Conformal anomalies of CFT’s with boundaries

The trace anomaly of conformal field theories in four dimensions is characterized by ‘a’ and ‘c’-functions. The scaling properties of the effective action of a CFT in the presence of boundaries is shown to be determined by a, c and two new functions (charges) related to boundary effects. The boundary charges are computed for different theories and different boundary conditions. One of the boundary charges depends on the bulk c charge.

. Charges in the anomaly of the effective action.
integral of the trace anomaly since new boundary terms may appear. An experience with two dimensional CFT's shows that boundary terms in some quantum quantities, such as a g-function in the entropy, may be interesting from the point of view of the renormalization group [3,4].
We study different CFT's in four dimensions with the boundary conditions which do not break the conformal invariance on the classical level and show that the integral anomaly has the following universal structure: (1.5) Quantities χ 4 , i, j 1 and j 2 are scale invariant functionals, χ 4 is the Euler characteristics of the background manifold, i is the integral of I over M, see (1.3). The boundaries result in two new terms with constants q 1 and q 2 (the boundary charges). The boundary functionals in (1.5) are We use the following notations: g µν is the metric of the background manifold M, the metric induced on the boundary ∂M of M is H µν = g µν − N µ N ν , N ν is a unit outward pointing normal vector to ∂M, K µν = H λ µ H ρ ν N λ;ρ is the extrinsic curvature tensor of ∂M,K µν = K µν − H µν K/3 is a traceless part of K µν . Conformal invariance of j 1 , j 2 follows from the fact thatK µν transforms homogeneously under conformal transformations. Definitions of the Riemann tensor, the Ricci tensor, the scalar curvature, respectively, are Our definitions of R and K µν coincide with those used by Dowker and Schofield in [5] and by Vassilevich in [6]. The only difference with those works is in the sign of R λ µνρ and direction of N µ . In our paper, in [5], and in [6] the scalar curvature on S n is positive. The curvatures constructed with the metric of ∂M will be denoted asR λ µνρ ,R µν ,R. Our results for the boundary charges in (1.5) are listed in table 1. Coefficient q 2 can depend on the type of the boundary conditions.
It should be noted that (1.5) for a scalar field with the Dirichlet boundary condition follows from results by Dowker and Schofield [5]. Table 1 agrees with [5] for this case.

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Functional j 2 was introduced for the first time in [7]. Anomalous rescalings of the effective action of Dirac fields and gauge bosons have been also studied by Moss and Poletti [8] for Einstein spaces with boundaries.
To our knowledge the model independent nature of eq. (1.5) has not been emphasized so far. It is important to note that eq. (2.1), which determines (1.5), can be derived in different regularization prescriptions, e.g. in the zeta-function method and the proper-time cutoff regularization, see [9]. Also, there are no local counterterms which can be added to the effective action to change the boundary terms in the integral anomaly. In this sense q 1 and q 2 are scheme independent.
In section 2 we relate the anomaly to known computations of boundary terms in the corresponding heat kernel coefficients. Anomalies in the models presented in table 1 are discussed in section 3. Concluding remarks are given in section 4. We point out a universal relation between c and q 1 (valid at least for all above models). It hints that q 1 is not an independent new charge.

The heat coefficient
We use the relation between the anomaly and the heat coefficient of a Laplace operator ∆ = −∇ 2 + X for the corresponding conformal theory where η = +1 for Bosons and η = −1 for Fermions. We ignore in (2.1) a possible contribution of zero modes of ∆. The heat coefficients for the asymptotic expansion of the heat kernel of ∆ are defined as If the classical theory is scale invariant the heat coefficient A 4 is a conformal invariant, see e.g. [9]. Therefore A 4 can be represented as a linear combination of conformal invariants constructed of the geometrical characteristics of M, ∂M and embedding of ∂M in M. On dimensional grounds, these invariants (in four dimensions) are χ 4 , i, j 1 and j 2 . The bulk part of A 4 is determined by χ 4 and i, and it is well known. We are interested in boundary terms in A 4 . To find them one should take into account the boundary term in χ 4 . The definition of the Euler number for a four-dimensional manifold with a boundary is as follows: 3)

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Derivation of these formulae can be found in [5]. Some values of χ 4 are: The boundary part of A 4 , therefore, is It is the aim of computations to check that the coefficient at S 4 does equal −2ηa, where a is the same constant which appears in the trace anomaly (1.1).
Appearance of S 4 among counter terms in a one-loop effective action dates back to works in 1980's, see [10]. If M is a domain of a flat Euclidean background and ∂M = S 3 one easily finds for (1.5) that A = −2a. In this case the anomaly is solely determined by S 4 . Mode-by-mode computations of the anomaly for the spinor and gauge fields have been done in [11,12].
After converting total derivatives in the bulk into surface terms and some algebra the boundary part A bd 4 of A 4 can be written as (2.10) Here we use 'flat' indices a, b in the tangent space to the boundary, see [6]. Ω µν is the field strength of the connection defined by (2.10) in [6]. The notations are TrK m = K µ 1 µ 2 . . . K µp µ 1 , K = TrK.

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In deriving (2.10)-(2.13) we used Gauss-Codazzi identities and the relations: KR , (2.14)  [8]. Calculations of A 4 in case of boundaries have been done by several authors. The key paper is by Branson and Gilkey [13]. A complete list of references can be found in [6].
The physical meaning of (3.1) is that the normal component of the spinor current vanishes on the boundary. Condition (3.1) does not break conformal invariance. The strength of the spin connection is The rest computation is straightforward. One finds 3)

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where r = 4 is the number of components of the Dirac spinor in four dimensions. In computing C +− 4 one uses the relation The data of table 1 follow from the sum of (3.2) and (3.3) and relations (2.14)-(2.16).

Gauge boson
By following [6] we consider the quantization of an Abelian gauge field V µ in the Lorentz gauge ∇V = 0. The results of table 1 are valid for a gauge invariant combination where (∆ (1) ) ν µ = −∇ 2 δ ν µ + R ν µ is the vector Laplacian, and ∆ (gh) = −∇ 2 is the Laplacian for ghosts.
We study two sorts of boundary conditions: the absolute (or electric [8]) boundary condition and relative (or magnetic [8]) boundary condition The both conditions are manifestly gauge and conformally invariant. Condition (3.5) requires that components of an electric field normal to ∂M and components of the magnetic field which are tangential to ∂M vanish on the boundary. This means that the boundary is a perfect conductor. In condition (3.6) the roles of electric and magnetic fields are interchanged.
In the Lorentz gauge the absolute boundary condition is reduced to the following conditions on the vector field and ghosts [6]: where V ± = Π ± V , and The corresponding boundary condition for a ghost field c is The relative condition in the same gauge is

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By taking into account that X µ ν = R µ ν , (Ω λρ ) ν µ = R ν µλρ , for the vector component with the absolute condition (3.7), (3.9) (S µ ν = −K µ ν ) we find (3.14) For the ghost part with condition (3.9) S = 0 and For the vector component with relative conditions (3.10), (3.11) (S µ ν = −Kδ µ ν ) For the ghost part with condition (3.12) In deriving (3.13), (3.16) we used the identity The first term in the right hand side of this identity is the total derivative on ∂M. Let us emphasize that the two sets of boundary conditions for the gauge field result in the same boundary terms in the anomalous scaling of the effective action.

Concluding remarks
The aim of this paper was to demonstrate a model independent form of the integral anomaly (1.5) in the presence of boundaries and obtain specific values of the boundary charges for some CFT's. It would be important now to study evolution of the boundary charges under the renormalization group.
Computations of boundary charges for other models can be continued along the lines of the present paper. Extensions of (1.5) to higher dimensional CFT's, say to 6 dimensions, are possible. Since the number of scale invariant structures is increasing we expect more boundary charges. A principal challenge here is the knowledge of boundary terms in the heat coefficient A 6 .

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An interesting issue is a possible relation between bulk charges a, c and boundary charges q 1 , q 2 . A conjecture of [14] is that q 1 and c may be related. Indeed, all models presented in table 1 satisfy a universal relation q 1 = 8c . (4.1) We leave a general proof and implications of (4.1) for a future analysis. Some arguments in the favour of (4.1) are presented in [14].
It should be noted that formula (1.5) was also discussed in a recent work [15] which appeared several days earlier of the present publication.