u(1) cs theory vs sl(2) cs formulation: boundary theory
TRANSCRIPT
U(1) CS Theory vs SL(2) CS Formulation:
Boundary Theory and Wilson Line
Xing Huanga,b,c 1, Chen-Te Mad,e,f,g 2, Hongfei Shuh,i 3, and
Chih-Hung Wuj 4
a Institute of Modern Physics, Northwest University, Xi’an 710069, China.b Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710069, China.c NSFC-SPTP Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China.
d Guangdong Provincial Key Laboratory of Nuclear Science,
Institute of Quantum Matter, South China Normal University, Guangzhou 510006,
Guangdong, China.e School of Physics and Telecommunication Engineering,
South China Normal University, Guangzhou 510006, Guangdong, China.f Guangdong-Hong Kong Joint Laboratory of Quantum Matter,
Southern Nuclear Science Computing Center, South China Normal University,
Guangzhou 510006, China.g The Laboratory for Quantum Gravity and Strings,
Department of Mathematics and Applied Mathematics,
University of Cape Town, Private Bag, Rondebosch 7700, South Africa.h Nordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden.i Department of Physics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan.j Department of Physics, University of California, Santa Barbara, CA 93106, USA.
1e-mail address: [email protected] address: [email protected] address: [email protected] address: [email protected]
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Abstract
We first derive the boundary theory from the U(1) Chern-Simons theory. We
then introduce the Wilson line and discuss the effective action on an n-sheet
manifold from the back-reaction of the Wilson line. The reason is that the U(1)
Chern-Simons theory can provide an exact effective action when introducing the
Wilson line. This study cannot be done in the SL(2) Chern-Simons formulation
of pure AdS3 Einstein gravity theory. It is known that the expectation value
of the Wilson line in the pure AdS3 Einstein gravity is equivalent to entangle-
ment entropy in the boundary theory up to classical gravity. We show that the
boundary theory of the U(1) Chern-Simons theory deviates by a self-interaction
term from the boundary theory of the AdS3 Einstein gravity theory. It provides
a convenient path to the building of “minimum surface=entanglement entropy”
in the SL(2) Chern-Simons formulation. We also discuss the Hayward term in
the SL(2) Chern-Simons formulation to compare with the Wilson line approach.
To reproduce the entanglement entropy for a single interval at the classical level,
we introduce two wedges under a regularization scheme. We propose the quan-
tum generalization by combining the bulk and Hayward terms. The quantum
correction of the partition function vanishes. In the end, we exactly calculate
the entanglement entropy for a single interval. The pure AdS3 Einstein gravity
theory shows a shift of central charge by 26 at the one-loop level. The U(1)
Chern-Simons theory does not have such a shift from the quantum effect, and the
result is the same in the weak gravitational constant limit. The non-vanishing
quantum correction shows the naive quantum generalization of the Hayward term
is incorrect.
2
1 Introduction
The goal of the study of emergent spacetime is to obtain the bulk geometry from other
equivalent descriptions. Since Einstein gravity theory is not renormalizable, people
expect that Einstein gravity theory may not be a fundamental theory. The gravity
theory should emerge from some fundamental theories. One promising approach is
the Holographic Principle [1, 2, 3, 4, 5]. This principle states that the physical de-
grees of freedom in Quantum Gravity is fully encoded by its boundary. One candidate
for perturbative Quantum Gravity, String Theory, provided a conjecture, the Anti-
de Sitter/Conformal Field Theory Correspondence (AdS/CFT correspondence). String
theory in the (d + 1)-dimensional AdS manifold is dual to CFTd. It is a surprising
fact that a gravitational theory can be studied from a non-gravitational theory while
being calculable and also realizable. The CFT becomes a probe of Emergent Spacetime.
Furthermore, the Ryu-Takayanagi prescription shows that the surface with a minimum
area (minimum surface) in AdSd+1 contains the information of entanglement entropy
in CFTd [6, 7]. Therefore, this seems to imply Quantum Entanglement generates the
bulk spacetime at least at the classical level.
The calculation of quantum entanglement quantities, in general, relies on the n-sheet
manifold with the analytical continuation of n [8], which is hard to calculate. The
observation of the minimum surface provides a simplified way to study entanglement
entropy [9, 10]. However, to confirm the observation, it is necessary to perform a per-
turbation in a boundary theory to probe the quantum regime of bulk gravity theory
[11, 12]. To obtain exact properties in Quantum Entanglement, we study Chern-Simons
(CS) Theory [13].
To probe Quantum Gravity from CS theory, we begin with the AdS3 Einstein gravity
theory. Because 3d pure Einstein gravity theory (metric formulation) does not have
local degrees of freedom [14], the action can be rewritten to an SL(2) CS formulation
(gauge formulation) [15]. Although the metric formulation is not equivalent to the
gauge formulation due to its measure, the renormalizable gauge formulation is conve-
nient for computation in the quantum regime [16].
The absence of local degrees of freedom also implies that the bulk action can be reduced
to a boundary action. The first derivation showed that the boundary theory is the Li-
ouville theory [17], which has conformal symmetry. Because the Liouville theory does
1
not have a normalizable vacuum, it contradicts the bulk theory [18]. Indeed, it is just
a classical action because the measure was not considered. Recently, due to a proper
consideration for including the measure into the boundary theory, the 2d Schwarzian
theory appears. This theory has the expected SL(2) reparametrization gauge redun-
dancy [19]. The 2d Schwarzian theory does not have conformal symmetry, but it is
expected due to conformal anomaly.
To study holographic entanglement entropy in the gauge formulation, the first question
is to find the gauge version of the minimum surface. Because all operators can be
rewritten in terms of the Wilson lines in the gauge formulation, Wilson line is shown
to be a probe of holographic entanglement [20, 21]. In the AdS/CFT correspondence,
it is not a simple task to study at an operator level, but CS theory is a simple system
for avoiding the difficulty. Hence studying holographic entanglement entropy from the
gauge formulation of pure AdS3 Einstein gravity theory should give the clearest clue to
emergent spacetime.
The co-dimension two Hayward term [22] was recently connected with the entanglement
entropy at the classical gravity level [23]. The basic idea is to introduce a Euclidean bulk
manifold with two boundaries that are not smoothly junctioned. When one glues two
surfaces at a co-dimension two wedge, the junction conditions of the two boundary sur-
faces induce an additional Hayward term for ensuring a well-posed variational principle.
The Hayward term shows a contribution of the wedge to the partition function [23, 24].
This contribution implies the equivalence with that introducing the co-dimension two
cosmic brane action with a suitable tension [23, 24]. The approach of the comic brane
effectively introduces the back-reaction on the replica orbifold that generates a conical
geometry [25]. The advantage of the Hayward term is that it is now a gravitational
action rather than an auxiliary tool (cosmic brane action). Due to the aforementioned
connection of the Hayward term with entanglement entropy at the classical level, it is
tempting to understand how the wedge term arises in the SL(2) CS formulation.
In this paper, we study holographic entanglement entropy in the gauge formulation
from the calculation of entanglement entropy and Wilson line including the quantum
correction. Although the SL(2) CS formulation has exact solutions to the boundary
effective action and the partition function [19], it does not have an exact description
when the Wilson lines are introduced. Introducing the Wilson line produces a back-
reaction to generate an n-sheet manifold [20], and the term related to the Wilson line
2
vanishes in the limit n → 1. Hence this does not produce a problem, but it relies
on whether the limit or a smooth solution exists. To properly understand the bulk-
boundary correspondence in the gauge formulation, we will first study a similar but
simpler system, U(1) CS theory. Furthermore, to understand whether the Hayward
term is a suitable proposal for obtaining entanglement entropy, we study the Hayward
term in the SL(2) CS formulation. To summarize our results:
• We derive the boundary action of U(1) CS theory. The result shows a combination
of left- and right-chiral non-interacting scalar field theories [26], which can be
rewritten from the non-interacting scalar field theory [27]. Introducing the Wilson
lines is equivalent to deforming the chiral scalar fields, and it does not change
the bulk theory. To obtain the non-interacting scalar field theory on an n-sheet
manifold [28, 29], it is necessary to deform the boundary condition. This study is
performed by first deforming the chiral scalar fields and then choosing the n-sheet
background, which then establishes an exact correspondence between the Wilson
line and entanglement entropy. This study shows that deformation is crucial for
the relation between the Wilson line and entanglement entropy.
• We derive the boundary action for the SL(2) CS formulation. The boundary
action can be written as a form that is similar to the U(1) case, but it has
additional terms for the interaction. Because we can choose the weak gravita-
tional constant limit to truncate the self-interaction, the non-trivial deformation
is again necessary for building an exact correspondence between the Wilson line
and entanglement entropy as in the U(1) CS theory. Indeed, this establishes the
“minimum surface=entanglement entropy” prescription with the quantum fluctu-
ation of gauge fields for studying the quantum deformation of minimum surface
[16].
• We use the boundary term to obtain the co-dimension two Hayward term. This
method is the same as in the metric formulation. By considering two wedges
forming a single interval on the boundary, reproducing the entanglement entropy
at the classical level. Combining the bulk and Hayward terms does not generate
quantum correction. It is inconsistent with the one-loop exact calculation of the
entanglement entropy. Hence we conclude that the quantum generalization does
not provide a suitable description of the entanglement entropy.
• We calculate the entanglement entropy for a single interval in the SL(2) CS for-
mulation, and it shows an exact shift of central charge by 26. The shift should
3
result from the additional interacting term since the U(1) CS theory does not
have the shifting. This result supports the equivalence of the two theories in the
weak gravitational constant limit. The U(1) CS theory and the SL(2) CS formu-
lation do not have the same number of gauge fields, but the boundary degrees of
freedom are the same. Hence showing equivalence is a non-trivial result.
The rest of the paper is organized as follows: We derive the boundary action and
study the dual between the Wilson lines and entanglement entropy for U(1) CS theory
in Sec. 2. We then consider the boundary action, the building of “minimum sur-
face=entanglement entropy” for SL(2) CS formulation as well as the set-up of the
co-dimension two Hayward term in Sec. 3. We calculate the entanglement entropy for
a single interval in SL(2) CS formulation at one-loop in Sec. 4. In the end, we have our
concluding remarks in Sec. 5. We give the details of the one-loop correction for Renyi
entropy in Appendix A.
2 U(1) CS Theory
To provide a clear picture of the bulk-boundary correspondence, we first introduce U(1)
CS theory. We derive the boundary action in U(1) CS theory and also introduce the
Wilson lines for a study of entanglement entropy.
2.1 Action
The action for the U(1) CS theory in the Lorentzian manifold is given by
SU(1)
=k
2π
∫d3x
(AtFrθ −
1
2
(Ar∂tAθ − Aθ∂tAr
))− k
2π
∫d3x
(AtFrθ −
1
2
(Ar∂tAθ − Aθ∂tAr
))− k
4π
∫dtdθ
E+t
E+θ
A2θ
+k
4π
∫dtdθ
E−tE−θ
A2θ. (1)
The boundary conditions become:
(E+θ At − E
+t Aθ)|r→∞ = 0; (E−θ At − E
−t Aθ)|r→∞ = 0. (2)
4
The t, r, θ are the polar coordinates, and the ranges are given by:
−∞ < t <∞; 0 < r <∞; 0 < θ ≤ 2π. (3)
The boundary zweibein is defined by:
E+ ≡ Eθ + Et; E− ≡ Eθ − Et. (4)
We also define the component of the boundary zweibein as the following:
E+θ dθ ≡ Eθ; E+
t dt ≡ Et; E−θ dθ ≡ Eθ; E−t dt ≡ −Et. (5)
The boundary metric is defined by the boundary zweibein
gµν ≡1
2(E+
µ E−ν + E−µ E
+ν ), (6)
in which the indices of boundary spacetime are labeled by µ = t, θ. The notation for
the x+ and x− is defined as in the following:
x+ = t+ θ; x− = t− θ. (7)
2.2 Bulk-Boundary Correspondence
Integrating out At and At respectively, we obtain:
Frθ = 0; Frθ = 0. (8)
The solutions are:
Ar = g∂rg−1; Aθ = g∂θg
−1; Ar = −g∂rg−1; Aθ = −g∂θg−1, (9)
where g and g are reparametrized by φ and φ respectively:
g ≡ e−φ; g ≡ eφ. (10)
Hence the solutions become:
Ar = ∂rφ; Aθ = ∂θφ; Ar = ∂rφ; Aθ = ∂θφ. (11)
5
By substituting the solutions into Eq. (1), the action becomes:
SU(1)
= − k
4π
∫d3x (∂rφ∂t∂θφ− ∂θφ∂t∂rφ) +
k
4π
∫d3x (∂rφ∂t∂θφ− ∂θφ∂t∂rφ)
− k
4π
∫dtdθ
E+t
E+θ
∂θφ∂θφ+k
4π
∫dtdθ
E−tE−θ
∂θφ∂θφ
= − k
4π
∫d3x (−∂θ∂rφ∂tφ− ∂θφ∂t∂rφ) +
k
4π
∫d3x (−∂θ∂rφ∂tφ− ∂θφ∂t∂rφ)
− k
4π
∫dtdθ
E+t
E+θ
∂θφ∂θφ+k
4π
∫dtdθ
E−tE−θ
∂θφ∂θφ
=k
4π
∫dtdθ ∂θφ∂tφ−
k
4π
∫dtdθ ∂θφ∂tφ
− k
4π
∫dtdθ
E+t
E+θ
∂θφ∂θφ+k
4π
∫dtdθ
E−tE−θ
∂θφ∂θφ
=k
4π
∫dtdθ ∂θφ
(∂t −
E+t
E+θ
∂θ
)φ− k
4π
∫dtdθ ∂θφ
(∂t −
E−tE−θ
∂θ
)φ, (12)
where in the second equality we have performed an integration by parts. Hence the
effective action for the U(1) CS theory is the combination of the left and right chiral
scalar field theories. Note that in the derivation, it is unnecessary to choose the asymp-
totic value for the gauge field. In the SL(2) CS formulation, we need to choose the
asymptotic gauge field for obtaining the AdS3 metric. It is the main point for obtain-
ing the equivalence in boundary degrees of freedom between the U(1) CS theory and
SL(2) CS formulation. We will discuss the SL(2) CS formulation later.
2.3 Non-Chiral Scalar Field Theory and n-Sheet Manifold
The n-sheet cylindrical manifold is:
ds2 = −dt2 + n2dθ2, E+t E
−t = −1, E+
θ E−θ = n2. (13)
We can redefine the
θ ≡ nθ, (14)
and then the metric becomes the flat background, but the range of the angular co-
ordinate changes. The non-interacting scalar field theory with a curved background
6
is:
SU(1)c = − k
8π
∫dtdθ
√− det gµν g
µν∂µΦ∂νΦ
= −nk8π
∫dtdθ
(− ∂tΦ∂tΦ +
1
n2∂θΦ∂θΦ
), (15)
which can be obtained by integrating out the auxiliary field Π from the following action
SU(1)c1 =nk
4π
∫dtdθ
(Π∂tΦ−
1
2Π2 − 1
2n2∂θΦ∂θΦ
). (16)
Now we apply the field redefinition:
Φ ≡ φ+ φ; Π ≡ 1
n∂θ(φ− φ) (17)
to obtain:
Π∂tΦ−1
2Π2 − 1
2n2∂θΦ∂θΦ
=1
n(∂θφ− ∂θφ)(∂tφ+ ∂tφ)
− 1
2n2(∂θφ∂θφ+ ∂θφ∂θφ− 2∂θφ∂θφ)
− 1
2n2(∂θφ∂θφ+ ∂θφ∂θφ+ 2∂θφ∂θφ)
=1
n∂θφ
(∂t −
1
n∂θ
)φ− 1
n∂θφ
(∂t +
1
n∂θ
)φ+
1
n∂θφ∂tφ−
1
n∂tφ∂θφ;
then
nk
4π
∫dtdθ
(Π∂tΦ−
1
2Π2 − 1
2n2∂θΦ∂θΦ
)=
k
4π
∫dtdθ ∂θφ
(∂t −
1
n∂θ
)φ− k
4π
∫dtdθ ∂θφ
(∂t +
1
n∂θ
)φ
+k
4π
∫dtdθ (∂θφ∂tφ− ∂tφ∂θφ)
=k
4π
∫dtdθ ∂θφ
(∂t −
1
n∂θ
)φ− k
4π
∫dtdθ ∂θφ
(∂t +
1
n∂θ
)φ, (18)
in which the final equality is up to a total derivative term. By comparing the action
with Eq. (1), we obtain the action SU(1) with the following choices of background:
E+t
E+θ
=1
n;
E−tE−θ
= − 1
n. (19)
Therefore, we can choose the following solution:
E+t = 1; E+
θ = n; E−t = −1; E−θ = n. (20)
7
2.4 Wilson Lines
We calculate the expectation value of the Wilson line in the U(1) CS theory
〈W 〉 ≡ 1
ZU(1)
exp(iSU(1) +
√2c2 lnW (P,Q)
), (21)
where ZU(1) is the partition function of the U(1) CS theory, and
√2c2 ≡
c
6(1− n), (22)
in which c is just a real-valued number, and
W (P,Q) ≡ P exp
(∫ s(P )
s(Q)
dsdxµ
dsAµ
)exp
(∫ s(Q)
s(P )
dsdxν
dsAν
), (23)
where the P denotes a path-ordering, and P,Q are the two end points of the Wilson
lines on a time slice. The equations of motion are:
k
2πFνρ = i
√2c2εµνρ
∫ s(Q)
s(P )
dsdxµ
dsδ(x− x(s)
);
k
2πFνρ = −i
√2c2εµνρ
∫ s(P )
s(Q)
dsdxµ
dsδ(x− x(s)
). (24)
The solution is:
A = gag−1 + gdg−1; A = −gag−1 − gdg−1, (25)
where
g = eφ; g = eφ; a ≡√c2
2
1
k
(dz
z− dz
z
); z ≡ θ + iψ; z ≡ θ − iψ.
(26)
The Euclidean time is defined as
ψ ≡ it. (27)
The solution corresponds to the following parametrization:
ρ(s) = s; z(s) = 0. (28)
Hence the solution can be rewritten as the following:
A = a+ dφ; A = −a+ dφ, (29)
8
in which the gauge field a has the consistent holonomy∫a = 2πi
√2c2
k. (30)
Writing the components of the gauge fields, the solution is given as in the following:
Ar = ∂rφ,
Aθ = ∂θφ+
√c2
2
1
k
(1
θ + iψ− 1
θ − iψ
),
Aψ = ∂ψφ+ i
√c2
2
1
k
(1
θ + iψ+
1
θ − iψ
);
Ar = ∂rφ,
Aθ = ∂θφ−√c2
2
1
k
(1
θ + iψ− 1
θ − iψ
),
Aψ = ∂ψφ− i√c2
2
1
k
(1
θ + iψ+
1
θ − iψ
). (31)
We can introduce the new scalar fields as:
φ = φ+
√c2
2
1
klnθ + iψ
θ − iψ; ¯φ = φ−
√c2
2
1
klnθ + iψ
θ − iψ. (32)
The gauge fields can be rewritten from the new scalar fields:
Ar = ∂rφ, Aθ = ∂θφ, Aψ = ∂ψφ;
Ar = ∂r¯φ, Aθ = ∂θ
¯φ, Aψ = ∂ψ¯φ. (33)
Therefore, the Wilson line gives a back-reaction or a redefinition of the fields. Because
the non-chiral scalar field theory on an n-sheet cylindrical manifold does not change the
form of the action, we also need to do the same deformation of the boundary conditions
as the following:
At = iAψ = Aθ; At = iAψ = −Aθ. (34)
We have already chosen the n-sheet cylindrical background in the above boundary con-
dition. The deformation of the chiral scalar fields for obtaining the non-chiral theory
is a crucial step for establishing the relation between the Wilson line and entanglement
entropy.
Now we want to discuss the relation between the geometry and the chiral scalar fields.
If we define the metric as in the SL(2) CS formulation [15]:
gµν ≡ 2eµeν ; Aµ ≡ e+ ω; Aµ ≡ e− ω, (35)
9
where eµ is vielbein, and ωµ is spin connection, it is easy to show that reproducing
the n-sheet cylinder manifold is impossible. This result means that the gauge fields
cannot build a similar relation to the geometry. Hence it implies why the n-sheet
cylinder manifold cannot be obtained from the back-reaction. Later we will show that
the n-sheet cylinder manifold can be obtained from the back-reaction in the SL(2)
CS formulation [20, 21]. This study demonstrates why the SL(2) CS formulation is a
non-trivial holographic model.
3 SL(2) CS Formulation
We first introduce the action in the SL(2) CS formulation, then the construction of the
AdS3 geometry from the gauge fields [15]. We then show that the boundary action is a
deformation of the U(1) case. In the end, we introduce the Wilson line to demonstrate
the building of the “minimum surface=entanglement entropy” [16]. This study clearly
shows that the back-reaction of the Wilson line produces the n-sheet cylindrical mani-
fold [20]. It exhibits the difference in the relation between the gauge fields and geometry
in the SL(2) CS formulation and U(1) CS theory. We also provide construction with
two wedges to realize the Hayward term [22, 23] in the SL(2) CS formulation.
3.1 Action
The action of the SL(2) CS formulation is given by [15]
SG
=k
2π
∫d3x Tr
(AtFrθ −
1
2
(Ar∂tAθ − Aθ∂tAr
))− k
2π
∫d3x Tr
(AtFrθ −
1
2
(Ar∂tAθ − Aθ∂tAr
))− k
4π
∫dtdθ Tr
(E+t
E+θ
A2θ
)+k
4π
∫dtdθ Tr
(E−tE−θ
A2θ
). (36)
The boundary conditions of the gauge fields are defined by:
(E+θ At − E
+t Aθ)|r→∞ = 0; (E−θ At − E
−t Aθ)|r→∞ = 0. (37)
10
To identify the SL(2) CS formulation by the 3d pure Einstein gravity theory, the con-
stant k is identified as
k ≡ l
4G3
, (38)
where
1
l2≡ −Λ. (39)
Note that the Λ is cosmological constant, the G3 is the three-dimensional gravitational
constant. The Frθ and Frθ are the r-θ components of the field strengths associated with
the gauge potential A and A, respectively. The gauge fields are still defined by the
vielbein eµ and spin connection ωµ, but the gauge group is now SL(2):
Aµ ≡ AaµJa ≡1
leµ + ωµ; Aν ≡ Aaν Ja ≡
1
leν − ων . (40)
The indices of bulk spacetime are labeled by µ and ν, and the indices of Lie algebra
are labeled by a. The spacetime indices are labeled by µ and ν. The indices of bulk
spacetime and Lie algebra are raised or lowered by
η ≡ diag(−1, 1, 1). (41)
Note that the bulk theory is locally equivalent to the standard CS theory with the
SL(2) gauge group. The measure in this gauge formulation is∫DADA, not the same
as in the metric formulation.
We now summarize our conventions here. The SL(2)×SL(2) generators are given by:
J0 ≡
(0 −1
212
0
); J1 ≡
(0 1
212
0
);
J2 ≡
(12
0
0 −12
),
J0 ≡
(0 −1
212
0
); J1 ≡
(0 −1
2
−12
0
);
J2 ≡
(12
0
0 −12
). (42)
The generators satisfy the below algebraic relations:
[Ja, J b] = εabcJc; Tr(JaJ b
)=
1
2ηab,
[Ja, J b] = −εabcJc; Tr(JaJ b
)=
1
2ηab. (43)
11
We will work within the AdS3 background, and the geometry is given by
ds2 = −(r2 + 1)dt2 +dr2
r2 + 1+ r2dθ2, (44)
in which the ranges of coordinates are defined by:
−∞ < t <∞; 0 < r <∞; 0 < θ ≤ 2π. (45)
Once we choose the unit
Λ = −1, (46)
the AdS3 solution corresponds to the following gauge fields:
A =√r2 + 1J0dx
+ + rJ1dx+ +
dr√r2 + 1
J2;
A =√r2 + 1J0dx
− + rJ1dx− +
dr√r2 + 1
J2. (47)
The configuration of gauge fields satisfies the equations of motion:
F = F = 0, (48)
and the solutions can be represented by the SL(2) transformation, g and g.
3.2 Boundary Effective Theory
We first show that the 2d Schwarzian theory appears on the boundary [19], and the
boundary theory is dual to another theory, which extended the U(1) case by the inter-
action.
3.2.1 2d Schwarzian Theory
To obtain the boundary effective theory, we first integrate out A0 and A0, which is
equivalent to using the following condition, respectively:
Frθ = 0; Frθ = 0. (49)
12
We substitute the condition into the action and consider AdS3 geometry with a bound-
ary, which can be obtained from the asymptotic behavior of the gauge fields:
Ar→∞ =
(dr2r
0
rE+ −dr2r
); Ar→∞ =
(−dr
2r−rE−
0 dr2r
),
(50)
The boundary manifold of AdS3 geometry is described by the E+ and E−. The SL(2)
transformations can be parametrized as the following:
gSL(2) =
(1 0
F 1
)(λ 0
0 1λ
)(1 Ψ
0 1
); gSL(2) =
(1 −F0 1
)(1λ
0
0 λ
)(1 0
−Ψ 1
). (51)
The asymptotic gauge fields provide the constraint on the boundary from the identifi-
cation:
g−1SL(2)∂θgSL(2)|r→∞ = Aθ|r→∞, g−1
SL(2)∂θgSL(2)|r→∞ = Aθ|r→∞. (52)
This leads to the following constraint on the boundary:
λ =
√rE+
θ
∂θF; Ψ = − 1
2rE+θ
∂2θF
∂θF, λ =
√rE−θ∂θF
; Ψ = − 1
2rE−θ
∂2θ F
∂θF. (53)
The boundary effective action can be obtained as
SGb
=k
2π
∫dtdθ
(3
2
(D−∂θF)(∂2θF)
(∂θF)2− D−∂
2θF
∂θF
)− k
2π
∫dtdθ
(3
2
(D+∂θF)(∂2θ F)
(∂θF)2− D+∂
2θ F
∂θF
), (54)
where we have defined:
F ≡ F
E+θ
; F ≡ F
E−θ, (55)
D+ ≡1
2∂t −
1
2
E−tE−θ
∂θ; D− ≡1
2∂t −
1
2
E+t
E+θ
∂θ. (56)
Note that the boundary theory loses the conformal symmetry but remains to be scale
invariant. The scale symmetry is necessary for mapping a single interval to a spherical
manifold or n-sheet cylinder manifold for the calculation of entanglement entropy [10].
13
3.2.2 Dual Theory on Boundary
Now we show that the boundary theory is dual to the following action
S2d1 =4k
π
∫dtdθ
((D−φ
)(∂θφ)
+ Π(∂θF − e4φ
)). (57)
The measure of the path integral is∫dφdFdΠ. Now we first integrate out the Π and
then integrate out the φ, equivalent to using the equalities:
ln ∂θF = 4φ; D−φ =1
4
D−∂θF
∂θF; ∂θφ =
1
4
∂2θF
∂θF;(
D−φ)(∂θφ)
=1
16
∂2θF
(∂θF )2
(D−∂θF
), (58)
and then we obtain
S2d2 =k
4π
∫dtθ
∂2θF
(∂θF )2
(D−∂θF
). (59)
The measure becomes∫
(dF/∂θF ). We can show that the dual theory is equivalent to
the 2d Schwarzian theory up to a total derivative term
k
2π
∫dtdθ
(3
2
(D−∂θF )(∂2θF )
(∂θF )2− D−∂
2θF
∂θF
)=
k
4π
∫dtdθ
∂2θF
(∂θF )2(D−∂θF ). (60)
Now we integrate out the F , and then the generic solution of Π is
Π = f(t). (61)
Then we can integrate out Π and perform a field redefinition
φ→ φ− 1
4ln f (62)
to obtain the following action
S2d3 =2k
π
∫dtdθ
((∂tφ)(∂θφ)− E+
t
E+θ
(∂θφ)(∂θφ)− e4φ
), (63)
up to a total derivative term. The measure is∫dφ.
Now we do a similar dual for the A part and begin from the following action
S2d1t =4k
π
∫dtdθ
((D+φ
)(∂θφ)
+ Π(∂θF − e4φ
)). (64)
14
The measure is∫dφdF dΠ. We first integrate out the field, F and then the generic
solution of Π becomes
Π = −f(t). (65)
Then we can do a field redefinition
φ→ φ− 1
4ln f (66)
to obtain the following action
S2d3t =2k
π
∫dtdθ
((∂tφ)(∂θφ)− E−tE−θ
(∂θφ)(∂θφ)
+ e4φ
). (67)
The measure is∫dφ.
Hence the 2d Schwarzian theory is dual to the theory:
SCB
= S2d3 + S2d3t
=4k
π
∫dtdθ
((D−φ
)(∂θφ)− e4φ
)− 4k
π
∫dtdθ
((D+φ
)(∂θφ)
+ e4φ
). (68)
The measure is∫dφdφ. Note that the U(1) case, it loses the interacting terms, exp(4φ)
and exp(4φ). The interacting term vanishes (or does not depend on φ) under the weak
gravitational constant limit k → ∞. Hence all new information should be encoded by
the interacting terms. We will do a one-loop exact calculation for entanglement entropy
[16] to show the difference between the SL(2) CS formulation and U(1) CS theory.
3.3 Wilson Line
We calculate the Wilson line [20]
WR(C) =
∫DUDPDλ exp
[ ∫C
ds
(Tr(PU−1DsU) + λ(s)
(Tr(P 2)− c2
))], (69)
where U is now an SL(2) element, P is its conjugate momentum,
√2c2 ≡
c
6(1− n), (70)
but c becomes the central charge of CFT2. The covariant derivative is defined as that:
DsU ≡d
dsU + AsU + UAs, As ≡ Aµ
dxµ
ds. (71)
15
The trace operation in the WR(C) acts on the representation. The equations of motion
are:
ik
2πFµ1µ2 = −
∫ds
dxµ3
dsεµ1µ2µ3δ
3(x− x(s)
)UPU−1;
ik
2πFµ1µ2 =
∫ds
dxµ3
dsεµ1µ2µ3δ
3(x− x(s)
)P. (72)
One solution of the above equations of motion can be represented by the SL(2) trans-
formations, g and g [20]:
A = g−1ag + g−1dg, g = exp(L1z) exp(ρL0);
A = −g−1ag − g−1dg, g = exp(L−1z) exp(−ρL0), (73)
where
a ≡ 1
k
√c2
2
(dz
z− dz
z
)L0. (74)
The equations of motion are similar to the U(1) case, and the generation of the holonomy
and solution are also similar. The SL(2) algebra satisfies the following relations:
[Lj, Lk] = (j − k)Lj+k, j, k = 0,±1; Tr(L20) =
1
2; Tr(L−1L1) = −1, (75)
and the traces of other bilinears vanish. We then choose:
z ≡ r exp(iΦ); z ≡ r exp(−iΦ) (76)
to obtain the spacetime [20]
ds2 = dρ2 + exp(2ρ)(dr2 + n2r2dΦ2). (77)
This solution corresponds to the following choice:
U(s) = 1; P (s) =√
2c2L0 (78)
with the curve:
z(s) = 0; ρ(s) = s. (79)
When n approaches to one, the geometry is AdS3. The back-reaction of Wilson line
generates the n-sheet manifold [20]. Using the redefinition
r ≡ exp(t), (80)
16
the boundary geometry becomes the n-sheet cylinder
ds2 = dt2 + n2dΦ2 (81)
up to a scale transformation. Because the boundary theory is scale-invariant, it is fine
to apply a scale transformation to the geometry.
Because WR vanishes under the limit n → 1, only the pure Einstein gravity theory
survives. One can see that the Wilson line serves as a useful auxiliary tool for analytical
continuation. The expectation value of the Wilson line is given by
〈WR〉 =ZnZn
1
+O(n− 1), (82)
where Zn is the n-sheet partition function of the boundary theory. When we do the
analytical continuation of n to one, the bulk calculation is equivalent to the boundary
calculation. The analytical continuation exactly shows the operator correspondence
between the bulk operator, Wilson line, and entanglement entropy of the boundary
theory
SEE = limn→1
1
1− nln〈WR〉, (83)
Because the classical solution gives the entanglement entropy of CFT2 [21], our result
provides the quantum deformation of a geodesic line.
If entanglement entropy is proportional to the 〈lnWR〉, the quantum contribution only
contributes to an area term. However, our result is instead ln〈WR〉. Hence holographic
entanglement entropy [6] cannot be given by the area term. Note that the difference
between the two expressions cannot be observed at the classical level. Therefore, the
study of quantum correction is a non-trivial task.
We discuss why the “minimum surface=entanglement entropy” can work in the SL(2)
CS formulation. We first introduce the Wilson line, and its back-reaction gives the
n-sheet geometry. The back-reaction can be seen as a deformation of the background
solution or gauge fields as in the U(1) case. Because the geometry is defined by the
gauge fields, the deformation changes the geometry, and then the n-sheet manifold
appears. Because the background changes, the boundary zweibein also needs to change
for preserving the boundary condition. Deforming the boundary zweibein also changes
the boundary term of the SL(2) CS formulation. Indeed, it is quite non-trivial. Now the
17
back-reaction of the Wilson line leads to a natural modification due to the connection
between the geometry and the gauge field. The boundary theory depends on the choice
of background. Here we cannot obtain an exact effective action when introducing the
Wilson line, but we can in the U(1) case. Because it is hard to know whether the
limit n → 1 limit is smooth without an exact solution, a similarly exact study should
be helpful. In the U(1) case, we exactly show that the Wilson line deforms the chiral
scalar fields. An additional change to the boundary condition is necessary for obtaining
entanglement entropy.
3.4 Hayward Term
Here we discuss an alternative approach, the co-dimension two Hayward term, for
computing entanglement entropy. The Hayward term is given by [22]
SH ≡1
8πG3
∫Γ
ds (π − θ), (84)
where s is the proper length of the curve
Γ ≡ Σ1 ∩ Σ2, (85)
with two co-dimension one boundary surfaces Σ1 and Σ2. The
−π < θ ≤ π (86)
is the angle between the normal vectors of two surfaces, Σ1 and Σ2, thus Γ represents
the co-dimension two wedge. The existence of the Hayward term is to ensure a well-
posed variational principle when a boundary manifold has the wedge.
We consider the setup in Fig. 1 and replace the joint Γ with a cap of radius r by
introducing a local smooth co-dimension one surface Σ′. Note that we take
r = ε, (87)
and eventually, we would shrink ε to zero. If we are only considering a thin slice of
the joint, one can always assume that the metric is locally flat around the joint. The
wedge term is then solved by the boundary integration in the co-dimension one surface
Σ′ under the limit
Σ′ → Γ (88)
18
S1
S2
S'G
Θ�
Figure 1: The Σ1 and Σ2 are the co-dimension one boundary surfaces. The θ is the angle between the
normal vectors of two surfaces, Σ1 and Σ2. The Γ = Σ1∩Σ2 is the co-dimension two wedge term. The
Σ′ is a smooth 2d surface.
that collapses into integration in the co-dimension two joint [22].
We introduce the boundary term
SHCS ≡ −k
4π
∫Σ′dtdθ Tr(A2
θ)−k
4π
∫Σ′dtdθ Tr(A2
θ) (89)
at
r = ε, (90)
instead of at infinity. The solutions of the gauge fields at the boundary are:
A = J0dx+ + J2dr; A = J0dx
− + J2dr. (91)
These solutions provide the following relation:
g−1∂θg|r=0
=
(1λ∂θλ−Ψλ2∂θF ∂θΨ + 2Ψ
λ∂θλ−Ψ2λ2∂θF
λ2∂θF λ2Ψ∂θF − 1λ∂θλ
)∣∣∣∣r=0
= Aθ|r=0
=
(0 −1
212
0
). (92)
19
This equality shows the following conditions:
1
λ∂θλ = Ψλ2∂θF ;
λ2∂θF =1
2;
∂θΨ + 2Ψ
λ∂θλ−Ψ2λ2∂θF = −1
2. (93)
Hence the conditions imply the following equations:
λ2 =1
2∂θF;
2
λ∂θλ = Ψ; ∂θΨ +
1
2Ψ2 = −1
2. (94)
After we substitute the boundary conditions to the bulk theory with the boundary
term, we obtain the 2-dimensional Schwarzian theory for the F . For the F , we also
obtain a similar term. Hence combining the bulk theory and the boundary term gives
SBH
=k
2π
∫dtdθ
(3
2
∂−∂θF∂2θF
(∂θF )2− ∂−∂
2θF
∂θF
)− k
2π
∫dtdθ
(3
2
∂+∂θF ∂2θ F
(∂θF )2− ∂+∂
2θ F
∂θF
).
(95)
However, substituting the classical solution into the bulk term and Hayward term, only
the boundary term survives. In the end, we obtain:
SBH = − k
2π
∫dtdθ Tr(e2
θ + ω2θ) =
1
2SH . (96)
We clearly see that in the SL(2) CS formulation, the boundary term does not lead
to the correct wedge term. This result can be traced back to the fact that the usual
boundary term (89), when recast in the gauge formulation, is already off by a factor of
two. One may think that certain information is lost when we fix the boundary metric
and rewrite the gravity theory using the CS formulation.
A way to fix this problem at the classical level is to consider double wedges. The setup
is given by Fig. 2. In this scenario, we again impose two co-dimension one surfaces
Σ1 and Σ2 with usual Dirichlet boundary conditions. However, an additional cut-off
surface is located at
r = ε. (97)
We immediately see that this would introduce two cusp-like co-dimension two wedges.
Following a similar procedure, we replace the two joints Γ1 and Γ2 with the two caps
20
Figure 2: A construction with double wedges. The Σ1 and Σ2 are the co-dimension one boundary
surfaces. The Σ′ is a cut-off surface that is located at r = ε. We replace the two joints Γ1 and Γ2 with
two caps of radius r1 and r2 respectively.
of the radius r1 and r2 respectively. Two limits are then taken in order, where we first
shrink the two caps to zero radius to obtain the correct Hayward term, and eventually
we take the cut-off surface
ε→ 0. (98)
This limit provides a natural interpretation that we are considering a classical gravity
dual of a CFT on a single interval. In the first approach, we first took the limits with
the inverse orders. It is why we only obtained a single wedge without reproducing the
expected result.
Classically, we have recovered the Hayward term. A natural question is whether one
can use the wedge picture to construct the n-sheet manifold and perform a well-defined
analytic continuation of n → 1 in the quantum regime. We solve the fluctuation field
in the equations of motion. Note that the equations of motion of the fluctuation field
F are:
λ2 =1
2∂θF;
2
λ∂θλ = Ψ; ∂θΨ +
1
2Ψ2 = −1
2. (99)
21
We can rewrite the equations as the following
∂2θ (ln ∂θF )− 1
2(∂θ(ln ∂θF ))2 =
1
2. (100)
It is easy to obtain the general solution by:
f ≡ ln ∂θF = −2 ln cos
(θ
2+ c1(t)
)+ c2(t), (101)
where c1(t) and c2(t) are arbitrary time-dependent functions. By doing the integration
for θ, we obtain the solution of the F (θ) as
F (θ) = 2 exp(c2(t)
)tan
(θ
2+ c1(t)
)+ c3(t), (102)
where c3(t) is also an arbitrary time-dependent function. The solution is periodic for θ
F (θ) = F (θ + 2π). (103)
We will discuss the result of the quantum contribution in the next section.
4 Entanglement Entropy at One-Loop
We calculate entanglement entropy for a single interval in the boundary theory of the
SL(2) CS formulation [16]. For the convenience of calculation, we choose another form
of the boundary theory on a spherical manifold [19]. Note that since the calculation on
the torus on is one-loop exact [19], the entanglement entropy is also one-loop exact. The
entanglement entropy shows a shift of center charge, 26 [16], and the shifting does not
appear in the U(1) case. In the end, we discuss the quantum correction by combining the
bulk and Hayward terms [22]. The result shows that the quantum correction vanishes.
Therefore, we have seen the difference between the approaches of the Wilson line [20]
and Hayward terms [23] in the quantum regime. The details of Renyi entropy for the
one-loop correction are given in Appendix A.
22
4.1 Boundary Theory
After we integrate out At and At, the effective theory can be rewritten as the SL(2)
transformations
SG1
= − k
4π
∫d3x Tr
(− g−1
(∂rg)g−1(∂tg)g−1∂θg + g−1
(∂rg)g−1(∂θg)g−1(∂tg))
+k
4π
∫d3x Tr
(− g−1
(∂rg)g−1(∂tg)g−1∂θg + g−1
(∂rg)g−1(∂θg)g−1(∂tg))
+k
2π
∫dtdθ Tr
(g−1(∂θg)g−1(D−g
))− k
2π
∫dtdθ Tr
(g−1(∂θg)g−1(D+g
)), (104)
By rewriting the SL(2) transformations in terms of other parameters, the effective action
can be written as:
SG1
=k
π
∫dtdθ
((∂θλ)(D−λ)
λ2+ 2λ2(∂θF )(D−Ψ)
)−kπ
∫dtdθ
((∂θλ)(D+λ)
λ2+ 2λ2(∂θF )(D+Ψ)
). (105)
Due to the following boundary conditions:
λ2∂θF = E+θ r; λ2∂θF = E−θ r, (106)
the following terms:
k
π
∫dtdθ
(2λ2(∂θF )(D−Ψ)
), −k
π
∫dtdθ
(2λ2(∂θF )(D+Ψ)
)(107)
are total derivative terms. Therefore, the boundary effective action becomes
SG1 =k
π
∫dtdθ
((∂θλ)(D−λ)
λ2− (∂θλ)(D+λ)
λ2
). (108)
With the definition:
F ≡ F
E+θ
; F ≡ F
E−θ, (109)
23
we obtain:
λ =
√r
∂θF; λ =
√r
∂θF. (110)
Therefore, we obtain the boundary action
SG1 =k
4π
∫dtdθ
((∂2θF)(D−∂θF)
(∂θF)2− (∂2
θ F)(D+∂θF)
(∂θF)2
). (111)
Finally, we choose the field redefinition:
F ≡ tan
(φ
2
); F ≡ tan
(φ
2
)(112)
to obtain
SG1
=k
4π
∫dtdθ
[(∂2θφ)(D−∂θφ)
(∂θφ)2− (∂θφ)(D−φ)
]− k
4π
∫dtdθ
[(∂2θ φ)(D+∂θφ)
(∂θφ)2− (∂θφ)(D+φ)
], (113)
up to a total derivative term.
4.2 Spherical Manifold
Using scale transformations, the Euclidean AdS3 metric approaching to a boundary
(r →∞) can be written as [10]
ds2 =dr2
r2+ r2ds2
s, (114)
where
ds2s ≡ dψ2 + sin2 ψdθ2, 0 ≤ ψ < π, 0 ≤ θ < 2π. (115)
The boundary zweibein is:
E+θ = E−θ = sinψ; E+
ψ = −iE+t = i; E−ψ = −iE−t = −i. (116)
24
4.3 Entanglement Entropy for Single Interval
To calculate entanglement entropy, we first identify the boundary conditions of fields.
We then calculate entanglement entropy and compare the result to the U(1) case. We
calculate the Renyi entropy
Sn ≡lnZn − n lnZ1
1− n(117)
from the replica trick [8], where Zn is the n-sheet partition function, and Z1 is equivalent
to the partition function. When n→ 1, the Renyi entropy gives entanglement entropy.
Because the calculation is one-loop exact, the quantum fluctuation of n-sheet partition
function only comes from the one-loop order. We then obtain that the n-sheet partition
function is a product of the classical n-sheet partition-function (Zc) and the one-loop
n-sheet partition-function (Zq)
Zn = Zn,c · Zn,q. (118)
Calculating the Renyi entropy is necessary to take the logarithm on the n-sheet partition
function
lnZn = lnZn,c + lnZn,q. (119)
Hence the contribution from the classical term and the one-loop term is not mixed.
4.3.1 n-Sheet Manifold
We first do a coordinate transformation on the unit sphere ds2s to obtain
ds2s = sech2(y)(dy2 + dθ2), (120)
where
sech y ≡ sinψ. (121)
The range of θ in the n-sheet manifold is extended to
0 < θ ≤ 2πn. (122)
The periodicity of θ is also extended to 2πn. The y-direction needs regularization, and
the range becomes
− lnL
ε< y ≤ ln
L
ε, (123)
where L is the length of an interval, and ε is a cut-off. Assuming the Dirichlet boundary
condition in the y-direction leads to the periodicity 4 ln(L/ε).
25
4.3.2 Boundary Condition
Now we map the sphere to the torus for the calculation of n-sheet manifold. We choose
the coordinates of an n-sheet torus
z ≡ θ + iy
n(124)
with the identification:
z ∼ z + 2π; z ∼ z + 2πτn. (125)
The identification leads to the boundary condition [16]:
φ
(y
n,θ
n+ 2π
)= φ
(y
n,θ
n
)+ 2π;
φ
(y
n+ 2π · Im(τn),
θ
n+ 2π · Re(τn)
)= φ
(y
n,θ
n
),
φ
(y
n,θ
n+ 2π
)= φ
(y
n,θ
n
)+ 2π;
φ
(y
n+ 2π · Im(τn),
θ
n+ 2π · Re(τn)
)= φ
(y
n,θ
n
). (126)
Due to the periodicity of y, the complex structure corresponds to the unit sphere is [16]
τn =
(2i
nπ
)ln
(L
ε
). (127)
Note that
Re(τn) = 0. (128)
The Fourier transformation of the fields gives:
φ =θ
n+ ε(y, θ); φ = − θ
n+ ε(y, θ), (129)
where
ε(y, θ) ≡∑j,k
εj,kei jnθ− k
τy; ε∗j,k ≡ ε−j,−k,
ε(y, θ) ≡∑j,k
εj,kei jnθ− k
τy; ε∗j,k ≡ ε−j,−k. (130)
The saddle-point is given by the θ/n in the φ and the −θ/n in the φ. Each Fourier
modes has three zero-modes:
εj,k = 0; εj,k = 0, j, k = −1, 0, 1 (131)
because of the SL(2) gauge symmetry.
26
4.3.3 Entanglement Entropy
To calculate the n-sheet partition function, we need to Wick rotate for the boundary
effective action:
SEG1
= − k
4π
∫ π
0
dψ
∫ 2πn
0
dθ
[(∂2θφ)(D−∂θφ)
(∂θφ)2− (∂θφ)(D−φ)
]+k
4π
∫ π
0
dψ
∫ 2πn
0
dθ
[(∂2θ φ)(D+∂θφ)
(∂θφ)2− (∂θφ)(D+φ)
]=
k
4π
∫ π2
Im(τ)
−π2
Im(τ)
dy
∫ 2πn
0
dθ sech(y)
[(∂2θφ)(D−∂θφ)
(∂θφ)2− (∂θφ)(D−φ)
]− k
4π
∫ π2
Im(τ)
−π2
Im(τ)
dy
∫ 2πn
0
dθ sech(y)
[(∂2θ φ)(D+∂θφ)
(∂θφ)2− (∂θφ)(D+φ)
], (132)
in which the covariant derivative becomes:
D+ =i
2∂ψ −
i
2
E−ψE−θ
∂θ =i
2∂ψ −
1
2 sinψ∂θ = − i
2cosh(y)∂y −
1
2cosh(y)∂θ,
D− =i
2∂ψ −
i
2
E+ψ
E+θ
∂θ =i
2∂ψ +
1
2 sinψ∂θ = − i
2cosh(y)∂y +
1
2cosh(y)∂θ. (133)
We substitute the saddle-point into the boundary effective action, and note that
c = 6k (134)
as the central charge of CFT2, we have:
− k
4π
∫ π2
Im(τ)
−π2
Im(τ)
dy
∫ 2πn
0
dθ1
2n2− k
4π
∫ π2
Im(τ)
−π2
Im(τ)
dy
∫ 2πn
0
dθ1
2n2
=−cπτ12n
= − c
6nlnL
ε, (135)
where
τ ≡ nτn. (136)
Therefore, we obtain
lnZn,c =c
6nlnL
ε. (137)
27
The Renyi entropy from the saddle-point is:
Sn,c =c
1− n
(1
6n− n
6
)lnL
ε=c(1 + n)
6nlnL
ε. (138)
When we take n→ 1, the saddle-point contributes to the entanglement entropy
S1,c =c
3lnL
ε. (139)
The Renyi entropy from the one-loop correction is given by:
Sn,q =1
1− n(
lnZn,q − n lnZ1,q
)=
13(n+ 1)
3nlnL
ε. (140)
Therefore, the summation of classical and one-loop terms gives the Renyi entropy
Sn =(c+ 26)(n+ 1)
6nlnL
ε(141)
and the entanglement entropy
S1 =c+ 26
3lnL
ε. (142)
The details of the one-loop correction are given in Appendix A.
The result shows that the quantum contribution does not change the form of Renyi
entropy and entanglement entropy. It shifts the value of the central charge by 26. Hence
the conformal anomaly does not take the Renyi entropy and entanglement entropy to
go beyond the result of CFT2. For the U(1) case, entanglement entropy does not have
such a shift from k0 term, and it is proportional to k. Because the difference between
the U(1) and SL(2) cases is the self-interaction, and the interaction vanishes in the
weak gravitational constant limit
k →∞, (143)
the shift of the central charge or the one-loop contribution is caused by the self-
interaction. Hence this shows that the SL(2) CS formulation should match with the
U(1) CS theory in the limit of the weak gravitational constant.
The fact that the U(1) CS theory matches the SL(2) CS formulation in a limit is not
trivial. The U(1) theory only has two degrees of freedom, but the SL(2) formulation has
six degrees of freedom. It is why the geometry can be related to the SL(2) formulation
28
but not related to the U(1) theory. However, their equivalence is on the boundary, and
the bulk does not have dynamical degrees of freedom. The asymptotic gauge fields for
showing the AdS3 metric gives the additional four required constraints. Its boundary
theory provides consistent degrees of freedom with the U(1) case. The appearance of
the self-interaction term shows the difference between the U(1) and SL(2) cases, which
do not affect the physical degrees of freedom. Hence studying the exact solution in the
U(1) case provides a better understanding of the SL(2) case.
4.4 Hayward Term
We discuss whether it is possible to study the quantum correction by computing the
bulk and Hayward terms here. From the solution (102), we choose the saddle point as
in the Wilson line:
Fsaddle ≡ 2 exp(c2(t)
)tan
(θ
2
)+ c3(t). (144)
By substituting the solution into the action with the field redefinition (112), the c2(t)
and c3(t) would not affect the result. Because the introduction of the n-sheet torus with
the boundary conditions for the φ would be exactly the same, the quantum fluctuation
only comes from the c1(t) term in Eq. (102). All situations are similar as in the Wilson
line, except that the quantum fluctuation does not depend on the θ. The non-vanishing
one-loop result relies on the non-trivial dependence of the θ. Therefore, the partition
function does not include quantum fluctuation. Introducing the n-sheet manifold to
the approach of the Hayward term [23], the quantum fluctuation still cannot generate
the non-trivial dependence of the θ. Therefore, the result implies the inconsistency
between the entanglement entropy and naive quantum generalization to the quantum
regime. This fact again shows why the classical result is too universal, and the inclusion
of quantum contribution is a non-trivial task.
5 Discussion and Conclusion
In this paper, we studied the holographic entanglement entropy [6] in the SL(2) CS
formulation [15] and compare the result to the U(1) CS theory. We showed that the
Wilson line is a suitable bulk operator for obtaining entanglement entropy of the bound-
ary theory [16, 20, 21]. This proof is quite non-trivial because it is non-perturbative.
We also showed that the Hayward term [22] in the SL(2) CS formulation reproduces
the entanglement entropy at the classical level by a double-wedges construction. By
29
combining the bulk and Hayward terms for a quantum generalization, the quantum
correction vanishes in the partition function. This result implies that the Wilson line
is a suitable candidate for studying entanglement entropy. The proof strengthens the
correspondence of “minimum surface=entanglement entropy”, but it relies on a smooth
limit of the analytical continuation [16]. Hence it is not clear why such equivalence can
be built, especially considering the lack of exact effective action after introducing the
Wilson line. This result motivates us to find a simple system to do a similar exact study.
By showing that the U(1) CS theory is equivalent to the SL(2) CS formulation in the
weak gravitational constant limit, we can study the Wilson line in the U(1) CS theory
to establish the equivalence. We can build the equivalence, but it is necessary to deform
the boundary after the back-reaction of the Wilson line. Because the chiral scalar fields
are not connected to the geometry in the boundary theory of the U(1) CS theory, the
deformation of boundary conditions and back-reaction cannot be done simultaneously.
Although the U(1) CS theory and the SL(2) CS formulation are equivalent in the weak
gravitational constant limit, it does not imply that the kinematic information, bound-
ary geometry, can be reconstructed by the boundary fields in the U(1) CS theory. It is
mainly because the numbers of gauge fields on the bulk are not the same, and the weak
gravitational constant limit does not play a role in defining a geometry. In the SL(2) CS
formulation, the back-reaction deforms the gauge fields and also the geometry [20, 21].
Because the boundary zweibein coupled to the boundary gauge fields, deforming the
geometry is equivalent to deforming the boundary condition and theory. In the end, we
calculated entanglement entropy in the SL(2) CS formulation, and it shows a shift of
the central charge by 26 [16]. By comparing the result to the U(1) case, it shows that
the shift should be caused by the self-interaction term.
The U(1) CS theory has two independent gauge fields, but the SL(2) CS formulation
has six gauge fields. It seems that their physical degrees of freedom do not match.
However, the bulk theory in a topological theory does not have dynamics. We need to
compare the physical degrees of freedom in the boundary theories. In the SL(2) CS
formulation, the AdS3 geometry provides the constraint to the asymptotic gauge fields.
The constraint leads the boundary theory to have the equivalent physical degrees of
freedom with the U(1) CS theory. The SL(2) CS formulation has one self-interacting
term in the boundary theory, and it does not appear in the U(1) case. However, the
local interacting term does not change the physical degrees of freedom. Because the
interacting term vanishes in the weak gravitational constant limit, the pure AdS3 Ein-
stein gravity theory reduces to the U(1) CS theory. The reduction itself is not trivial,
30
and the various exact results also complement the SL(2) CS formulation.
In the end, we comment on the holographic principle in the SL(2) CS formulation. Here
we can build the correspondence between the Wilson line and entanglement entropy
because the physical degrees of freedom in the CS theory are only constituted by the
Wilson line. For obtaining entanglement entropy, the bulk calculation, in the end,
reduces to the boundary calculation. The study of quantum correction provides a
better understanding of the holographic principle since the classical contribution itself
is usually not enough to justify different proposals or conjectures. It is also hard to
have the bulk and boundary actions simultaneously to check the equivalence. The
holographic principle works here because it requires one necessary condition that the
boundary fields and the asymptotic metric field do not decouple. By studying the U(1)
CS theory, we know that the condition is not trivial. In this case, changing to the
boundary condition (or Lagrangian) after introducing the geodesic operator (Wilson
lines) is necessary.
Acknowledgments
We would like to thank Chuan-Tsung Chan, Bartlomiej Czech, Jan de Boer, Kristan
Jensen, and Ryo Suzuki for their useful discussion.
Xing Huang acknowledges the support of the NSFC Grant No. 11947301 and the
Double First-class University Construction Project of Northwest University. Chen-Te
Ma was supported by the Post-Doctoral International Exchange Program; China Post-
doctoral Science Foundation, Postdoctoral General Funding: Second Class (Grant No.
2019M652926); Science and Technology Program of Guangzhou (Grant No. 2019050001)
and would like to thank Nan-Peng Ma for his encouragement. Hongfei Shu was sup-
ported by the JSPS Research Fellowship 17J07135 for Young Scientists from Japan
Society for the Promotion of Science (JSPS); the grant “Exact Resultsin Gauge and
String Theories” from the Knut and Alice Wallenberg foundation. Chih-Hung Wu was
supported by the National Science Foundation under Grant No. 1820908 and the fund-
ing from the Ministry of Education, Taiwan (R. O. C).
We would like to thank the National Tsing Hua University and Institute for Advanced
Study at the Tsinghua University.
31
Discussions during the workshop, “East Asia Joint Workshop on Fields and Strings
2019”, was useful to complete this work.
A Details of Renyi Entropy for One-Loop
Now we consider the quantum fluctuation from the ε(y, θ) and ε(y, θ) to obtain the
one-loop term [16]. Because the contributions from the sector of φ and φ are the same,
we only show the calculation related to the φ field. The expansion from the ε in the
boundary effective action is [16]
k
4π
∫ π2
Im(τ)
−π2
Im(τ)
dy
∫ 2πn
0
dθ
(n2(∂2θ ε(y, θ)
)(∂∂θε(y, θ)
)−(∂θε(y, θ)
)(∂ε(y, θ)
))= −i k
4πnτ ·
{n2∑j,k
[(− j2
n2· 1
2·(ik
τ+ i
j
n
)(ij
n
)]|εj,k|2
−∑j,k
[(ij
n
)· 1
2·(ik
τ+ i
j
n
)]|εj,k|2
}= −i k
8π
∑j,k
j(j2 − 1)
(k +
j
nτ
)|εj,k|2, (145)
where
∂ ≡ 1
2(−i∂y + ∂θ). (146)
Doing the derivative on the logarithm of the n-sheet one-loop partition shows
∂τ lnZn,q = −∑j 6=0,±1
∞∑k=−∞
jn
k + jnτ. (147)
Applying the following useful relation of the digamma function, we obtain
ψ(1− x)− ψ(x) = π cot(πx), (148)
in which the digamma function is defined by
ψ(a) ≡ −∞∑n=0
1
n+ a. (149)
Therefore, the complicated summation in the n-sheet partition function can be simpli-
fied as:∞∑
m=−∞
1
m− x= −
∞∑m=0
1
m+ x+∞∑m=1
1
m− x= −
∞∑m=0
1
m+ x+∞∑m=0
1
m+ 1− x
= ψ(x)− ψ(1− x) = −π · cot(πx). (150)
32
Hence we obtain:
∂τ lnZn,q = −∑j 6=0,±1
∞∑k=−∞
jn
k + jnτ
= −∑j 6=0,±1
(j
nπ
)· cot
(j
nπτ
)
= −2π∞∑j=2
(j
n
)· cot
(jπτ
n
). (151)
To obtain a universal term, we use the re-summation:
∂τ lnZn,q = −2π∞∑j=2
(j
n
)· cot
(jπτ
n
)
= −2π∞∑j=2
j
n·[
cot
(jπτ
n
)+ i
]+ 2πi
∞∑j=2
j
n.
(152)
We can perform a regularization for the divergent series
∞∑j=1
j → − 1
12. (153)
Hence we obtain:
∂τ lnZn,q = −2π∞∑j=2
j
n·[
cot
(jπτ
n
)+ i
]+ 2πi
∞∑j=2
j
n
→ −2π∞∑j=2
j
n·[
cot
(jπτ
n
)+ i
]− i13π
6n. (154)
After integrating out the τ , we obtain
lnZn,q = −2∞∑j=2
[ln sin
(πjτ
n
)+ i
jπτ
n
]− i13πτ
6n+ · · · , (155)
where · · · is independent of the τ . The first series is convergent for the
Im(τ) > 0. (156)
When we consider the limit
L
ε→∞, (157)
33
we obtain
lnZn,q =13
3nlnL
ε, (158)
and the Renyi entropy for the one-loop correction is [16]:
Sn,q =1
1− n(
lnZn,q − n lnZ1,q
)=
13(n+ 1)
3nlnL
ε. (159)
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