aa r X i v : . [ h e p - t h ] F e b Prepared for submission to JHEP
Free energy and defect C -theorem in free fermion Yoshiki Sato
Physics Division, National Center for Theoretical Sciences, National Tsing-Hua University,Hsinchu 30013, Taiwan
Abstract:
We describe a p -dimensional conformal defect of a free Dirac fermion on a d -dimensional flat space as boundary conditions on a conformally equivalent space H p +1 × S d − p − . We classify allowed boundary conditions and find that the Dirichlet type of boundaryconditions always exists while the Neumann type of boundary condition exists only for a two-codimensional defect. For the two-codimensional defect, a double trace deformation triggers arenormalization group flow from the Neumann boundary condition to the Dirichlet boundarycondition, and the free energy at UV fixed point is always larger than that at IR fixed point.This provides us with further support of a conjectured C -theorem in DCFT. ontents H p +1 × S q − S d and HS d S d d d a and F HS d H p +1 × S q − with Dirichlet boundary condition 12 H d d d H p +1 × S q − p p C -theorem 25 A renormalization group (RG) is a fundamental concept in quantum field theory (QFT). Bya deformation by a relevant operator O , I CFT + λ Z d d x O , (1.1)– 1 – conformal field theory (CFT) at a UV fixed point flows to a CFT at an IR fixed point. TheRG flow is not reversible, and this irreversibility implies the existence of a monotonic function,known as a C -function. A C -theorem which states the existence of the C -function is pioneeredby Zamolodchikov [1] in two-dimensional QFTs and is extended to higher dimensions [2–7].Combining a -theorem [2, 3] which is a C -theorem in d = 4 and F -theorem [4, 5] which is a C -theorem in d = 3, a generalised F -theorem is conjectured [8]: A free energy on a sphere S d , ˜ F = sin (cid:18) πd (cid:19) log Z [ S d ] , (1.2)is a C -function and satisfies the monotonic relation˜ F UV ≥ ˜ F IR . (1.3)The generalised F -theorem states monotonicity of an anomaly coefficient in the free energyfor even d , while monotonicity of a finite part in the free energy for odd d . The C -theoremcan also be proved by using an information theoretical method for d ≤ p -dimensional defect, it is possible to consider an RG flow triggered by arelevant operator localising on the defect, I DCFT + ˆ λ Z d p x ˆ O . (1.4)In [12], we conjectured that the defect free energy on a sphere, which is an increment of thefree energy due to the defect,log hD ( p ) i = log Z DCFT [ S d ] − log Z CFT [ S d ] , (1.5)decreases under the defect RG flow. More precisely, the universal part of the defect freeenergy, ˜ D = sin (cid:16) πp (cid:17) log |hD ( p ) i| (1.6)is a C -function, and it decreases under the defect RG flow,˜ D UV ≥ ˜ D IR . (1.7)For BCFTs with p = d −
1, a slight modification is needed since BCFTs is defined ona hemisphere HS d (a half space of the original CFTs), and the boundary free energy isintroduced as log hD ( p ) i = log Z BCFT [ HS d ] −
12 log Z CFT [ S d ] . (1.8) It is pointed out that the boundary free energy does not decrease monotonically under the bulk RG flow[13, 14]. – 2 –or BCFT, our conjecture reproduces a proved C -theorem in BCFT , known as g -theorem[15–17], and a C -theorem in higher dimensional BCFT [18–21]. In particular, the C -theoremin BCFT is proved in [18] by extending [3] and in [22] by extending [17]. In the context ofholography, a C -theorem in BCFT is investigated in [23–27]. For DCFT with p < d − H p +1 × S q − , where q = d − p is a codimensionof the defect, and impose boundary conditions at the boundary of H p +1 . The idea of mappingBCFTs on a flat space R d to H d has appeared in [37, 38], and the similar idea for DCFTshas appeared in [35, 39, 40]. It enables us to classify allowed boundary conditions which areconsistent with a recent classification [41]. The Dirichlet type boundary condition is alwaysallowed while the Neumann type boundary condition is allowed only in q = 1 , , ,
4. An RGflow from the Neumann boundary condition to the Dirichlet boundary condition realised bya double trace deformation as is familiar with the AdS/CFT setup [42–48], and the defectfree energy with the Neumann boundary condition is always larger than that of the Dirichletboundary condition.In this paper, we extend our analysis [34] to a free fermion to provide a further check ofour conjecture. To compare with a scalar field, a defect free energy of a free fermion in higherdimensions has been studied only in BCFT [36, 37] and DCFT with a two-codimensionaldefect in the context of entanglement entropy [49, 50]. In particular, the existence of anontrivial boundary condition is unclear in a free fermion since a unique boundary conditionis allowed in H d [36, 51].The organisation of this paper is as follows. In the next section, we summarise coordinatesystems and Weyl transformations among the coordinate systems. Furthermore, we classifyboundary conditions of a massive fermion on H d and a massless fermion on H p +1 × S q − . Inparticular, for a massless fermion on H p +1 × S q − , we show that a boundary condition of aDirichlet type always exists while a boundary condition of a Neumann type exists only in q = 2. In section 3, we compute free energies of a massless fermion on S d and a hemisphere HS d using a zeta-function regularisation for a warm-up of the next section. In section 4, wecompute free energies of a massive fermion on H d and free energies of a massless fermion H p +1 × S q − with Dirichlet boundary condition by using a zeta-function regularisation. Insection 5, we obtain free energies with Neumann boundary condition by analytical continu-ation and confirm the validity of our conjecture. The final section is devoted to discussion. In [26], it is argued that the defect entropy, which is an increment of the entanglement entropy of thespherical region where the defect sits at the centre due to the presence of the defect, is a candidate of a C -function in DCFT. The defect entropy is not a C -function for p < d − – 3 –ppendix A contains the list of anomaly parts and finite parts of free energies on S d , HS d , H d and H p +1 × S q − with Dirichlet boundary condition, and appendix B is a technical detailsof a computation. In this section, we summarise coordinate systems for a sphere S d , a hemisphere HS d , ahyperbolic space H d and H p +1 × S q − and conformal maps among a flat space R d and them.Let us consider DCFT d on R d ,d s = d x a + d y i , ( a = 1 , · · · , p, i = p + 1 , · · · , d ) (2.1)where a p -dimensional defect sits at y i = 0. For later convenience, we introduce a codimensionof the defect, q = d − p . (2.2)By using the polar coordinate for the y i -coordinate,d y i = d z + z d s S q − , (2.3)the metric of the flat space becomesd s = d x a + d z + z d s S q − = z (cid:18) d x a + d z z + d s S q − (cid:19) . (2.4)After a Weyl rescaling, the metric (2.4) reduces to a 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) . (2.5)Now the defect sits at the boundary of H p +1 . We can also use the global coordinate for H p +1 ,d s = R (cid:0) d ρ + sinh ρ d s S p + d s S q − (cid:1) , (2.6)where the defect sits at ρ = ∞ . Introducing a new variable ϕ , tan ϕ = sinh ρ , the metric (2.6)becomes d s = R cos ϕ (cid:0) d ϕ + sin ϕ d s S p + cos ϕ d s S q − (cid:1) . (2.7)After a Weyl rescaling, the metric (2.7) can be mapped to the sphere metric with radius R d s = R (cid:0) d ϕ + sin ϕ d s S p + cos ϕ d s S q − (cid:1) , (2.8)where 0 ≤ ϕ < π and the defect sits at ϕ = π/
2. The hemisphere HS d has the same metricof the sphere (2.8). A different point is that the range of ϕ is 0 ≤ ϕ ≤ π/ ϕ = π/
2. – 4 – .2 Boundary condition of fermion on H p +1 × S q − In this section, we classify allowed boundary conditions of a massless fermion on H p +1 × S q − .A free fermion on a curved background which is conformally equivalent to R d is studied ine.g. [37, 49, 52–54]. See [40] for the notation of a fermion on H × S . The action of a massiveDirac fermion is given by I = Z d d x √ g (cid:16) i ψ † Γ a ∇ a ψ + M ψ † ψ (cid:17) , (2.9)where we assume M ≥
0. The rank of the gamma matrix for d ≥ r d = 2 ⌊ d ⌋ , (2.10)and the gamma matrix satisfies the anti-commutation relation, { Γ a , Γ b } = 2 δ ab . (2.11)The covariant derivative ∇ a and the spin connection ω µbc are defined as ∇ a = e µa ∇ µ , ∇ µ = ∂ µ + 12 σ bc ω µbc ,σ ab = 14 [Γ a , Γ b ] , ω µbc = e νb ( ∂ µ e cν − Γ ανµ e cα ) (2.12)using a frame field e µa , which satisfies e µa e νb g µν = δ ab . (2.13)The covariant derivative satisfies a relation,(Γ a ∇ a ) = ∇ − R , (2.14)where R is a Ricci scalar.For H d , a solution of the equation of motion for the massive fermion behaves ψ ∼ z ∆ ± (2.15)near the boundary at z = 0, where ∆ ± is given by∆ ± − d −
12 = ± M R . (2.16)Then, the two boundary conditions become degenerate in the massless limit, and the allowedboundary condition is unique for a massless fermion. The massive fermion is not conformal. However, we introduce a mass term for later convenience. – 5 –or the product space H p +1 × S q − , we consider a massless fermion from the beginning.We first decompose the fermionic field as ψ ( z, x, θ ) = X ℓ ψ H p +1 ( z, x ) ⊗ ψ ℓ, S q − ( θ ) , (2.17)and the gamma matrices are also decomposed similarly. For even p and even q ≥
4, thedecomposition of the fermion (2.17) has an additional U (1) as is clear from that the rank ofthe spinor representation are different in both sides of (2.17). The spherical part ψ ℓ, S q − ( θ )satisfies the equation, (Γ · ∇ ) S q − ψ ℓ, S q − ( θ ) = ± i ℓ + q − R ψ ℓ, S q − ( θ ) , (2.18)where we decomposed the covariant derivative on H p +1 × S q − into the covariant derivativeson H p +1 and S q − appropriately. Then, the equation of motion reduces to i (Γ · ∇ ) H p +1 ± ℓ + q − R ! ψ H p +1 ( z, x ) = 0 . (2.19)The solution of this equation of motion behaves as ψ H p +1 ( z, x ) ∼ z ∆ ℓ ± (2.20)near the boundary, z = 0, and ∆ ℓ ± is given by∆ ℓ ± = p ± (cid:18) ℓ + q − (cid:19) . (2.21)The parameter ∆ ℓ ± can be understood as the conformal dimensions of operators localising ona p -dimensional conformal defect at the boundary of H p +1 . Then not all the operators withconformal dimensions (2.21) are allowed to exist due to the unitarity bound in p dimensions[55]: ∆ ≥ p − . (2.22)∆ ℓ + is always above the unitarity bound, while ∆ ℓ − is not necessary to satisfy the unitaritybound unless ℓ ≤ − q . (2.23)Hence the mode with ℓ = 0 for q = 2 is allowed to have the boundary conditions correspondingto ∆ ℓ − .The allowed boundary conditions for the massless fermion are classified as follows andare listed in table 1. We are indebted to D. Rodriguez-Gomez and J. G. Russo for the decomposition of the fermion. – 6 – = 1 case:
In this case, the geometry is the hyperbolic space H d , and the allowed boundarycondition is unique, Mixed b.c. : ∆ = d − . (2.24)This boundary condition is called a mixed boundary condition. q = 2 case: Only the ℓ = 0 mode is allowed, resulting in a nontrivial boundary conditionwith ∆ ℓ =0 − = p − for p ≥
2. Note that ∆ ℓ =0 − saturates the unitarity bound. Then, twodifferent boundary conditions are allowed,Dirichlet b.c. : ∆ D = ∆ ℓ + for all ℓ , Neumann b.c. : ∆ N = ( ∆ ℓ − for ℓ = 0 , ∆ ℓ + for ℓ = 0 . (2.25)In particular, ∆ ℓ =0 − vanishes for p = 1 case, and this implies that the defect operator is theidentity operator. Thus, we exclude p = 1 case of a nontrivial boundary condition. q ≥ case: Only the Dirichlet type boundary condition is allowed,Dirichlet b.c. : ∆ D = ∆ ℓ + for all ℓ . (2.26) q = 1 q = 2 q = 3 q = 4 q = 5 · · · p = 1 ∆ ∆ D ∆ D ∆ D ∆ D · · · p = 2 ∆ ∆ D / ∆ ℓ =0 − ∆ D ∆ D ∆ D p = 3 ∆ ∆ D / ∆ ℓ =0 − ∆ D ∆ D ∆ D · · · p = 4 ∆ ∆ D / ∆ ℓ =0 − ∆ D ∆ D ∆ D p = 5 ∆ ∆ D / ∆ ℓ =0 − ∆ D ∆ D ∆ D ... ... ... . . . Table 1 . Classification of the allowed boundary conditions in the free massless fermion. The Neumannboundary conditions exist in the shaded cells and the allowed modes differ from the Dirichlet ones areshown in the right side. For q = 1, the boundary condition is unique. For q = 2, ∆ ℓ =0 − saturates theunitarity bound (2.22). S d and HS d In this section, we compute free energies of a free massless Dirac fermion on S d and HS d usinga zeta-function regularisation for a warm-up of the next section.– 7 – .1 Free energy on S d For a massless fermion on S d , the free energy is given by F [ S d ] = −
12 tr log h − ˜Λ − (Γ a ∇ a ) i = − ∞ X ℓ =0 g ( d ) ( ℓ ) log ν ( d ) ℓ ˜Λ R ! , (3.1)where we use the equation, Γ a ∇ a ψ ℓ = ± i ν ( d ) ℓ R ψ ℓ (3.2)with the eigenvalues, ν ( d ) ℓ = ℓ + d , ℓ = 0 , , , · · · . (3.3)The degeneracy for each sign is given by g ( d ) ± ( ℓ ) = r d Γ( ℓ + d )Γ( d )Γ( ℓ + 1) , (3.4)where r d (2.10) is the rank of the gamma matrix as before. For later convenience, we introducea notation g ( d ) ( ℓ ) = 2 g ( d ) ± ( ℓ ) = 2 r d Γ( ℓ + d )Γ( d )Γ( ℓ + 1) . (3.5)We write the free energy (3.1) in the Schwinger representation, F [ S d ] = Z ∞ d tt ∞ X ℓ =0 g ( d ) ( ℓ ) e − tν ( d ) ℓ / (˜Λ R ) , (3.6)and we introduce the regularised free energy [57] to remove the divergence in the integral, F s [ S d ] = Z ∞ d tt − s ∞ X ℓ =0 g ( d ) ( ℓ ) e − tν ( d ) ℓ / (˜Λ R ) = 12 ( ˜Λ R ) s Γ( s ) ζ S d ( s ) , (3.7) As noted in [34, 56], there is an ambiguity to decompose the logarithmic function into two parts. Werequire two conditions to remove the ambiguity: the free energy in the Schwinger representation should beconvergent and the resulting zeta function does not depend on the cutoff scale ˜Λ. In (3.1), the two decomposedlogarithmic functions are the same and the free energy in the Schwinger representation (3.6) is convergent inthe ℓ → ∞ limit. – 8 –here the zeta function ζ S d ( s ) is defined by ζ S d ( s ) ≡ ∞ X ℓ =0 g ( d ) ( ℓ ) (cid:16) ν ( d ) ℓ (cid:17) − s . (3.8)Then the (unregularised) free energy is obtained in the s → F s [ S d ] = 12 (cid:18) s − γ E + log( ˜Λ R ) (cid:19) ζ S d (0) + 12 ∂ s ζ S d (0) + O ( s ) , (3.9)which is divergent due to the pole at s = 0. Here γ E is the Euler constant. By removing thepole, the remaining part becomes the renormalized free energy F ren [ S d ] ≡ ∂ s ζ S d (0) + 12 log(Λ R ) ζ S d (0) , (3.10)where Λ = e − γ E ˜Λ.To compute the zeta function, we expand the gamma functions in the degeneracy (3.5),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 +1 (cid:16) ν ( d ) ℓ (cid:17) n − d : even , d − X n =0 ( − d − + n β n,d +1 (cid:16) ν ( d ) ℓ (cid:17) n d : odd , (3.11)where we used the awkward suffix for α n,d +1 and β n,d +1 to use the same notation in the scalarcase [34]. Since α ,d +1 = 0, we can omit the n = 0 term in the summation for even d wheneverwe want. Using the asymptotic expansion ((5.11.14) in [58])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) , (3.12)where B ( m ) k ( x ) is the generalized Bernoulli polynomial, and comparing both sides, we findEven d : α n,d +1 = ( − d + n (cid:18) d − d − n (cid:19) B ( d ) d − n (cid:18) d (cid:19) , (3.13)Odd d : β n,d +1 = ( − d − + n (cid:18) d − d − − n (cid:19) B ( d ) d − − n (cid:18) d (cid:19) . (3.14) d When d is odd, the zeta function reduces to ζ S d ( s ) = 4 r d Γ( d ) d − X n =0 ( − d − + n β n,d +1 ζ H (cid:18) s − n, (cid:19) , (3.15)– 9 –here we use the identity ζ H (cid:18) s − n, d (cid:19) = ζ H (cid:18) s − n, (cid:19) − d − X m =0 (cid:18) m + 12 (cid:19) n − s (3.16)and the expansion (3.11) with the replacement ν ( d ) ℓ → m + 1 /
2. Since the zeta function at s = 0 vanishes, ζ S d (0) = 4 r d Γ( d ) d − X n =0 ( − d − + n β n,d +1 ζ H (cid:18) − n, (cid:19) = 0 , (3.17)due to the fact ζ H ( − n, /
2) = 0, there is no conformal anomaly, and the finite part remainsin the free energy: F ren [ S d ] = 12 ∂ s ζ S d (0)= r d Γ( d ) ( − d +12 β ,d +1 log 2 + 2 r d Γ( d ) d − X n =1 ( − d − + n β n,d +1 (2 − n − ζ ′ ( − n ) . (3.18)This formula correctly reproduces the known results [5, 59]. The finite parts of the free energyfor d ≤ d When d is even the zeta function is given by ζ S d ( s ) = 4 r d Γ( d ) d X n =0 ( − d + n α n,d +1 ζ ( s − n + 1) , (3.19)where we use the identity ζ H (cid:18) s − n + 1 , d (cid:19) = ζ ( s − n + 1) − d − X m =0 ( m + 1) n − − s (3.20)and the expansion (3.11) with the replacement ν ( d ) ℓ → m + 1. The renormalized free energyis given by F ren [ S d ] = − A [ S d ] log(Λ R ) + F fin [ S d ] , (3.21)with the anomaly coefficient A [ S d ] = − ζ S d (0)= r d Γ( d ) d X n =1 ( − d + n α n,d +1 B n n , (3.22)– 10 –here B n is the Bernoulli number, and the finite part F fin [ S d ] = 12 ∂ s ζ S d (0)= 2 r d Γ( d ) d X n =1 ( − d + n α n,d +1 ζ ′ ( − n + 1) . (3.23)There is a logarithmic divergent term associated with the conformal anomaly in the freeenergy. The free energy (3.21) with (3.22) correctly reproduces the known conformal anomaly[8]. The anomaly coefficients and the finite parts of the free energy for d ≤
10 are listed intables 2 and 3 in appendix A. a and F The finite part of the free energy (3.18) and the anomaly coefficient of the free energy (3.22)do not depend on the choice of the cutoff scale and are universal in this sense. Thus, weintroduce 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 ǫ is used for the cutoff instead of Λ. In [8], it is pointed out that the universal freeenergy has an integral representation: F univ [ S d ] = − r d sin (cid:0) πd (cid:1) Γ( d + 1) Z d u cos ( πu ) Γ (cid:18) d + 12 + u (cid:19) Γ (cid:18) d + 12 − u (cid:19) . (3.25)For odd d , the prefactor of the universal free energy (3.25) is finite. However, for even d , theprefactor in (3.25) is divergent due to the sine function. This divergence may be replacedwith the logarithmic divergence by introducing a small cutoff ǫ , − (cid:0) πd (cid:1) = ( − d +12 d : odd , ( − d π log (cid:18) Rǫ (cid:19) d : even . (3.26)Using the replacement (3.26), the universal free energy in the integral representation (3.25)has the same behaviour of (3.24). A proof of the equivalence of the two expressions (3.24)and (3.25) is presented in [60]. HS d Next, let us consider the free energy on the hemisphere. At the boundary of HS d , a mixedboundary condition is imposed [51], P + ψ = 0 . (3.27)– 11 –ere P + is a projection operator, P + = 12 (1 − i Γ ∗ Γ a e µa n µ ) (3.28)with a chirality matrix Γ ∗ and an incoming normal vector n µ . See appendix A in [51] for thedetail of a construction of the chirality matrix Γ ∗ . The mixed boundary condition preservesa conformal symmetry.A degeneracy with the mixed boundary condition is given by (3.4) which is just a half ofthe degeneracy of S d (3.5). Then, the free energy on HS d is nothing but half of that on S d , F ren [ HS d ] = 12 F ren [ S d ] . (3.29)This formula correctly reproduces the known results [36]. H p +1 × S q − with Dirichlet boundary condition In this section, we first compute a free energy of a massive fermion on H d for the later purpose,although we are interested in a massless (conformal) fermion. After that, we compute a freeenergy of a massless fermion on H p +1 × S q − . H d In this section, we extend a computation of the zeta function for a massive fermion on H d for d = 3 , M = m/R is given by F [ H d ]( m ) = −
12 tr log h − ˜Λ − (cid:0) (Γ a ∇ a ) − M (cid:1)i = − Z ∞ d ω µ ( d ) ( ω ) (cid:20) log (cid:18) ω + i m ˜Λ R (cid:19) + log (cid:18) ω − i m ˜Λ R (cid:19)(cid:21) , (4.1)where ˜Λ is the UV cutoff scale introduced to make the integral dimensionless. The parameter ω is an eigenvalue of the equation,Γ a ∇ a ψ ω = i ωR ψ ω , ω ≥ , (4.2) In [36], anomaly coefficients of a ball B d which is conformally equivalent to H d , is obtained. – 12 –nd the Plancherel measure of a fermion on H d of unit radius takes the form [53] µ ( d ) ( ω ) = c d r d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:0) d + i ω (cid:1) Γ (cid:0) + i ω (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = c d r d d − Y j = ( ω + j ) d : odd ,ω coth( πω ) d − Y j =1 ( ω + j ) d : even , (4.3)with the coefficient c d = Vol( H d )2 d − π d/ Γ( d/
2) (4.4)and the rank of the gamma matrix r d (2.10). The regularised volume of the hyperbolic spaceis given by Vol( 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) . (4.5)The hyperbolic volume is finite for even d but divergent for odd d due to the pole of the sinefunction. By introducing a small cutoff parameter ǫ , the sine function can be replaced by thelogarithmic divergence, − (cid:0) π d − (cid:1) = ( − d − π log (cid:18) Rǫ (cid:19) d : odd , ( − d d : even . (4.6)Then, the coefficient becomes c d = − (cid:0) π d − (cid:1) Γ( d ) = 1Γ( d ) ( − d − π log (cid:18) Rǫ (cid:19) d : odd , ( − d d : even . (4.7)Since the Plancherel measure (4.3) needs to satisfy the square integrability condition, thefree energy is defined for m ≥
0. That is, equation (4.1) represents a free energy with aboundary condition, ∆ = ( d − / m . We add a mass term in the free energy (4.1) for theregularisation of the zero mode, and we take a massless limit to obtain free energies of theconformal fermion.By using the Schwinger representation, the free energy can be written as F [ H d ]( m ) = 12 Z ∞ d tt Z ∞ d ω µ ( d ) ( ω ) (cid:16) e − t ( ω +i m ) / ˜Λ R + e − t ( ω − i m ) / ˜Λ R (cid:17) . (4.8)– 13 –o remove the divergence in the integral, we introduce the regularised free energy F s [ H d ]( m ) = 12 Z ∞ d tt − s Z ∞ d ω µ ( d ) ( ω ) (cid:16) e − t ( ω +i m ) / ˜Λ R + e − t ( ω − i m ) / ˜Λ R (cid:17) = 12 ( ˜Λ R ) s Γ( s ) ζ H d ( s, m ) , (4.9)where the zeta function is defined as ζ H d ( s, m ) = Z ∞ d ω µ ( d ) ( ω ) (cid:0) ( ω + i m ) − s + ( ω − i m ) − s (cid:1) . (4.10)Then, the (unregularised) free energy is obtained in the s → F s [ H d ]( m ) = 12 (cid:18) s − γ E + log( ˜Λ R ) (cid:19) ζ H d ( s, m ) + 12 ∂ s ζ H d ( s, m ) + O ( s ) . (4.11)By removing the pole at s = 0, we obtain the renormalized free energy F ren [ H d ]( m ) = 12 ∂ s ζ H d (0 , m ) + 12 log(Λ R ) ζ H d (0 , m ) , (4.12)where Λ = e − γ E ˜Λ.In the following, we will compute the renormalized free energy by evaluating the zetafunction based on the method used in [62, 63]. d Using the expansion of the Plancherel measure (4.3) µ ( d ) ( ω ) = c d r d d − X k =0 β k,d +1 ω k , (4.13)the zeta function is convergent for Re s > k + 1, ζ H d ( s, m ) = c d r d d − X k =0 β k,d +1 Z ∞ d ω ω k (cid:0) ( ω + i m ) − s + ( ω − i m ) − s (cid:1) = 2 c d r d d − X k =0 β k,d +1 ( − k sin (cid:16) πs (cid:17) m k +1 − s Γ(2 k + 1) k +1 Y i =1 s − i . (4.14)We immediately obtain ζ H d (0 , m ) = 0 , (4.15) ∂ s ζ H d (0 , m ) = c d r d d − X k =0 β k,d +1 πm k +1 k + 1 ( − k +1 . (4.16)– 14 –or the massive fermion, the renormalized free energy is given by F ren [ H d ]( m ) = 12 ∂ s ζ H d (0 , m )= ( − d − Γ( d ) d − X k =0 β k,d +1 m k +1 k + 1 ( − k +1 log (cid:18) Rǫ (cid:19) . (4.17)In the massless limit, the renormalized free energy vanishes, F ren [ H d ] = 0 . (4.18)Here and hereafter, we omit (0) in free energies for the massless fermion. Equation (4.18)implies that the boundary anomaly does not exist. d Using the expansion of the Plancherel (4.3) µ ( d ) ( ω ) = c d r d coth( πω ) d X k =1 α k,d +1 ω k − , (4.19)and the identity coth( πω ) = 1 + 2e πω − , (4.20)the zeta function can be decomposed into two parts, ζ H d ( s, m ) = ζ (1) H d ( s, m ) + ζ (2) H d ( s, m ) , (4.21) ζ (1) H d ( s, m ) = c d r d d X k =1 α k,d +1 Z ∞ d ω ω k − (cid:0) ( ω + i m ) − s + ( ω − i m ) − s (cid:1) , (4.22) ζ (2) H d ( s, m ) = 2 c d r d d X k =1 α k,d +1 Z ∞ d ω ω k − e πω − (cid:0) ( ω + i m ) − s + ( ω − i m ) − s (cid:1) . (4.23)The first term of the zeta function can be calculated as ζ (1) H d ( s, m ) = 2 c d r d d X k =1 α k,d +1 m k − s ( − k cos (cid:16) πs (cid:17) Γ(2 k ) k Y i =1 s − i , (4.24)and we can easily read off ζ (1) H d (0 , m ) = c d r d d X k =1 α k,d +1 ( − k k m k , (4.25) ∂ s ζ (1) H d (0 , m ) = c d r d d X k =1 α k,d +1 ( − k k m k ( H k − log m ) , (4.26)– 15 –here H k is a harmonic number.On the other hand, it is difficult to perform the integral in ζ (2) H d ( s, m ), hence we compute ζ (2) H d (0 , m ) and ∂ s ζ (2) H d (0 , m ) instead. ζ (2) H d (0 , m ) is independent on the mass term, ζ (2) H d (0 , m ) = c d r d d X k =1 α k,d +1 ( − k +1 B k k , (4.27)and the derivative can be computed as ∂ s ζ (2) H d (0 , m ) = − c d r d d X k =1 α k,d +1 f k ( m ) (4.28)with f k ( m ) = Z ∞ d ω ω k − e πω − ω + m )= ( − k " m k − k −
1) + m k k (cid:18) k − log( m ) (cid:19) + k − X l =1 B l l m k − l k − l − B k H k − k + k − X r =0 ( − r (cid:18) k − r (cid:19) m k − − r (cid:18) ζ ′ ( − r, m ) − B r +1 ( m ) H r r + 1 (cid:19) , (4.29)where B r +1 ( m ) is a Bernoulli polynomial. See appendix B for the detailed derivation of(4.29).We obtain the renormalized free energy for the massive fermion, F ren [ H d ]( m ) = − A [ H d ]( m ) log(Λ R ) + F fin [ H d ]( m ) (4.30)with the coefficient of the logarithmic divergent part A [ H d ]( m ) = − (cid:16) ζ (1) H d (0 , m ) + ζ (2) H d (0 , m ) (cid:17) = c d r d d X k =1 α k,d +1 ( − k B k − m k k , (4.31)and the finite part F fin [ H d ]( m ) = 12 (cid:16) ∂ s ζ (1) H d (0 , m ) + ∂ s ζ (2) H d (0 , m ) (cid:17) = c d r d d X k =1 α k,d +1 ( − k +1 (cid:18) − m k H k − k − B k H k − k + m k − k − k − X l =1 B l l m k − l k − l + k − X r =0 ( − r (cid:18) k − r (cid:19) m k − − r (cid:18) ζ ′ ( − r, m ) − B r +1 ( m ) H r r + 1 (cid:19)! . (4.32)– 16 –n the massless limit, m →
0, the renormalized free energy has a simple expression F ren [ H d ] = − A [ H d ] log(Λ R ) + F fin [ H d ] , (4.33)with the anomaly coefficient and the finite part A [ H d ] = c d r d d X k =1 α k,d +1 ( − k +1 B k k , (4.34) F fin [ H d ] = c d r d d X k =1 α k,d +1 ( − k ζ ′ (1 − k ) . (4.35)To see the massless limit of the finite term, it is convenient to use (B.7) instead of (4.29).The free energy on H d is just the half of that on S d , F ren [ H d ] = 12 F ren [ S d ] , (4.36)and this reproduces the known anomaly coefficients in literature [36].The anomaly coefficients and the finite parts of the free energy for d ≤
10 are listed intables 2 and 3 in appendix A. H p +1 × S q − The free energy on H p +1 × S q − for the massless Dirac fermion except for even p and even q ≥ F [ H p +1 × S q − ] = − ∞ X ℓ =0 g ( q − ( ℓ ) Z ∞ d ω µ ( p +1) ( ω ) log ω + (cid:16) ν ( q − ℓ (cid:17) ˜Λ R , (4.37)with the Plancherel measure (4.3) and the degeneracy (3.5). For even p and even q ≥
4, thereare two fermions of opposite U (1) charge as the additional U (1) appeared in (2.17). Then,the free energy on H p +1 × S q − with even p and even q ≥ F [ H p +1 × S q − ] = − ∞ X ℓ =0 g ( q − ( ℓ ) Z ∞ d ω µ ( p +1) ( ω ) log ω + (cid:16) ν ( q − ℓ (cid:17) ˜Λ R . (4.38)To treat these two cases simultaneously, we introduce a notation, d U (1) = ( p and even q ≥ , . (4.39)In the following, we compute the free energy using the zeta-function regularisation dividedinto two cases: even p case and odd p case. An important notice is that we use differentdecompositions for the logarithmic function depending on the evenness of p .– 17 – .2.1 Even p Performing similar computations in section 4.1.1, the renormalized free energy is given by F ren [ H p +1 × S q − ] = 12 ζ H p +1 × S q − (0) log(Λ R ) + 12 ∂ s ζ H p +1 × S q − (0) (4.40)with the zeta function ζ H p +1 × S q − ( s ) = d U (1) ∞ X ℓ =0 g ( q − ( ℓ ) ζ H p +1 (cid:16) s, ν ( q − ℓ (cid:17) . (4.41)Here ζ H p +1 (cid:16) s, ν ( q − ℓ (cid:17) is the zeta function of H d with mass ν ( q − ℓ (4.14). By using theexpansion of the degeneracy (3.11) with the replacement d → q −
1, the zeta function becomes ζ H p +1 × S q − ( s ) = 2 d U (1) c p +1 r p +1 p X k =0 β k,p +2 ( − k sin (cid:16) πs (cid:17) Γ(2 k + 1) k +1 Y i =1 s − i · r q − Γ( q − q − X n =0 ( − q − + n α n,q ζ ( s − n − k ) q : odd , q − X n =0 ( − q − n β n,q ζ H (cid:18) s − n − k − , (cid:19) q : even . (4.42)We immediately find ζ H p +1 × S q − (0) = 0 , (4.43)and ∂ s ζ H p +1 × S q − (0) = − d U (1) c p +1 r p +1 p X k =0 β k,p +2 ( − k π k + 1 · r q − Γ( q − q : odd , q − X n =0 ( − q − n β n,q ζ H (cid:18) − n − k − , (cid:19) q : even . (4.44)For even p , we find the following: • For odd q , the renormalized free energy vanishes, F ren [ H p +1 × S q − ] = 0 . (4.45)This implies that both bulk and defect anomalies vanish.– 18 – For even q , the renormalized free energy has a defect anomaly which comes from thevolume of H p +1 , F ren [ H p +1 × S q − ] = −A [ H p +1 × S q − ] log (cid:18) Rǫ (cid:19) (4.46)with A [ H p +1 × S q − ] = − ( − p + q d U (1) Γ( p + 1)Γ( q − p X k =0 β k,p +2 ( − k k + 1 · q − X n =0 ( − n β n,q ζ H (cid:18) − n − k − , (cid:19) . (4.47) • For q = p + 2, the defect anomaly is proportional to the bulk anomaly on S p +2 , A [ S p +2 ] = 2 A [ H p +1 × S p +1 ] , (4.48)as in the scalar case [34, 35] and the holographic case [35]. It is expected that theholographic result should be the same as the free fermion result because the anomalycoefficient does not depend on the strength of a coupling constant and the holographycan be applied if the number of fermions is large.For q = 2, our result correctly reproduces the free energy obtained in [49, 50] (with anappropriate dimension of the spinor).The anomaly coefficients of the free energy are listed in table 2 in appendix A. p For odd p , the Plancherel measure (4.19) is decomposed into two parts using the identity(4.20), and the free energy consists of two parts, F [ H p +1 × S q − ] = − c p +1 r p +1 p +12 X k =1 α k,p +2 ∞ X ℓ =0 g ( q − ( ℓ ) Z ∞ d ω ω k − · " log ω + i ν ( q − ℓ ˜Λ R ! + log ω − i ν ( q − ℓ ˜Λ R ! − c p +1 r p +1 p +12 X k =1 α k,p +2 Z ∞ d ω ω k − e πω − ∞ X ℓ =0 g ( q − ( ℓ ) · " log ν ( q − ℓ + i ω ˜Λ R ! + log ν ( q − ℓ − i ω ˜Λ R ! . (4.49)In contrast to section 4.1.2, we decompose the logarithmic function differently for each term,and the ordering of the summation over ℓ and the integral over ω is important.– 19 –he Schwinger representation of the free energy (4.49) is given by F [ H p +1 × S q − ] = c p +1 r p +1 p +12 X k =1 α k,p +2 Z ∞ d tt ∞ X ℓ =0 g ( q − ( ℓ ) Z ∞ d ω ω k − · h e − t ( ω +i ν ( q − ℓ ) / (˜Λ R ) + e − t ( ω − i ν ( q − ℓ ) / (˜Λ R ) i + c p +1 r p +1 p +12 X k =1 α k,p +2 Z ∞ d tt Z ∞ d ω ω k − e πω − ∞ X ℓ =0 g ( q − ( ℓ ) · h e − t ( ν ( q − ℓ +i ω ) / (˜Λ R ) + e − t ( ν ( q − ℓ − i ω ) / (˜Λ R ) i . (4.50)For the first term, we first perform the integral over ω , and hence the first term is convergentin the ω → ∞ limit in the Schwinger representation. On the other hand, for the second term,we first perform the summation over ℓ , and hence the second term is convergent in the ℓ → ∞ limit in the Schwinger representation.The renormalized free energy is given by F ren [ H p +1 × S q − ] = 12 ζ H p +1 × S q − (0) log(Λ R ) + 12 ∂ s ζ H p +1 × S q − (0) , (4.51)where the zeta function is a sum of 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 ) , (4.52)with ζ (1) H p +1 × S q − ( s ) = c p +1 r p +1 p +12 X k =1 α k,p +2 ∞ X ℓ =0 g ( q − ( ℓ ) · Z ∞ d ω ω k − (cid:20)(cid:16) ω + i ν ( q − ℓ (cid:17) − s + (cid:16) ω − i ν ( q − ℓ (cid:17) − s (cid:21) , (4.53) ζ (2) H p +1 × S q − ( s ) = 2 c p +1 r p +1 p +12 X k =1 α k,p +2 Z ∞ d ω ω k − e πω − · ∞ 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) . (4.54)– 20 –he first term in the zeta function is convergent for Re s > k , ζ (1) H p +1 × S q − ( s ) = 2 c p +1 r p +1 p +12 X k =1 α k,p +2 ( − k cos (cid:16) πs (cid:17) Γ(2 k ) k Y i =1 s − i · r q − Γ( q − q − X n =0 ( − q − + n α n,q ζ ( s − n − k + 1) q : odd , q − X n =0 ( − q − n β n,q ζ H (cid:18) s − n − k, (cid:19) q : even . (4.55)It follows that ζ (1) H p +1 × S q − (0) = c p +1 r p +1 r q − Γ( q − p +12 X k =1 α k,p +2 ( − k k · q − X n =0 ( − q − + n α n,q ζ ( − n − k + 1) q : odd , q : even , (4.56)and ∂ s ζ (1) H p +1 × S q − (0)= c p +1 r p +1 r q − Γ( q − p +12 X k =1 α k,p +2 ( − k k · q − X n =0 ( − q − + n α n,q (cid:2) H k ζ ( − n − k + 1) + ζ ′ ( − n − k + 1) (cid:3) q : odd , q − X n =0 ( − q − n β n,q ∂ s ζ H (cid:18) − n − k, (cid:19) q : even . (4.57)Using the expansion of the gamma functions in the degeneracy,Γ (cid:16) ν ( q − ℓ + q − (cid:17) Γ (cid:16) ν ( q − ℓ − q − + 1 (cid:17) = q − X n =0 ( − q − + n α n,q n − X i =0 (cid:18) n − i (cid:19) (cid:16) ν ( q − ℓ + i ω (cid:17) i ( − i ω ) n − − i q : odd , q − X n =0 ( − q − n β n,q n X i =0 (cid:18) ni (cid:19) (cid:16) ν ( q − ℓ + i ω (cid:17) i ( − i ω ) n − i q : even , (4.58)– 21 –nd its complex conjugate, we obtain ζ (2) H p +1 × S q − ( s ) = 2 c p +1 r p +1 p +12 X k =1 α k,p +2 Z ∞ d ω ω k − e πω − r q − Γ( q − · q − X n =0 ( − q − + n α n,q n − X i =0 (cid:18) n − i (cid:19) ( − i) n − − i ω n − − i · (cid:2) ζ H ( s − i, i ω ) + ( − i +1 ζ H ( s − i, − i ω ) (cid:3) q : odd , q − X n =0 ( − q − n β n,q n X i =0 (cid:18) ni (cid:19) ( − i) n − i ω n − i · (cid:20) ζ H (cid:18) s − i,
12 + i ω (cid:19) + ( − i ζ H (cid:18) s − i, − i ω (cid:19)(cid:21) q : even . (4.59)Although it is difficult to perform the integral over ω analytically, it is possible to simplify ζ (2) H p +1 × S q − (0) and its derivative at s = 0 furthermore. After a bit of calculation, we obtain ζ (2) H p +1 × S q − (0) = c p +1 r p +1 r q − Γ( q − p +12 X k =1 α k,p +2 q − X n =0 ( − q − α n,q · n − X i =0 (cid:18) n − i (cid:19) ( − i + k + n +1 i + 1 ⌊ i +12 ⌋ X l =0 (cid:18) i + 12 l (cid:19) B l B k +2 n − l k + n − l q : odd , q : even . (4.60)In addition, the derivative of the zeta function at s = 0 can be computed as ∂ s ζ (2) H p +1 × S q − (0) = 2 c p +1 r p +1 p +12 X k =1 α k,p +2 Z ∞ d ω ω k − e πω − r q − Γ( q − · q − X n =0 ( − q − + n α n,q n − X i =0 (cid:18) n − i (cid:19) ( − i) n − − i ω n − − i · (cid:2) ∂ s ζ H ( − i, i ω ) + ( − i +1 ∂ s ζ H ( − i, − i ω ) (cid:3) q : odd , q − X n =0 ( − q − n β n,q n X i =0 (cid:18) ni (cid:19) ( − i) n − i ω n − i · (cid:20) ∂ s ζ H (cid:18) − i,
12 + i ω (cid:19) + ( − i ∂ s ζ H (cid:18) − i, − i ω (cid:19)(cid:21) q : even . (4.61)– 22 –or even q , the combination of the Hurwitz zeta functions can be simplified using a formula ∂ s ζ H (cid:18) − i,
12 + i ω (cid:19) + ( − i ∂ s ζ H (cid:18) − i, − i ω (cid:19) = Γ( i + 1)(2 π i) i Li i +1 ( − e − πω ) + π i i + 1 B i +1 (cid:18)
12 + i ω (cid:19) . (4.62)See e.g. (B.19) in [34] for the derivation. Unfortunately, it is difficult to simplify the equationsanymore, so we perform the integral numerically.For odd p , we find the following: • For odd q , we numerically find the relation among free energies F ren [ S d ] = F ren [ H d − k × S k ] (4.63)for k = 1 , , · · · , d/ −
1. That is, the anomaly coefficients and the finite parts satisfythe relations, A [ S d ] = A [ H d − k × S k ] , (4.64) F fin [ S d ] = F fin [ H d − k × S k ] . (4.65) • For even q , the anomaly parts vanish since the bulk dimension d = p + q is odd. Wenumerically find the relation between the universal parts of the free energy F fin [ S d ] = F fin [ H d − k × S k ] . (4.66)The equivalence of the anomaly coefficients between H d − k × S k and S d follows from a relationof Euler characteristic χ [ H d − k × S k ] = χ [ S d ] as pointed out [34, 35] because the bulk anomalyis related to the Euler characteristic. In section 4, we computed the free energy on H p +1 × S q − with Dirichlet boundary condition.In this section, we compute the free energy with Neumann boundary condition. Neumannboundary condition exists only when q = 2 as we saw in section 2. We decompose the free energy into a sum of ℓ , F [ H p +1 × S q − ] = ∞ X ℓ =0 g ( q − ( ℓ ) F ℓ (cid:16) ν ( q − ℓ (cid:17) , (5.1)– 23 –here 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) = − d U (1) Z ∞ d ω µ ( p +1) ( ω ) " log ω + i ν ( q − ℓ ˜Λ R ! + log ω − i ν ( q − ℓ ˜Λ R ! . (5.2)Since the ω integral is performed before the summation over ℓ , the logarithmic function isdecomposed in this way.From now on, we concentrate on q = 2 because Neumann boundary condition is allowedonly when q = 2. The difference of free energies between the two boundary conditions comesfrom the ℓ = 0 mode. The Dirichlet boundary condition has a positive value ν (1) ℓ =0 = 12 , (5.3)while the Neumann boundary condition has a negative value ν (1) ℓ =0 = − . (5.4)In section 4.1, we obtained the zeta functions as analytical functions of positive m . It ispossible to analytically continue the zeta functions to a m < Even p From (4.17) the free energy of the ℓ = 0 mode is given by F ℓ =0 (cid:16) ν (1) ℓ =0 (cid:17) = ( − p +1 p Γ( p + 1) p X k =0 β k,p +2 (cid:16) ν (1) ℓ =0 (cid:17) k +1 k + 1 ( − k log (cid:18) Rǫ (cid:19) . (5.5)Then, the difference of the free energies becomes F ∆ D [ H p +1 × S ] − F ∆ N [ H p +1 × S ] = ( − p +1 Γ( p + 1) p X k =0 β k,p +2 ( − k k (2 k + 1) log (cid:18) Rǫ (cid:19) . (5.6)By using 12 k (2 k + 1) = 2 Z d u u k , (5.7)(3.11) with the replacement ν ( d ) ℓ → u , and the reflection formula of the gamma functionΓ( z )Γ(1 − z ) = π sin( πz ) (5.8)with z = u − p +12 + 1, the coefficient of the logarithmic divergent part becomes( − p +1 Γ( p + 1) p X k =0 β k,p +2 ( − k k (2 k + 1) = 2( − p +1 π Γ( p + 1) Z d u cos( πu )Γ (cid:18) p + 12 + u (cid:19) Γ (cid:18) p + 12 − u (cid:19) . (5.9)We find a relation F ∆ D [ H p +1 × S ] − F ∆ N [ H p +1 × S ] = − F univ [ S p ] , (5.10)where the universal free energy on S p is given by (3.24).– 24 – dd p From (4.31) and (4.32) with (B.7), the difference of the free energies is given by F ∆ D [ H p +1 × S ] − F ∆ N [ H p +1 × S ]= − c p +1 r p +1 p +12 X k =1 α k,p +2 (cid:18) f k (cid:18) (cid:19) − f k (cid:18) − (cid:19)(cid:19) = − c p +1 r p +1 p +12 X k =1 α k,p +2 ( − k k − (2 k −
1) + Z − d u u k − ψ ( u ) ! . (5.11)By using the identity ψ ( u ) − ψ ( − u ) = 1 u + π cot( πu ) , (5.12)(3.11) with the replacement ν ( d ) ℓ → u and the reflection formula of the gamma function with z = u − p +12 + 1, we find F ∆ D [ H p +1 × S ] − F ∆ N [ H p +1 × S ] = 2 c p +1 r p +1 Z d u cos( πu )Γ (cid:18) p + 12 + u (cid:19) Γ (cid:18) p + 12 − u (cid:19) = − F univ [ S p ] , (5.13)where the universal free energy on S p is given by (3.24).Independent of the evenness of p , the difference of the free energies between the twoboundary conditions is proportional to the universal free energy of the fermion on S p , F ∆ D [ H p +1 × S ] − F ∆ N [ H p +1 × S ] = − F univ [ S p ] . (5.14)This fact implies that the Neumann boundary condition for q = 2 is trivial in the sense thatthe defect operator saturates the unitarity bound (2.23) and becomes a free field. C -theorem We are now in a position to compare the result in section 5.1 with our proposed conjecture in[12]. The defect free energy (1.5) may not be invariant under the Weyl transformation, whilewe expect that the difference of the defect free energies is invariant. An RG flow from theNeumann boundary condition to the Dirichlet boundary condition is triggered by a doubletrace deformation as is familiar in the AdS/CFT setup [42–48], and the difference of theuniversal part of the defect free energies is given by˜ D UV − ˜ D IR = − sin (cid:16) πp (cid:17) (cid:0) F ∆ N [ H p +1 × S ] − F ∆ D [ H p +1 × S ] (cid:1) = 2 ˜ F [ S p ] , (5.15)– 25 –here ˜ F [ S p ] = 2 r p Γ( p + 1) Z d u cos( πu )Γ (cid:18) p + 12 + u (cid:19) Γ (cid:18) p + 12 − u (cid:19) (5.16)is positive for any p . The positivity of the sphere free energy leads to the positivity of thedifference of the free energies between at UV fixed point and at IR fixed point. In this case,our proposed defect C -theorem holds. In this paper, we studied a free Dirac fermion on H p +1 × S q − as a DCFT. In section 2, weclassified the allowed boundary conditions, and we found that a nontrivial boundary conditionis allowed only in q = 2. In sections 3 and 4, we computed the free energy on S d , HS d , H d and H p +1 × S q − with the Dirichlet boundary condition using the zeta-function regularisation. Inparticular, we obtained relations of free energies, which hold also in a conformally coupledscalar field [34, 35] in section 4.2. In section 5, we computed the difference of the free energiesbetween the Dirichlet boundary condition and the Neumann boundary condition in q = 2and confirmed the validity of our proposed defect C -theorem.We obtained various results similar to a conformally coupled scalar case [34, 35]. However,there are several differences between the fermion case and the scalar case. The first differ-ence comes from the codimension of the defect which allows nontrivial boundary conditions.Nontrivial boundary conditions in the conformally coupled scalar are allowed in q = 1 , , , q = 2. The seconddifference is that we rigorously derive the equivalence of the free energies between HS d and H d for arbitrary d . However, for the conformally coupled scalar [34], the equivalence of thefree energies between HS d and H d is checked only numerically because a nontrivial identityamong Bernoulli polynomials are required for a proof of the equivalence for arbitrary d .In section 2.2, we gave a classification of boundary conditions in a free fermion, andthis means that we constructed a concrete model of a DCFT. A task to derive the sameclassification (or defect operator) of boundary conditions by using the method in [41] remains.The concrete model would be useful for a study of DCFTs. Acknowledgments
The author thanks T. Nishioka for the collaboration of the related work, useful discussionand various comments on the draft of this paper. The author thanks D. Rodriguez-Gomezand J. G. Russo for comments on the draft of this paper and useful discussions. The authoralso thanks Y. Abe, Y. Okuyama and M. Watanabe for useful communication. The work issupported by the National Center of Theoretical Sciences (NCTS).– 26 –
List of tables · S S S S · − H − H − H − − H − H − H − H − − H − H
10 147972993760 − 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 the Dirichlet boundary condition. – 27 – F fin [ M ] S ζ ′ ( − S log 2 + π ζ (3) S − ζ ′ ( −
1) + ζ ′ ( − S − log 2 − π ζ (3) − π ζ (5) S ζ ′ ( − − ζ ′ ( −
3) + ζ ′ ( − S log 2 + π ζ (3) + π ζ (5) + π ζ (7) S − ζ ′ ( −
1) + ζ ′ ( − − ζ ′ ( −
5) + ζ ′ ( − S − log 2 − π ζ (3) − π ζ (5) − π ζ (7) − π ζ (9) S
10 32315 ζ ′ ( − − ζ ′ ( −
3) + ζ ′ ( − − ζ ′ ( −
7) + ζ ′ ( − d F fin [ HS d ] = F fin [ S d ]Even d F fin [ HS d ] = F fin [ S d ]Odd d d F fin [ H d ] = F fin [ S d ]Even 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 [ HS d ], F fin [ H d ], and F fin [ H p +1 × S q − ] with theDirichlet boundary condition. B Detail derivation of (4.29)
In this appendix, we give a detailed derivation of (4.29).To perform the integral f k ( m ) = Z ∞ d ω ω k − e πω − ω + m ) , (B.1)we first take the derivative respect to m , ∂ m f k ( m ) = 2 mg k ( m ) , (B.2) g k ( m ) = Z ∞ d ω ω k − (e πω − ω + m ) . (B.3)Since g k ( m ) satisfies the recursion relation g k +1 ( m ) = − m g k ( m ) + ( − k +1 B k k , (B.4)– 28 –ith the initial condition g ( m ) = 12 (cid:18) log m − m − ψ ( m ) (cid:19) , (B.5)the solution is given by g k ( m ) = ( − k − m k − g ( m ) − k − X l =1 m − l B l l ! . (B.6)By integrating 2 mg k ( m ) from 0 to m , f k ( m ) can be evaluated as f k ( m ) = ( − k (cid:20) − ζ ′ (1 − k ) + m k − k −
1) + m k k (cid:18) k − log( m ) (cid:19) + Z m d µ µ k − ψ ( µ ) + k − X l =1 B l l m k − l k − l , (B.7)where we use f k (0) = 2 Z ∞ d ω ω k − e πω − ω = ( − k − ζ ′ (1 − k ) . (B.8)The remaining integral is performed using a formula in [64, 65] Z m d µ µ n ψ ( µ ) = ( − n (cid:18) B n +1 H n n + 1 − ζ ′ ( − n ) (cid:19) + n X r =0 ( − r (cid:18) nr (cid:19) m n − r (cid:18) ζ ′ ( − r, m ) − B r +1 ( m ) H r r + 1 (cid:19) . (B.9) References [1] A.B. Zamolodchikov,
Irreversibility of the Flux of the Renormalization Group in a 2D FieldTheory , JETP Lett. (1986) 730.[2] J.L. Cardy, Is There a c Theorem in Four-Dimensions? , Phys. Lett.
B215 (1988) 749.[3] Z. Komargodski and A. Schwimmer,
On Renormalization Group Flows in Four Dimensions , JHEP (2011) 099 [ ].[4] 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 [ ].[5] I.R. Klebanov, S.S. Pufu and B.R. Safdi, F -Theorem without Supersymmetry , JHEP (2011) 038 [ ].[6] R.C. Myers and A. Sinha, Seeing a c-theorem with holography , Phys. Rev.
D82 (2010) 046006[ ]. – 29 –
7] R.C. Myers and A. Sinha,
Holographic c-theorems in arbitrary dimensions , JHEP (2011) 125 [ ].[8] S. Giombi and I.R. Klebanov, Interpolating between a and F , JHEP (2015) 117 [ ].[9] H. Casini and M. Huerta, A Finite entanglement entropy and the c-theorem , Phys. Lett.
B600 (2004) 142 [ hep-th/0405111 ].[10] H. Casini and M. Huerta,
On the RG running of the entanglement entropy of a circle , Phys. Rev.
D85 (2012) 125016 [ ].[11] H. Casini, E. Test´e and G. Torroba,
Markov Property of the Conformal Field Theory Vacuumand the a Theorem , Phys. Rev. Lett. (2017) 261602 [ ].[12] N. Kobayashi, T. Nishioka, Y. Sato and K. Watanabe,
Towards a C -theorem in defect CFT , JHEP (2019) 039 [ ].[13] D.R. Green, M. Mulligan and D. Starr, Boundary Entropy Can Increase Under Bulk RG Flow , Nucl. Phys. B (2008) 491 [ ].[14] Y. Sato,
Boundary entropy under ambient RG flow in the AdS/BCFT model , Phys. Rev. D (2020) 126004 [ ].[15] I. Affleck and A.W.W. Ludwig,
Universal noninteger ‘ground state degeneracy’ in criticalquantum systems , Phys. Rev. Lett. (1991) 161.[16] D. Friedan and A. Konechny, On the boundary entropy of one-dimensional quantum systems atlow temperature , Phys. Rev. Lett. (2004) 030402 [ hep-th/0312197 ].[17] H. Casini, I.S. Landea and G. Torroba, The g-theorem and quantum information theory , JHEP (2016) 140 [ ].[18] K. Jensen and A. O’Bannon, Constraint on Defect and Boundary Renormalization GroupFlows , Phys. Rev. Lett. (2016) 091601 [ ].[19] M. Nozaki, T. Takayanagi and T. Ugajin,
Central Charges for BCFTs and Holography , JHEP (2012) 066 [ ].[20] D. Gaiotto, Boundary F-maximization , .[21] Y. Wang, Defect a -Theorem and a -Maximization , .[22] H. Casini, I. Salazar Landea and G. Torroba, Irreversibility in quantum field theories withboundaries , JHEP (2019) 166 [ ].[23] S. Yamaguchi, Holographic RG flow on the defect and g theorem , JHEP (2002) 002[ hep-th/0207171 ].[24] T. Takayanagi, Holographic Dual of BCFT , Phys. Rev. Lett. (2011) 101602 [ ].[25] M. Fujita, T. Takayanagi and E. Tonni,
Aspects of AdS/BCFT , JHEP (2011) 043[ ].[26] J. Estes, K. Jensen, A. O’Bannon, E. Tsatis and T. Wrase, On Holographic Defect Entropy , JHEP (2014) 084 [ ].[27] R.-X. Miao, C.-S. Chu and W.-Z. Guo, New proposal for a holographic boundary conformal fieldtheory , Phys. Rev. D (2017) 046005 [ ]. – 30 –
28] S.P. Kumar and D. Silvani,
Holographic flows and thermodynamics of Polyakov loop impurities , JHEP (2017) 107 [ ].[29] S.P. Kumar and D. Silvani, Entanglement of heavy quark impurities and generalizedgravitational entropy , JHEP (2018) 052 [ ].[30] M. Beccaria, S. Giombi and A. Tseytlin, Non-supersymmetric Wilson loop in N = 4 SYM anddefect 1d CFT , JHEP (2018) 131 [ ].[31] 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 [ ].[32] Y. Wang,
Surface Defect, Anomalies and b -Extremization , .[33] R. Rodgers, Holographic entanglement entropy from probe M-theory branes , JHEP (2019) 092 [ ].[34] T. Nishioka and Y. Sato, Free energy and defect C -theorem in free scalar theory , .[35] D. Rodriguez-Gomez and J.G. Russo, Free energy and boundary anomalies on S a × H b spaces , JHEP (2017) 084 [ ].[36] D. Rodriguez-Gomez and J.G. Russo, Boundary Conformal Anomalies on Hyperbolic Spacesand Euclidean Balls , JHEP (2017) 066 [ ].[37] C.P. Herzog and I. Shamir, On Marginal Operators in Boundary Conformal Field Theory , JHEP (2019) 088 [ ].[38] S. Giombi and H. Khanchandani, CFT in AdS and boundary RG flows , JHEP (2020) 118[ ].[39] A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality , Phys. Rev.
D74 (2006) 025005 [ hep-th/0501015 ].[40] S.M. Chester, M. Mezei, S.S. Pufu and I. Yaakov,
Monopole operators from the − ǫ expansion , JHEP (2016) 015 [ ].[41] E. Lauria, P. Liendo, B.C. Van Rees and X. Zhao, Line and surface defects for the free scalarfield , JHEP (2021) 060 [ ].[42] E. Witten, Multitrace operators, boundary conditions, and AdS/CFT correspondence , hep-th/0112258 .[43] M. Berkooz, A. Sever and A. Shomer, ‘Double trace’ deformations, boundary conditions andspace-time singularities , JHEP (2002) 034 [ hep-th/0112264 ].[44] 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 ].[45] S.S. Gubser and I.R. Klebanov, A universal result on central charges in the presence of doubletrace deformations , Nucl. Phys. B (2003) 23 [ hep-th/0212138 ].[46] T. Hartman and L. Rastelli,
Double-trace deformations, mixed boundary conditions andfunctional determinants in AdS/CFT , JHEP (2008) 019 [ hep-th/0602106 ].[47] D.E. Diaz and H. Dorn, Partition functions and double-trace deformations in AdS/CFT , JHEP (2007) 046 [ hep-th/0702163 ]. – 31 –
48] S. Giombi, I.R. Klebanov, S.S. Pufu, B.R. Safdi and G. Tarnopolsky,
AdS Description ofInduced Higher-Spin Gauge Theory , JHEP (2013) 016 [ ].[49] I.R. Klebanov, S.S. Pufu, S. Sachdev and B.R. Safdi, Renyi Entropies for Free Field Theories , JHEP (2012) 074 [ ].[50] M. Beccaria and A. Tseytlin, C T for higher derivative conformal fields and anomalies of (1, 0)superconformal 6d theories , JHEP (2017) 002 [ ].[51] J.S. Dowker, J.S. Apps, K. Kirsten and M. Bordag, Spectral invariants for the Dirac equationon the d ball with various boundary conditions , Class. Quant. Grav. (1996) 2911[ hep-th/9511060 ].[52] R. Camporesi, The Spinor heat kernel in maximally symmetric spaces , Commun. Math. Phys. (1992) 283.[53] R. Camporesi and A. Higuchi,
On the Eigen functions of the Dirac operator on spheres and realhyperbolic spaces , J. Geom. Phys. (1996) 1 [ gr-qc/9505009 ].[54] A. Lewkowycz, R.C. Myers and M. Smolkin, Observations on entanglement entropy in massiveQFT’s , JHEP (2013) 017 [ ].[55] S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories , Adv. Theor. Math. Phys. (1998) 783 [ hep-th/9712074 ].[56] A. Monin, Partition function on spheres: How to use zeta function regularization , Phys. Rev. D (2016) 085013 [ ].[57] D. Vassilevich, Heat kernel expansion: User’s manual , Phys. Rept. (2003) 279[ hep-th/0306138 ].[58] “
NIST Digital Library of Mathematical Functions .” http://dlmf.nist.gov/, Release 1.1.0 of2020-12-15.[59] M. Marino,
Lectures on localization and matrix models in supersymmetric Chern-Simons-mattertheories , J. Phys. A (2011) 463001 [ ].[60] J. Dowker, On a - F dimensional interpolation , .[61] 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 [ hep-th/9505061 ].[62] R. Camporesi and A. Higuchi,
Spectral functions and zeta functions in hyperbolic spaces , J. Math. Phys. (1994) 4217.[63] A. Bytsenko, E. Elizalde and S. Odintsov, The Conformal anomaly in N-dimensional spaceshaving a hyperbolic spatial section , J. Math. Phys. (1995) 5084 [ gr-qc/9505047 ].[64] V.S. Adamchik, Polygamma functions of negative order , Journal of computational and applied mathematics (1998) 191.[65] O.R. Espinosa and V.H. Moll,
On some integrals involving the hurwitz zeta function: Part 2 , The Ramanujan Journal (2002) 449 [ math/0107082 ].].