Conformal Janus on Euclidean sphere

Abstract

Abstract

We interpret Janus as an interface in a conformal field theory and study its properties. The Janus is created by an exactly marginal operator and we study its effect on the interface conformal field theory on the Janus. We do this by utilizing the AdS/CFT correspondence. We compute the interface free energy both from leading correction to the Euclidean action in the dual gravity description and from conformal perturbation theory in the conformal field theory. We find that the two results agree each other and that the interface free energy scales precisely as expected from the conformal invariance of the Janus interface.

Published for SISSA by Springer

Received: May 26, 2016 Revised: August 31, 2016 Accepted: November 30, 2016

Published: December 7, 2016

Conformal Janus on Euclidean sphere

Dongsu Bak,a,e Andreas Gustavssonb and Soo-Jong Reyc,d,e

aPhysics Department, University of Seoul,

Seoul 02504, Korea

bSchool of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea

cSchool of Physics & Astronomy and Center for Theoretical Physics, Seoul National University,Seoul 08826, Korea

dCenter for Theoretical Physics, College of Physical Sciences, Sichuan University, Chengdu 610064 P.R. China

eB.W. Lee Center for Fields, Gravity & Strings, Institute for Basic Sciences, Daejeon 34047, Korea

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We interpret Janus as an interface in a conformal eld theory and study its properties. The Janus is created by an exactly marginal operator and we study its e ect on the interface conformal eld theory on the Janus. We do this by utilizing the AdS/CFT correspondence. We compute the interface free energy both from leading correction to the Euclidean action in the dual gravity description and from conformal perturbation theory in the conformal eld theory. We nd that the two results agree each other and that the interface free energy scales precisely as expected from the conformal invariance of the Janus interface.

Keywords: AdS-CFT Correspondence, Conformal and W Symmetry

ArXiv ePrint: 1605.00857

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP12(2016)025

Web End =10.1007/JHEP12(2016)025

JHEP12(2016)025

Contents

1 Introduction 2

2 Summary of results 4

3 Janus on Euclidean sphere 5

4 Gravity dual 6

5 Renormalized free energy 95.1 Free energy in Fe erman-Graham coordinates 105.2 Free energy in other coordinates 135.3 Relations between cuto s 14

6 Free energy of Janus CFT2 15

7 Free energy of Janus CFT3 17

8 Stress tensor one-point function 19

9 Conformal perturbation theory 219.1 Related Janus solutions 22

10 Interface degrees of freedom 23

A Coordinates on AdSd+1 24

A.1 Global, AdS slice, Poincare patches 24A.2 The conformal boundary 25

B Computation of Isurface 26

C Computation of Ibulk for ICFT2 27

D Computation of Isurface for ICFT2 28

E General expressions for ICFTd to rst order in 2 29E.1 The bulk integral 29E.2 The surface term 30

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Figure 1. An interface produced by a coupling parameter varying in vertical direction: (a) At o -criticality, the interface has a characteristic thickness. (b) At criticality, the interface is a codimension-one geometric surface.

1 Introduction

An interface refers to a (d 1)-dimensional subsystem (interface system) immersed inside

a d-dimensional bulk system. It is known that critical behavior of the combined system is rich and highly nontrivial. The bulk system may be at a critical point or at o -critical point. At each cases, the interface system may separately be at critical or at o -critical point. As the parameters of bulk and interface systems are varied, the interface would undergo variety of phase transitions.

In this paper, we study a setup that both bulk and interface systems are at criticality and that allow us to study its behavior via AdS/CFT correspondence. This is a typical situation that interactions in the bulk and in the boundaries are of the same order. Equivalently, the interactions that drive the bulk into criticality also drive the interface into criticality as depicted in gure 1. The critical behavior of bulk system is described by d-dimensional conformal eld theory (CFT) and the critical behavior of interface system is described by (d 1)-dimensional conformal eld theory. The total system is described

by the interface CFT immersed inside the bulk CFT.

In the setup, the interface is given by a Janus deformation of bulk CFT, whose gravity dual is described by a Janus solution [1].1 This has the special feature that the Janus interface is constructible out of exactly marginal deformation of the bulk CFT, where the deformation parameter is interpreted as a varying coupling parameter of the exactly marginal operator. Across the interface, the coupling parameter interpolates from one asymptotic value to another, whose characteristic scale sets the thickness of the interface. In the regime the d-dimensional system is at criticality, this characteristic scale is driven to zero, so the deformation parameter jumps from one coupling constant to another across the interface. Moreover, given the argument a paragraph above, we expect that the system is

1For recent discussions of Janus (related) systems, see [2]{[19] and references therein. There are also related studies of interface/domain-wall partition function in supersymmetric gauge theories using localization [20{22].

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described by a CFTd coupled to a CFTd1. This is somewhat surprising since one generally

expects that the marginal deformation neither introduces any new degrees of freedom nor opens a new mass gap.

In the setup above, the interface is a domain wall whose thickness is varied with the deformation of bulk CFT. In other words, Janus interface is a thick domain wall expanded around the thin wall limit. It should be noted, however, that there are also interfaces in conformal system whose thickness does not vary with deformation of the bulk CFT, i.e. intrinsically thin domain wall. In AdS/CFT setup, such interface can be engineered by D5-branes intersecting D3-branes with co-dimension one, providing a string theory setup for two-dimensional graphene interacting with strongly coupled three-dimensional conformal gauge system [23, 24]. There are also situations in which an interface CFT is realized inside non-conformal bulk in which the conformal symmetry is realized dynamically [25].

An interesting point of the Janus deformation is that it is a hybrid of bulk CFTd and interface CFTd1, whose spacetime dimension di ers by one. Systems in even and odd

dimensions di er each other for their physical properties. Weyl and chiral symmetries are anomalous in even dimensions but intact in odd dimensions. Helmholtz free energy behaves very di erently in even and odd dimensions. The Janus interface we study in this paper combines CFTs of even and odd dimensions. As such, one expects it furnishes a concrete setup for hybrid physical characteristics, exhibiting in one part even-dimensional behavior and in other part odd-dimensional behavior. We con rm such expectation from several one-point functions including renormalized free energy and stress tensor. We use the AdS/CFT correspondence and compute these physical observables, rst from the gravity dual and then from dual CFT. With respect to the deformation parameter, we nd complete agreement of both computations.

This work is organized as follows. In section 3, we rst Wick-rotate the system to Euclidean space and then put it on a d-dimensional sphere. We then turn on the exactly marginal coupling so that the Janus interface is located at the equator. We assume that the system admits large N holography. In section 4, we construct the AdSd+1 gravity dual, in which the Janus is obtained by exciting a scalar eld. In sections 5, 6 and 7, using the gravity dual of section 4 and the AdS/CFT correspondence, we compute the free energy of the system. This physical observable diverges in the bulk infrared, and requires a regularization. By the AdS/CFT correspondence, in section 5, we relate this gravitational regularization to the regularization in the dual CFT. In sections 6 and 7, we extract the free energy of Janus interface for even and odd dimensions, respectively. We can also compute in section 8 the one-point function of the stress tensor. In section 9, we compute the interface free energy from the dual CFT by conformal perturbation theory. In this section, we also construct some related Janus solutions whose boundary spacetimes are conformal to the d-sphere. In section 10, we comment on the g-theorem [26] regarding renormalization group ow of the interface entropy. We argue that the interface free energy is interpreted as the negative of the interface entropy. The change of multiple interfaces may either increase or decrease, depending on the signs of the deformation [19] though the total entropy is always positive de nite. We also relegate technical computations in the appendices.

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2 Summary of results

We will let parametrize the interface deformation, so that = 0 corresponds to no interface. Then for a 2d CFT on S2 with a Janus 1d interface on the equator, we get from the AdS side the following result for the partition function

Z(r; ; ) = ea2

r2

[epsilon1]2 a1 r[epsilon1] a log FI (2.1)

where is a cuto near the boundary where the CFT lives in Fe erman-Graham coordinates and = [epsilon1]

2r as introduced below. The coe cients a2 and a are independent of , whereas a1 and FI depend on . This re ects the fact that a2 and a have 2d origin, while a1 and

FI arise due to the 1d interface. As explained in [27], by forming the ratio

Z(r; ; )

Z(r; ; 0) = eb1

r

[epsilon1] FI (2.2)

where b1 = a1( )a1(0) and FI = FI( )FI(0), we see that the divergences corresponding

to a2 and a of the 2d CFT cancel. In particular the log divergences cancel out. We can then isolate the 1d interface theory contribution in a non-ambiguous way, and cancel the divergences corresponding to b1 by adding a counter-term in the 1d interface theory, leaving us with a physical interface free energy FI. We nd that

FI( ) = 4G log

p1 2 2 (2.3)

where is the radius of AdS and G is Newtons constant. In particular FI(0) = 0 as one would expect when there is no interface. We also notice that FI( ) < 0 and that our

interface Janus deformation breaks supersymmetry.For 3d CFT on S3 with 2d Janus interface on the equator we get

Z(r; ; ) = ea3

r3

[epsilon1]3 a2 r

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[epsilon1]2 a1 r[epsilon1] a0 log [epsilon1]FI (2.4)

where a3 and a1 are constants independent of , while a2, a0 and FI depend on . Again we may isolate the contribution from the interface degrees of freedom by forming the ratio

Z(r; ; )

Z(r; ; 0) = ea2

r2

2

[epsilon1]2 a0 log [epsilon1]FI (2.5)

In this case we may cancel the quadratic divergence and the log divergence by adding counterterms. The log divergence will then give rise to a conformal anomaly given by minus of the coe cient of the log divergence,

a0 = 22

16G + O( 4) (2.6)

and the free energy is ambiguous. Still one should be able to, if one can identify the Fe erman-Graham cuto with corresponding cuto in the interface CFT, use this result to compare with an equally ambiguous eld theory computation of the free energy, along similar lines as was done in four-dimensional super Yang-Mills theory in [28, 29].

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Figure 2. The Janus system on a sphere is depicted. On the northern/southern hemisphere, we turn on an exactly marginal scalar operator with a coupling +I= I respectively.

3 Janus on Euclidean sphere

Begin with the bulk CFTd on Rd and the planar Janus interface Rd1. By a conformal map, one can put the bulk CFTd on a d-dimensional sphere, Sd and an interface CFTd1

on the equator, which is a (d 1)-dimensional sphere. Parametrize the Sd by

ds2d( ) = r2 d 2 + sin2 ds2Sd1(!)

; (3.1)

where r is the radius of the Sd, ds2Sd1 is the metric of the Sd1 with unit radius. The bulk

angular coordinates is split into the altitude angle ranging over [0; ] and the interface angular coordinates !. We then introduce an interface that halves the Sd at the equator, = =2 as depicted in gure 2. In general, this requires to introduce localized degrees of freedom on Sd1 that couple to the two sides Sd[notdef].

A distinguishing feature of the Janus interface is that it can be arranged from the bulk CFTd by simply turning on an exactly marginal scalar operator. Denote undeformed

CFTd Lagrangian L0( ) and its exactly marginal scalar operator O( ). The deformed

bulk CFTd is then de ned by the action

I =Z

: (3.3)

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Z [B( )] O( ) : (3.2)

Here, in the spirit of AdS/CFT correspondence, we expressed the deformation parameter in terms of boundary value of the bulk scalar eld B( ) which is dual to the operator

O. The simplest and well-known Janus construction is when the operator O is given

by the Lagrange density operator L0 of the bulk CFTd. More generally, the Janus can be

constructed with any scalar operators as long as they are exactly marginal. Indeed, in our considerations below, we only utilize the fact that the operator is exactly marginal scalar operator. The identi cation of precise functional form of the coupling parameter (B) is

a complicated problem in a speci c AdS/CFT correspondence. To the leading order in the

bulk eld expansion, one has in general

(B( )) = B( ) + O 2B( )

L0( ) +

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The leading order is universal, while higher orders change with reparametrization of the bulk eld.2

For the Janus deformation with the interface at the equator, the gravity dual eld B( ) takes the form

B( ) = I

2 ; (3.4)

where (x) is the sign function and I is the deformation amplitude representing the jump of coupling parameter across the interface. The coupling parameter depends only on , and so retains the stabilizer subgroup SO(d) of the bulk SO(d + 1) isometry group. The interface is distinguished by the topological quantum number, sign I = [notdef]1. For the case

of a single interface, without loss of generality, one can take the sign positive-de nite.

4 Gravity dual

It is known that the Janus interface geometry arises as a classical solution to the system of Einstein gravity coupled to negative cosmological constant and a minimal massless scalar eld

Igravity =

1 16G

I@Md+1K (4.1)

ZMd+1

R gab@a@b +d(d 1)2 1 8G

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where the second term is the Gibbons-Hawking boundary action [30]. The boundary @Md+1

on which the CFTd lives belongs to the conformal equivalence class of Sd. The interface geometry can be found for arbitrary dimensions. For (d + 1) = 3 and 5, this system can be consistently embedded into the Type IIB supergravity and hence, via the AdS/CFT correspondence, microscopic understanding of dual interface CFTd system can be obtained [1, 8].

The scalar eld here originates from the dilaton eld of the underlying Type IIB supergavity and hence it is holographically dual to the CFTd Lagrangian density. The equations of motion read

gabra@b = 0 Rab =

d2 gab + @a@b (4.2)

The vacuum solution is the AdSd+1 space with curvature radius and an everywhere constant scalar eld. The Janus geometry is a nontrivial domain-wall solution in which the scalar eld and metric approach those of the vacuum solutions.

For instance, in three dimensions, (d + 1) = 3, the Euclidean Janus geometry is given by [8]

ds2 = 2

dy2 + f(y) ds2M2

(y) = 0 + 1p2 log

1 +

p1 2 2 + p2 tanh y 1 +

p1 2 2 p2 tanh y !

(4.3) 2For detailed discussion for the three-dimensional case, see refs. [8, 15, 16]. In this case, the bulk scalar describes the size modulus deformation of the target space. The latter two references also include the discussion of half-BPS Janus system.

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Figure 3. The Janus pro le of minimal scalar eld. The vertical axis covers (y) 0, the

horizontal axis covers y = [10; +10], and the depth covers entire domain of the deformation

parameter = [0; 1=p2 = 0:705 : : :).

where

f (y) = 12

1p2 . The metric of M2 has to satisfy

Rpq( g) =

gpq (4.5)

Below, we shall choose the metric of M2 as a global Euclidean AdS2:

ds2(M2) = 1 cos2

where the ber coordinate ranges over [0; 2 ].

The pro le of the metric function f(y) and the scalar eld (y) are plotted over the entire range of the deformation parameter in gures 3 and 4.

The Janus geometry preserves the SO(2,1) isometry out of the SO(3,1) isometry, viz. Euclidean AdS2 hypersurface inside Euclidean AdS3 space. The conformal compacti cation of y = [notdef]1 is given by the boundary geometry of two hemispheres S2[notdef], joined at the equator,

= 2 . Hence, the entire boundary forms a full sphere S2, with an interface at the equator. We shall take the boundary metric as (3.1). Without loss of generality, 0 can be set to zero, so that the scalar eld asymptotes to

([notdef]1) = [notdef]

1 p2arctanh p

(4.7)

By the AdS/CFT correspondence, we identify these boundary values ([notdef]1) of the scalar

eld with [notdef]I. The scalar eld is massless, so it sources an exactly marginal scalar operator

in the dual CFT2.

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1 +

(4.4)

p1 2 2 cosh 2y

and 0 <

d 2 + sin2 d2

(4.6)

2

Figure 4. The Janus pro le of metric function f(y). The vertical axis covers f(y), the horizontal axis covers y = [5; +5], and the depth encompasses entire domain of the deformation parameter

= [0; 1=p2 = 0:705 : : :).

More generally, in arbitrary dimensions, the gravity dual of an exactly marginal deformation is again described by the action (4.1). The Euclidean Janus geometry is given by two patches, labeled by [notdef] [31],

ds2[notdef] =

2 q2[notdef]

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dq2[notdef]

P (q[notdef])

+ ds2Md

[notdef](q[notdef]) = 0 [notdef]

Z

q

q

dx xd1

pP (x); (4.8)

where P (x) is dimension-speci c polynomial

P (x) = 1 x2 +

2 d(d 1)

x2d (4.9)

and q denotes the smallest positive root of P (q ) = 0. Here, q[notdef] are parameters ranging over [0; q ] and is the deformation parameter ranged over [ 0; pd 1

d1

d

d12 ). The

metric of the hypersurface Md has to satisfy the hyperbolicity

Rpq( g) = (d 1) gpq : (4.10)

We choose the metric of Md as

ds2(Md) = 1cos2 [d 2 + sin2 ds2(Sd1)] : (4.11)

To cover the entire space, we need to choose the coordinates q[notdef] of the two respective

patches ranged over the same interval. To see this clearer, we revisit the three-dimensional

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solution, (d + 1) = 3, and rename the y coordinate as

q+ = 1

pf(y) (y 2 [0; +1])

q =

1

pf(y) (y 2 [1; 0]) : (4.12)

We see that q[notdef] ! 0 correspond to y ! [notdef]1 and q[notdef] = q corresponds to y = 0. The entire

domain of y = (1; +1) is covered by two identical copies of q[notdef] = [0; q ). For general

dimension d, provided the scalar eld is di erentiable, the two patches are smoothly joined at q[notdef] = q . In the asymptotic regions, q[notdef] ! 0, the scalar eld takes the asymptotic

values ([notdef]1) = 0 [notdef] R

q

0 dg

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pP (g) where the sign depends on the respective coordinate patch used. The Janus geometry has SO(d,1) isometry of the AdSd hypersurface out of the SO(d + 1,1) isometry of the Euclidean AdSd+1 space. By the AdS/CFT correspondence, the holographic dual CFTd is deformed by an interface that preserves (d 1)-dimensional

conformal invariance. We shall refer to the latter system as interface CFT (ICFT). In the gravity dual, the minimal scalar eld is massless, so it couples to an exactly marginal scalar operator in dual CFTd. We have just shown that the Janus geometry provides an elegant construction of interface while preserving the conformal invariance both at the bulk and the interface.

5 Renormalized free energy

To understand the ICFT better, we now compute physical observables. The simplest observable is the free energy, the expectation value of an identity operator. In this section, we compute the free energy of the ICFT by computing the classical, on-shell Euclidean action of the gravity dual. The classical Euclidean action is infrared divergent, so we shall be computing it using the method of holographic renormalization [32]. In this method, the rst step is to regularize the action of gravity dual by introducing an infrared cut-o at timelike in nity, where the geometry asymptotes to AdSd+1. For the vacuum solution (in which the Janus deformation parameter is put to zero), the cuto will be chosen such that it retains the stability isometry subgroup maximal, namely, the induced boundary metric is a d-dimensional sphere, Sd. The second step is to add counter-terms to cancel divergences as the infrared cuto is removed. Such a cuto can be chosen in any coordinate system one adopts and di erent choices correspond to di erent subtraction schemes. In general, these schemes di er from one another by the amount of nite subtractions in addition to the infrared divergences. Among them, the minimal subtraction scheme, viz. the scheme that only subtracts the infrared divergence, is provided by the Fe erman-Graham (FG) coordinates.

Therefore, as a preliminary step, we rst exercise out the computation of renormalized free energy for the undeformed (without Janus interface) CFTd on Sd, described both in FG coordinate system and in other coordinate system. From the computation, we explicitly nd that we obtain the nite subtraction di erent in the two coordinate systems. We shall

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gd1

hence adopt the FG coordinate system in this section and extract the free energy in the minimal subtraction scheme.

5.1 Free energy in Fe erman-Graham coordinates

Start with the undeformed CFTd on Sd of radius r. Its gravity dual is described by the Euclidean AdSd+1 spaceds2 = 2

hd2 + sinh2 ds2(Sd)

i

It turns out that the FG coordinate has a complication for the evaluation of the free energy.

We compute the free energy of the undeformed CFTd on Sd in the FG scheme, viz. by regularizing infrared divergences in the FG coordinate system. In general, one can always put the FG coordinate system in the form

ds2 := gabdxadxb = 2

The on-shell action of the gravity dual diverges in the infrared. We regularize it in the FG scheme by cutting o the FG geometry at u = with u . Taking into account the

Gibbons-Hawking boundary action at the cut-o , the full regularized action reads

Ireg =

1 8G

ddx

Z

(5.1)

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du2u2 +1u2 hij(x; u2) dxidxj

( 0 u ): (5.2)

1 16G

ZM[epsilon1] dd+1x pg

R gab@a@b +d(d 1) 2

1 8G

Z@M[epsilon1] ddx p K (5.3)

where ij is the induced metric at the surface u = . K is the trace of the extrinsic curvature Kab. Recall that, with the surface normal unit vector n speci ed by nu = [lscript]u; ni = 0, the

extrinsic curvature Kab is de ned by

Kab = 12Ln(gab nanb) ; (5.4)

where Ln is the Lie derivative along n. Using the Einstein eld equation (4.2), the regu

larized on-shell action becomes

Ireg = Ibulk + Isuface ; (5.5)

where

Ibulk = d8G 2

ZM[epsilon1] dd+1x pg

Isuface =

Z@M[epsilon1] ddx p K : (5.6)

In the above FG coordinates system, we nd that

Ibulk = dd1

8G

Z[epsilon1]du ud+1 ph

ddx

Isuface =

d1

8G d

1 1 du@u

ph

u=[epsilon1]

: (5.7)

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The regularized action Ireg in general has an expansion3

Ireg = d1 16G

Z

ddx

a(0) d +a(2) d2 + [notdef] [notdef] [notdef] 2 log( )a(d) + O( 0); (5.8)

where the logarithmic contribution exists only when d is even. In the holographic renormalization, one chooses the counter-term as

Ict =

d1

16G

qh(0)

Z

ddx

qh(0)

a(0) d +a(2) d2 + [notdef] [notdef] [notdef] 2 log( )a(d)

(5.9)

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such that in the limit ! 0 of the renormalized action Iren = Ibulk + Isurface + Ict all the singular divergences are subtracted, while leaving the nite contribution intact.

We now apply the above general consideration to the metric (5.1). The FG coordinates are identi ed to be

u = 2re (5.10)

and

hij dxidxj =

1 u2 4r2

2r2 ds2(Sd) : (5.11)

The regularized action takes the form

Ireg = Vol(Sd)

16G d (=2)d1 (Ad + Bd) (5.12)

where Ad and Bd are contributions from Ibulk and Isurface, respectively. For the above metric, they take the forms

Ad =

Z

1

dz

1zd+1 (1 z2)d

Bd =

(1 z2)d z= ; (5.13)

where the parameter is related to the cut-o in the FG coordinate u by

=

2r : (5.14)

It is illuminating to work out explicitly for lower dimensions.

For d = 2, one nds

A2 = 12

1 2 2

+ 2 log

1 d

1 1 dz@z

B2 =

1 2 2

: (5.15)

3With Janus deformation below, the singular terms of remaining powers, b(1)[epsilon1]d+1 +b(3)[epsilon1]d+3 +[notdef] [notdef] [notdef] may

appear and those should be subtracted in addition by counter-terms.

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The renormalized action reads4

Iren =

c3 log(2r) : (5.16)

Here, we identi ed the central charge of the CFTd with the Brown-Henneaux [35] or Henneaux-Rey [36] central charge c = 3[lscript]

2G . This identi cation also agrees with the central charge derived by the AdS/CFT correspondence from the Weyl anomaly [37]. We can extract the Weyl anomaly integrated over the boundary sphere from the coe cient of the log term in the regularized action above.5 Adding the renormalization point scale appropriately, we nally obtain

Iren =

c3 log(r ): (5.17)

Hence, the renormalized partition function is obtained as

Zren := exp(Iren) = (r )

c3 : (5.18)

For d = 3, one nds

A3 = 16

3 +

1 3 3

3

3 +

3 3

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1

B3 =

1 3 +

+ 3 : (5.19)

Therefore, the renormalized action reads [38]

Iren = 2

2G : (5.20)

The renormalized partition function

Zren = exp

2 2G

(5.21)

is independent of scale and hence ambiguity-free.

For d = 4, one nds

A4 = 1

4 4

2 2 + 2 2

4

4 6 log

2 2 2 2 + 4 : (5.22)

4For d even, the renormalized action includes regularization-dependent, non-universal contribution. To resolve any ambiguity in the correspondence, one has to specify how the regularization is done from the view point of the both sides. Here, our choice is in such a way that the regularization is independent of the couplings. See [28, 29, 33, 34] for discussions in this context.

5To compare our result with [37], one should note the di erent conventions. First we use a FG coordinate u that has a double pole, while they use a FG coordinate that has a simple pole at the boundary. This accounts for a factor of 2 in the de nition of the Weyl anomaly. Second, they use a convention where Riemann tensor has opposite sign compared to us. By noticing this, we nd that our result can be reproduced from their more general result by specializing to a round two-sphere boundary with curvature scalar R = 2/r2 in our convention.

B4 =

1 4 +

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The renormalized action reads

Iren = 32G log(2r) : (5.23)

The result ts perfectly with the Weyl a-anomaly of four-dimensional N = 4 SU(N) gauge

theories [39] and N = 2 (SU(N)) k quiver gauge theories [40] on S4

a = 3

2G = aoN2 ; (5.24) with ao = 1 and ao = k, respectively. Reinstating the renormalization point scale , the renormalized partition function reads

Zren = ( r)aoN2: (5.25)

5.2 Free energy in other coordinates

We now compute the renormalized free energy of the undeformed CFTd in other scheme, viz. in other coordinate system. The metric in (5.1) can be written as

ds2 = 2

dy2 + cosh2 y cos2

d 2 + sin2 ds(Sd1)2 : (5.26)

We shall cut o at the infrared along the hypersurface

r cos

cosh y = 1 (5.27)

This choice of the cuto turns out to agree with the FG cuto described in the subsection 5.1 provided the cuto 1 here is identi ed with an appropriate function of the FG cuto . To see this explicitly, we note that the coordinates in (5.1) are related to the ones in (5.26) by

cos cosh y =

1cosh and sin = tanh sin : (5.28)

Thus, it is clear that the cuto hypersurface (5.27) describes the same hypersurface as constant in the coordinate system (5.1). The precise relation between 1 and will be relegated to the next subsection.

For the explicit computation, let us focus on the three-dimensions, d = 3. With the cuto (5.27), the bulk action becomes

Ibulk = 14G 2

ZM[epsilon1] d3x pg =

GZy00 dy cosh2 y Z (y)0d [notdef] sin [notdef]cos2 ; (5.29)

where cosh y0 = r

[epsilon1]1 and cos (y) = [epsilon1]1r cosh y. The result is

Ibulk = 2G

2

4

r2

21

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r1 21r2 + log

0

@

1

A

3

5

[epsilon1]1 r

: (5.30)

1 +

q1 [epsilon1] 21 r2

Suppose we identify [epsilon1]1 r

1 +

q1 [epsilon1]21r2= : (5.31)

{ 13 {

Then, one nds that the bulk action

Ibulk = 2G

14 1 2 2 + log

(5.32)

agrees with the bulk action in the FG coordinate system. In the present coordinate system, one also nds that Isurface is given by

Isurface =

G

r2

21

r1 21 r2

!

=

4G

1 2 2

: (5.33)

This again agrees with the surface action contribution computed in the FG coordinate system. We relegate details of the computation to appendix B.

We draw the conclusion that, while di erent coordinate system gives in general di erent subtraction schemes, appropriate relation between cuto s can be speci ed to prescribe identical subtraction scheme and hence the same renormalized free energy. This prompts us to understand precise relation between infrared cuto s in the holographic renormalization, to which we now turn in the next subsection.

5.3 Relations between cuto s

In the holographic renormalization, the FG scheme is considered the most convenient as it subtracts power divergences only. The lesson of the last subsection was that the FG scheme can be made not only in FG coordinate system but also in any other coordinate systems provided each respective cuto is correspondingly related to each other. Below, we nd explicit relations between cuto s in di erent coordinate system that all lead to the minimal subtraction.

We rst de ne the cuto of FG minimal subtraction scheme by the following hyper-surface in the global AdS coordinate system (5.1) and the FG coordinate system (5.2)

2re1 =

u0 = : (5.34)

We also de ne the cuto 1 in the other coordinate system (5.26) rcosh 1

= 1 : (5.35)

As explained in the last subsection, this cuto is not independent but leads to the same cuto as the FG scheme. As such, we used the cuto position 1 the same value for either

choices of the cuto .

To relate the two cuto s, we nd it convenient to introduce dimensionless cuto parameters by (as was done for the rst in (5.14))

=

2r 1 = 1r : (5.36)

{ 14 {

JHEP12(2016)025

From (5.34) and (5.35), one nds the relation between the two cuto parameters as

1

+ =

2 1 : (5.37)

This can be inverted. We nd the desired relation for the FG minimal subtraction scheme as

= 1

1 +

p1 21: (5.38)

6 Free energy of Janus CFT2

Having understood schemes for holographic renormalization, we now extract the renormalized free energy of the ICFTd. In this section, via the AdS/CFT correspondence, we shall rst compute the free energy from the gravity dual. We shall focus on the three-dimensional gravity dual, the Janus geometry (4.3).

The subtraction scheme and renormalization thereof must preserve all symmetries the system retains (apart from the Weyl anomaly for d even and nontrivial curvature background). The infrared cuto needs to be chosen accordingly. For the gravity dual of undeformed CFTd, we saw in the previous section that the FG scheme works perfectly, since it simply corresponds to a foliation of the Euclidean AdSd+1 space by Sd hypersurfaces, for which the SO(d + 1) isometries are manifest. For the Janus interface, however, these

SO(d + 1) isometries are broken to SO(d). Accordingly, the cuto hypersurfaces must be chosen such that the SO(d + 1)/SO(d) coset is nonlinearly realized. Indeed, for the Janus geometry dual to the ICFTd on R1,d1, the requisite FG coordinate system was constructed in [2]. There, it was pointed out that the FG coordinate system does not cover the entire bulk region: the wedge-shaped bulk region emanating from the boundary location of the interface is not covered. This implies that the FG coordinate system constructed in [2] is not globally well-de ned and it has to be further analytically extended.

In this work, we do not attempt to construct a FG coordinate system and its analytic extension thereof. Instead, we introduce a coordinate v by

v = r cos

pf(y) (6.1) and simply declare our cuto for the minimal subtraction scheme by the hypersurface v = 1. We are motivated to adopt this scheme since this choice simply replaces the cosh y factor in the undeformed geometry in (5.26) by the square-root of the scale function f(y) in the Janus geometry (4.3). We can further justify this choice by the following observation. At a short distance away from the interface, the corresponding bulk geometry takes near the cuto hypersurface the same form as the undeformed one once we ignore the higher-order terms that do not contribute to the renormalized action. Later, we will show that this observation holds for arbitrary dimensions.

Thus, for the computations below, we shall adopt the coordinate system (v; ; )+ (v; ; ) by eliminating the coordinate y using the above relation (6.1). This coordinate

system consists of two branches, [notdef], coming from the region with positive/negative y,

respectively.

{ 15 {

JHEP12(2016)025

We nd that

Ibulk = I0bulk + Ibulk ; (6.2)

where

I0bulk =

2G

141 2 + log + O( )

Ibulk( ) =

2G

1

"1

p

1 2 2

1

2 log

p1 2 2+ O( )

#

: (6.3)

We introduce the function (z) by

(z) = 1 z

pz

Z

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1

dx

0 1 + x2 +

q1 + x4 + 2zx2

= p1 + z

1 z1 + z ; (6.4)

where K(k2) and E(k2) are, respectively, the rst kind and the second kind of the complete elliptic integral de ned by

K(k2) =

Z

p2

K

1 z1 + z

E

1

1 p1 x2p1 k2x2

E(k2) =

Z

1 p1 k2x2 p1 x2

: (6.5)

For small , the function (

p1 2 2) is expanded as

(

p1 2 2) =8 2 +15128 4 [phone] + O( 8) : (6.6)

For the surface action, we also obtain

Isurface = I0surface + Isurface ; (6.7)

where

I0surface =

2G

121 2 + O( )

Isurface( ) =

2G

2

p

1 2 2

+ O( )

: (6.8)

As a consistency check, we nd in the limit approaches zero that both I0bulk and I0surface agree with those without the Janus deformation. One also note that Ibulk(0) =

Isurface(0) = 0, as required. The details of the computation are again relegated to appen

dices C and D.

Thus, the renormalized free energy is found to be

Iren = F = F(0) + FI ; (6.9)

{ 16 {

with

F(0) =

2G log( r)

FI =

4G log

p1 2 2 : (6.10) where F(0) is the renormalized free energy of undeformed CFT2 and FI the interface free energy. Note that the latter is independent of renormalization scheme as discussed in section 2. We also recall that, for 1, the massless minimal scalar eld is expanded as

(y) 0 = tanh y +

1

2 3

tanh y + 13 tanh3 y

+ [notdef] [notdef] [notdef] : (6.11)

Interestingly, the total free energy is monotonically lowered by introducing the Janus interface of deformation . The renormalized partition function is given by Z = Z(0) [notdef] ZI, where

Z(0) = ( r)

c 3

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ZI =

"

1

p1 2 2

#c6: (6.12)

We note that FI and hence ZI are independent of r. This re ects that the interface is an odd-dimensional conformal eld theory, preserving the SO(2,1) conformal invariance. We can also relate, by a suitable conformal transformation, the free energy FI to the

interface entropy SI, as was done in [18, 41{43]. In turn, from the exponential of interface entropy, we also learn about the degeneracy of ground-states newly created by the presence of the interface.

7 Free energy of Janus CFT3

For arbitrary dimension d + 1 of the gravity dual, we use the metric in the form

ds2 = 2 q2

dq2P (q) +

d 2 + sin2 ds2(Sd) : (7.1)

To get the d-dimensional boundary as Sd, we choose the infrared cuto hypersurface as

q( ) = 1

cos ; (7.2) where 1 is a dimensionless cuto parameter introduced previously. Expressions for the bulk action and the surface action in arbitrary dimensions are complicated, so we relegate them in appendix E. Rather, we specialize our computation to lowest even dimension, (d + 1) = 4. At zeroth order in , we nd that

I(0)bulk =

2

4G

1 cos2

1 31

3 1 + 2 + O( 1)

= 2

32G

1 3

9

+ 16 + O( )

(7.3)

{ 17 {

and

I(0)surface =

32

4G

1 31+ 1 1 + O( 1)

= 32 32G

1 3 +1 + O( ) : (7.4)

These are in perfect agreement with the previous computation based on the FG coordinate system.

For the corrections to rst-order in 2, we nd that

I(1)bulk = 2

32

32G

1 21

+ 23 log 1 1 + O( 1)

= 2 32 32G

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1 4 2

1

2 +

2

3 log 2 +

2

3 log + O( )

(7.5)

and

I(1)surface = 2

2

64G

13 21 4 + O( 1)

= 2

2

64G

13 4 2 +

52 + O( )

: (7.6)

Summing up all the contributions, we get

Ibulk + Isurface = 2

4G

2 31+ 2 + 2 2 64G

7 21 2 + 4 log 1 + O( 4) (7.7)

= 2

4G

14 3 34 + 2 + 2 2 64G

74 2 + 4 log 112 + 4 log 2

+ O( 4) :

Thus the renormalized free energy is given by

Iren = F = F(0) + FI ; (7.8)

with

F(0) = 2

2G

FI = 2

2

16G log( r) + O( 4) (7.9)

whose detailed identi cation is discussed in section 2. We see again that the interface contribution to the free energy shows the structure of CFT2. In particular, being described by even-dimensional conformal eld theory, the interface free energy exhibits log r dependence, indicating the Weyl anomaly of the interface.6

6See [44{47] for related studies of anomaly in the presence of boundaries.

{ 18 {

8 Stress tensor one-point function

To further probe the interface, we study another physical observable, the one-point function of the stress tensor in CFTd and its change under the Janus deformation. We will again extract this observable from the gravity dual and compare the result with the CFTd. We

shall begin with (d + 1) = 3. As explained in the last section, the construction of FG coordinate system for the Janus geometry faces a di culty in the wedge of the bulk region that emanates from the boundary interface. So we will rst determine the stress tensor one-point function at an in nitesimal distance away from the interface.

Note that, in the coordinate system in (4.8), the cuto hypersurface may be introduced by

q[notdef] cos =

cos

in nitesimal but with ~ = 1 1. This condition implies that the region of

interest is at least an in nitesimal distance away from the interface from the viewpoint of the boundary space. In this region of the cuto hypersurface, q[notdef] becomes in nitesimal: q[notdef] =

1r cos

hh(2)ij h(0)ijh(2)klh(0)kli+ ij ; (8.5)

where ij is the contribution of the minimal scalar eld to the stress tensor

ij =

8G

@iB@jB 12h(0)ijh(0)kl@kB@lB : (8.6)

Here, we denote by B( ) the boundary value of the minimal scalar eld . As we are away from the interface, gradients of B vanishes and hence ij = 0.7 Using the metric in (5.1), one nds for [negationslash]= 2 that

hTij[angbracketright] =

{ 19 {

JHEP12(2016)025

1r (8.1)

This generalizes the d = 2 case of the previous section. We then consider the region of the surface speci ed by

~

r (8.2)

where we take ~

1

~

1 : (8.3) Then, in the metric of (4.8), the function P (q[notdef]) can be replaced by 1 q2[notdef] while the terms

ignored are of su ciently higher order that they can be dropped o when evaluating the holographic stress tensor. With such replacement, the metric becomes the undeformed one. Now, in the FG coordinate system of the metric given in (5.2), the metric hij(x; u) is

expandable in general in the form

hij = h(0)ij + u2h(2)ij + [notdef] [notdef] [notdef] + udh(d)ij + ud log u2

(d)ij + [notdef] [notdef] [notdef] (8.4) where, for the undeformed case, the logarithmic term is present only when d is even. Here, h(0)ij is the metric for the boundary space which will also be denoted by hBij.

For d = 2, the one-point function of the boundary stress tensor is given by [48]

hTij[angbracketright] =

8G

c24r2 hBij ; (8.7)

7Along the interface, this expression becomes singular and one needs some other method to x it (see below).

where hBij is the boundary value of metric eld hij. This expression coincides with that of the undeformed case. We shall x possible contribution at the interface location from our expression of the free energy. Since the above expression is independent of our deformation parameter, one may compare the known result of the CFT on a sphere. One has in general

hT ii[angbracketright]CFT =

c12r2 ; (8.8) which follows from the Weyl anomaly of the CFT2. By the Lorentz invariance, this implies that

hTij[angbracketright]CFT =

which agrees with our computation above.

One can also check the stress tensor one-point function from our expression of the renormalized free energy. Varying the free energy with respect to r, one gets

I(0)ren =

Z

ce ( )

3 log r ; (8.12) where, to leading order in , we read o from (7.9)

ce ( ) = 32

16G 2 + O( 4) : (8.13)

From these expressions together with the unbroken SO(3) symmetry of our Janus solution, one obtains that [angbracketleft] Tij[angbracketright] is given by

h T [angbracketright] = [angbracketleft] T [angbracketright] = 0

h T [angbracketright] =

ce

24r2 hB

c24 R(hB) =

c24r2 hBij ; (8.9)

phB hijB[angbracketleft]Tij[angbracketright]CFT = c3 log r : (8.10)

This is consistent with the trace of the stress tensor in (8.8). On the other hand, as Iren

is independent of r, its variation with respect to r implies that the interface contribution

h T ii[angbracketright] vanishes for d = 2, which further implies that

h Tij[angbracketright] = 0 : (8.11) For d = 3, it is straightforward to show that the stress tensor one-point function vanishes away from the interface location, which agrees with the stress tensor of the unde-formed case. In fact, one can show that the undeformed holographic stress tensor from (5.1) vanishes for any odd-dimensional sphere. On the other hand, for d = 3, the interface contribution is non-vanishing,

FI =

JHEP12(2016)025

1

2

d2q

2 ; (8.14)

where ; denote the directions along the interface. This again demonstrates that the interface contribution to the stress tensor is consistent with the expected structure for CFT on a sphere of one lower dimensions.

For completeness, we also record here the expression for the one-point function of the exactly marginal scalar operator:

[angbracketleft]O[angbracketright]ICFT =

d1

8G

2

rd [notdef] cos [notdef]d

; (8.15)

which we obtained from the gravity dual by the rules of AdS/CFT correspondence.

{ 20 {

9 Conformal perturbation theory

So far, we computed one-point functions of ICFTd1 from the gravity dual. In this section,

we compute them directly from the dual CFTd in the regime the deformation is weak. In this regime, we can use the conformal perturbation theory. Here again, we focus on d = 2, but the method is applicable straightforwardly in arbitrary dimensions.

Consider a CFTd perturbed by a local operator O. The Lagrangian density of the

theory is given by

L( ) = L0( ) + B( ) O( ) ; (9.1) where the deformation coupling parameter (3.3) is expanded to the leading order in B. We

shall compute the one-point functions perturbatively in terms of the correlation functions of undeformed CFTd. We assume that [angbracketleft]O[angbracketright]CFT = 0. Then, the leading-order correction

to the free energy is identi able as second order e ect of the deformation

F =

1

2!

JHEP12(2016)025

ZZ

d2

phB( ) B( ) d2 [prime]

qhB( [prime]) B( [prime]) [angbracketleft]O( )O( [prime])[angbracketright]CFT + [notdef] [notdef] [notdef] (9.2)

Here, hBij denotes the metric of the boundary space, and the ellipse denotes higher-order correction O(3B). Taking advantage of the conformal invariance, we can map the computation to R2. We start from S2 and describe it in terms of R2 variables q = (q1; q2) via stereographic projection

ds2B = 4r2

(1 + q2)2 dq2 : (9.3)

Under the Weyl transformation

hBij ! 2(q) hBij ; (9.4) the exactly marginal scalar operator with = 2 transforms as

O

!

1 2(q)O : (9.5)

With the choice of

(q) = 1 + q2

2r ; (9.6) we are mapping the two-sphere S2 to a two-plane R2 charted by the Cartesian coordinates q. The equator of S2 is conformally mapped to a circle of unit radius on R2

q2 = 1 : (9.7)

On R2, the two-point function of the exactly marginal scalar operator O is given by

[angbracketleft]O(q)O(q[prime])[angbracketright]CFT = N

[(q q[prime])2 + 2 2]2

(9.8)

where we introduce a ultraviolet regulator 2 2. As the scalar eld dual to the operator O

is normalized in the gravity side as in (4.1), the normalization factor in (9.8) is xed as

N =

42G (9.9)

{ 21 {

S

N

S

a)

b)

Figure 5. a) The Janus system on Rd with an interface of spherical shape is depicted. b) The Janus system on R [notdef] Sd1 with an interface at = 0 is depicted where is a coordinate along

R direction. On the N/S region, we turn on an exactly marginal scalar operator with a coupling +I= I respectively.

by the standard dictionary of the AdS/CFT correspondence in [49]. We then further perform a coordinate transformation the two-plane from R2 to a cylinder R[notdef]S1 de ned by

q1 + iq2 = exp(1 + i2) : (9.10)

Here, 1 < 1 < 1 and 0 2 2. The equator in S2, where the Janus interface is

to be placed, is mapped in this coordinates to 1 = 0. Thus, with B = I

2

, the

JHEP12(2016)025

interface contribution to the free energy becomes

F =

2I 8

Z

L 2 d1

Z

2

0 d2Z

L

2

L2

d[prime]1

Z

2

0 d[prime]2 N

(1) ([prime]1)

[cosh(1[prime]1)cos(2[prime]2)+ 2]2

+O(3I) ; (9.11)

where we also introduce the infrared regulator L by putting the system to a box of size L. This infrared regulator corresponds in the holographic gravity dual to an ultraviolet cuto around the north and the south poles of the original S2. This integral can be carried out [11, 18], and the result reads

Freg =

2G

L 4 2

L

2 + O( )

1 p2 +

1

2 coth

2I + O(3I) : (9.12)

Thus, after renormalization, we get

FI =

4G 2 + O( 3) (9.13)

where we used the fact I = + O( 2) for d = 2 which can be identi ed from (3.4) and (4.7). We see that this agrees with our result (6.10) in the gravity dual side, con rming the correspondence.

9.1 Related Janus solutions

Motivated by the above maps for the eld theory side in this section, we would like to obtain coordinates for the bulk Janus solution corresponding to the latter two ICFTs discussed above. First we consider the ICFT on Rd with a spherical-shaped interface. The

{ 22 {

corresponding ICFT is depicted in gure 5a. To obtain this conformal boundary, we start by expressing the metric ds2Md in (4.8) as

ds2Md =

4(1 2)2

d2 + 2ds2(Sd1) ; (9.14)

which is obtained by the coordinate transformation

sin = 2

+ 1 : (9.15)

Here, 0 1 for the q+ patch and 1 < 1 for the q patch. At q[notdef] = q , one needs

an inversion coordinate transformation ! 1 to join the two patches in a smooth manner.

To get the metric of the conformal boundary as a plane, we choose a de ning function as

v(q[notdef]) = q[notdef] [notdef]

1 2[notdef]

2 (9.16)

which has a simple zero at the boundary q[notdef] = 0. We then multiply the bulk metric by

v(q[notdef])2 and take the limit q[notdef] = 0. We then get the boundary metric as that of Rd,

ds2B = d2 + 2 ds2(Sd1) : (9.17)

An alternative way to derive the bulk metric is to let v be the new bulk coordinate replacing q[notdef]. We then re-express the bulk metric in the new coordinate v and nd a double pole in

the metric at v = 0. We then multiply the bulk metric by the de ning function v2 and take the limit v ! 0. Again we will arrive at the same boundary metric. In these new

boundary coordinates, the interface is located at the unit sphere Sd1 Rd at = 1. Of

course, the radius of Sd1 can take any positive value by a rescaling of v in the above.

One can further make the coordinate transformation = e and get

ds2(Md) =

1 sinh2

JHEP12(2016)025

d2 + ds2(Sd1) : (9.18)

The coordinate is ranged over (1; 0] for the q+ patch while [0; 1) for the q patch.

If the boundary is identi ed with the coordinate v = q[notdef] sinh with v = 0, the boundary

space becomes R [notdef] Sd1 with the metric ds2B = 2

d2 + ds2(Sd1) : (9.19)

The interface is located at = 0, as depicted in gure 5b.

10 Interface degrees of freedom

In a CFT on a torus, extra ground-state degeneracy is produced by the presence of a boundary (or a defect), whose number is denoted by g. Then, ln g is identi ed with the boundary entropy counting degrees of freedom localized on the boundary. The g-theorem states that ddl g(l) 0 (10.1)

{ 23 {

with some length scale l, which is rst suggested in ref. [26].8 In the situation the RG- ow is triggered by operators localized at the boundary, the g-theorem was proven in [50]. However, as was shown in [51] (see also explicit computation [52, 53] in the free eld theory contexts), the g-function may either increase or decrease when the RG ow is triggered by operators of the bulk CFT. Here we examine interface counterparts of ground-state degenerarcy and g-theorem. On the two-sphere S2, the interface is localized along a circle, which may be interpreted as a circle of Euclidean time. Then F of the interface

contribution can be interpreted as an interface entropy SI. Consider two interfaces [19] that are separated by a small distance l around the equator. Let their interface coe cients are given by I and [prime]I which may have either signatures. In such circumstance, it is clear that

SI(I; [prime]I; l=r) ! SI(I + [prime]I) (10.2) as l=r ! 0. The entropy may either increase if sign I = sign [prime]I or decrease if sign I =

sign [prime]I. Therefore, as for the defect, we do not expect monotonicity property for the interface. Of course, this is expected since the perturbation induced by the interface is controlled by the bulk CFT operators. The same argument holds for higher-dimensional interfaces. We do not have any monotonicity of the interfaces degrees of freedom.

Acknowledgments

We thank John Cardy, Yu Nakayama and Adam Schwimmer for useful discussions. We also acknowledge the \Liouville, Integrability and Branes (11)" Focus Program at the Asia-Paci c Center for Theoretical Physics for excellent collaboration environment. SJR acknowledges the NORDITA workshop \Holography and Dualities 2016: new Advances in String and Gauge Theory", where this work was completed. DB was supported in part by the National Research Foundation grant 2014R1A1A2053737. SJR was supported in part by the National Research Foundation of Korea grants 2005-0093843, 2010-220-C00003 and 2012K2A1A9055280.

A Coordinates on AdSd+1

The Euclidean AdSd+1 is described in R1,d+1 of Cartesian coordinates (x0; x) by the hypersurface

XMXN MN = (X0)2 + X2 = 2: (A.1) We want to introduce coordinates of this space. Depending on how we foliate the space, there are three independent coordinate systems.

A.1 Global, AdS slice, Poincare patches

We may slice the Euclidean AdSd+1 by foliations of Sd. This leads to global patch coordinates, given by

X0 = cosh 2 [1; 1)

X = ^

e sinh 2 Rd+1 :

8See also the discussion on boundary/interface F-theorem in [27].

{ 24 {

JHEP12(2016)025

(A.2)

Here, 2 [0; 1) and ^

e is a Euclidean vector on Sd of unit radius. The Sd may be further parametrized by ^

e = (e sin ; cos ), where e is a vector on Sd1 of unit radius. We can alternatively introduce a compact coordinate 2 [0; =2] such that sinh = tan . In

terms of either variable, the Euclidean AdSd+1 metric is given by

ds2 = 2

d2 + sinh2 ds2(Sd)

= 2

cos2

d 2 + sin2 ds2(Sd) : (A.3)

We may also slice the Euclidean AdSd+1 by foliations of AdSd. The leads to AdS slicing coordinates, given by

X[notdef] = ^

n cosh y 2 [1; 1) [notdef] Rd

Xd+1 = sinh y 2 R

: (A.4)

Here, y 2 [1; +1] and ^

n is a Lorentzian vector on AdSd of unit radius. The AdSd may be further parametrized by ^

n = (sec ; n tan ), where n is a vector on AdSd1 of unit radius.

The metric is given in terms of these coordinates by

ds2 = 2 dy2 + cosh2 y ds2(AdSd)

: (A.5)

The above two coordinate systems are related by the di eomorphism

cosh = cosh y sec sinh sin = cosh y tan sinh cos = sinh y (A.6)

Thus, we see that 2 [0; 2 ] and 2 [2 ; ] intervals in the global coordinate is mapped to

y 2 [1; 0] and y 2 [0; 1] intervals in the AdS slice coordinate, respectively.

Finally, we may also slice the Euclidean AdSd+1 by foliation of Rd. This leads to the Poincare patch coordinates given by

X0 = (=2)

z + 1 + xixi

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2 R+

X = (x=z) 2 Rd

Xd+1 = (=2)

=z

(A.7)

z + 1 + xixi

=z

2 R

The metric is

ds2 = 2z2 dz2 + dx2

: (A.8)

A.2 The conformal boundary

We want to take the conformal boundary as a sphere. Clearly, this is most naturally described in the global coordinate. We want to know how this boundary looks like in the AdS slice coordinate. To do so, we de ne the boundary with the following infrared cuto

e1 = and cos

cosh y = 1 : (A.9)

{ 25 {

By the di eomorphism (A.6), they are related as

1 1 =

1

2

+ 1

(A.10)

and also as

1

sin =

1 +

sin : (A.11)

On the other hand, using the relation

1

2( 1 + )2= 1 21; (A.12)

we see that the metric of cuto Sd in the global coordinate

ds2 = 24 ( 1 )2

d 2 + sin2 ds2(Sd1)

JHEP12(2016)025

(A.13)

becomes in the AdS slice coordinate

ds2 =

1 21 cos2

1 d 2 + 11 21sin2 ds2(Sd1): (A.14)

B Computation of Isurface

In terms of the coordinate

v = r cos

cosh y ; (B.1) the boundary surface is speci ed by the hypersurface v = 1, where 1 is a cuto . On the surface, the y coordinate ranges over [y1; y1], where cosh y1 =

r

[epsilon1]1 . We also nd that

cos 2

[epsilon1]1r ; 1

and we de ne cos 0 = [epsilon1]1r . We use ( ; ) as boundary surface coordinates. For a xed , the coordinate y has double roots, as seen from (B.1). Thus, the boundary surface can be covered by two branches of coordinates xi = ( ; )+ [ ( ; ) where +=

refers to the part of surface with positive/negative y. We also denote bulk coordinates as xa = (v; xi). Tangent vectors to the boundary are @i. The normal vector is orthogonal to the tangent vectors with respect to the bulk metric gab. This means that

giana = ni = 0 : (B.2)

The bulk metric can be expressed as

ds2 = N2dv2 + ij(dxi + Nidv)(dxj + Njdv) : (B.3)

By matching this with the metric of Euclidean AdS3, we can identify shift and lapse functions N; Ni as

N =

1

v q1 v2r2

N = vr2 v2

tan (B.4)

{ 26 {

Moreover, ij can be identi ed with the induced metric on the boundary. It has the non-vanishing components

= 21

v2

r2 cos2

r2v2 1

= 2r2v2 sin2 : (B.5)

In these coordinates, the unit normal vector na obeying gabnanb = 1 is given by

na = (N; 0; 0)

na =

1N (1; Ni) (B.6)

We can then compute the extrinsic curvature

K = 1

pg @a(pg na) (B.7)

with the result

p K = 2r2

v2

sin

q1

v2r2 cos2 : (B.8)

We then obtain the integral of the extrinsic curvature as

1 8G 2

JHEP12(2016)025

r1 21r2 ; (B.9)

where the extra factor 2 in the left-hand side comes from the fact that we have two branches of boundary coordinates.

C Computation of Ibulk for ICFT2

With the cuto surface de ned in section 6, the bulk integral takes the form

Ibulk = G

Z

y0

Z

0

0 d Z

2

0 dp K =

G

r2

21

0 dy f(y) Z

y

d sin cos2 ; (C.1)

where y0 (> 0) is de ned by the relation

pf(y0) = r 1 (C.2)

and cos y = [epsilon1]1r

0

pf(y). From this, e2y0 can be solved in terms of 1 by

e2y0 = 4

p1 2 2

r2 21 1 2

+ O( 21) : (C.3)

For the regularization, the O( 21) contribution is not needed. Carrying out the integration, one gets

Ibulk =

G

r 1

Z

y0

0 dy

y0

pf(y)

Z

0 dy f(y)

: (C.4)

{ 27 {

In this expression, the rst integral can be rearranged as

Z

y0

0 dy

y0

0 dy cosh y

pf(y) =

p

1 2 2

+ (1 2 2)

1 4

Z

1

Z

y0 dy hp

f(y) (1 2 2)

14 cosh y

i

; (C.5)

where

(

p1 2 2)

Z

1

0 dy hp

f(y) (1 2 2)

14 cosh y

i

: (C.6)

From this, it is straightforward to get

r 1

Z

y0

0 dy

JHEP12(2016)025

pf(y) = r 1 p1 2 2

+ r2

21

1

2 + O( 21) : (C.7)

Carrying out the second integral in (C.4) explicitly, we get

Ibulk =

G

"

1

2

r2 21 1 2

+ r 1 p 12 2

+ 12 log 12r 14 log1

p12 2 +O( 21)

#

(C.8)

Finally, (z) can be expressed in terms of the complete elliptic integrals and the relation (5.31) is used to rewrite the above in terms of instead of 1.

D Computation of Isurface for ICFT2

In this appendix, we work in the coordinates (v; ; )[notdef] introduced in section 6. To simplify

our presentation, we shall introduce two quantities Dd and ~Dd , respectively, de ned by

Dd(v; )

s1

vr cos

2+ 2d(d 1) vr cos

2d

~Dd(v; ) s

1

v2r2 +

2 d(d 1)

vr cos

2d: (D.1)

The description here is in parallel with the treatment of appendix B, which is for the undeformed case. In the metric given in (B.3), the lapse and shift can be identi ed as

N =

v

1 ~D22

N = v r2

tan ~D22

(D.2)

and the nonvanishing components of ij are given by

= 2r2 v2

~D22 D22

= 2r2v2 sin2 (D.3)

{ 28 {

Then, the unit normal vector is given by the form in (B.6). Thus, the extrinsic curvature contribution is identi ed as

p K = 2r2

v2 sin

"

1D2 +

2 2

vr cos

4 D2 ~D22

#

: (D.4)

The integral over the boundary of the extrinsic curvature is given by

1 8G2

Z

0

0 d Z

2

0 dp K ; (D.5)

where again we have an extra factor 2 and cos 0 = [epsilon1]1

rq . Thus, by carrying out integral

explicitly, one is led to

Isurface =

G

r2 21 12 +2r 1 p1 2 2

+ O( 21)

: (D.6)

Again, this result can be written in terms of the cuto .

E General expressions for ICFTd to rst order in 2

E.1 The bulk integral

For a d-dimensional ICFT on Sd, we have the following bulk metric

ds2 = q2

dq2P (q) +

JHEP12(2016)025

d 2 + sin2 ds2Sd1 cos2

!

: (E.1)

Therefore, the bulk term is

Ibulk = 2dd1

8G

ZSd1 d d1

0

0 d

sind1

cosd

q

q( ) dq

1 qd+1

pP (q) (E.2)

Here, the metric function

P (q) = 1 q2 + Eq2d ; (E.3) the expansion parameter

E =

2 d(d 1)

(E.4)

related to the Janus deformation. The integration bounds are speci ed by

P (q ) = 0

q =

1 cos 0

q( ) = 1

cos (E.5)

We rst evaluate the last integral,

(E; q( )) :=

Z

q

q( ) dq

1 qd+1

pP (q) (E.6)

{ 29 {

to linear order in E. We eliminate the q2 term from P (q) usingq2 = 1 + Eq2d (E.7)

and change the integration variable as q = q t. We get

P (q t) = (1 t2)

1 Eq2d t2

t2(d1) 1 t2 1

!

: (E.8)

We then Taylor expand the integrand to rst order in E,

(E; q( )) =

1 qd

Z

1 dt 1

td+1p1 t2

1 + 12Eq2d t2

t2(d1) 1 t2 1

+ O(E2)!

JHEP12(2016)025

: (E.9)

Finally, we expand the prefactor 1=qd and the lower boundary of integration in powers of E, where the following expansion

q = 1 +

1

2E + O(E2) (E.10)

for the smallest root is used. We nd [31]

(E; q( )) = (0; q( )) + 2

F1 12; 2d2; 2; 1 q( )2

p1 q( )2 E+ O(E2) (E.11)

where

(0; q( )) =

Z

1

q( ) dt

1 td+1p1 t2

: (E.12)

Next, we integrate over to obtain the bulk term

Ibulk = 2dd1

8G Vd1

Z

0

0 d

sind1

cosd (E; q( )) : (E.13)

Here, we may expand the bound as 0(E) = 0(0)+E [prime]0(0)+O(E2) and pick up a boundary

term E [prime]0(0) (E; q ) but to linear order in E this is zero because (0; 1) = 0. We get

Ibulk = dd1

4G Vd1

Z

0(0)

0 d

sind1

cosd (E; q( )) ; (E.14)

where cos 0(0) = 1. Changing variable of integration to x = cos , we nally get

Ibulk = dd1

4G Vd1

Z

1 (1 x2)

d2

2

xd

E; 1x : (E.15)

E.2 The surface term

Using the same technique as we used for the case d = 2, we nd

p K = d1d

r v

dsind1 1

Dd + E

vr cos

2d Dd ~D2d

!

: (E.16)

{ 30 {

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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