on M

connecting Σ

to Σ

is modeled in the right side. We show however later

that a twist is needed to be considered for the WCFT/WAdS case.

A relation similar to [30] in the form of

C(M

)

. Ψ

[ϕ

)] =

r∈M

Dϕ(r)

−S

WCFT

[ϕ]

x∈Σ

δ(ϕ(t

) − ϕ

)), (3.1)

for the case of warped CFT states should be derived . In the above relation, S

WCFT

[ϕ]

should be replaced by the action for WCFT theories, which would be the chiral Liouville

action as discussed previously or other Lagrangians for warped CFTs which we will in-

troduce later. The optimization procedure then would be performed by minimizing the

functional C

) with respect to the two ﬁelds of ρ and h.

The plot of the manifold and its boundary is shown in ﬁgure 1. Note that in the WCFT

case we don’t have the complete conformal invariance, therefore, the state |Ψ

i after path

integration could depend not only on Σ but the whole M

. However, in the warped CFT

case, there is still a warped analogue of the conformal transformation connecting the metrics

of the two surfaces M

(1)

and M

(2)

which will be used to simplify the process.

So ﬁrst, deriving the corresponding equation of motion of (2.10) in [12], for the case of

warped CFT and chiral Liouville action would lead to

∂

−

ρ +

h + h(∂

−

ρ)

2ρ

= 0, (3.2)

where we also have ∂

−

h = 0. One could see from this equation how these two ﬁelds are

coupled, behave diﬀerently in left versus right, and how central charge and Kac-Moody level

simultaneously play a role. After choosing a boundary condition, one needs to solve these

coupled equations for the ﬁeld h and ρ which then lead to the time-slice of warped AdS

Then, the renormalization group ﬂow for the case of WCFTs could be considered in

order to understand further its path-integral optimization complexity. In fact, for this case,

there should be two position dependent cut oﬀs which would be the functions of coordinates

, t

−

) as

= e

ρ(t

−

)

, Λ

= e

h(t

−

)

. (3.3)

– 9 –

JHEP02(2020)052

If we want to have a theory that would not change after optimization, both cutoﬀs should

satisfy the RG equation,

dλ

(Λ

)

dΛ

= β(λ

(Λ

)), (3.4)

where β is the beta function.

The Callan-Symmanzik equation for the partition function Z and also the correlation

function would satisfy the following equations



dΛ

+ β(λ

)

dλ

+ ξ(λ

)



Z[Λ

, λ

] = 0,



dΛ

+ β(λ

)

dλ

+ ξ(λ

)



Z[Λ

, λ

] = 0. (3.5)

The ξ(λ) functions here would break the conformal symmetry leading to the

WCFT case.

We then need to understand how to couple these two RG equations for these two ﬁelds.

As for a general partition function Z for the whole system, we could have the following

relation



dΛ

+ Λ

dΛ

+ β(λ

)

dλ

+ β(λ

)

dλ

+ ξ(λ

) + ξ(λ

)



Z[Λ

, Λ

, λ

] = 0. (3.6)

In [44, 45], the renormalization group ﬂow equations for ﬁeld theories with quench

disorder as an example of non-relativistic, coupled RG ﬂow have been studied where their

couplings could vary randomly in space. In the case they have studied, the couplings

were also mixed together, similar to the situation for WCFTs. So their methods could be

applied here as well. So in the next parts, using various methods, we try to understand

these coupled renormalization group ﬂows for the case of WCFTs better.

3.2 Local RG ﬂow in the locally chiral gauge

As mentioned, Liouville action is always written in the conformal gauge, it is local and has

a left and right Virasoro algebra. However, in the chiral gauge, the chiral Liuoville theory

has only the right-moving Virasoro Kac-Moody algebra and it is in fact semi-local.

Also, for the case of CFT we have the ground state and its Virasoro descendants. For

the case of warped CFTs, however, we just have a left Virasoro and a right U(1) Kac-Moody

algebra. As the right and left descendants behave diﬀerently, the Euclidean path integral

optimization process that prepares the left and right moving states would be diﬀerent.

For the ground state of CFT we have the relation

Ψ( ˜ϕ(x)) =

−S

CFT(ϕ)

δ<z<∞

Dϕ(z, x)



ϕ(δ,x)= ˜ϕ(x)

, (3.7)

which computes the weight of a ﬁeld conﬁguration ˜ϕ(x). The Virasoro descendants could

be computed by changing the cutoﬀ surface to z = δ. For the chiral Liouville case, there

would be two parameters, one free continuos parameter k which determines the central

– 10 –

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