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首页三维重力中的相变:轻量标量与全息超导
本文主要探讨的是"3D重力中的相变与分形维数"这一主题,针对具有高斯边界条件的三维量子引力理论。作者们发现,当理论中存在足够轻的标量场时,会引发一个二阶相变,即所谓的"全息超导相变"。这种相变发生在纯重力解仍然是局部AdS3的情况下,与纯重力解中的局部AdS3结构之间的第一阶相变(类似Hawking-Page相变)并存。 在这个三维重力模型中,关键的概念是标量场的凝聚,它改变了系统的全局性质。值得注意的是,尽管背景是局部AdS3,但相变的存在表明了非平凡的物理现象,即通过全息原理映射到CFT(共形场论)中的Rényi熵会发生相应变化。Rényi熵与系统的热力学行为密切相关,其随Rényi参数的改变而发生相变,这在全息理论中是一个重要标志。 作者们通过数学上的优雅解释,将临界维数与AdS3商组的Hausdorff维联系起来,这是一种描述复杂几何结构维度的有效方法。他们通过数值计算,分别在模空间的边界附近和内部进行了详细的分析。这种方法不仅有助于理解理论的物理特性,还为计算临界维数提供了实用工具。 此外,他们的研究与Belin、Keller和Zadeh等人的最新CFT工作进行了对比,这些工作利用高阶共形块来确定临界尺寸。令人惊讶的是,他们的计算结果显示出良好的一致性,这进一步验证了理论的准确性和有效性。 这篇论文深入探讨了三维重力中的相变现象,不仅揭示了分形维度的重要性,而且展示了如何通过全息理论将这些概念应用于实际的物理系统。这项工作不仅对量子引力理论的发展有重要贡献,也为理解复杂系统中的相变机制提供了新的视角。
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JHEP05(2018)080
general bulk geometry M. We then specialize to the main case of interest, for which M is a
quotient of hyperbolic space H
3
such as a handlebody, and find that the critical dimension
has a rather beautiful mathematical interpretation. The quotient is by a group of M¨obius
maps, which has a limit set, a subset of the Riemann sphere (the boundary of Euclidean
AdS
3
). This limit set has a finite Hausdorff dimension δ > 0, which is sometimes referred
to as the fractal dimension of the limit set. This Hausdorff dimension is precisely equal to
the critical dimension of the scalar field, ∆
c
= δ. In other words, a scalar is unstable if and
only if its dimension is less than the Hausdorff dimension of the limit set. The calculation
in section 3 can therefore be regarded as a CFT estimate of this Hausdorff dimension,
which provides explicit lower bounds on δ.
In section 5 we turn to the explicit computation of the critical dimension, using an
algorithm of McMullen for computing the Hausdorff dimension. We will use the algorithm
to compute the critical dimension analytically, finding the asymptotic behaviour of δ as the
boundary of moduli space is approached, for the handlebody which dominates the partition
function. We also describe what happens to the instability at large genus. We will also use
the algorithm to efficiently compute the critical dimension numerically. For example, for
the genus two surface described above, the Hausdorff dimension is δ = 0.189124 ···, close
to our CFT bound. We use the numerical data to provide plots of this critical dimension
as a function of moduli, and as a function of genus.
2 Review of higher genus partition functions in 3D gravity and 2D CFT
In this section we will review the description of higher genus Riemann surfaces, and the
construction of solutions to three-dimensional gravity with such boundaries, which can be
interpreted as saddle points for the higher genus partition function of a holographic CFT.
In particular, we describe a class of symmetric surfaces that we will use as examples. We
will also review the interpretation of these partition functions in terms of R´enyi entropies.
2.1 Moduli spaces and handlebodies
We are interested in studying holographic two-dimensional conformal field theories, dual to
three dimensional AdS gravity in Euclidean signature, in particular on a Riemann surface
B of genus g ≥ 2. The partition function of the theory on such a surface, denoted Z
g
(τ),
will depend on the conformal structure of the surface B. Here τ is a collection of 3g − 3
complex coordinates which parameterize the moduli space M
g
of conformal structures on
B. At genus one, τ can be identified with the usual torus modulus. At higher genus there
are various different coordinates which can be used to describe the moduli τ , some of which
we will now describe.
9
9
Because of the conformal anomaly the partition function will in addition depend on a choice of metric
within a given conformal class. Thus Z
g
should not — strictly speaking — be regarded a function of τ
alone. This dependence, however, involves only the central charge and not any of the other dynamical data
of the CFT (such as operator dimensions or structure constants), so will not be important for us here.
We will therefore suppress this dependence and simply indicate the dependence on the conformal structure
moduli τ.
– 7 –
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JHEP05(2018)080
For many purposes in CFT and gravity, the most convenient way to realize a Riemann
surface B is as a quotient of the Riemann sphere C
∗
by a Schottky group Γ, B = Ω(Γ)/Γ.
Here Γ is a discrete subgroup of PSL(2, C), which acts on C
∗
in the usual way by M¨obius
transformations, and Ω(Γ) is the set of points on the Riemann sphere where this group ‘acts
nicely’. More precisely, Ω(Γ) is the set of points z ∈ C
∗
which have some neighborhood U
containing no other images of z under the group: γ · z ∈ U for γ ∈ Γ implies that γ is the
identity. Equivalently, if we define the limit set Λ(Γ) to be the set of accumulation points of
the action of Γ on C
∗
(a set about which we will have much more to say later), then Ω(Γ) is
just the Riemann sphere with those points removed: Ω(Γ) = C
∗
−Λ(Γ). More specifically,
a Schottky group Γ of genus g is a subgroup of PSL(2, C) that is freely generated by g
loxodromic
10
elements of PSL(2, C), having as a fundamental domain the exterior of 2g
closed curves (usually circles), such that each of the g generators of Γ maps one of these
boundaries to another in pairs. Intuitively, to obtain a Schottky representation, we can
cut the surface along g disjoint closed loops such that it stays in one piece and becomes
a sphere with 2g holes, flatten it onto the complex plane, and build the Schottky group
from the M¨obius maps that glue the surface back together along its g seams. A given
Riemann surface can be written as a Schottky group in many different ways, depending on
the choice of g cycles to cut along. The presentation as a Schottky group is equivalent to
the plumbing construction used in [37]. A more detailed review of Schottky uniformization
can be found in [11, 28, 38]. A slightly different approach to calculations in the Schottky
coordinates was used in [39].
A rather different presentation of the Riemann surface B is as an algebraic curve. In
this case we represent B as the set of solutions to an equation such as
y
n
=
N
Y
k=1
z − u
k
z − v
k
. (2.1)
Here, B is a genus g = (N − 1)(n − 1) surface, represented as n-fold branched cover over
the Riemann sphere parameterized by the z-plane, with 2N branch points (u
k
, v
k
). As
the resulting Riemann surface automatically possesses a Z
n
symmetry (usually referred
to as replica symmetry) where one permutes the n sheets, one cannot describe a general
point in moduli space M
g
using this parameterization. Instead, this equation describes
only a 2N − 3 dimensional slice of moduli space, a family of surfaces with an enhanced
(Z
n
) automorphism group. Except in special cases one cannot map out the full moduli
space this way.
11
The advantage of this approach is that the moduli of this surface are
easy to describe — they are the locations (u
k
, v
k
) of the branch points. For physics, this
presentation is natural for describing R´enyi entropies (or certain correlation functions in
orbifold theories), with the branch points u
k
and v
k
corresponding to the insertion points
of twist and anti-twist operators.
10
A loxodromic element γ of PSL(2, C) is one which is conjugate to
q 0
0 q
−1
!
for some q with 0 < |q| < 1.
11
One important special case is the genus two moduli space. Every genus two curve is hyperelliptic, so
can be represented as a 2-fold cover of the sphere branched over 6 points.
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