Free energy and defect C-theorem in free scalar theory
aa r X i v : . [ h e p - t h ] J a n Prepared for submission to JHEP
YITP-21-04
Free energy and defect C -theorem in free scalar theory Tatsuma Nishioka a and Yoshiki Sato b a Yukawa Institute for Theoretical Physics, Kyoto University,Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan b Physics Division, National Center for Theoretical Sciences, National Tsing-Hua University,Hsinchu 30013, Taiwan
Abstract:
We describe conformal defects of p dimensions in a free scalar theory on a d -dimensional flat space as boundary conditions on the conformally flat space H p +1 × S d − p − .We classify two types of boundary conditions, Dirichlet type and Neumann type, on theboundary of the subspace H p +1 which correspond to the types of conformal defects in the freescalar theory. We find Dirichlet boundary conditions always exist while Neumann boundaryconditions are allowed only for defects of lower codimensions. Our results match with arecent classification of the non-monodromy defects, showing Neumann boundary conditionsare associated with non-trivial defects. We check this observation by calculating the differenceof the free energies on H p +1 × S d − p − between Dirichlet and Neumann boundary conditions.We also examine the defect RG flows from Neumann to Dirichlet boundary conditions andprovide more support for a conjectured C -theorem in defect CFTs. ontents S d d d a and F HS d d d H d d d HS d H p +1 × S p p H p +1 × S q − p p H p +1 × S q − C -theorem 43 – i – Derivation of (4.21) 51
A conventional view of quantum field theories (QFTs) relies on particle picture of quantumfields as a fundamental description of the theories, but the importance of non-local objectshas been increasingly recognized in recent studies to discriminate theories from those havingthe same local descriptions but different global structures [1, 2]. Extended observables suchas Wilson-’t Hooft loops rarely have concrete realizations in terms of fundamental fields inLagrangian and are typically defined as boundary conditions, hence called defects in general.Various types of defects are pervasive in physics: loop and surface operators in condensedmatter and high energy physics, cosmic string and domain wall in cosmology, D-branes instring theory, to name just a few. (Refer to [3] for recent progress in diverse fields.)Theoretical aspects of defects are less scrutinized as opposed to local operators due tothe lack of their fundamental descriptions in QFT as well as their intricate dependence on theshapes. On the other hand, focusing on a class of defects with a large amount of symmetrywe have better understanding of their universal characters. In particular the kinematics ofplanar and spherical defects (conformal defects) in conformal field theories (CFTs) are highlyconstrained by a large subgroup of conformal group [4–13]. To be concrete let D ( p ) be a p -dimensional conformal defect in Euclidean CFT on R d . The conformal group SO(1 , d + 1) isbroken by the presence of the defect to the subgroup SO(1 , p + 1) × SO( q ) where SO(1 , p + 1)is the conformal group on the p -dimensional worldvolume of D ( p ) while SO( q ) is the rotationgroup around D ( p ) with q = d − p . As a special case CFTs on a manifold with boundary(BCFTs) are also regarded as defect CFTs (DCFTs) with q = 1.It should be noted that there are two kinds of local operators in DCFT: local operators O in the bulk CFT and defect local operators ˆ O with support only on the worldvolume of D ( p ) . This is easily seen in a simple example of DCFT consisting of a bulk CFT d and alower-dimensional CFT p without interaction in between. The relation between O and ˆ O isdetermined by the bulk-to-defect operator product expansion, which thereby defines possibletypes of defects in a given bulk CFT.CFTs occupy distinguished positions as fixed points of renormalization group (RG) flowswhere scale invariance is believed to enhance to conformal invariance [14]. One can perturb aCFT by a relevant operator O and let it flow to another CFT at the IR fixed point. RG flowscan be geometrized by adding a “height” function which measures the degrees of freedom ofthe theories on a space of QFTs. Then theories are expected to flow only from the UV to theIR. A C -theorem elevates this belief to the statement for the existence of such a monotonicfunction known as a C -function. Zamolodchikov proved the c -theorem for the first time in two– 1 –imensions [15], which was generalized to the a -theorem as a conjecture in four dimensions[16] and proved more recently [17]. These theorems state the type A central charges for theconformal anomalies play the role of a C -function in even dimensions. On the other hand,the F -theorem asserts that the sphere free energy be a C -function in odd dimensions [18–21]. Despite the difference of the structures in even and odd dimensions, the dimensionaldependence of the C -functions is beautifully unified by the generalized F -theorem [22] thatproposes an interpolating function between the type A anomaly and sphere free energy˜ F ≡ sin (cid:18) π d (cid:19) log Z [ S d ] , (1.1)decreases along any RG flow: ˜ F UV ≥ ˜ F IR . (1.2)An information theoretic proof of the theorem was given by [23–25] for d ≤
4, but at themoment of writing it remains open whether the generalized F -theorem holds in higher di-mensions.Now for DCFTs one can trigger an RG flow by perturbing the theory using a defectlocal operator ˆ O in addition to O , thus DCFTs allow for a wider class of deformation thanCFTs without defect. If one is concerned with the dynamics of defect operators it will beconvenient to focus on defect RG flows triggered by defect localized operators while keepinga bulk CFT fixed. Given the success of the C -theorems in CFTs it is tempting to ask ifthere exists a monotonic function which decreases along any defect RG flow in DCFTs. Afew concrete proposals were put forwarded in the case of BCFTs and named the g -theoremswhich employ either the boundary entropy [26] or the hemisphere free energy [27, 28] as a C -function. In BCFT , the g -theorem was given two proofs: one by a field theoretic method[29] and the other by an information-theoretic method [30]. In BCFT two proofs are givenby [31] and [32] with different means, showing the boundary central charge b for conformalanomaly becomes a C -function. In higher-dimensional BCFTs, there are no general proofsof the g -theorem, but some holographic calculations based on a probe brane model [33], theAdS/BCFT construction [34–36] and supergravity solutions [37], support the validity of theproposal. The situation is less clear for general DCFTs, but when p = 2 (and for any d ) the b -theorem [31] states that the central charge of surface operators is shown to be a C -function.(See also [40–44] for further investigations.)In the previous work [9], we proposed a C -theorem in DCFT stating that the defect freeenergy on the sphere S d defined by an increment of the sphere free energy due to the defectlog hD ( p ) i ≡ log Z DCFT [ S d ] − log Z CFT [ S d ] , (1.3) In contrast, the boundary free energy does not necessarily decrease under a bulk RG flow [38, 39]. – 2 –s a C -function. More precisely, we introduced an interpolating function of the defect freeenergy in an analogous way to (1.1) by˜ D = sin (cid:16) π p (cid:17) log |hD ( p ) i| , (1.4)and conjectured ˜ D decreases under any defect RG flow:˜ D UV ≥ ˜ D IR . (1.5)When defects do not interact with the bulk theory our proposal simply reduces to the general-ized F -theorem (1.2) for the p -dimensional defect theories while it incorporates the g -theoremfor BCFTs when p = d −
1. The relation (1.5) passes several checks for defects in field theories[9, 41, 45] and holographic models [42, 46, 47].The main purpose of this paper is to examine the conjectured relation (1.5) in moredetail in the simplest theory: a free conformally coupled scalar field. To this end we willdescribe conformal defects in the theory on flat space as boundary conditions on a conformallyequivalent space, where a defect RG flow is caused by changing the boundary condition by thedouble trace deformation as is familiar in the AdS/CFT setup [48–54]. The idea of mappingBCFTs on flat space to the hyperbolic space H d and studying the boundary RG flow hasappeared in the recent works [55, 56]. Similarly, line operators in four dimensions (i.e., p = 1and q = 3) can also be characterized as boundary conditions on H × S [57], and monopoleoperators which are codimension three defects can be characterized as boundary conditions on H d − × S [58]. We will extend these ideas to more general DCFTs by employing a conformalmap from flat space to H p +1 × S q − where defects are located at the boundary of H p +1 (seefigure 1).We will introduce Neumann type boundary conditions on H p +1 × S q − and considerthe defect RG flow from Neumann to Dirichlet type. For BCFT on the hemisphere HS d ,Neumann and Dirichlet boundary conditions can be realized by imposing parity conditionson the eigenmodes. Then, the free energies for each boundary condition can be obtainedfrom those on the sphere by truncation (see e.g. [28, 31] for the detail). On the other hand,the Neumann/Dirichlet boundary condition on H d , which is conformally equivalent to HS d though, cannot be described by a parity condition as the spectrum of the eigenfunctions iscontinuous. The boundary conditions on H d are rather dictated by the asymptotic behavior ofthe field near the boundary as in the AdS/CFT. This approach has an advantage that we canview the defect theory as a “holographic” dual of the bulk field on the Euclidean AdS space H d , which allows us to classify types of conformal defects through the boundary conditionson H p +1 × S q − in the conformally coupled free scalar theory. We will show it is alwayspossible to impose Dirichlet boundary conditions for any p and q while Neumann boundaryconditions are allowed only for special cases if we require the defect theory to be unitary.Reassuringly our results conform with the classification of the non-monodromy defects for afree massless scalar theory carried out in [59] by other means. It leads us to speculate that– 3 –irichlet boundary condition corresponds to trivial (or no) defects while Neumann boundarycondition to non-trivial defects.The free energy of a conformally coupled scalar field on H p +1 × S q − has been calculatedin literature [19, 60–64] (see also [65] for a related work) and shown to have a logarithmicdivergence: F [ H p +1 × S q − ] = · · · − A [ H p +1 × S q − ] log (cid:18) Rǫ (cid:19) + · · · , (1.6)where R is a radius of the hyperbolic space and the sphere. The small parameter ǫ serves as aUV cutoff for the sphere as well as an IR cutoff for the hyperbolic space. The explicit valuesof the coefficients A [ H p +1 × S q − ] are obtained for some p and q either using the heat kernelmethod or by summing over eigenvalues. It is however not straightforward to apply one ofthese methods to the cases when both p and q are odd [62]. To overcome this difficulty, wewill use the zeta function regularization throughout this paper and complete the calculationof the free energy on H p +1 × S q − . As we will see in the main text, our approach is notonly applicable to any p and q , but also makes it easy to compare the free energies on theconformally equivalent spaces. For instance, we will check the equality between the universalparts of the free energies on H d and HS d for Dirichlet boundary condition conjectured by[63] by combining numerical and analytic ways. We will also verify a few other relationsfor the free energies and prove or conjecture new ones which will be summarized below.Another advantage of our approach than the other methods is to make manifest the differencebetween the anomalies from the bulk theory and defect. Actually, there are two sources ofthe logarithmic divergences: one from the bulk anomaly when d = p + q even and the otherfrom the defect anomaly when p even. In our approach, the bulk anomaly depends on thecutoff introduced for the zeta regularization while the defect anomaly depends on anothercutoff that arises from the renormalized volume of the hyperbolic space H p +1 for even p .We will leverage our results to test if the conjectured relation (1.5) holds for the defectRG flow from Neumann to Dirichlet when the former is allowed. Strictly speaking, we will notdirectly check our proposal that employs the defect free energy on S d as a C -function. Insteadwe assume that the difference of the free energy is invariant under the conformal map from S d to H p +1 × S q − . We will calculate the difference of the free energies on H p +1 × S q − betweenNeumann and Dirichlet boundary conditions in two ways: the residue method [53, 54] and When both p and q are odd, it is expected that only bulk anomaly exists and the bulk anomaly is thesame as that of S p + q . However, it is technically difficult to confirm this expectation. It is not clearly specified what type of boundary conditions is imposed on the hyperbolic space in [63], buttheir boundary condition is of Dirichlet type in our terminology. When defects have conformal anomaly the free energy may not be invariant, but when defects are sphericalthe anomaly is of type A which depends only on the Euler characteristic of the worldvolumes. In our setup,defects on S d and H d are always spherical, so should have the same anomaly. When there are no defectanomalies, we need not be worried about this issue as the free energy is invariant by definition. – 4 –nalytic continuation method. We find both methods give the same result consistent withthe defect C -theorem (1.5).The organization of this paper is as follows. In section 2, we review several useful co-ordinates for DCFT and conformal maps among them. Furthermore, we discuss boundaryconditions of Dirichlet type and Neumann type for a conformally coupled scalar field on H p +1 × S q − . The Neumann type boundary conditions fall into two classes, free boundarycondition and the other. We show that the Dirichlet boundary condition always exists butthe Neumann boundary condition exists only in q = 1 , , , q = p + 2 for q ≥
3. Section 3 begins as a warm-up with the calculationof the free energy on S d and HS d . The purpose of this section is twofold: to illustrate thezeta regularization method and to provide analytic results for the boundary free energy on HS d in arbitrary dimensions. In section 4, we proceed to compute the free energies on H d and H p +1 × S q − with the Dirichlet boundary conditions. Along the way we find variousidentities for the free energies on the conformally equivalent spaces. In section 5, we calculatethe difference of the free energies between the Neumann and Dirichlet boundary conditionson H p +1 × S q − in two ways and confirm (1.5) holds for all the cases. Finally section 6 isdevoted to discussion and future directions. Appendices include the lists of the free energieson S d , HS d , H d and H p +1 × S q − obtained in the main text, various formulas and technicaldetails of some calculations. Since the body of the paper is rather lengthy and technical, in what follows we will summarizethe main results.In section 2 we classify the boundary condition of a conformally coupled scalar fieldtheory on H p +1 × S q − , which preserves a defect conformal symmetry. Neumann boundarycondition is allowed only for q = 1 , , , q = p + 2for q ≥
3. Our result matches the classification of non-monodromy defects in a free scalartheory given by [59] by other means.In section 3, using the zeta-function regularization, we compute the free energies on S d and HS d . • For S d , the renormalized free energy takes the following form: F ren [ S d ] = ( F fin [ S d ] d : odd − A [ S d ] log(Λ R ) + F fin [ S d ] d : even (1.7)where Λ is a UV cutoff introduced in the zeta regularization. We reproduce knownanomaly coefficients A [ S d ] for even d and known universal finite terms F fin [ S d ] for odd d [19, 22, 66–68]. – 5 – For HS d , we find the renormalized free energy takes the form: F ren [ HS d ] = − A [ HS d ] log(Λ R ) + F fin [ HS d ] d : odd − A [ S d ] log(Λ R ) + F fin [ HS d ] d : even (1.8)After subtracting the half of the free energy on S d , we obtain the boundary free energywith the Dirichlet or Neumann boundary condition. This reproduces known results in[27, 28, 31, 63, 69].In section 4 we examine the case for H d and H p +1 × S q − with the Dirichlet boundarycondition for the hyperbolic space in the zeta regularization. • For H d , we find the renormalized free energy takes the following form: F ren [ H d ] = −A [ H d ] log (cid:18) Rǫ (cid:19) d : odd − A [ H d ] log(Λ R ) + F fin [ H d ] d : even (1.9)where log( R/ǫ ) arises from the regularized volume of H d and only appears for odd d . We obtain the universal parts of the free energy and reproduce known results in[62, 63, 70, 71]. We confirm the equivalence of the free energies between H d with theDirichlet boundary condition and HS d with the Dirichlet boundary condition: A [ H d ] = A [ HS d ] d : odd F ren [ H d ] = F ren [ HS d ] d : even (1.10)(We also confirm similar results hold for the Neumann boundary conditions.) • For H p +1 × S q − we find the renormalized free energy takes the form: F ren [ H p +1 × S q − ] = −A [ H p +1 × S q − ] log (cid:18) Rǫ (cid:19) p : even − A [ H p +1 × S q − ] log(Λ R ) + F fin [ H p +1 × S q − ] p : odd(1.11)We obtain the following results:1. For even p and even q , A [ H p +1 × S q − ] = 0, indicating the presence of defectanomaly.2. For even p and q = p + 2, we numerically check A [ S p +2 ] = 2 A [ H p +1 × S p +1 ] . (1.12)3. For even p and odd q , A [ H p +1 × S q − ] = 0 or equivalently F ren [ H p +1 × S q − ] = 0.– 6 –. For odd p , we numerically verify the relation: F ren [ S d ] = F ren [ H k × S d − k ] . (1.13)These relations were conjectured in [62] from the calculations in free scalar and holo-graphic theories. Our results provide more evidence for their conjectures at least in theconformally coupled free scalar in arbitrary dimensions. Especially we perform system-atic computations in the zeta regularization including the cases with odd p and odd q which was missing in [62] due to some technical difficulty.These results lead us to speculate that A [ H p +1 × S q − ] is associated with the bulkanomaly while A [ H p +1 × S q − ] is with the defect anomaly.In section 5 we obtain the free energies with the Neumann boundary condition for thehyperbolic space using two different methods: (1) an analytic continuation, and (2) theresidue method which is conjectured in [54]. We give a proof of the conjecture and apply itto H p +1 × S q − . We confirm that the interpolated defect free energy ˜ D with the Dirichletboundary condition is always smaller than that with Neumann boundary condition, andour results are consistent with the conjectured C -theorem (1.5) in DCFT. Specifically, thedifference of the free energies on H p +1 × S q − between the two boundary conditions equalsthe free energy on S p for q = 2 , H p +1 between thetwo boundary conditions for q = 3, and the monotonicity follows from the positivity of theinterpolated sphere free energy ˜ F or the monotonicity of the free energy on H p +1 . We first review coordinate systems and conformal maps between them which are suitable fordescribing conformal defects in DCFTs. We then proceed to classify conformal boundaryconditions for a conformally coupled free scalar field, and show that they correspond to aclassification of conformal (non-monodromy) defects in the same theory.
Let us consider DCFT d on flat space with the metricd s = d x a + d y i , ( a = 1 , · · · , p, i = p + 1 , · · · , d ) , (2.1)where a p -dimensional defect sits at the origin y i = 0 in the transverse directions. For laterconvenience we introduce q = d − p , which represents a codimension of the defect. By usingthe polar coordinate for the y i -coordinates,d y i = d z + z d s S q − , (2.2) For S , we can compute the both sides analytically but we can not prove the equality for arbitrary d . – 7 –ith the metric d s S q − for a unit ( q − s = z (cid:18) d x a + d z z + d s S q − (cid:19) . (2.3)By a Weyl transformation, the above metric reduces to the geometry H p +1 × S q − with radius R , d s = R (cid:18) d x a + d z z + d s S q − (cid:19) . (2.4)Now the defect is located at the boundary of the hyperbolic space. We can also use the globalcoordinate for the hyperbolic space part:d s = R (cid:0) d ρ + sinh ρ d s S p + d s S q − (cid:1) , (2.5)where the defect becomes a p -sphere at ρ = ∞ . Introducing a new variable ϕ by tan ϕ = sinh ρ ,the metric (2.5) becomesd s = R cos ϕ (cid:0) d ϕ + sin ϕ d s S p + cos ϕ d s S q − (cid:1) , (2.6)which the metric can be mapped by a further Weyl transformation to the d -sphere metric :d s = R (cid:0) d ϕ + sin ϕ d s S p + cos ϕ d s S q − (cid:1) , (2.7)where the defect is mapped to a p -sphere at ϕ = π/
2. See figure 1 for the illustration of theresulting conformal map.It will also be convenient to introduce the standard representation of the sphere metric,d s = R (cid:0) d ϕ + sin ϕ d s S d − (cid:1) , (2.8)where 0 ≤ ϕ < π for the sphere and 0 ≤ ϕ ≤ π/ Next, let us consider a conformally coupled real scalar field on (2.4) (or (2.5)). The action isgiven by I = − Z d d x √ g (cid:2) ( ∂ µ φ ) + ξ R φ (cid:3) , (2.9)with the parameter ξ and the Ricci scalar R : ξ = d − d − , R = ( q − q − − p ( p + 1) R . (2.10)– 8 – ( p ) R d conformal map −−−−−−−−−−−−→ D ( p ) × H p +1 S q − Figure 1 . Conformal map from flat space R d with a p -dimensional planar defect to H p +1 × S q − . Now we would like to investigate the boundary condition for the scalar field near the boundary, z = 0 (or ρ = ∞ ), of the hyperbolic space. For this purpose, we decompose the scalar fieldinto the eigenfunctions by the spherical harmonics on S q − : φ ( z, x, θ ) = X ℓ φ H p +1 ( z, x ) Y ℓ, S q − ( θ ) , (2.11)where ( z, x ) are the coordinates of the hyperbolic space in Poincar´e coordinate and θ standsfor those of the sphere S q − . The spherical harmonics Y ℓ, S q − ( θ ) satisfies the equation: −∇ S q − Y ℓ, S q − ( θ ) = ℓ ( ℓ + q − R Y ℓ, S q − ( θ ) . (2.12)Here ℓ is an integer whose range is from −∞ to ∞ for q = 2 and from 0 to ∞ for q ≥ H p +1 × S q − (cid:0) −∇ H p +1 − ∇ S q − + ξ R (cid:1) φ ( z, x, θ ) = 0 , (2.13)reduces to the equation of motion of a massive scalar field on H p +1 : (cid:0) −∇ H p +1 + M (cid:1) φ H p +1 ( z, x ) = 0 , (2.14)with the mass given by M R = ℓ ( ℓ + q −
2) + ( q − − p . (2.15)Then the solution to the equation of motion behaves as φ H p +1 ∼ z ∆ ℓ ± , (2.16)near the boundary, z = 0, as is well-known in the AdS/CFT correspondence. Here ∆ ℓ ± arethe roots of the equation: ∆(∆ − p ) = M R , (2.17)– 9 –nd are explicitly given by∆ ℓ ± = p ± | ℓ | ( q = 2) p ± (cid:18) ℓ + q − (cid:19) ( q > . (2.18)For q = 1, the spherical part does not exist, so there is no ℓ -dependence in ∆ ± :∆ ± = p ± , ( q = 1) . (2.19)While we are only concerned with a QFT of a scalar field on H p +1 × S q − , it may beviewed as a bulk system in a holographic setup as shown by the above consideration. Theparameters ∆ ℓ ± can be understood as the conformal dimensions of operators localizing on a p -dimensional conformal defect at the boundary of H p +1 . (See also section 2.3 in [72] for arelated discussion.) Then not all the operators with dimensions (2.18) (or (2.19) for q = 1)are allowed to exist due to the unitarity bound in p dimensions: ∆ ℓ ± ≥ p − , ( p > , ∆ ℓ ± > , ( p ≤ . (2.22)It follows that ∆ ℓ + is always above the bound, while ∆ ℓ − is not necessarily so unless | ℓ | ≤ , ( q = 2) ,ℓ ≤ − q , ( q > . (2.23)Hence the modes with small ℓ are allowed to have sensible boundary conditions correspondingto ∆ ℓ − . For clarity we define the Dirichlet and Neumann boundary conditions for q ≥ D = ∆ ℓ + for all ℓ , Neumann b. c. : ∆ N = ( ∆ ℓ − > ℓ ∆ ℓ + otherwise . (2.24) Restricting to normalizable boundary conditions on the hyperbolic space H p +1 the mass of the scalar fieldis subject to the so-called Breitenlohner-Freedman (BF) bound: M R ≥ − p . (2.20)When this condition is met there are two real solutions to (2.17), ∆ ± . While the solution with the larger root∆ + is always square-integrable, the solution with the smaller root ∆ − is not necessarily so with respect to theKlein-Gordon inner product. Thus, requiring the square integrability leads to the bound for ∆ − :∆ − ≥ p − . (2.21)From the viewpoint of the AdS/CFT correspondence, this matches with the unitarity bound for scalar primaryoperators in a p -dimensional CFT. – 10 –n accordance with the case for q = 1 where ∆ D = ∆ + and ∆ N = ∆ − .In addition to them there are boundary conditions with a constant solution (zero mode)on H p +1 : φ H p +1 ∼ const , (2.25)which corresponds to defect operators of dimension ∆ ℓ − = 0. Among them is the specialboundary condition ∆ ℓ =0 − = 0 associated with the excitation of the identity operator on thedefect. We call them “free” boundary conditions following [57]:Free b. c. : ∆ F = ( ∆ ℓ − = 0 ℓ = 0∆ ℓ + ℓ = 0 , (2.26)for q ≥
2, and ∆ F = ∆ − = 0 for p = q = 1. On the other hand, the zero modes with ℓ = 0are termed charged dimension zero operators and excluded in [59] on the basis of the clusterdecomposition which assures the dimension zero mode must be the defect identity operator.Thus, we will also take into account the free boundary conditions while excluding the chargedzero modes ∆ ℓ =0 − = 0 from the classification.It follows from (2.18) and (2.19) that the free boundary conditions are allowed only when p = q = 1 and q = p + 2 ( p ≥ H p +1 × S q − is associated with a p -dimensional scalar Wilson surface in d = 2 p +2dimension: W Σ p = e g R Σ p φ , (2.27)where g is a dimensionless coupling, Σ p the worldvolume of a p -dimensional surface and φ abulk scalar field of dimension p .Having this caveat in mind, we obtain the classification of the boundary conditions for∆ ℓ − : q = 1 case: It follows from (2.19) that there exists the Neumann boundary condition for p ≥ p = 1. q = 2 case: The bound (2.23) becomes ℓ ≤
1, so the Neumann boundary conditions with ℓ =0 , ± ℓ = 0 mode, however, does not give a new boundary conditionas ∆ ℓ =0+ = ∆ ℓ =0 − = p . Note that ∆ ℓ = ± − = p − p >
2. In[59] it is argued that the ℓ = ± These modes should be treated with case as they are the source of the IR divergence in the free energy. The defect identity operators are taken into account in the v2 of [59]. – 11 –hus, there are two types of Neumann boundary conditions for p > N1 = ( ∆ ℓ + for ℓ = 1∆ ℓ − for ℓ = 1 , ∆ N2 = ( ∆ ℓ + for ℓ = ± ℓ − for ℓ = ± . (2.28)The mode with (∆ ℓ = − , ∆ ℓ = − − ) is essentially the same as the ∆ N1 boundary condition becausewe can change the label of ℓ without changing physics. q = 3 case: Only the ℓ = 0 mode satisfies the unitarity bound and gives us a nontrivialNeumann boundary condition with ∆ ℓ =0 − = p − for p ≥
2. Hence there is only one type ofthe Neumann boundary condition:∆ N = ( ∆ ℓ + for ℓ ≥ ℓ − for ℓ = 0 . (2.29)The free boundary condition can be imposed only when p = 1, which describes a scalar Wilsonloop in the four-dimensional free scalar field theory [57]. q = 4 case: Only the ℓ = 0 mode is allowed, resulting in the Neumann boundary conditionwith ∆ ℓ =0 − = p − p ≥ p = 2, which corresponds to a scalar Wilson surface in the six-dimensional freescalar field theory (see also [73]). q ≥ case: In this case, there are no Neumann boundary conditions satisfying the unitaritybound (2.22), but there still exists the free boundary condition when q = p + 2 associatedwith a p -dimensional scalar Wilson surface in d = 2 p + 2 dimensions.Our results are consistent with the classification of the non-monodromy defects in a freescalar theory in [59], which are summarized in table 1. The aim of this section is to demonstrate the zeta function regularization through the calcu-lation of free energies on the sphere S d and the hemisphere HS d . They have been extensivelystudied in literature in various methods (see e.g., [74, 75] for reviews), and we do not claimany originality of our results except for giving their explicit expressions. The main results are(3.19), (3.22) and (3.23) for S d , and (3.45), (3.46), (3.50), (3.51) for HS d .– 12 – = 1 q = 2 q = 3 q = 4 q = 5 q = 6 · · · p = 1 ∆ D / ∆ F ∆ D ∆ D / ∆ F ∆ D ∆ D ∆ D · · · p = 2 ∆ D / ∆ N ∆ D ∆ D / ∆ ℓ =0 − ∆ D / ∆ F ∆ D ∆ D p = 3 ∆ D / ∆ N ∆ D / ∆ ℓ = ± − ∆ D / ∆ ℓ =0 − ∆ D / ∆ ℓ =0 − ∆ D / ∆ F ∆ D · · · p = 4 ∆ D / ∆ N ∆ D / ∆ ℓ = ± − ∆ D / ∆ ℓ =0 − ∆ D / ∆ ℓ =0 − ∆ D ∆ D / ∆ F p = 5 ∆ D / ∆ N ∆ D / ∆ ℓ = ± − ∆ D / ∆ ℓ =0 − ∆ D / ∆ ℓ =0 − ∆ D ∆ D ... ... ... . . . Table 1 . Classification of the allowed boundary conditions in the free scalar theory. The Neumannboundary conditions exist in the shaded cells and the allowed modes differ from the Dirichlet ones areshown in the right side. Our table is the same as the classification of the non-monodromy defects in[59] except that ours has additional column for q = 1 and boundary conditions ∆ ℓ = ± − for q = 2 and p ≥
3, and ∆ ℓ =0 − for q = 4 and p ≥ p = q = 1 and q = p + 2. S d Let us first consider the free energy on S d as a warm-up. For a conformally coupled scalar on S d , the free energy is given by F [ S d ] = 12 tr log (cid:20) ˜Λ − (cid:18) −∇ S d + d ( d − R (cid:19)(cid:21) = 12 tr log (cid:20) ˜Λ − (cid:18) ℓ ( ℓ + d − R + d ( d − R (cid:19)(cid:21) = 12 ∞ X ℓ =0 g ( d ) ( ℓ ) " log ν ( d ) ℓ + ˜Λ R ! + log ν ( d ) ℓ − ˜Λ R ! , (3.2)where R is the radius of S d and ˜Λ is the UV cutoff scale introduced to make the integraldimensionless. The degeneracy g ( d ) ( ℓ ) and the parameter ν ( d ) ℓ are defined by g ( d ) ( ℓ ) = (2 ℓ + d −
1) Γ( ℓ + d − d ) Γ( ℓ + 1) , ν ( d ) ℓ = ℓ + d − . (3.3) When we decompose log h (( ν ( d ) ℓ ) − / / (˜Λ R ) i into the two logarithmic functions in the third line, thereis an ambiguity, log (cid:16) ν ( d ) ℓ (cid:17) − / R ) = log e i θ ν ( d ) ℓ + 1 / R ) − ρ ! + log e − i θ ν ( d ) ℓ − / R ) ρ ! , (3.1)where ρ is a real number and 0 ≤ θ < π (see also [76]). It leads to the ambiguities of the anomaly term andthe finite term of the free energy. We fix the ambiguity of the phase by demanding a good convergence in theSchwinger representation of the free energy (3.4) at large ℓ . The ambiguity of the scale can also be fixed byrequiring that the zeta function be independent of the parameter ˜Λ R . – 13 –ne can rewrite the free energy (3.2) in the Schwinger representation: F [ S d ] = − Z ∞ d tt ∞ X ℓ =0 g ( d ) ( ℓ ) (cid:20) e − t (cid:16) ν ( d ) ℓ + (cid:17) / (˜Λ R ) + e − t (cid:16) ν ( d ) ℓ − (cid:17) / (˜Λ R ) (cid:21) . (3.4)This is divergent, implying the UV divergence of the free energy. To make the integral finitewe introduce the regularized free energy [77]: F s [ S d ] = − Z ∞ d tt − s ∞ X ℓ =0 g ( d ) ( ℓ ) (cid:20) e − t (cid:16) ν ( d ) ℓ + (cid:17) / (˜Λ R ) + e − t (cid:16) ν ( d ) ℓ − (cid:17) / (˜Λ R ) (cid:21) = −
12 ( ˜Λ R ) s Γ( s ) ζ S d ( s ) , (3.5)where the zeta function ζ S d ( s ) is defined by ζ S d ( s ) ≡ ∞ X ℓ =0 g ( d ) ( ℓ ) "(cid:18) ν ( d ) ℓ + 12 (cid:19) − s + (cid:18) ν ( d ) ℓ − (cid:19) − s . (3.6)Then the (unrenormarized) free energy is obtained in the s → F s [ S d ] = − (cid:18) s − γ E + log( ˜Λ R ) (cid:19) ζ S d (0) − ∂ s ζ S d (0) + O ( s ) , (3.7)which is divergent due to the pole at s = 0. After removing the pole term, the remainingpart becomes the renormalized free energy F ren [ S d ] ≡ − ζ S d (0) log(Λ R ) − ∂ s ζ S d (0) , (3.8)where Λ = e − γ E ˜Λ.In calculating the zeta function (3.6), we find it convenient to expand the degeneracy as: g ( d ) ( ℓ ) = 2Γ( d ) d − X k =0 γ k,d ( c ) (cid:16) ν ( d ) ℓ + c (cid:17) k . (3.9)To fix γ k,d ( c ) we introduce coefficients α n,d and β n,d as follows [74]: g ( d ) ( ℓ ) = d ) d − Y j =0 (cid:20)(cid:16) ν ( d ) ℓ (cid:17) − j (cid:21) = 2Γ( d ) d − X n =0 ( − d − + n α n,d (cid:16) ν ( d ) ℓ (cid:17) n d : odd2 ν ( d ) ℓ Γ( d ) d − Y j = (cid:20)(cid:16) ν ( d ) ℓ (cid:17) − j (cid:21) = 2Γ( d ) d − X n =0 ( − d − n β n,d (cid:16) ν ( d ) ℓ (cid:17) n +1 d : even(3.10)– 14 –ote that we use a slightly different notation from [74] and include the n = 0 contribution inodd d although α ,d = 0 for some convenience. Comparing (3.9) with (3.10) we find γ k,d ( c ) = d − X n = ⌈ k ⌉ ( − d − + n + k (cid:18) nk (cid:19) α n,d c n − k d : odd d − X n = ⌊ k ⌋ ( − d + n + k (cid:18) n + 1 k (cid:19) β n,d c n +1 − k d : even (3.11)With this expansion we can perform the summation over ℓ in (3.6) and obtain a summationof the Hurwitz zeta functions: ζ S d ( s ) = 2Γ( d ) d − X k =0 (cid:20) γ k,d (cid:18) (cid:19) ζ H (cid:18) s − k, d (cid:19) + γ k,d (cid:18) − (cid:19) ζ H (cid:18) s − k, d − (cid:19)(cid:21) . (3.12)It remains to determine the coefficients α n,d and β n,d to calculate the renormalized free energy.We fix them by comparing the two representations, (3.3) and (3.10), of g ( d ) ( ℓ ). Using theasymptotic expansion in [78, 5.11.14],Γ( x + a )Γ( x + b ) = ∞ X k =0 (cid:18) x + a + b − (cid:19) a − b − k (cid:18) a − b k (cid:19) B ( a − b +1)2 k (cid:18) a − b + 12 (cid:19) , (3.13)where B ( m ) k ( x ) is the generalized Bernoulli polynomial which reduces to the Bernoulli poly-nomial B k ( x ) = B (1) k ( x ) when m = 1, and comparing the both sides, we find α n,d = ( − d − + n (cid:18) d − d − − n (cid:19) B ( d − d − − n (cid:18) d − (cid:19) , (3.14)for odd d and β n,d = ( − d − + n (cid:18) d − d − − n (cid:19) B ( d − d − − n (cid:18) d − (cid:19) , (3.15)for even d . d When d is odd the zeta function (3.12) reduces to ζ S d ( s ) = 2Γ( d ) d − X k =0 d − X n = ⌈ k ⌉ ( − d − + n + k α n,d (cid:18) nk (cid:19) k − n (cid:20) ζ H (cid:18) s − k, d (cid:19) + ( − k ζ H (cid:18) s − k, d − (cid:19)(cid:21) . (3.16)– 15 –sing the identity (B.9) for the Hurwitz zeta functions the terms in the bracket becomes ζ H (cid:18) s − k, d (cid:19) + ( − k ζ H (cid:18) s − k, d − (cid:19) = ζ H (cid:18) s − k, (cid:19) + ( − k ζ H (cid:18) s − k, (cid:19) − ⌊ d ⌋− X m =0 (cid:18) m + 12 (cid:19) k − s − ⌊ d ⌋− X m =0 ( − k (cid:18) m + 12 (cid:19) k − s . (3.17)Rearranging the summations P d − k =0 P d − n = ⌈ k ⌉ = P d − n =0 P nk =0 the zeta function becomes ζ S d ( s ) = 2Γ( d ) d − X n =0 2 n X k =0 ( − d − + n + k α n,d (cid:18) nk (cid:19) k − n (1 + ( − k ) ζ H (cid:18) s − k, (cid:19) − d ) ⌊ d ⌋− X m =1 "(cid:18) m + 12 (cid:19) − s + (cid:18) m − (cid:19) − s d − X n =0 ( − d − + n α n,d m n = 4Γ( d ) d − X n =0 n X l =0 ( − d − + n α n,d (cid:18) n l (cid:19) − n (2 s − l ) ζ ( s − l ) (3.18)where we used the identity (B.12) and removed the summation over m by resorting to thedefinition (3.10) of α n,d in the third equality.Taking the s → ζ S d (0) = 0 as (1 − k ) ζ ( − k ) = 0 holds for a non-negativeinteger k . Thus, there is no conformal anomaly in the free energy and only the universal finitepart remains in the free energy (3.8): F ren [ S d ] = F fin [ S d ]= Γ( d )2 Γ( d ) Γ(2 − d ) log 2+ d − X k =1 d − X n = k ( − d +12 + n + k α n,d (1 − k ) (2 n − k + 1) k k +2 n π k Γ( d ) ζ (2 k + 1) , (3.19)where we used (B.2) and (B.6). We also used the Pochhammer symbol ( n ) k ≡ Γ( n + k ) / Γ( n )to simplify the expression. The explicit values of F fin [ S d ] up to d = 9 are shown in table 3 inappendix A. – 16 – .1.2 Even d Performing a similar reduction for odd d using the identity (B.9) with a = 1, the zeta function(3.12) for even d can be written as ζ S d ( s ) = 2Γ( d ) d − X k =0 d − X n = ⌊ k ⌋ ( − d + n + k k − n − (cid:18) n + 1 k (cid:19) β n,d · h (1 + ( − k − ) ζ ( s − k ) + ( − k − δ d, ( ζ H ( s − k, − ζ ( s − k )) i . (3.20)In contrary to the odd-dimensional case, there is a logarithmic divergent term associated withthe conformal anomaly in the free energy (3.8): F ren [ S d ] = − A [ S d ] log(Λ R ) + F fin [ S d ] , (3.21)where the anomaly coefficient A [ S d ] can be read off from (3.12) as A [ S d ] = 1Γ( d ) d − X n =0 β n,d ( − d + n n + 1 (cid:18) B n +2 (cid:18) (cid:19) + 2 n + 12 n +2 (cid:19) + 12 δ d, (3.22)while the universal term becomes F fin [ S d ] = 1Γ( d ) d − X n =0 n X m =0 ( − d + n m − n +1 (cid:18) n + 12 m + 1 (cid:19) β n,d ζ ′ ( − m − − δ d, X k =0 k − (cid:0) ∂ s ζ H ( − k, − ζ ′ ( − k ) (cid:1) . (3.23)For d = 2, ∂ s ζ H (0 ,
0) is ill-defined, which reflects the IR divergence due to the zero mode.Tables 2 and 3 in appendix A show the anomaly coefficients and the finite parts of thefree energies on S d for even d up to d = 10. Our result (3.22) correctly reproduces theconformal anomaly of the free scalar theory obtained in literature (e.g. [19, 22, 66–68, 79]).The finite part is less known as it depends on the regularization scheme (i.e., the choice ofthe UV cutoff Λ) when there exists a conformal anomaly. When d = 4 (3.23) agrees with theresult in [28] which uses the same zeta regularization as ours. a and F The finite parts of the free energy (3.19) for odd d and the anomaly parts of the free energy(3.22) for even d are universal in the sense that they are independent of the cutoff choice. We used the Taylor expansion of the Bernoulli polynomials B n ( x + y ) = P nk =0 (cid:0) nk (cid:1) B k ( x ) y n − k . When we compare our results with (3.25), the divergent factor, (cid:0) sin (cid:0) πd (cid:1)(cid:1) − , should be replaced with thelogarithmic term according to (4.7). – 17 –hus it will be convenient to introduce the “universal” free energy: F univ [ S d ] = F fin [ S d ] d : odd − A [ S d ] log (cid:18) Rǫ (cid:19) d : even (3.24)where we use ǫ for the cutoff instead of Λ. While the structure of the universal free energyappears to depend on the dimensionality it is shown in [22] to have an integral representationwhich smoothly interpolates between even and odd d : F univ [ S d ] = − (cid:0) πd (cid:1) Γ( d + 1) Z d ν ν sin( πν ) Γ (cid:18) d ± ν (cid:19) . (3.25)Here we used the shorthand notation Γ( x ± y ) ≡ Γ( x + y ) Γ( x − y ). The prefactor is finitefor odd d , but divergent for even d due to the pole from the zeros of the sine function. Thisdivergence may be replaced with the logarithmic divergence by introducing a small cutoffparameter ǫ : − (cid:0) πd (cid:1) = ( − d +12 d : odd( − d π log (cid:18) Rǫ (cid:19) d : even (3.26)See [80, 81] for a proof of the equivalence of the two expressions (3.24) and (3.25). HS d The free energies on hemisphere are obtained in [28, 31, 63, 69, 82]. Here we extend them tohigher dimensions by using the zeta function regularization. See also [55] for a related work.In the coordinate system (2.8), the Dirichlet boundary condition is given by φ (cid:16) π (cid:17) = 0 , (3.27)and the Neumann boundary condition is given by ∂ ϕ φ (cid:16) π (cid:17) = 0 . (3.28)If we put the theory on the sphere (2.8) with the defect at ϕ = π/
2, the Dirichlet boundarycondition is equivalent to imposing an anti-symmetric condition at ϕ = π/ φ ( ϕ ) = − φ ( π − ϕ ) , (3.29) We thank J. S. Dowker for drawing our attention to these works. To derive the boundary condition from the action, one needs to add a boundary term to the action. SeeAppendix C in [31]. – 18 –nd the Neumann boundary condition is equivalent to imposing a symmetric condition at ϕ = π/ φ ( ϕ ) = φ ( π − ϕ ) . (3.30)If the scalar field is expanded into the spherical harmonics on S d − , φ ( ϕ, θ ) = X m f m ( ϕ ) Y m, S d − ( θ ) , (3.31) Y m, S d − ( θ ) with odd (even) ℓ − m are odd (even) functions about ϕ = π/ g ( d )+ ( ℓ ) = (cid:18) ℓ + d − d − (cid:19) = ℓ Γ( ℓ + d − d )Γ( ℓ + 1) , Neumann: g ( d ) − ( ℓ ) = (cid:18) ℓ + d − d − (cid:19) = ( ℓ + d −
1) Γ( ℓ + d − d )Γ( ℓ + 1) . (3.32)We use a subscript + ( − ) for the Dirichlet (Neumann) boundary condition. The degeneracycan also be written as g ( d ) ± ( ℓ ) = 12 g ( d ) ( ℓ ) ∓
12 Γ( d −
1) Γ (cid:16) ν ( d ) ℓ + d − (cid:17) Γ (cid:16) ν ( d ) ℓ − d − + 1 (cid:17) , (3.33)by using ν ( d ) ℓ defined by (3.3).For a conformally coupled scalar on HS d , the free energy is given by (3.25) with thedegeneracy replaced by those for the Dirichlet/Neumann boundary conditions: F [ HS d ± ] = 12 ∞ X ℓ =0 g ( d ) ± ( ℓ ) " log ν ( d ) ℓ + ˜Λ R ! + log ν ( d ) ℓ − ˜Λ R ! . (3.34)Here we added the suffix to manifest boundary conditions explicit. Since the sum of thedegeneracies satisfies the relation, g ( d )+ ( ℓ ) + g ( d ) − ( ℓ ) = g ( d ) ( ℓ ) , (3.35)the sum of the free energies on a hemisphere equals the free energy on a sphere, F [ HS d + ] + F [ HS d − ] = F [ S d ] . (3.36) We adopt this unusual convention because we use + for the Dirichlet boundary condition on H d . We choose this normalization for the free energy on HS d . Hence our free energy satisfies (3.36) contraryto Appendix C in [31]. – 19 –he zeta functions of each boundary condition are given by ζ HS d ± ( s ) = 12 ζ S d ( s ) ∓
12 Γ( d − ∞ X ℓ =0 Γ (cid:16) ν ( d ) ℓ + d − (cid:17) Γ (cid:16) ν ( d ) ℓ − d − + 1 (cid:17) "(cid:18) ν ( d ) ℓ + 12 (cid:19) − s + (cid:18) ν ( d ) ℓ − (cid:19) − s , (3.37)and the renormalized free energies become F ren [ HS d ± ] ≡ − ζ HS d ± (0) log(Λ R ) − ∂ s ζ HS d ± (0) , (3.38)as in the previous section.In the following, we compute the zeta functions by using the relation,Γ (cid:16) ν ( d ) ℓ + d − (cid:17) Γ (cid:16) ν ( d ) ℓ − d − + 1 (cid:17) = d − X n =0 ( − d − + n α n,d n − X k =0 (cid:18) n − k (cid:19) x k c n − − k d : odd d − X n =0 ( − d − n β n,d n X k =0 (cid:18) nk (cid:19) x k c n − k d : even (3.39)where x = ν ( d ) ℓ − c . d Using the expansion (3.39) and performing a similar computation in section 3.1.1, the zetafunctions can be written as ζ HS d ± ( s ) − ζ S d ( s )= ∓ d − d − X n =1 n − X l =0 ( − d − + n α n,d (cid:18) n − l + 1 (cid:19) − n (2 s − l +1 ) ζ ( s − l − . (3.40)Since ζ S d (0) = 0 for odd dimension, we find ζ HS d ± (0) = ± d − d − X n =1 ( − d − + n α n,d (cid:18) B n n − − n − n (cid:19) , (3.41) This follows from (3.10) and the binomial theorem for ν ( d ) ℓ . – 20 –here we again used the Taylor expansion of the Bernoulli polynomials. The derivative ofthe zeta functions reduces to ∂ s ζ HS d ± (0) − ∂ s ζ S d (0) = ∓ d − d − X n =1 n − X l =0 ( − d − + n − n α n,d (cid:18) n − l + 1 (cid:19) · h (1 − l +1 ) ζ ′ ( − l −
1) + ζ ( − l −
1) log 2 i . (3.43)It follows that the renormalized free energies on HS d is F ren [ HS d ± ] = − A [ HS d ± ] log(Λ R ) + F fin [ HS d ± ] , (3.44)where A [ HS d ± ] = ±
12 Γ( d − d − X n =1 ( − d − + n α n,d (cid:18) B n n − − n − n (cid:19) , (3.45)and F fin [ HS d ± ] = 12 F fin [ S d ] ±
12 Γ( d − d − X n =1 n − X l =0 ( − d − + n − n α n,d (cid:18) n − l + 1 (cid:19) · h (1 − l +1 ) ζ ′ ( − l −
1) + ζ ( − l −
1) log 2 i . (3.46)The renormalized free energies on HS d for odd d have logarithmic divergences due to thepresence of the boundary. The anomaly parts of the boundary free energies for the Neumannboundary condition are always greater than those of the Dirichlet boundary condition, andthis is consistent with C -theorems in BCFT d as we will see in section 5.1. The anomaly partsand the finite parts of the renormalized free energies are listed in table 4 in appendix A. Inthe presence of the boundary anomaly the finite terms depend on the regularization schemeand are not universal. d Repeating a similar computation to section 3.1.2, the zeta functions reduce to ζ HS d ± ( s ) − ζ S d ( s ) = ∓ d − d − X n =0 n X l =0 ( − d − n β n,d (cid:18) n l (cid:19) l − n ζ ( s − l ) ∓ δ d, ( ζ H ( s, − ζ ( s )) . (3.47) The term including log 2 can be simplified further to ∓ d − d − X n =1 ( − d − + n − n α n,d − B n n − Γ (cid:0) d (cid:1) (cid:0) − d (cid:1) log 2 . (3.42) It is still meaningful to consider the difference of the finite parts in the same regularization as it no longerdepends on the scheme. – 21 –hen, at s = 0 we find ζ HS d ± (0) − ζ S d (0) = 0 ,∂ s ζ HS d ± (0) − ∂ s ζ S d (0) = ∓ d − d − X n =1 n X l =1 ( − d − n β n,d (cid:18) n l (cid:19) l − n ζ ′ ( − l ) ∓ δ d, (cid:0) ∂ s ζ H (0 ,
0) + ζ ′ (0) (cid:1) . (3.48)It follows that the renormalized free energies on HS d are F ren [ HS d ± ] = − A [ HS d ± ] log(Λ R ) + F fin [ HS d ± ] , (3.49)where A [ HS d ± ] = 12 A [ S d ] , (3.50)and F fin [ HS d ± ] = 12 F fin [ S d ] ±
12 Γ( d − d − X n =1 n X l =1 ( − d − n β n,d l − n (cid:18) n l (cid:19) ζ ′ ( − l ) ± δ d, (cid:0) ∂ s ζ H (0 ,
0) + ζ ′ (0) (cid:1) . (3.51)The bulk anomaly of HS d is just a half of the bulk anomaly on S d as is consistent with thefact that the type A anomaly coefficient is fixed by the Euler characteristic of the manifold.The boundary free energy with Neumann boundary condition is always greater than thatwith Dirichlet boundary condition. The anomaly parts and finite parts of the free energiesare listed in table 4 in appendix A. We move onto the calculation of the free energies on the hyperbolic space H d and a productspace H p +1 × S q − in the zeta regularization. Most parts of the calculations are the same asbefore, but only the difference from section 3 is the continuous spectrum of the conformallaplacian on the hyperbolic space. The main results of this section can be found in (4.14),(4.15), (4.22), (4.23) and (4.24) for H d , (4.37), (4.38) and (4.51) for H p +1 × S , and (4.60),(4.61) and (4.75) for H p +1 × S q − . H d Next we consider the case on the hyperbolic space H d . Since H d is non-compact the conformallaplacian of the free scalar has a continuous spectrum ω = [0 , ∞ ): −∇ H d φ ω = ω R + (cid:18) d − R (cid:19) ! φ ω . (4.1)– 22 –hus, the free energy on H d of radius R is given by F [ H d ] = 12 Z ∞ d ω µ ( d ) ( ω ) (cid:20) log (cid:18) ω + i ν ˜Λ R (cid:19) + log (cid:18) ω − i ν ˜Λ R (cid:19)(cid:21) , (4.2)where we introduced the parameter ν as ν = ∆ − d − , ∆(∆ − d + 1) = − d ( d − . (4.3)There are two solutions to the above equation, which correspond to the Dirichlet and Neu-mann boundary conditions on H d . They are given by ∆ + = d and ∆ − = d − ν = and ν = − in terms of ν . The Plancherel measure µ ( d ) ( ω ) on H d of unit radius takes the form [74, 75]: µ ( d ) ( ω ) = c d (cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:0) d − + i ω (cid:1) Γ(i ω ) (cid:12)(cid:12)(cid:12)(cid:12) = c d d − Y j =0 ( ω + j ) d : odd ω tanh( πω ) d − Y j = ( ω + j ) d : even (4.4)where the product should be omitted for d = 2. The coefficient c d ≡ Vol( H d )2 d − π d Γ (cid:0) d (cid:1) , (4.5)is proportional to the volume Vol( H d ) of the hyperbolic space of unit radius, which may begiven in dimensional regularization byVol( H d ) = π d − Γ (cid:18) − d (cid:19) = − π d +12 sin (cid:0) π d − (cid:1) Γ (cid:0) d +12 (cid:1) . (4.6)The hyperbolic volume is finite for even d , but divergent for odd d due to the pole in thegamma function, which may be replaced by the logarithmic divergence by introducing a smallcutoff parameter ǫ : − (cid:0) π d − (cid:1) = ( − d − π log (cid:18) Rǫ (cid:19) d : odd( − d d : even (4.7) As noted in the footnote 9, there are ambiguities to decompose log( ω + ν ) into a sum of logarithmicfunctions. The free energy (4.2) appears to be invariant under ν → − ν , but it should be understood to be definedonly for ν > – 23 –fter the regularization, the coefficient takes the form: c d = − (cid:0) π d − (cid:1) Γ( d ) = ( − d − Γ( d ) 2 π log (cid:18) Rǫ (cid:19) d : odd( − d Γ( d ) d : even (4.8)The free energy (4.2) is divergent and we regularize it by introducing the renormalizedfree energy with the zeta regularization as in section 3: F ren [ H d ] = − ζ H d (0 , ν ) log(Λ R ) − ∂ s ζ H d (0 , ν ) , (4.9)where the zeta function is defined by ζ H d ( s, ν ) ≡ Z ∞ d ω µ ( d ) ( ω ) (cid:2) ( ω + i ν ) − s + ( ω − i ν ) − s (cid:3) . (4.10)In what follows, we will compute the renormalized free energy for the Dirichlet boundarycondition with ν = by evaluating the zeta function (4.10) following the method in [71, 83]. d When d is odd we expand the Plancherel measure µ ( d ) ( ω ) in an analogous way as in (3.10): d − Y j =0 ( ω + j ) ≡ d − X k =1 α k,d ω k . (4.11)Then we can perform the integration over ω in the zeta function (4.10) which is convergentfor Re s > d/ ζ H d ( s, ν ) = c d d − X k =1 α k,d Z ∞ d ω (cid:20) ω k ( ω + i ν ) s + ω k ( ω − i ν ) s (cid:21) = 2 c d d − X k =1 ( − k α k,d ν k +1 − s sin (cid:16) πs (cid:17) Γ (2 k + 1) k +1 Y i =1 s − i . (4.12)It follows that ζ H d (0 , ν ) and ∂ s ζ H d (0 , ν ) become ζ H d (0 , ν ) = 0 ,∂ s ζ H d (0 , ν ) = c d d − X k =1 ( − k +1 π α k,d ν k +1 k + 1 . (4.13) The integral with respect to ω produces a factor Γ( s − k − / Γ( s ), which equals the product Q k +1 i =1 ( s − i ) − for integer k . – 24 –ence the renormalized free energy (4.9) does not have a logarithmic divergence dependingon the UV cutoff Λ, but has a logarithmic divergence that arises from the regularized volumeof the hyperbolic space (4.6): F ren [ H d ] = −A [ H d ] log (cid:18) Rǫ (cid:19) , (4.14)where the anomaly coefficient is given by A [ H d ] = ( − d − Γ( d ) d − X k =1 ( − k +1 α k,d k +1 (2 k + 1) . (4.15)Since there are no bulk anomalies when d is odd, we interpret A [ H d ] as defect anomaly fromthe boundary theory in ( d −
1) dimensions.The anomaly coefficients for d ≤ d When d is even, we expand the product in the Plancherel measure as d − Y j = ( ω + j ) ≡ d − X k =0 β k,d ω k . (4.16)We decompose the zeta function (4.10) into two parts using the identitytanh( π ω ) = 1 − πω + 1 , (4.17)and perform a similar integration to (4.12) for the first part to obtain ζ H d ( s, ν ) = 2 c d d − X k =0 ( − k +1 β k,d " ν k +2 − s cos (cid:16) πs (cid:17) Γ(2 k + 2) k +2 Y i =1 s − i − Z ∞ d ω ω k +1 e πω + 1 (cid:0) ( ω + i ν ) − s + ( ω − i ν ) − s (cid:1)(cid:21) . (4.18)While we do not know how to perform the remaining integral in the square bracket it isanalytic in s and convergent in the s → ζ H d (0 , ν ) = c d d − X k =0 ( − k +1 β k,d k + 1 h ( ν ) k +1 + (1 − − k − ) B k +2 i ,∂ s ζ H d (0 , ν ) = c d d − X k =0 β k,d (cid:20) ( − ν ) k +1 ( H k +2 − log ν ) k + 1 + 2 f k ( ν ) (cid:21) , (4.19)– 25 –here B k +2 ≡ B k +2 (0) and H k +2 are Bernoulli and Harmonic numbers respectively andwe introduced f k ( ν ) by f k ( ν ) ≡ Z ∞ d ω ω k +1 e πω + 1 log( ω + ν ) , (4.20)whose detail may be found in appendix C.For ν = 1 /
2, the derivative of the zeta function can be simplified to ∂ s ζ H d (cid:18) , (cid:19) = c d d − X k =0 β k,d ( − k k +1 X j =1 ( − j k − j (cid:18) k + 1 j (cid:19) ζ ′ ( − j ) − δ d, ζ ′ (0) , (4.21)where we use an identity (C.12) which we conjecture to hold in appendix C. The renormalized free energy (4.9) has the UV logarithmic divergence which reflects thebulk conformal anomaly: F ren [ H d ] = − A [ H d ] log(Λ R ) + F fin [ H d ] , (4.22)where the anomaly coefficient is given by A [ H d ] = ( − d d ) d − X k =0 ( − k +1 β k,d k + 1 h − k − + (1 − − k − ) B k +2 i (4.23)The finite term follows from (4.21): F fin [ H d ] = ( − d Γ( d ) d − X k =0 2 k +1 X j =1 ( − k + j +1 k +1 − j β k,d (cid:18) k + 1 j (cid:19) ζ ′ ( − j ) + 12 δ d, ζ ′ (0) . (4.24)The anomaly coefficients and finite parts for d ≤
10 are summarized in tables 2 and 3 inappendix A. From the table we argue without proof the following identity holds for even d : A [ H d ] = 12 A [ S d ] . (4.25) HS d Let us compare the anomaly coefficients and finite parts of the free energies on H d with thoseon HS d with Dirichlet boundary condition. The odd j terms are the same as the half of the derivative of the zeta function on S d , (3.23). – 26 – dd d From (3.45) and (4.15), the difference of the anomaly coefficients is given by A [ H d ] − A [ HS d + ] = ( − d − Γ( d ) d − X k =1 ( − k +1 α k,d (cid:20) k +1 (2 k + 1) + d − k (cid:18) B k − k (cid:19)(cid:21) . (4.26)We argue the right hand side always vanishes, thus A [ H d ] = A [ HS d + ] (4.27)holds for arbitrary d . This relation is difficult for us to prove analytically but we check itexplicitly for a number of d . However, by employing an alternative expression for the anomalycoefficients for HS d + in odd d given by [84], A [ HS d + ] = 12 Γ( d + 1) B ( d ) d (cid:18) d − (cid:19) , (4.28)one can prove the identity (4.27) using the integral representation (see e.g. [82]) B ( d ) d (cid:18) d − (cid:19) = − d Z d ν d − Y j =0 (cid:0) ν − j (cid:1) , (4.29)and the expansion (3.10). For the finite parts of the free energies, F fin [ H d ] is different from F fin [ HS d + ]. However, thefinite parts are not universal in the sense that they depend on the choice of the cutoffs Λ and ǫ in the presence of the boundary anomalies, so we can always make F fin [ H d ] equal F fin [ HS d ]by tuning ǫ appropriately in comparing the two. Even d For the anomaly parts, both of the anomaly coefficients (3.50) and (4.23) on HS d and H d are given by a half of the bulk anomaly on S d . Thus, the difference of the free energiesequals to that of the finite parts (3.51) and (4.24): F ren [ H d ] − F ren [ HS d + ] = − d ) d − X k =1 ǫ k ζ ′ ( − k ) , (4.30)where we rearrange the ranges of n and k , P d − n =1 P nk =1 = P d − k =1 P d − n = k , to find ǫ k ≡ d − X n = k β n,d ( − d + n n − k )+1 (cid:20)(cid:18) n + 12 k (cid:19) − ( d − (cid:18) n k (cid:19)(cid:21) = γ k,d (cid:18) (cid:19) − ( d − d − X n = k β n,d ( − d + n n − k )+1 (cid:18) n k (cid:19) . (4.31) We are indebted to J. S. Dowker for providing us a proof of the identity (4.27) and valuable correspondences. – 27 –e check that ǫ k ≥ = 0 for 4 ≤ d ≤
20, and further speculate it holds for any even d . To sumup we observe that the free energy on H d with Dirichlet boundary condition coincides withthat on HS d with Dirichlet boundary condition. The values of the free energies for d ≤ H d can be identified with the Dirichlet boundary condition on HS d for any d as mentionedin [63]. H p +1 × S We treat the free energy on H p +1 × S separately from the case on H p +1 × S q − as thedegeneracy of S is different from S q − with q ≥
3. This space is associated with a codimension q = 2 defect and has been a focus of research due to the relation to entanglement entropy[19, 64, 85].Expanding the eigenfunctions with respect to the angular modes ℓ ∈ Z , the free energy H p +1 × S is given by F [ H p +1 × S ] = 12 tr log (cid:20) ˜Λ − (cid:18) −∇ H p +1 − ∇ S − p R (cid:19)(cid:21) = 12 ∞ X ℓ = −∞ Z ∞ d ω µ ( p +1) ( ω ) log (cid:18) ω + ℓ ˜Λ R (cid:19) , (4.32)with the Plancherel measure µ ( p +1) ( ω ) (4.4). It will be convenient to decompose the logarith-mic function into two logarithmic functions. Here we employ two different decompositionsdepending on the ordering of the integral over ω and the summation over ℓ . p Since the Plancherel measure µ ( p +1) ( ω ) (4.4) for even p is a polynomial, the integral over ω can be performed before the summation over ℓ . Requiring the convergence in the ω → ∞ limit fixes the decomposition of the free energy: F [ H p +1 × S ] = 12 ∞ X ℓ = −∞ Z ∞ d ω µ ( p +1) ( ω ) (cid:20) log (cid:18) ω + i | ℓ | ˜Λ R (cid:19) + log (cid:18) ω − i | ℓ | ˜Λ R (cid:19)(cid:21) = − Z ∞ d tt ∞ X ℓ = −∞ Z ∞ d ω µ ( p +1) ( ω ) h e − t ( ω +i | ℓ | ) / (˜Λ R ) + e − t ( ω − i | ℓ | ) / (˜Λ R ) i . (4.33)Repeating the same regularization as in section 4.1.1, the renormalized free energy on H p +1 × S is given by F ren [ H p +1 × S ] ≡ − ζ H p +1 × S (0) log(Λ R ) − ∂ s ζ H p +1 × S (0) , (4.34)– 28 –here ζ H p +1 × S ( s ) ≡ ∞ X ℓ = −∞ ζ H p +1 ( s, | ℓ | )= 4 c p +1 p X k =1 α k,p +1 ( − k sin (cid:16) πs (cid:17) Γ (2 k + 1) ζ ( s − k − k +1 Y i =1 s − i . (4.35)To regularize the divergence from the zero mode from ℓ = 0 we introduce a mass m for thescalar. This IR regularization amounts to replacing | ℓ | with p ℓ + ( mR ) above. Then itcan be shown that the resulting zeta function for ℓ = 0 vanishes in the m →
0. From thisexpression, we immediately obtain ζ H p +1 × S (0) = 0 ,∂ s ζ H p +1 × S (0) = 2 c p +1 p X k =1 α k,p +1 ( − k +1 π k + 1 ζ ( − k − . (4.36)Since both bulk and defect are even dimensional when p is even the renormalized free energy(4.34) may have both types of anomalies: F ren [ H p +1 × S ] = − A [ H p +1 × S ] log(Λ R ) − A [ H p +1 × S ] log (cid:18) Rǫ (cid:19) , (4.37)but from (4.36) we find A [ H p +1 × S ] = 0 , A [ H p +1 × S ] = ( − p Γ( p + 1) p X k =1 α k,p +1 − k +1 k + 1 ζ ( − k − . (4.38)Thus defect anomalies are there while bulk anomalies vanish in this case.The explicit values of the anomaly coefficients for d ≤
10 are listed in table 2 in appendixA. p For odd p we use the identity (4.17) to the Plancherel measure µ ( p +1) ( ω ) (4.4) and applydifferent decompositions to each term to derive F [ H p +1 × S ] = c p +1 p − X k =0 β k,p +1 ∞ X ℓ = −∞ Z ∞ d ω ω k +1 (cid:20) log (cid:18) ω + i | ℓ | ˜Λ R (cid:19) + log (cid:18) ω − i | ℓ | ˜Λ R (cid:19)(cid:21) − c p +1 p − X k =0 β k,p +1 Z ∞ d ω ω k +1 e πω + 1 ∞ X ℓ = −∞ (cid:20) log (cid:18) | ℓ | + i ω ˜Λ R (cid:19) + log (cid:18) | ℓ | − i ω ˜Λ R (cid:19)(cid:21) . (4.39)– 29 –ere the ordering between the integral and the summation is important in this expression.In the Schwinger representation of (4.39), the first term is convergent in the ω → ∞ limit,while the second term is convergent in the | ℓ | → ∞ limit. Repeating similar computations as in section 3 and section 4.1.2, the renormalized freeenergy on H p +1 × S is given by the same form as (4.34) with the zeta function consisting oftwo parts: ζ H p +1 × S ( s ) = ζ (1) H p +1 × S ( s ) + ζ (2) H p +1 × S ( s ) ,ζ (1) H p +1 × S ( s ) = c p +1 p − X k =0 β k,p +1 ∞ X ℓ = −∞ Z ∞ d ω ω k +1 (cid:2) ( ω + i | ℓ | ) − s + ( ω − i | ℓ | ) − s (cid:3) ,ζ (2) H p +1 × S ( s ) = − c p +1 p − X k =0 β k,p +1 Z ∞ d ω ω k +1 e πω + 1 ∞ X ℓ = −∞ (cid:2) ( | ℓ | + i ω ) − s + ( | ℓ | − i ω ) − s (cid:3) . (4.42)The first term can be computed as ζ (1) H p +1 × S ( s ) = 4 c p +1 p − X k =0 β k,p +1 ( − k +1 cos (cid:16) πs (cid:17) Γ(2 k + 2) ζ ( s − k − k +2 Y i =1 s − i , (4.43)where we regulate the ℓ = 0 mode in the same way as for the even p case. It follows that ζ (1) H p +1 × S ( s ) and its derivative at s = 0 are given by ζ (1) H p +1 × S (0) = 0 ,∂ s ζ (1) H p +1 × S (0) = 2 c p +1 p − X k =0 β k,p +1 ( − k +1 k + 1 ζ ′ ( − k − . (4.44) For odd p , it is possible to use a different representation of the zeta function ζ H p +1 × S ( s ) = Z ∞ d ω µ ( p +1) ( ω ) ∞ X ℓ = −∞ (cid:2) ( | ℓ | + i ω ) − s + ( | ℓ | − i ω ) − s (cid:3) = Z ∞ d ω µ ( p +1) ( ω ) [ ζ H ( s, − i ω ) + ζ H ( s, − i ω ) + ζ H ( s, i ω ) + ζ H ( s, ω )] . (4.40)In the s → ζ H p +1 × S (0) = 0 ,∂ s ζ H p +1 × S (0) = − Z ∞ d ω µ ( p +1) ( ω ) log(2 sinh( πω )) , (4.41)where we use (4.46) and (4.48). Hence we obtained the same expression of the free energy in [62]. However,the derivative of the zeta function still diverges. Hence we need to use the same regularization as in [19, 62]. – 30 –he second one can be written as ζ (2) H p +1 × S ( s ) = − c p +1 p − X k =0 β k,p +1 Z ∞ d ω ω k +1 e πω + 1 · [ ζ H ( s, − i ω ) + ζ H ( s, − i ω ) + ζ H ( s, i ω ) + ζ H ( s, ω )] . (4.45)Although it is difficult to perform the integration over ω , it is possible to compute ζ (2) H p +1 × S (0)and ∂ s ζ (2) H p +1 × S (0). Since the combination of the Hurwitz zeta function vanishes, ζ H (0 , − i ω ) + ζ H (0 , − i ω ) + ζ H (0 , i ω ) + ζ H (0 , ω ) = 0 , (4.46)in the s → ζ (2) H p +1 × S (0) = 0 . (4.47)Using the derivative of the combination of the Hurwitz functions,lim s → ∂ s [ ζ H ( s, − i ω ) + ζ H ( s, − i ω ) + ζ H ( s, i ω ) + ζ H ( s, ω )] = − πω )) , (4.48)which is the same as the regularization of P ℓ log( ω + | ℓ | ) [19, 62], the derivative of ζ (2) H p +1 × S ( s )reduces to ∂ s ζ (2) H p +1 × S (0) = 4 c p +1 p − X k =0 β k,p +1 Γ(2 k + 2)(2 π ) k +2 " k X m =1 (1 − m − k − ) ζ (2 m ) ζ ( − m + 2 k + 3) − (1 − − k − ) ζ (2 k + 3) + 2(1 − − k − ) ζ (2 k + 2) log 2 i , (4.49)where we use formulas eq. (4) and eq. (14) in [86] after changing a variable u = e − πω .Since ζ H p +1 × S (0) = ζ (1) H p +1 × S (0) + ζ (2) H p +1 × S (0) = 0, neither bulk nor defect anomalyappear: A [ H p +1 × S ] = A [ H p +1 × S ] = 0 . (4.50)This is consistent with the fact that both p + q and p are odd. Hence the renormalized freeenergy has only a finite term: F ren [ H p +1 × S ] = − c p +1 p − X k =0 β k,p +1 Γ(2 k + 2)(2 π ) k +2 " k X m =1 (1 − m − k − ) ζ (2 m ) ζ ( − m + 2 k + 3) − − − k − ζ (2 k + 3) + 2(1 − − k − ) ζ (2 k + 2) log 2 (cid:21) . (4.51)– 31 –et us compare the free energy on H p +1 × S with the free energy on S p +2 (3.19). Weargue the equivalence between (4.51) and (3.19) for arbitrary p : F ren [ H p +1 × S ] = F ren [ S p +2 ] . (4.52)We are not aware of an analytic proof of this identity, but we check it holds up to p of order O (100). Given the equivalence of the free energies one can derive a number of mathematicalidentities for the Riemann zeta functions which appear to be unknown in literature. H p +1 × S q − Expanding the scalar field by the spherical harmonics labeled by ℓ on S q − the free energyfor Dirichlet boundary condition on H p +1 × S q − is given by F [ H p +1 × S q − ] = 12 tr log (cid:20) ˜Λ − (cid:18) −∇ H p +1 − ∇ S q − + ( q − p − d − R (cid:19)(cid:21) = 12 ∞ X ℓ =0 g ( q − ( ℓ ) Z ∞ d ω µ ( p +1) ( ω ) log ω + (cid:16) ν ( q − ℓ (cid:17) ˜Λ R , (4.53)with the Plancherel measure µ ( p +1) ( ω ) given by (4.4), the degeneracy g ( q − ( ℓ ) and ν ( q − ℓ for q > F ren [ H p +1 × S q − ] ≡ − ζ H p +1 × S q − (0) log(Λ R ) − ∂ s ζ H p +1 × S q − (0) , (4.54)where ζ H p +1 × S q − ( s ) is the summation of the zeta function on H p +1 over the angular modes: ζ H p +1 × S q − ( s ) ≡ ∞ X ℓ =0 g ( q − ( ℓ ) ζ H p +1 ( s, ν ( q − ℓ ) . (4.55)As in section 4.2, we use different expressions for the decompositions of the logarithmicfunctions depending on the cases so that the resulting forms have good convergent behaviorsin the Schwinger representation. p The regularized volume of H p +1 (4.6) has a logarithmic divergence after the regularization(4.7) and the renormalized free energy (4.54) has two types of logarithmic divergences: F ren [ H p +1 × S q − ] = − A [ H p +1 × S q − ] log(Λ R ) − A [ H p +1 × S q − ] log (cid:18) Rǫ (cid:19) . (4.56)– 32 –sing the degeneracy (3.9) and the expansion (3.11), we can perform the sum over ℓ toget ζ H p +1 × S q − ( s ) = 4 c p +1 Γ( q − p X k =1 ( − k α k,p +1 sin (cid:16) πs (cid:17) Γ (2 k + 1) k +1 Y i =1 s − i · q − X n =0 ( − q − n α n,q − ζ H (cid:18) s − k − n − , q − (cid:19) q : even q − X n =0 ( − q − + n β n,q − ζ H (cid:18) s − k − n − , q − (cid:19) q : odd (4.57)Using the identities (B.9), (B.12), (B.2) and the relations (3.10) we obtain ζ H p +1 × S q − (0) = 0 ,∂ s ζ H p +1 × S q − (0) = 2 π c p +1 Γ( q − p X k =1 α k,p +1 ( − k +1 k + 1 · q − X n =0 ( − q + n k + 2 n + 2 α n,q − B k +2 n +2 q : even0 q : odd (4.58)Hence we find the followings: • For even q , the renormalized free energy has the logarithmic divergence F ren [ H p +1 × S q − ] = −A [ H p +1 × S q − ] log (cid:18) Rǫ (cid:19) , ( p : even , q : even) , (4.59)where the anomaly coefficient is given by A [ H p +1 × S q − ] = ( − p + q Γ( p + 1) Γ( q − p X k =1 q − X n =1 α k,p +1 α n,q − ( − k + n +1 (2 k + 1)( k + n + 1) B k +2 n +2 . (4.60) • For odd q , we find A [ H p +1 × S q − ] ∝ ζ H p +1 × S q − (0) = 0 and A [ H p +1 × S q − ] ∝ ∂ s ζ H p +1 × S q − (0) = 0, and there are no conformal anomalies. From (4.58) the finiteterm of the renormalized free energy also vanishes, so F ren [ H p +1 × S q − ] = 0 , ( p : even , q : odd) . (4.61)This is consistent with the results obtained by [62].– 33 – .3.2 Odd p As in section 4.2.2, we decompose the zeta function into two parts: ζ H p +1 × S q − ( s ) = ζ (1) H p +1 × S q − ( s ) + ζ (2) H p +1 × S q − ( s ) , (4.62)where ζ (1) H p +1 × S q − ( s ) and ζ (1) H p +1 × S q − ( s ) are defined by ζ (1) H p +1 × S q − ( s ) = c p +1 p − X k =0 β k,p +1 ∞ X ℓ =0 g ( q − ( ℓ ) · Z ∞ d ω ω k +1 (cid:20)(cid:16) ω + i ν ( q − ℓ (cid:17) − s + (cid:16) ω − i ν ( q − ℓ (cid:17) − s (cid:21) ,ζ (2) H p +1 × S q − ( s ) = − c p +1 p − X k =0 β k,p +1 Z ∞ d ω ω k +1 e πω + 1 · ∞ X ℓ =0 g ( q − ( ℓ ) (cid:20)(cid:16) ν ( q − ℓ + i ω (cid:17) − s + (cid:16) ν ( q − ℓ − i ω (cid:17) − s (cid:21) . (4.63)By performing the integration first for ζ (1) H p +1 × S q − ( s ), we obtain ζ (1) H p +1 × S q − ( s ) = 4 c p +1 Γ( q − p − X k =0 ( − k +1 β k,p +1 cos (cid:16) πs (cid:17) Γ(2 k + 2) k +2 Y i =1 s − i · q − X n =1 ( − q − n α n,q − ζ ( s − k − n − q : even q − X n =0 ( − q − + n β n,q − (2 s − k − n − − ζ ( s − k − n − q : odd(4.64)where we used (3.10) with d → q −
1. For ζ (2) H p +1 × S q − ( s ) we sum over ℓ first and use theexpansion (3.9) with d → q − c → i ω and the identity (B.9) as in section 3 to write ζ (2) H p +1 × S q − ( s ) = − c p +1 Γ( q − p − X k =0 β k,p +1 Z ∞ d ω ω k +1 e πω + 1 · q − X n =0 [ γ n,q − (i ω ) ζ H ( s − n, i ω ) + γ n,q − ( − i ω ) ζ H ( s − n, − i ω )] q : even q − X n =0 (cid:20) γ n,q − (i ω ) ζ H (cid:18) s − n,
12 + i ω (cid:19) + γ n,q − ( − i ω ) ζ H (cid:18) s − n, − i ω (cid:19)(cid:21) q : odd(4.65) We use the regularization scheme for ζ (2) H p +1 × S q − ( s ) which is different from that of ζ (1) H p +1 × S q − ( s ). – 34 –or odd p the regularized volume of H p +1 is finite and the coefficient c p +1 given by (4.8)does not give rise to a logarithmic divergence in the zeta function. It follows from (4.54) thatthe logarithmic divergence of the free energy is determined by F ren [ H p +1 × S q − ] = − A [ H p +1 × S q − ] log(Λ R ) + F fin [ H p +1 × S q − ] , (4.66)where the anomaly part and the finite part are A [ H p +1 × S q − ] = 12 (cid:16) ζ (1) H p +1 × S q − (0) + ζ (2) H p +1 × S q − (0) (cid:17) ,F fin [ H p +1 × S q − ] = − (cid:16) ∂ s ζ (1) H p +1 × S q − (0) + ∂ s ζ (2) H p +1 × S q − (0) (cid:17) . (4.67)From (4.64) we read ζ (1) H p +1 × S q − (0) = 2 c p +1 Γ( q − p − X k =0 ( − k +1 β k,p +1 k + 1 · q : even q − X n =0 ( − q − + n β n,q − (2 − k − n − − ζ ( − k − n − q : odd(4.68)while from (4.65) and after a bit of calculation, we obtain ζ (2) H p +1 × S q − (0)= 8 c p +1 Γ( q − p − X k =0 β k,p +1 · q : even q − X m =0 2 m +1 X n =0 ( − q − + n n + 1 (cid:18) m + 1 n (cid:19) β m,q − · ⌊ n +12 ⌋ X r =0 ( − r (1 − − r ) (cid:18) n + 12 r (cid:19) B r · − − m − k − r (2 π ) m +2 k +4 − r Γ(2 m + 2 k + 4 − r ) ζ (2 m + 2 k + 4 − r ) q : odd(4.69)– 35 –rom (4.64) and (4.65), we read ∂ s ζ (1) H p +1 × S q − (0)= 2 c p +1 Γ( q − p − X k =0 ( − k +1 k + 1 β k,p +1 · q − X n =1 ( − q − n α n,q − ζ ′ ( − k + n + 1)) q : even q − X n =0 ( − q − + n β n,q − (cid:20) (2 − n − k − − ζ ′ ( − k − n − (cid:16) (2 − n − k − − H k +2 + 2 − n − k − log 2 (cid:17) ζ ( − k − n − (cid:21) q : odd(4.70)and ∂ s ζ (2) H p +1 × S q − (0)= − c p +1 Γ( q − p − X k =0 β k,p +1 Z ∞ d ω ω k +1 e πω + 1 q − X n =0 · q − X m = ⌈ n ⌉ ( − q − m + n (cid:18) mn (cid:19) α m,q − (i ω ) m − n · [ ∂ s ζ H ( − n, i ω ) + ( − n ∂ s ζ H ( − n, − i ω )] q : even q − X m = ⌊ n ⌋ ( − q − + m + n (cid:18) m + 1 n (cid:19) β m,q − (i ω ) m +1 − n · (cid:20) ∂ s ζ H (cid:18) − n,
12 + i ω (cid:19) + ( − n − ∂ s ζ H (cid:18) − n, − i ω (cid:19)(cid:21) q : odd (4.71)For even q , the bracket can be written by using the polylogarithm functions Li n ( x ), ∂ s ζ (2) H p +1 × S q − (0)= − c p +1 Γ( q − p − X k =0 β k,p +1 Z ∞ d ω ω k +1 e πω + 1 q − X n =0 q − X m = ⌈ n ⌉ ( − q − n (cid:18) mn (cid:19) α m,q − · ω m − n ( − n Γ( n + 1)(2 π ) n Li n +1 (e − πω ) + ⌊ n +12 ⌋ X l =0 ( − l − πn + 1 (cid:18) n + 1 n + 1 − l (cid:19) B l ω n +1 − l , (4.72)– 36 –here we use (B.19) and the facts B n +1 − r vanishes for even n − r > m vanishes for n = r in the third line. Our method reproduces the known regularization of P ℓ g ( q − ( ℓ ) log( ω + ( ν ( q − ℓ ) ) ((3.38) and (3.50) in [62]) and is easily generalized to higherdimensions.For odd p and even q the anomaly parts vanish while for odd p and odd q they turnout to equal the bulk anomaly of S p + q . We thus find a set of identities relating the anomalycoefficients on the conformally equivalent spaces: A [ H k × S d − k ] = A [ S d ] . (4.73)Combined with the result in section 4.2.2, the above relation holds for k = 1 , · · · , ⌈ d/ ⌉ − F fin [ H k × S d − k ] = F fin [ S d ] . (4.74)The equality between the finite parts is pointed out in [62] for odd d ≤ F ren [ H k × S d − k ] = F ren [ S d ] , ( k = 1 , · · · , ⌈ d/ ⌉ − . (4.75)We checked them either by analytically or numerically for d of order O (10). They shouldhold for any d on physical ground as there are no defect anomalies when p is odd while H k × S d − k has the same Euler characteristic as S d , thus has the same bulk anomaly as S d .Substituting various values for p and q to the anomaly parts and the finite parts of thefree energies, we obtain tables 2 and 3 in appendix A. In section 4, we computed the free energies on H d and H p +1 × S q − for Dirichlet boundarycondition. In this section, we turn to the case for Neumann boundary condition. We calculatethe difference of the free energies between Neumann and Dirichlet boundary conditions in twoways. First we calculate the free energy on H d for Neumann boundary condition from theresult for Dirichlet boundary condition by analytically continuing the dimension ∆ from theDirichlet value ∆ + to the Neumann value ∆ − . Next we use the residue method for the samecalculation. These two methods turn out to give the same answer. We then apply the residuemethod to the calculation of the free energy on H p +1 × S q − . Finally, we check the defect C -theorem (1.5) holds for all the cases. While we do not write explicitly, this equality leads to the evaluated form of integrals including thepolygamma function, which to our best knowledge has not appeared in literature. – 37 – .1 Analytic continuation
The free energy for Neumann boundary condition can be derived from the Dirichlet value on H d obtained in section 4 as the latter is analytic in ν , so can be analytically continued fromthe positive ν to negative ν region. Odd d From (4.13), we read off the free energy as a function of ν , F ren [ H d ]( ν ) = ( − d − Γ( d ) d − X k =1 ( − k α k,d ν k +1 k + 1 log (cid:18) Rǫ (cid:19) . (5.1)The free energy with Neumann boundary condition, ν = − /
2, is given by F ren [ H d ]( − /
2) = − F ren [ H d ](1 / . (5.2)The anomaly parts of the Neumann boundary condition are the minus of those for the Dirichletboundary condition. The difference of the free energies between the two boundary conditionsis given by F ∆ + [ H d ] − F ∆ − [ H d ] = − A [ H d ] log (cid:18) Rǫ (cid:19) , (5.3)where we introduced new notations F ∆ + [ H d ] = F ren [ H d ](1 /
2) and F ∆ − [ H d ] = F ren [ H d ]( − / H d with the Neumann boundarycondition coincides with that on HS d − provided the two cutoffs ǫ and Λ are appropriatelyidentified. Even d The free energy for even d is F ren [ H d ]( ν ) = − ζ H d (0 , ν ) log(Λ R ) − ∂ s ζ H d (0 , ν ) . (5.4)The zeta function and its derivative defined by (4.19) are analytical functions of ν , and theycan be analytically continued to the ν < ν should be understood as (1 /
2) log ν ).Then most of the terms are canceled out in the difference of the free energy except for f k ( ν ),resulting in F ∆ + [ H d ] − F ∆ − [ H d ] = − c d Z d µ µ sin( πµ ) Γ (cid:18) d − ± µ (cid:19) , (5.5)where we use the identities ψ ( µ + 1 / − ψ ( − µ + 1 /
2) = π tan( πµ ) and d − X k =0 ( − k β k,d µ k +1 = µπ Γ (cid:18) d − ± µ (cid:19) cos( πµ ) , (5.6)– 38 –hich follows from (3.3) and (3.10).In lower dimensions, the difference of the free energy becomes F ∆ + [ H d ] − F ∆ − [ H d ] = − log(2 π ) + ∂ s ζ H (0 , d = 2 − π ζ (3) d = 4 π ζ (3) + π ζ (5) d = 6 − π ζ (3) − π ζ (5) − π ζ (7) d = 8 (5.7)where we use the identity that follows from (C.9) for d = 2. We find perfect agreement withthe difference of the free energies on HS d listed in table 4 while both of the differences on H and HS have the same IR divergences from ∂ s ζ H (0 , F ∆ + [ H d ] − F ∆ − [ H d ] = F fin [ HS d + ] − F fin [ HS d − ] , (5.8)should hold for even d ≥ In total, we conclude from the agreements of the universal parts of the free energies thatthe Dirichlet/Neumann boundary conditions on H d are one-to-one correspondence with thoseon HS d for any d . We will derive the same conclusion from a more indirect method in thefollowing. In the previous section, we obtained the difference of the free energies via a naive analyticalcontinuation in terms of the parameter ν . The same result can be derived by using the residuemethod [54, 56], which argues that the difference of the derivatives of the free energies is givenby the residue of µ ( d ) ( ω ) / (2 ω ( ω − i ν )) at ω = i ν with suitable normalization: ∂ ν F [ H d ]( ν ) − ∂ ν F [ H d ]( − ν ) = 2 π i ν Res ω =i ν µ ( d ) ( ω )2 ω ( ω − i ν ) (5.10)= − c d ν sin( πν ) Γ (cid:18) d − ± ν (cid:19) . (5.11)By integrating the above expression (5.11) from ν = 0 to ν = , we obtain F ∆ + [ H d ] − F ∆ − [ H d ] = 1sin( π d − ) Γ( d ) Z d ν ν sin( πν ) Γ (cid:18) d − ± ν (cid:19) . (5.12)For even d , this expression is the same as (5.5) derived from the analytic continuation. Also,by replacing the pole from the sine function for odd d with the logarithmic divergence using(4.7) this also reproduces the boundary anomaly given in (5.3). With (3.51) and (5.5) this equality leads to the identity for even d ≥ Z d µ µ sin( πµ ) Γ (cid:18) d − ± µ (cid:19) = ( d − d − X n =1 n X l =1 ( − n β n,d l − n n l ! ζ ′ ( − l ) . (5.9) – 39 –e( ω )Im( ω ) i − i C + C − Figure 2 . The contours for the Green’s functions for the Dirichlet (the blue real line) and Neumannboundary conditions (the blue real line and two orange circles C + and C − ). Derivation of the residue method
The Green’s function of a massive scalar field withDirichlet boundary condition has the integral representation [87]: G ∆ + ( x , x ) = 1 R d − Z ∞−∞ d ω ω + (cid:0) ∆ + − d − (cid:1) Ω ( d ) ω ( x , x ) . (5.13)The Green’s function with Neumann boundary condition can be obtained by changing thecontour as in figure 2: G ∆ − ( x , x ) = 1 R d − Z R + C + + C − d ω ω + (cid:0) ∆ + − d − (cid:1) Ω ( d ) ω ( x , x ) , (5.14)where C + is a clockwise circle around a pole at ω = i (cid:0) ∆ + − d − (cid:1) and C − is a counter-clockwise circle around a pole at ω = − i (cid:0) ∆ + − d − (cid:1) .In the following, we need the expression of Ω ( d ) ω ( x , x ) at the coincident point,Ω ( d ) ω (0) = Ω ( d ) ω ( x, x ) = Γ (cid:0) d − (cid:1) π d − +1 Γ( d −
1) Γ (cid:0) d − ± i ω (cid:1) Γ( ± i ω ) . (5.15)The Plancherel measure (4.4) can be written by using Ω ( d ) ω (0) as µ ( d ) ( ω ) = 2 Vol( H d ) Ω ( d ) ω (0) . (5.16)Now the derivative of the free energy can be expressed as ∂ ν F [ H d ]( ν ) = Vol( H d ) ν G ∆ + ( x, x ) ,∂ ν F [ H d ]( − ν ) = Vol( H d ) ν G ∆ − ( x, x ) . (5.17)– 40 –hen, the difference between them is given by ∂ ν F [ H d ]( ν ) − ∂ ν F [ H d ]( − ν ) = − ν Z C + + C − d ω µ ( d ) ( ω ) ω + ν = π i ν ω =i ν µ ( d ) ( ω ) ω ( ω − i ν ) − π i ν ω = − i ν µ ( d ) ( ω ) ω ( ω + i ν )= π µ ( d ) (i ν ) . (5.18)In the last line, we used µ ( d ) (i ν ) = µ ( d ) ( − i ν ). This completes the derivation of the residuemethod (5.10). H p +1 × S q − Now let us apply the residue method 5.2 to the free energy calculation on H p +1 × S q − withNeumann boundary condition. Since the Neumann boundary condition has negative ν (1) ℓ = ± for q = 2 or negative ν ( q − ℓ =0 for q ≥
3, it is convenient to express the free energy as a sum ofeach mode F [ H p +1 × S q − ] = ∞ X ℓ =0 g ( q − ( ℓ ) F ℓ (cid:16) ν ( q − ℓ (cid:17) , (5.19)where F ℓ (cid:16) ν ( q − ℓ (cid:17) is the free energy for the ℓ -th mode on H p +1 : F ℓ (cid:16) ν ( q − ℓ (cid:17) ≡ Z ∞ d ω µ ( p +1) ( ω ) log ω + (cid:16) ν ( q − ℓ (cid:17) ˜Λ R , (5.20)and ν ( q − ℓ = ∆ ℓ − p as before. For q = 2, we have to take a summation from ℓ = −∞ to ∞ .By applying the residue method (5.11), we obtain ∂ ν ℓ F ℓ ( ν ℓ ) − ∂ ν ℓ F ℓ ( − ν ℓ ) = − c p +1 ν ℓ sin( πν ℓ ) Γ (cid:18) d − ± ν ℓ (cid:19) . (5.21)Hereafter we omit the superscript ( q −
1) in ν ( q − ℓ to simplify expressions. In the following,we will compute the difference of the free energies between ∆ D and ∆ N . q = 2 case: The allowed boundary conditions are classified in (2.28). The difference of thefree energies between ∆ + and ∆ N1 comes from the ν mode, where the former has ν = 1(∆ ℓ =1+ = p + 1) while the latter has ν = − ℓ =1 − = p − ν mode and integrating from ν = 0 to ν = 1, we obtain F ( ν = 1) − F ( ν = −
1) = 1sin (cid:0) π p (cid:1) Γ( p + 1) Z d ν ν sin( πν ) Γ (cid:16) p ± ν (cid:17) = − F univ [ S p ] . (5.22) It is also possible to apply the analytic continuation method in section 5.1, which gives the same result.Here we use the residue method due to its simplicity. – 41 –n the last line, we use the integral expression of the sphere free energy (3.25). This implies F ∆ D [ H p +1 × S ] − F ∆ N1 [ H p +1 × S ] = − F univ [ S p ] . (5.23)In the same way, the difference of the free energies between the ∆ N1 and ∆ N2 boundaryconditions reads F ∆ N1 [ H p +1 × S ] − F ∆ N2 [ H p +1 × S ] = − F univ [ S p ] . (5.24)Note that the difference of the free energies equals the sphere free energy of a p -dimensionalfree scalar field. This result conforms with the fact that the Neumann boundary conditionsfor q = 2 are trivial in the sense that the defect operator saturates the unitarity bound andbecomes a free field. q = 3 case: In this case the difference of the free energies between the two boundaryconditions (2.29) comes from the ν mode only, where the Dirichlet boundary condition has ν = (∆ ℓ =0+ = p +12 ) while the Neumann boundary condition has ν = − (∆ ℓ =0 − = p − ).By applying the residue method (5.21) to the ν mode and integration from ν = 0 to , thedifference of the free energies is F (cid:18) ν = 12 (cid:19) − F (cid:18) ν = − (cid:19) = 1sin (cid:0) π p (cid:1) Γ( p + 1) Z d ν ν sin( πν ) Γ (cid:16) p ± ν (cid:17) = F ∆ + [ H p +1 ] − F ∆ − [ H p +1 ] . (5.25)In the final line, we use the integral expression of the difference of the free energy on H p +1 (5.12). Hence we conclude that F ∆ D [ H p +1 × S ] − F ∆ N [ H p +1 × S ] = F ∆ + [ H p +1 ] − F ∆ − [ H p +1 ] . (5.26) q = 4 case: The difference of the free energies between the two boundary conditions (2.29)comes from the ν mode for p ≥
2, where the Dirichlet condition has ν = 1 (∆ ℓ =0+ = p + 1)while the Neumann condition has ν = − ℓ =0 − = p − ν mode and integrating from ν = 0 to 1, we find F ( ν = 1) − F ( ν = −
1) = 1sin (cid:0) π p (cid:1) Γ( p + 1) Z d ν ν sin( πν ) Γ (cid:16) p ± ν (cid:17) = − F univ [ S p ] . (5.27)Hence we conclude that F ∆ D [ H p +1 × S ] − F ∆ N [ H p +1 × S ] = − F univ [ S p ] . (5.28)As in q = 2 the difference if given by the sphere free energy of a p -dimensional scalar field.This also conforms with the saturation of the unitarity bound for the Neumann conditionwhen q = 4. – 42 – ree boundary condition: In this case we see from (2.18) that the free boundary conditionassociated with a p -dimensional scalar Wilson loop exists for q = p +2 and q ≥ ℓ =0 − = 0while the Dirichlet condition has ∆ ℓ =0+ = p . Thus, the difference of the free energies betweenthe two is given by F ∆ D [ H p +1 × S p +1 ] − F ∆ F [ H p +1 × S p +1 ] = F (cid:16) ν = p (cid:17) − F (cid:16) ν = − p (cid:17) = 1sin (cid:0) π p (cid:1) Γ( p + 1) Z p d ν ν sin( πν ) Γ (cid:16) p ± ν (cid:17) . (5.29)The integral of the right hand side suffers from the IR divergences due to the zero mode forthe free boundary condition with ν = − p . C -theorem Let us compare the results in sections 5.1, 5.2 and 5.3 with the proposed C -theorem (1.5) inDCFT. We anticipate that the difference of the defect free energies is invariant under Weyltransformations. Since one can trigger the defect RG flow from the Neumann to Dirichletboundary condition by the double trace deformation [48–54] the difference of the free energiesbetween the UV and IR fixed points is given by˜ D UV − ˜ D IR = − sin (cid:16) π p (cid:17) (cid:0) F ∆ N [ H p +1 × S q − ] − F ∆ D [ H p +1 × S q − ] (cid:1) . (5.30)For the free boundary condition, however, the double trace deformation does not leadto a defect RG flow to the Dirichlet boundary condition as the double trace operator hasdimension zero and is proportional to the defect identity operator. Presumably there aredefect RG flows between the free and Dirichlet boundary conditions which we are not awareof, but we concentrate on our consideration to the flow driven by the double trace deformationof the Neumann boundary condition with positive dimension ∆ N >
0. This leaves us the caseswith q ≤ C -theorem (1.5) forthe case with the free boundary conditions in section 6. q = 1 case: In this case we consider the defect RG flow on the boundary of H d ( p = d − D UV − ˜ D IR = 1Γ( d ) Z d ν ν sin( πν ) Γ (cid:18) d − ± ν (cid:19) , (5.31)which is always positive and the monotonicity of the defect free energy holds for p ≥ d ≥ The defect free energy of a scalar Wilson loop in four dimension has a similar IR divergence, but is shownto be zero after an IR regularization in [88]. We do not know if such a regularization can be applied to ourcase. – 43 – = 2 case:
From (5.23) and (5.24) we obtain˜ D ∆ N1 − ˜ D ∆ D = ˜ D ∆ N2 − ˜ D ∆ N1 = ˜ F [ S p ] , (5.32)where we used ∆ N1 , ∆ N2 , ∆ D instead of UV and IR and˜ F [ S p ] ≡ p + 1) Z d ν ν sin( πν ) Γ (cid:16) p ± ν (cid:17) (5.33)is positive for any p ≥ p ≥ C -theorem is satisfied inthis case. q = 3 case: The relation (5.26) can be translated to˜ D UV [ H p +1 × S ] − ˜ D IR [ H p +1 × S ] = ˜ D UV [ H p +1 ] − ˜ D IR [ H p +1 ] , (5.34)where we make explicit the dependence of ˜ D on the space. Hence the monotonicity of thedefect free energy amounts to that of the q = 1 case. q = 4 case: Since (5.28) can be translated to˜ D UV − ˜ D IR = ˜ F [ S p ] , (5.35)the monotonicity of the defect free energy holds for the same reason as in the q = 2 case for p ≥ In this paper we classified a certain class of conformal defects in the free scalar theory asboundary conditions on H p +1 × S q − . Our results are consistent with the classification ofthe non-monodromy defects in [59] carried out by other means. We believe our methods forcharacterizing conformal defects as boundary conditions on H p +1 × S q − can be applied to themonodromy defects classified in [59] as well. As a special case twist operators of codimension-two were studied as a boundary condition on H p +1 × S in [64]. It is also worthwhile to revisitthe O ( N ) model discussed in a recent paper [56] that admits various non-trivial boundaryconditions and supersymmetric theories with defects [43, 89–107] from the viewpoint of thispaper. (See also [108–111] for related works.)It should be possible to extend our analysis to fields with spin. For fermion, a non-trivialboundary condition is allowed, and we can consider an RG flow from Neumann to Dirichletboundary condition. We will report this result in [112]. For a symmetric traceless tensor withspin s , ∆ + always satisfies the unitarity bound∆ ≥ d + s − , (6.1)– 44 –nd ∆ + for H d also saturates the unitarity bound. However, ∆ − always violates the unitaritybound, which implies that a non-trivial Neumann boundary condition never exists for higherspin fields.By comparing our results with the classification by [59] we observe that Dirichlet bound-ary condition corresponds to trivial (or no) defects while Neumann boundary condition tonon-trivial defects. Indeed, we verified this observation through the free energy calculationsin some cases, which leads us to speculate that defects with Dirichlet boundary conditionhave a least ˜ D under any RG flow.In section 5.3 we examined the defect RG flow triggered by the double trace deformationof the Neumann boundary condition with ∆ N >
0. This restriction excludes the flows fromthe free boundary condition with zero mode (∆ F = 0) from our consideration. While we arenot aware of any defect RG flow between the free and Dirichlet boundary conditions one canstill calculate the difference of the defect free energy from (5.29):˜ D F − ˜ D D = 1Γ( p + 1) Z p d ν ν sin( πν ) Γ (cid:16) p ± ν (cid:17) . (6.2)The integral diverges for odd p due to the IR divergence from the zero mode while it is positiveand finite for p = 4 m − m = 1 , , · · · ) and negative and finite for p = 4 m . In this case thedefect C -theorem implies that the free boundary condition may be a UV fixed point of somedefect RG flow for p = 4 m − p = 4 m . (One cannotdraw any implication for odd p due to the IR divergence.) We leave further investigation onthis direction as a future work.The entanglement entropy of a spherical region on flat space R d can be mapped to thefree energy on H d − × S by the Casini-Huerta-Myers map [85]. In this context, the boundarycondition on the entangling surface, or equivalently the boundary condition on H d − × S ,has not been clarified explicitly. However, our results show that the boundary condition on H d − × S changes the universal parts of the free energy, and this implies that we should becareful in the boundary condition in the entanglement entropy. In [9] we derived a universalrelation between the defect free energy and defect entropy. They differ by a term proportionalto the one-point function of the stress tensor in the presence of defects, so one can calculatethe defect entropy from our results in this paper given the one-point function, without relyingon conventional methods such as the replica trick.The free energies for the Neumann boundary conditions were obtained somewhat indi-rectly as differences from the Dirichlet cases. This was enough for us to check if the defect C -theorem (1.5) holds under the defect RG flow as we assumed the difference of the freeenergies is invariant under conformal maps from S d to H p +1 × S q − . Nonetheless it is desir-able to have a precise relation between the defect free energy on S d and the free energy on– 45 – p +1 × S q − . A most naive guess would be D = log |hD ( p ) i| ? = F [ H p +1 × S q − ] − F [ S d ] . (6.3)However, this relation does not hold in general as the bulk anomalies are canceled out inthe left hand side while there can remain a bulk anomaly term in the right hand side. Forinstance, F [ S d ] should have bulk anomalies when d is even for any d . On the other hand,there are no bulk anomalies in F [ H p +1 × S q − ] when p is even as seen from (1.11) (only defectanomalies are there). Hence there remains the bulk anomaly in the right hand side of (6.3).Finding a correct relation between D and F [ H p +1 × S q − ] should be of interest.In BCFT d with even d , the conformal anomalies in the bulk theory have boundary termsdictated by boundary central charges [55, 113–117]. The free energy has a logarithmic di-vergence whose coefficient is completely fixed by the geometry of the boundary such as theextrinsic curvature at least in lower dimensions [113, 114]. In DCFT, we regard confor-mal defects as boundary conditions on H p +1 × S q − , so we may view ( ∂ H p +1 ) × S q − asthe codimension-one boundary and are tempted to apply the boundary anomaly formula[113, 114] to the present case. In a few cases, we computed the defect anomaly coefficientsfrom the boundary anomaly formula, but we were not able to reproduce our results correctly.We suspect the boundary anomaly formula may not be applicable to manifolds with boundarywhich is a product manifold. We hope to address this issue in future. Acknowledgments
We would like to thank A. O’Bannon with his research group, C. P. Herzog, D. Rodriguez-Gomez, J. G. Russo and M. Watanabe for valuable discussion and correspondences. We alsothank J. S. Dowker for providing us proofs of several identities conjectured in the earlierversion of the paper. The work of T. N. was supported in part by the JSPS Grant-in-Aidfor Scientific Research (C) No.19K03863 and the JSPS Grant-in-Aid for Scientific Research(A) No.16H02182. The work of Y. S. was supported by the National Center of TheoreticalSciences (NCTS). The absence of the bulk anomaly on H p +1 × S q − follows from the fact that the anomaly coefficient isproportional to the Euler characteristic of the manifold and χ [ H p +1 × S q − ] = 0 as χ [ H p +1 ] = 0 for even p . – 46 – List of tables · S S S S · − H − H − − H − − H − H − H − − H − − H − H
10 26314968800 − S S S S · − H − H − H − H − H − H − H − H − H − Table 2 . The bulk anomalies A [ H p +1 × S q − ] and the defect anomalies A [ H p +1 × S q − ] (shaded) on H p +1 × S q − with Dirichlet boundary conditions. – 47 – F fin [ M ] S − log(2 π ) − ζ ′ ( − − ∂ s ζ H (0 ,
0) (IR divergent) S log 2 − π ζ (3) S − ζ ′ ( − − ζ ′ ( − S − log 2 − π ζ (3) + π ζ (5) S ζ ′ ( − − ζ ′ ( − S log 2 + π ζ (3) − π ζ (5) − π ζ (7) S − ζ ′ ( −
1) + ζ ′ ( −
3) + ζ ′ ( − − ζ ′ ( − S − log 2 − π ζ (3) + π ζ (5) + π ζ (7) + π ζ (9) S
10 12520 ζ ′ ( − − ζ ′ ( − − ζ ′ ( −
5) + ζ ′ ( − − ζ ′ ( − H − log(2 π ) − ζ ′ ( − H H − ζ ′ ( −
1) + ζ ′ ( − − ζ ′ ( − H H ζ ′ ( − − ζ ′ ( −
2) + ζ ′ ( − − ζ ′ ( − H H − ζ ′ ( −
1) + ζ ′ ( −
2) + ζ ′ ( − − ζ ′ ( − ζ ′ ( −
5) + ζ ′ ( − − ζ ′ ( − H H
10 15040 ζ ′ ( − − ζ ′ ( − − ζ ′ ( −
3) + ζ ′ ( − − ζ ′ ( − − ζ ′ ( −
6) + ζ ′ ( −
7) + ζ ′ ( − − ζ ′ ( − p F fin [ H p +1 × S q − ] = 0Odd p F fin [ S d ] = F fin [ H k × S d − k ] for k = 1 , · · · , ⌈ d/ ⌉ − Table 3 . Table of the finite parts of F fin [ S d ], F fin [ H d ], and F fin [ H p +1 × S q − ] with Dirichlet boundaryconditions. – 48 – A [ HS d ± ] F fin [ HS d ± ] − F fin [ S d ]2 ∓ (cid:0) log(2 π ) − ∂ s ζ H (0 , (cid:1) (IR divergent)3 ∓ ∓ (cid:0) log 2 + ζ ′ ( − (cid:1) − ∓ π ζ (3)5 ± ± (cid:0) log 2 + ζ ′ ( − − ζ ′ ( − (cid:1) ± (cid:0) π ζ (3) + π ζ (5) (cid:1) ∓ ∓ (cid:0) log 2 + ζ ′ ( − − ζ ′ ( −
3) + ζ ′ ( − (cid:1) − ∓ (cid:0) π ζ (3) + π ζ (5) + π ζ (7) (cid:1) ± ± (cid:0) log 2 + ζ ′ ( − − ζ ′ ( −
3) + ζ ′ ( − − ζ ′ ( − (cid:1) Table 4 . The anomaly and the finite parts of the free energies on HS d . B Useful formulas
In this appendix, we summarize useful formulas of the zeta function and the Hurwitz zetafunction. Throughout this Appendix, we assume that n be a non-negative integer ( n =0 , , , , · · · ) and m be a positive integer ( m = 1 , , , · · · ). Zeta function
At specific points the Riemann zeta function takes the values: ζ (0) = − , (B.1) ζ ( − m ) = 0 , (B.2) ζ (2 m ) = ( − m − m − π m (2 m )! B m . (B.3)More generally the zeta function satisfies the relation: ζ ( s ) = 2 s π s − sin (cid:16) πs (cid:17) Γ(1 − s ) ζ (1 − s ) . (B.4)The derivative of the zeta function at non-positive integer points are given by ζ ′ (0) = −
12 log(2 π ) , (B.5) ζ ′ ( − m ) = ( − m m +1 π m Γ(2 m + 1) ζ (2 m + 1) , (B.6) ζ ′ (1 − m ) = ( − m +1 m )(2 π ) m (cid:2) ( ψ (2 m ) − log(2 π )) ζ (2 m ) + ζ ′ (2 m ) (cid:3) . (B.7) Hurwitz zeta function
The Hurwitz zeta function ζ H ( s, a ) has two arguments, s and a .To distinguish derivatives of the Hurwitz zeta function, we explicitly write the differentiationvariable such as ∂ s ζ H ( s, a ). – 49 –or specific values of a the Hurwitz zeta function reduces to the Riemann zeta function: ζ H ( s,
1) = ζ ( s ) ,ζ H ( s,
0) = ζ ( s ) s < s = 0 . (B.8)The argument a of the Hurwitz zeta functions can be shifted by a positive integer m by therelation: ζ H ( s, m + a ) = ζ H ( s, a ) − m − X k =0 ( k + a ) − s . (B.9)At special values of s it derives ζ H ( − n, m ) = ζ ( − n ) − m − X k =1 k n , (B.10) ζ H ( − n, m −
1) = ζ ( − n ) − m − X k =1 k n + δ m, δ n, . (B.11)Other useful identities are ζ H (cid:18) s, (cid:19) = (2 s − ζ ( s ) , (B.12) ζ H ( − n, a ) = − B n +1 ( a ) n + 1 . (B.13)The derivative with respect to s at special values are given by ∂ s ζ H (0 , a ) = log Γ( a ) −
12 log(2 π ) , (B.14) ∂ s ζ H (cid:18) s, (cid:19) = 2 s log 2 ζ ( s ) + (2 s − ζ ′ ( s ) , (B.15) ∂ s ζ H (cid:0) − m, (cid:1) = − B m m · m log 2 − m − − m − ζ ′ (1 − m ) . (B.16) Computation of the derivative of the Hurwitz zeta function
From the formula [78,25.12.13], Li s (e π i a ) + e π i s Li s (e − π i a ) = (2 π ) s e π i s Γ( s ) ζ H (1 − s, a ) , (B.17)which holds for Re s >
0, 0 < Re a ≤
1, Im a >
0, or Re s >
1, 0 < Re a ≤
1, Im a = 0, theHurwitz zeta function can be written as ζ H ( s, a ) = Γ(1 − s )(2 π ) − s e π i( s − (cid:16) Li − s (e π i a ) + e π i(1 − s ) Li − s (e − π i a ) (cid:17) , (B.18)– 50 –hich holds for Re s <
1, Im a >
0, or Re s <
0, Im a = 0. By taking the derivative withrespect to s and replace s with a negative integer − n ( n ≥ ∂ s ζ H ( − n, a ) + ( − n ∂ s ζ H ( − n, − a ) = Γ( n + 1)(2 π i) n Li n +1 (e π i a ) + π i B n +1 ( a ) n + 1 . (B.19) C Derivation of (4.21)
In this appendix, we give a detailed derivation of (4.21). Instead of performing the integralover ω directly, we take a derivative with respect to ν and integrate the obtained derivativein terms of ν . The same calculation can be found in Appendix A of [118] (see also [119] andAppendix A of [79]). The derivative of f k ( ν ) is given by ∂ ν f k ( ν ) = 2 νg k ( ν ) , (C.1) g k ( ν ) ≡ Z ∞ d ω ω k +1 (e πω + 1)( ω + ν ) . (C.2)Since g k ( ν ) satisfies the recursion relation, g k ( ν ) = − ν g k − ( ν ) + 2 k − π ) k Γ(2 k ) ζ (2 k ) , (C.3) g ( ν ) = 12 ψ (cid:18) ν + 12 (cid:19) −
12 log ν , (C.4)where ψ ( x ) is a polygamma function, the general solution can be easily obtained: g k ( ν ) = ( − ν ) k " ψ (cid:18) ν + 12 (cid:19) −
12 log ν + k X m =1 m − π ) m ( − ν ) − m Γ(2 m ) ζ (2 m ) . (C.5)By integrating g k ( ν ) from 0 to ν , we obtain f k ( ν ) as f k ( ν ) = ( − k Z ν d µ µ k +1 ψ (cid:18) µ + 12 (cid:19) + ( − k +1 k + 1) ν k +2 (2( k + 1) log ν − k X m =1 ( − k − m k − m + 1 2 m − π ) m Γ(2 m ) ζ (2 m ) ν k − m +2 + f k (0) . (C.6)For k = 0, the term involving the summation of m should be omitted. The remaining term f k (0) can be computed as f k (0) = ( − k (1 − − k − ) ζ ′ ( − k −
1) + 2 − k − log 2 | B k +2 | k + 2 , (C.7)where we use (B.3), (B.5), (B.7) and H k +1 − γ − ψ (2 k + 2) = 0 . (C.8)– 51 –sing Theorem 4.3 in [120], the remaining integral in (C.6) can be performed Z ν d µ µ k +1 ψ (cid:18) µ + 12 (cid:19) = k +1 X j =0 ( − j (cid:18) k + 1 j (cid:19) ν k +1 − j ζ ′ H (cid:18) − j, ν + 12 (cid:19) − H j B j +1 (cid:0) ν + (cid:1) j + 1 ! − − k − log 2 B k +2 k + 2 − (1 − − k − ) (cid:18) ζ ′ ( − k − − H k +1 B k +2 k + 2 (cid:19) , (C.9)where ζ ′ H ( − j, ν +1 /
2) = ∂ s ζ ′ H ( s, ν +1 / | s →− j . Specifically, the integral with ν = 1 / Z d µ µ k +1 ψ (cid:18) µ + 12 (cid:19) = k +1 X j =0 ( − j k +1 − j (cid:18) k + 1 j (cid:19) (cid:0) ζ ′ ( − j ) + H j ζ ( − j ) (cid:1) − − k − log 2 B k +2 k + 2 − (1 − − k − ) (cid:18) ζ ′ ( − k − − H k +1 B k +2 k + 2 (cid:19) , (C.10)and the derivative of the zeta function is given by ∂ s ζ H d (cid:18) , (cid:19) = c d d − X k =0 ( − k β k,d " − − k − k + 1 H k +1 − k X m =1 − k − (2 m − k − m + 1 B m m + k +1 X j =0 ( − j k − j (cid:18) k + 1 j (cid:19) (cid:0) ζ ′ ( − j ) + H j ζ ( − j ) (cid:1) +(1 − − k − ) H k +1 B k +2 k + 1 (cid:21) . (C.11)Now we would like to show a sum of the terms except ζ ′ ( − j ) in the bracket vanishes, − − k − k + 1 H k +1 − k X m =1 − k − (2 m − k − m + 1 B m m + k +1 X j =0 ( − j k − j (cid:18) k + 1 j (cid:19) H j ζ ( − j ) + (1 − − k − ) H k +1 B k +2 k + 1 = 0 . (C.12)For k = 0, the summation term P k − m =0 should be omitted. We confirmed (C.12) up to k = 100numerically. But, we do not know a proof of (C.12). The coefficient of ζ ′ (0) also vanishes for d ≥ d/ / Γ(2 − d/ ∂ s ζ H d (cid:18) , (cid:19) = c d d − X k =0 ( − k β k,d k +1 X j =1 ( − j k − j (cid:18) k + 1 j (cid:19) ζ ′ ( − j ) − δ d, ζ ′ (0) . (C.13)– 52 – eferences [1] A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality , JHEP (2014) 001,[ ].[2] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries , JHEP (2015) 172, [ ].[3] N. Andrei et al., Boundary and Defect CFT: Open Problems and Applications , J. Phys. A (2020) 453002, [ ].[4] D. M. McAvity and H. Osborn, Conformal field theories near a boundary in generaldimensions , Nucl. Phys.
B455 (1995) 522–576, [ cond-mat/9505127 ].[5] P. Liendo, L. Rastelli and B. C. van Rees,
The Bootstrap Program for Boundary CFT d , JHEP (2013) 113, [ ].[6] M. Bill`o, V. Goncalves, E. Lauria and M. Meineri, Defects in conformal field theory , JHEP (2016) 091, [ ].[7] A. Gadde, Conformal constraints on defects , JHEP (2020) 038, [ ].[8] M. Fukuda, N. Kobayashi and T. Nishioka, Operator product expansion for conformal defects , JHEP (2018) 013, [ ].[9] N. Kobayashi, T. Nishioka, Y. Sato and K. Watanabe, Towards a C -theorem in defect CFT , JHEP (2019) 039, [ ].[10] S. Guha and B. Nagaraj, Correlators of Mixed Symmetry Operators in Defect CFTs , JHEP (2018) 198, [ ].[11] M. Isachenkov, P. Liendo, Y. Linke and V. Schomerus, Calogero-Sutherland Approach toDefect Blocks , JHEP (2018) 204, [ ].[12] E. Lauria, M. Meineri and E. Trevisani, Spinning operators and defects in conformal fieldtheory , JHEP (2019) 066, [ ].[13] C. P. Herzog and A. Shrestha, Two Point Functions in Defect CFTs , .[14] Y. Nakayama, Scale invariance vs conformal invariance , Phys. Rept. (2015) 1–93,[ ].[15] A. B. Zamolodchikov,
Irreversibility of the Flux of the Renormalization Group in a 2D FieldTheory , JETP Lett. (1986) 730–732.[16] J. L. Cardy, Is There a c Theorem in Four-Dimensions? , Phys. Lett.
B215 (1988) 749–752.[17] Z. Komargodski and A. Schwimmer,
On Renormalization Group Flows in Four Dimensions , JHEP (2011) 099, [ ].[18] D. L. Jafferis, I. R. Klebanov, S. S. Pufu and B. R. Safdi, Towards the F -Theorem: N =2 FieldTheories on the Three-Sphere , JHEP (2011) 102, [ ].[19] I. R. Klebanov, S. S. Pufu and B. R. Safdi, F -Theorem without Supersymmetry , JHEP (2011) 038, [ ].[20] R. C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions , JHEP (2011) 125, [ ]. – 53 –
21] R. C. Myers and A. Sinha,
Seeing a c-theorem with holography , Phys. Rev.
D82 (2010) 046006, [ ].[22] S. Giombi and I. R. Klebanov,
Interpolating between a and F , JHEP (2015) 117,[ ].[23] H. Casini and M. Huerta, A Finite entanglement entropy and the c-theorem , Phys. Lett.
B600 (2004) 142–150, [ hep-th/0405111 ].[24] H. Casini and M. Huerta,
On the RG running of the entanglement entropy of a circle , Phys. Rev.
D85 (2012) 125016, [ ].[25] H. Casini, E. Test´e and G. Torroba,
Markov Property of the Conformal Field Theory Vacuumand the a Theorem , Phys. Rev. Lett. (2017) 261602, [ ].[26] I. Affleck and A. W. W. Ludwig,
Universal noninteger ‘ground state degeneracy’ in criticalquantum systems , Phys. Rev. Lett. (1991) 161–164.[27] M. Nozaki, T. Takayanagi and T. Ugajin, Central Charges for BCFTs and Holography , JHEP (2012) 066, [ ].[28] D. Gaiotto, Boundary F-maximization , .[29] D. Friedan and A. Konechny, On the boundary entropy of one-dimensional quantum systems atlow temperature , Phys. Rev. Lett. (2004) 030402, [ hep-th/0312197 ].[30] H. Casini, I. S. Landea and G. Torroba, The g-theorem and quantum information theory , JHEP (2016) 140, [ ].[31] K. Jensen and A. O’Bannon, Constraint on Defect and Boundary Renormalization GroupFlows , Phys. Rev. Lett. (2016) 091601, [ ].[32] H. Casini, I. Salazar Landea and G. Torroba,
Irreversibility in quantum field theories withboundaries , JHEP (2019) 166, [ ].[33] S. Yamaguchi, Holographic RG flow on the defect and g theorem , JHEP (2002) 002,[ hep-th/0207171 ].[34] T. Takayanagi, Holographic Dual of BCFT , Phys. Rev. Lett. (2011) 101602, [ ].[35] M. Fujita, T. Takayanagi and E. Tonni,
Aspects of AdS/BCFT , JHEP (2011) 043,[ ].[36] R.-X. Miao, C.-S. Chu and W.-Z. Guo, New proposal for a holographic boundary conformalfield theory , Phys. Rev. D (2017) 046005, [ ].[37] J. Estes, K. Jensen, A. O’Bannon, E. Tsatis and T. Wrase, On Holographic Defect Entropy , JHEP (2014) 084, [ ].[38] D. R. Green, M. Mulligan and D. Starr, Boundary Entropy Can Increase Under Bulk RGFlow , Nucl. Phys. B (2008) 491–504, [ ].[39] Y. Sato,
Boundary entropy under ambient RG flow in the AdS/BCFT model , Phys. Rev. D (2020) 126004, [ ].[40] K. Jensen, A. O’Bannon, B. Robinson and R. Rodgers,
From the Weyl Anomaly to Entropy ofTwo-Dimensional Boundaries and Defects , Phys. Rev. Lett. (2019) 241602, [ ]. – 54 –
41] J. Estes, D. Krym, A. O’Bannon, B. Robinson and R. Rodgers,
Wilson Surface CentralCharge from Holographic Entanglement Entropy , JHEP (2019) 032, [ ].[42] R. Rodgers, Holographic entanglement entropy from probe M-theory branes , JHEP (2019) 092, [ ].[43] A. Chalabi, A. O’Bannon, B. Robinson and J. Sisti, Central charges of 2d superconformaldefects , JHEP (2020) 095, [ ].[44] Y. Wang, Surface Defect, Anomalies and b -Extremization , .[45] M. Beccaria, S. Giombi and A. Tseytlin, Non-supersymmetric Wilson loop in N = 4 SYM anddefect 1d CFT , JHEP (2018) 131, [ ].[46] S. P. Kumar and D. Silvani, Holographic flows and thermodynamics of Polyakov loopimpurities , JHEP (2017) 107, [ ].[47] S. P. Kumar and D. Silvani, Entanglement of heavy quark impurities and generalizedgravitational entropy , JHEP (2018) 052, [ ].[48] E. Witten, Multitrace operators, boundary conditions, and AdS/CFT correspondence , hep-th/0112258 .[49] M. Berkooz, A. Sever and A. Shomer, ‘Double trace’ deformations, boundary conditions andspace-time singularities , JHEP (2002) 034, [ hep-th/0112264 ].[50] S. S. Gubser and I. Mitra, Double trace operators and one loop vacuum energy in AdS/CFT , Phys. Rev. D (2003) 064018, [ hep-th/0210093 ].[51] S. S. Gubser and I. R. Klebanov, A universal result on central charges in the presence ofdouble trace deformations , Nucl. Phys. B (2003) 23–36, [ hep-th/0212138 ].[52] T. Hartman and L. Rastelli,
Double-trace deformations, mixed boundary conditions andfunctional determinants in AdS/CFT , JHEP (2008) 019, [ hep-th/0602106 ].[53] D. E. Diaz and H. Dorn, Partition functions and double-trace deformations in AdS/CFT , JHEP (2007) 046, [ hep-th/0702163 ].[54] S. Giombi, I. R. Klebanov, S. S. Pufu, B. R. Safdi and G. Tarnopolsky, AdS Description ofInduced Higher-Spin Gauge Theory , JHEP (2013) 016, [ ].[55] C. P. Herzog and I. Shamir, On Marginal Operators in Boundary Conformal Field Theory , JHEP (2019) 088, [ ].[56] S. Giombi and H. Khanchandani, CFT in AdS and boundary RG flows , JHEP (2020) 118,[ ].[57] A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality , Phys. Rev.
D74 (2006) 025005, [ hep-th/0501015 ].[58] S. M. Chester, M. Mezei, S. S. Pufu and I. Yaakov,
Monopole operators from the − ǫ expansion , JHEP (2016) 015, [ ].[59] E. Lauria, P. Liendo, B. C. Van Rees and X. Zhao, Line and surface defects for the free scalarfield , JHEP (2021) 060, [ ].[60] R. B. Mann and S. N. Solodukhin, Universality of quantum entropy for extreme black holes , Nucl. Phys. B (1998) 293–307, [ hep-th/9709064 ]. – 55 –
61] S. N. Solodukhin,
Entanglement entropy of round spheres , Phys. Lett. B (2010) 605–608,[ ].[62] D. Rodriguez-Gomez and J. G. Russo,
Free energy and boundary anomalies on S a × H b spaces , JHEP (2017) 084, [ ].[63] D. Rodriguez-Gomez and J. G. Russo, Boundary Conformal Anomalies on Hyperbolic Spacesand Euclidean Balls , JHEP (2017) 066, [ ].[64] A. Belin, L.-Y. Hung, A. Maloney, S. Matsuura, R. C. Myers and T. Sierens, HolographicCharged R´enyi Entropies , JHEP (2013) 059, [ ].[65] A. Lewkowycz, R. C. Myers and M. Smolkin, Observations on entanglement entropy inmassive QFT’s , JHEP (2013) 017, [ ].[66] J. Quine and J. Choi, Zeta regularized products and functional determinants on spheres , Rocky Mountain J. Math. (1996) 719–729.[67] H. Kumagai, The determinant of the laplacian on the n-sphere , Acta Arithmetica (1999) 199–208.[68] A. Cappelli and G. D’Appollonio, On the trace anomaly as a measure of degrees of freedom , Phys. Lett. B (2000) 87–95, [ hep-th/0005115 ].[69] J. Dowker,
The boundary F-theorem for free fields , .[70] S. Giombi, A. Maloney and X. Yin, One-loop Partition Functions of 3D Gravity , JHEP (2008) 007, [ ].[71] A. Bytsenko, E. Elizalde and S. Odintsov, The Conformal anomaly in N-dimensional spaceshaving a hyperbolic spatial section , J. Math. Phys. (1995) 5084–5090, [ gr-qc/9505047 ].[72] H. Dorn, M. Salizzoni and C. Sieg, On the propagator of a scalar field on AdS × S and on theBMN plane wave , JHEP (2005) 047, [ hep-th/0307229 ].[73] A. Gustavsson, Conformal anomaly of Wilson surface observables: A Field theoreticalcomputation , JHEP (2004) 074, [ hep-th/0404150 ].[74] R. Camporesi, Harmonic analysis and propagators on homogeneous spaces , Phys. Rept. (1990) 1–134.[75] A. A. Bytsenko, G. Cognola, L. Vanzo and S. Zerbini,
Quantum fields and extended objects inspace-times with constant curvature spatial section , Phys. Rept. (1996) 1–126,[ hep-th/9505061 ].[76] A. Monin,
Partition function on spheres: How to use zeta function regularization , Phys. Rev. D (2016) 085013, [ ].[77] D. Vassilevich, Heat kernel expansion: User’s manual , Phys. Rept. (2003) 279–360,[ hep-th/0306138 ].[78] “
NIST Digital Library of Mathematical Functions .” http://dlmf.nist.gov/, Release 1.1.0 of2020-12-15.[79] S. Giombi, I. R. Klebanov and B. R. Safdi,
Higher Spin AdS d +1 /CFT d at One Loop , Phys. Rev. D (2014) 084004, [ ].[80] J. Dowker, Sphere Renyi entropies , J. Phys. A (2013) 225401, [ ]. – 56 –
81] J. Dowker, On a - F dimensional interpolation , .[82] J. Dowker, Determinants and conformal anomalies of GJMS operators on spheres , J. Phys. A (2011) 115402, [ ].[83] R. Camporesi and A. Higuchi, Spectral functions and zeta functions in hyperbolic spaces , J. Math. Phys. (1994) 4217–4246.[84] J. S. Dowker, Entanglement entropy for odd spheres , .[85] H. Casini, M. Huerta and R. C. Myers, Towards a derivation of holographic entanglemententropy , JHEP (2011) 036, [ ].[86] M. H. Zhao, On logarithmic integrals, harmonic sums and variations , .[87] D. Carmi, L. Di Pietro and S. Komatsu, A Study of Quantum Field Theories in AdS at FiniteCoupling , JHEP (2019) 200, [ ].[88] A. Lewkowycz and J. Maldacena, Exact results for the entanglement entropy and the energyradiated by a quark , JHEP (2014) 025, [ ].[89] S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric LanglandsProgram , hep-th/0612073 .[90] S. Gukov and E. Witten, Rigid Surface Operators , Adv. Theor. Math. Phys. (2010) 87–178,[ ].[91] J. Gomis, T. Okuda and D. Trancanelli, Quantum ’t Hooft operators and S-duality in N=4super Yang-Mills , Adv. Theor. Math. Phys. (2009) 1941–1981, [ ].[92] J. Gomis and T. Okuda, S-duality, ’t Hooft operators and the operator product expansion , JHEP (2009) 072, [ ].[93] J. Gomis, T. Okuda and V. Pestun, Exact Results for ’t Hooft Loops in Gauge Theories on S , JHEP (2012) 141, [ ].[94] A. Kapustin, B. Willett and I. Yaakov, Exact results for supersymmetric abelian vortex loopsin 2+1 dimensions , JHEP (2013) 099, [ ].[95] N. Drukker, T. Okuda and F. Passerini, Exact results for vortex loop operators in 3dsupersymmetric theories , JHEP (2014) 137, [ ].[96] T. Nishioka and I. Yaakov, Supersymmetric R´enyi Entropy , JHEP (2013) 155, [ ].[97] D. Gaiotto and H.-C. Kim, Surface defects and instanton partition functions , JHEP (2016) 012, [ ].[98] T. Nishioka and I. Yaakov, Supersymmetric R´enyi entropy and defect operators , JHEP (2017) 071, [ ].[99] K. Hosomichi, S. Lee and T. Okuda, Supersymmetric vortex defects in two dimensions , JHEP (2018) 033, [ ].[100] N. Drukker, I. Shamir and C. Vergu, Defect multiplets of N = 1 supersymmetry in 4d , JHEP (2018) 034, [ ].[101] S. M. Hosseini, C. Toldo and I. Yaakov, Supersymmetric R´enyi entropy and charged hyperbolicblack holes , JHEP (2020) 131, [ ]. – 57 – Superconformal surfaces in four dimensions , JHEP (2020) 056,[ ].[103] N. B. Agmon and Y. Wang, Classifying Superconformal Defects in Diverse Dimensions Part I:Superconformal Lines , .[104] N. Drukker, M. Probst and M. Tr´epanier, Defect CFT techniques in the 6d N = (2 , theory , .[105] K. Goto, L. Nagano, T. Nishioka and T. Okuda, Janus interface entropy and Calabi’s diastasisin four-dimensional N = 2 superconformal field theories , JHEP (2020) 048, [ ].[106] Y. Wang, Taming defects in N = 4 super-Yang-Mills , JHEP (2020) 021, [ ].[107] R. K. Gupta, A. Ray and K. Sil, Supersymmetric Graphene on Squashed Hemisphere , .[108] J. R. David, E. Gava, R. K. Gupta and K. Narain, Localization on AdS × S , JHEP (2017) 050, [ ].[109] E. Gava, K. Narain, M. Muteeb and V. Giraldo-Rivera, N = 2 gauge theories on thehemisphere HS , Nucl. Phys. B (2017) 256–297, [ ].[110] J. R. David, E. Gava, R. K. Gupta and K. Narain,
Boundary conditions and localization onAdS. Part I , JHEP (2018) 063, [ ].[111] J. R. David, E. Gava, R. K. Gupta and K. Narain, Boundary conditions and localization onAdS. Part II. General analysis , JHEP (2020) 139, [ ].[112] Y. Sato, in preparation , .[113] D. Fursaev, Conformal anomalies of CFT’s with boundaries , JHEP (2015) 112,[ ].[114] S. N. Solodukhin, Boundary terms of conformal anomaly , Phys. Lett. B (2016) 131–134,[ ].[115] C. P. Herzog, K.-W. Huang and K. Jensen,
Universal Entanglement and Boundary Geometryin Conformal Field Theory , JHEP (2016) 162, [ ].[116] D. V. Fursaev and S. N. Solodukhin, Anomalies, entropy and boundaries , Phys. Rev. D (2016) 084021, [ ].[117] C. P. Herzog and K.-W. Huang, Boundary Conformal Field Theory and a Boundary CentralCharge , JHEP (2017) 189, [ ].[118] M. M. Caldarelli, Quantum scalar fields on anti-de Sitter space-time , Nucl. Phys. B (1999) 499–515, [ hep-th/9809144 ].[119] R. Camporesi, zeta function regularization of one loop effective potentials in anti-de Sitterspace-time , Phys. Rev. D (1991) 3958–3965.[120] O. R. Espinosa and V. H. Moll, On some integrals involving the hurwitz zeta function: Part 2 , The Ramanujan Journal (2002) 449–468, [ math/0107082 ].].