三维重力中的相变：轻量标量与全息超导

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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

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

). 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, ∆

= 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

(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

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.

Because of the conformal anomaly the partition function will in addition depend on a choice of metric

within a given conformal class. Thus Z

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 –

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

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

k=1

z − u

z − v

. (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

, v

). As

the resulting Riemann surface automatically possesses a Z

symmetry (usually referred

to as replica symmetry) where one permutes the n sheets, one cannot describe a general

point in moduli space M

using this parameterization. Instead, this equation describes

only a 2N − 3 dimensional slice of moduli space, a family of surfaces with an enhanced

) automorphism group. Except in special cases one cannot map out the full moduli

space this way.

The advantage of this approach is that the moduli of this surface are

easy to describe — they are the locations (u

, v

) 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

and v

corresponding to the insertion points

of twist and anti-twist operators.

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.

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.

– 8 –

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