二维扭曲共形场理论的局部性与异常研究

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"Locality and Anomalies in Warped Conformal Field Theory" 本文详细探讨了在二维空间中扭曲共形场理论（Warped Conformal Field Theories, WCFTs）的各种特性。WCFTs是一种扩展了共形对称性的理论，它在量子场论和弦理论中扮演着重要角色，尤其是在AdS/CFT对应（Anti-de Sitter/Conformal Field Theory Correspondence）中。这篇研究由Kristan Jensen发表于JHEP杂志，详细阐述了该领域的最新发现。首先，文章揭示了新的拉格朗日WCFTs的存在。拉格朗日描述通常意味着一个理论可以通过局部相互作用的势来表述，这在物理学中是非常重要的。然而，作者指出所有已知的拉格朗日WCFTs实际上都需要无限的微调才能真正实现局部性。这意味着在这些理论中，要保持物理过程的局部性，需要对理论的参数进行精细调整，这在实践中可能是非常复杂的。接下来，作者推导了WCFT的异常。在量子场论中，异常是指理论在量子水平上失去经典对称性的情况，这通常是由于量子效应引起的。WCFT的异常被关联到了中心电荷，这是共形场论中一个关键的量子数，它与理论的拓扑性质和边界条件有关。这种关系的建立对于理解和计算WCFTs的物理性质至关重要。此外，通过分析WCFT的静水力学，作者重推导了Cardy公式的变形版本。Cardy公式在共形场论中用于估算边界态的数量，是理解和计算临界系统的重要工具。在WCFT的背景下，这个变形公式对于理解非平凡的边界行为和热力学性质有着深远的影响。最后，论文讨论了全息WCFTs的半局部性。在全息原理下，WCFTs与特定的AdS几何背景相关联，比如WAdS3黑洞。作者指出，这些系统的物质场显示出类似于Gregory-Laflamme不稳定性，这是一种在黑洞物理中常见的现象，表明某些情况下黑洞的内部结构可能会变得不稳定。同时，他们还发现WAdS3的“喉咙”区域不存在解耦极限，这意味着在这个区域内，引力和其他力无法分离处理，这对于理解AdS/CFT对应中的信息丢失问题提供了新的视角。关键词包括AdS-CFT对应、弦理论中的黑洞以及共形和WS对称性，表明这篇工作涵盖了广泛的理论物理领域。由于是开放获取的文章，这篇研究为感兴趣的学者提供了深入理解扭曲共形场理论的新窗口。

JHEP12(2017)111

and h

subject to two local redundancies: a “Weyl” rescaling and a “boost”. Under these

transformations, n

and h

transform as

“Weyl” : n

→ e

Ω

, h

→ h

“Boost” : n

→ n

, h

→ h

− ψn

(2.6)

where Ω and ψ are functions which characterize the rescaling and boost. Roughly speaking,

n is the more general version of the “right-moving” direction dx

−

, and h the more general

version of dx

From n

and h

we algebraically obtain two vector ﬁelds v

and w

satisfying

= w

= 1 , v

= w

= 0 . (2.7)

They transform under Weyl rescaling and boosts as

“Weyl” : v

→ e

−Ω

, w

→ w

“Boost” : v

→ v

+ ψw

, w

→ w

(2.8)

We also construct a boost-invariant measure d

√

γ with

√

γ =

det(γ

µν

) and

µν

≡ n

+ h

, (2.9)

along with a boost-invariant epsilon tensor

µν

= v

− v



µν

√

, ε

µν

= n

− n

√

γ 

µν

, (2.10)

where 

µν

and 

µν

are epsilon symbols. The measure transforms homogeneously as

√

γ →

Ω

√

γ under Weyl rescaling.

A few words are in order regarding the global structure of the geometry. We have

in mind a geometry in which (n

, h

) are globally-deﬁned and everywhere non-vanishing,

and that the WCFT in question is coupled to these ﬁelds in a Weyl and boost-invariant

way. M must then have the topology of a cylinder or a torus. Perhaps one may deﬁne the

geometry in such a way that (n

, h

) are not globally-deﬁned, but instead are sections of

some bundle.

The relation to ordinary NC geometry is the following. The basic building blocks

of NC geometry are the tensors (n

, h

νρ

), where n

is nowhere-vanishing, h

µν

is a spatial

metric, and the two are constrained so that n

µν

is positive-deﬁnite. In two spacetime

dimensions, h

µν

can be expressed in terms of a one-form via h

µν

= h

. The one-forms

and h

are those described above.

As for the Weyl and boost symmetries, these also have counterparts in NC geometry.

The NC geometry to which Galilean ﬁeld theories couple possesses a boost symmetry. This

transformation, known as a Milne boost, leaves n

invariant but shifts h

µν

→ h

µν

− (n

νρ

+ n

µρ

) ψ

+ n

ρσ

, ψ

= 0 . (2.11)

The Milne symmetry is intimately related to Galilean invariance as described in [25]. In

two spacetime dimensions where h

µν

= h

, the Milne boost (2.11) becomes the boost

– 6 –

JHEP12(2017)111

in (2.6) with ψ = h

. For Weyl, recall that non-relativistic scale-invariant theories are

characterized by a dynamical exponent z. When such a theory is coupled to NC geometry,

the theory is invariant under local “Weyl” rescalings of n

and h

νρ

→ e

Ω

, h

µν

→ e

2Ω/z

µν

. (2.12)

In two spacetime dimensions with z = ∞, this becomes the Weyl rescaling in (2.6).

From the one-forms (n

, h

) we deﬁne a covariant derivative D

via the connection

νρ

= v

∂

+ w

∂

. (2.13)

This connection is neither boost nor Weyl invariant. Under an inﬁnitesimal reparameter-

ization ξ

, Weyl rescaling ω, and boost ψ, which we colletively denote as χ = (ξ

, ω, ψ),

the connection one-form Γ

= Γ

νρ

varies as

= £

+ d∂

+ v

dω − w

dψ , (2.14)

where in the ﬁrst term we deﬁne the action of the Lie derivative £

along ξ

as though

were a tensor. The derivative D

has nonzero torsion but zero curvature,

= dΓ

+ Γ

∧ Γ

νρσ

∧ dx

= 0 ,

νρ

∧ dx

= −

νρ

∧ dx

= −v

dn − w

dh .

(2.15)

The torsion is determined in terms of the two scalars

N ≡ ε

µν

∂

, H ≡ ε

µν

∂

. (2.16)

The derivative also ﬁxes (n

, h

= D

= 0 . (2.17)

There is a ﬁrst-order version of this geometry. Because (n

, h

) are globally deﬁned,

the frame bundle is topologically trivial. We trivialize it by restricting the frame to be

= (v

, w

). There is no spin connection, and (2.15) follows.

There is another ﬁrst-order formulation of NC geometry which naturally appears in

Lifshitz holography (see e.g. [21, 27]). Given n

and the inverse spatial metric h

µν

, one

takes n

to be the temporal component of the coframe and decomposes h

µν

into inverse

spatial vielbeins. This decomposition is invariant under local spatial rotations as well as

under local boosts. Fixing the temporal component of the frame to be v

, these boosts act

in the same way as a Milne boost. In this way, the local boost symmetry is “absorbed”

into the local redeﬁnitions of frame.

Another form of NC geometry was described in [26], which naturally arises in the

Newtonian limit of General Relativity. Its d = 2 version (which of course has no relation to

General Relativity) consists of (n

, h

, Ω, T

) where T

is the (dualized) torsion satisfying

= −N , and Ω is a scalar which transforms under inﬁnitesimal boosts as

Ω = ε

µν

∂

(ψh

) − ψh

. (2.18)

One can use Ω to obtain a boost-invariant connection (although not a Weyl-invariant one).

This geometry is interesting and useful, but is unrelated to WCFT for reasons we explain

in the next subsection.

– 7 –

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