全息扭曲CFT的模态引导与负规范：理解和应用
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"自举全息扭曲的CFT，或者：我如何学会不再担心并忍受负面规范"
这篇学术文章深入探讨了扭曲的共形场理论（WCFT），这是一种非相对论量子场理论，由手性Virasoro和U（1）Kac-Moody代数描述。研究者利用模态不变性来分析这种理论的谱约束，特别是在全息WCFT的背景下，这些理论与三维重力理论中的Compère-Song-Strominger边界条件相关联。
在全息WCFT中，存在一个关键特征，即负U（1）级别，这可能导致后代态的负规范，通常会违反统一性原理。然而，作者们证明，只要（Virasoro-Kac-Moody）原语具有正范数，模态引导程序仍然是有效的。这意味着尽管存在负规范，但理论的数学结构仍然允许其保持一定的稳定性和一致性。
具体来说，他们证明了全息WCFT需要有负规范的基态或具有虚U（1）电荷的态。这种状态在全息框架下有自然的解释，因为全息对应原则将 bulk 空间中的物理现象与边界上的CFT对应起来。此外，对于具有大中心电荷和任意水平的WCFT，文章指出第一个激发主态满足Hellerman界限，这是对CFT谱的一种限制。
在水平为正的情况下，他们进一步指出，那些具有内部U（1）对称性的共形场理论（CFT）的已知范围可以直接应用到整体WCFT中。这为理解这类非相对论理论提供了新的视角和工具。
文章发表在JHEP期刊的2018年7月刊上，表明了在理论物理和弦理论领域，研究者正在积极探索和理解这些看似违反直觉的负规范现象，以及它们如何适应和扩展现有的物理理论框架。这项工作不仅加深了对WCFT的理解，也为未来的全息对应研究提供了重要的理论基础。
JHEP07(2018)112
J
|α=0
AdS vacuum
-
c
12
(Dressed) BTZ black holes
M
|α=0
P
0
=
¯
L
L
0
|α→0
L
0
|α→∞
L
0
L
Figure 1. Spectrum of 3D gravity with CSS boundary conditions where L
0
and P
0
denote the
zero modes of the Virasoro-Kac-Moody algebra. As α → 0, the L
0
axis approaches the L axis and
we recover the Brown-Henneaux identification of mass and angular momentum.
while descendants of primary states with P
0
< 0, like those of the vacuum, are healthy. Also
note that the
¯
L-dependent (second) term in eq. (2.5) is almost the Sugawara contribution
of the U(1) algebra, except that it is missing the appropriate powers of P
0
modes.
As with Brown-Henneaux boundary conditions, the zero modes charges of this algebra
correspond to linear combinations of mass and angular momentum, namely L
0
=
1
2
(M +J)
and P
0
=
1
2
(M −J). These modes may be written as
L
0
= L − α
2
¯
L, P
0
=
¯
L, (2.7)
where L and α
2
are constants defined in terms of L(x
+
) and B(x
+
) by
L =
1
2π
Z
dφL(x
+
), α
2
=
1
2π
Z
dφ[B
0
(x
+
)]
2
. (2.8)
It is worth noting that if
¯
L 6= 0 then α
2
= (2
¯
L)
−2
P
n6=0
P
n
P
−n
and hence α
2
is non-
vanishing for backgrounds with nontrivial P
n
charges. These backgrounds shift the energy
and angular momentum of the corresponding undressed (Brown-Henneaux) solution as
illustrated in figure 1. In particular, the fact that L
0
rotates clockwise with respect to L
for increasing values of α
2
, i.e. that the mass M = L
0
+P
0
decreases when
¯
L > 0, is related
to the fact that the level is negative whenever
¯
L is positive. On the other hand if
¯
L = 0,
α
2
remains finite but B(x
+
) drops out from all charges.
Another important feature of CSS boundary conditions is that the zero mode of the
U(1) charge is often assumed to be a fixed constant. This follows from consistency of the
asymptotic symmetry group, namely the integrability of the charges (2.5) and (2.6), which
prevents
¯
L from varying. A fixed value of P
0
seems to be too restrictive, however, as it
removes many of the solutions in eq. (2.3). Furthermore, the holographic derivation of
entropy in the dual WCFT uses a Cardy-like formula that requires a chemical potential
for the U(1) charge [12]. This implies that P
0
should be allowed to vary. Relatedly, if the
chemical potential is turned on, then modular invariance requires states with different P
0
– 6 –
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