Interface Conformal Anomalies

We consider two $d \geq 2$ conformal field theories (CFTs) glued together along a codimension one conformal interface. The conformal anomaly of such a system contains both bulk and interface contributions. In a curved-space setup, we compute the heat kernel coefficients and interface central charges in free theories. The results are consistent with the known boundary CFT data via the folding trick. In the $d=4$ case, two interface invariants generally allowed as anomalies turn out to have vanishing interface charges. These missing invariants are constructed from components with odd parity with respect to flipping the orientation of the defect. We conjecture that all invariants constructed from components with odd parity may have vanishing coefficient, even in the case of interacting interface CFT.


Introduction
We study boundaries and defects in conformal field theory (CFT) because they have broad applications and because they are fundamental to our understanding of CFT and quantum field theory more generally. Boundaries and defects are generally present in most experimental realizations of critical systems, and they bring with them the potential for a wide variety of experimentally verifiable consequences, for example surface critical exponents. Beyond that, however, there are fundamental questions about the classification and "space" of quantum field theories that can be answered through a careful study of defects. While it is often stated that a conformal field theory is defined -through operator product expansion -by its local operator spectrum and set of three-point correlation functions, in fact there are often extended operators, such as Wilson lines, which must be included for a proper definition of the CFT (see e.g. [1]). These extended operators carry with them an additional defect interpretation.
The complete classification of conformal defects or, equivalently, universality classes of critical behavior at the interfaces of CFTs, however, remains a challenging open problem. In particular, while earlier studies mostly focused on d = 2 critical systems [2][3][4][5], 1 there has been much recent interest in understanding d > 2 CFTs with boundaries or defects. 2 For CFTs without a boundary, the trace anomaly coefficients -which we call central charges -provide a useful classification [15]. To organize CFTs in the presence of general defects, it is thus natural to look for similar central charges. In this note, our defect example is a codimension-one conformal interface. We shall focus on d = 4 free CFT. Our main results will be the trace anomalies of d = 4 free interface CFT (ICFT). The computation relies on the heat kernel technique performed in generally curved spacetime; see [16] for a review. Alternatively, defect central charges may be computed by supersymmetric localization -but in superconformal theories only [17]. We focus on identical free CFTs with spin zero, one half, and one on both sides of the interface. These results generalize anomaly computation in boundary CFT (BCFT) [18][19][20], which can be recovered via the folding trick.
One might expect that the anomaly structure of ICFT can be fixed by that of BCFT through the folding trick. However, the moral of the present work is that the general interface story can be much richer. An interesting observation is that we find two interface invariants that are consistent with all the requirements to be a part of the anomaly but that have zero coefficient in d = 4 free ICFT. We do not have a simple argument for this vanishing. We conjecture this vanishing remains true for general ICFTs, but it is possible that interactions could generate new non-zero charges.

Interface Setup
Let Σ be an interface hypersurface where two manifolds or two parts of the same manifold are glued together along their common boundary. The bulk manifold will be denoted by M, dim M = d. We assume that the metric is continuous across Σ, but not necessarily smooth. Let us mark in an arbitrary way two sides of Σ by + and −. Let n + and n − be unit normals to the boundary pointing to + and − sides, respectively. Let x µ be local coordinates on M. The coordinates on Σ will be denoted by x j , j = 1, . . . , n − 1. The induced metric on Σ will be denoted by h ij . The extrinsic curvatures of Σ defined from two sides of Σ do not need to agree. The case of differentiable metric corresponds to K + ij = −K − ij . Let V be some vector bundle over M. Consider an operator L of Laplace type that acts on smooth sections of this bundle. L can be written as where E is a smooth endomorphism (a matrix valued function), and ∇ µ = ∂ µ + ω µ is a covariant derivative. We shall also need a bundle curvature We do not assume any continuity conditions for E, ω and Ω on Σ. To have a well defined spectral problem, one has to impose on sections φ of V some matching conditions on Σ. A natural choice is to request that φ is continuous, but the normal derivative jumps, The superscripts ± denote the limiting values that various quantities take when they approach Σ from different sides. 3 We also define a gauge/diffeomorphism vector as the difference between two connections on Σ.
Our next step is to define conformal matching conditions for various spins. It is instructive to compare them with conformal boundary conditions adopted in, for instance, [19].

Scalars:
For a conformally coupled scalar field ϕ, the operator L reads .
Thus, E = −ξR while ω, Ω and B vanish. Under the Weyl rescaling g µν →ḡ µν = e −2σ g µν the operator L and the field ϕ transform as Also,n It is easy to check that the conditions (4) with are Weyl invariant. The Euclidean action with an interface is The interface contribution is introduced to make the variational problem self-consistent.

Spinors:
The massless Dirac operator reads Here A, B, . . . are flat indices. Further, {A, B, . . . } = {a, b, . . . , n} so that e n j = 0 and e n n + = 1 on the + side of Σ and e n n − = −1 on the − side. This implies γ n = γ n + = −γ n − . We have In this case, As / D is a first order operator, the matching condition The Weyl invariance of these conditions can be easily checked. We shall need also The bulk Dirac action is standard and the interface theory does not require a surface term.

U (1) gauge field:
The matching conditions for abelian gauge fields and ghosts in the Lorentz gauge are particular cases of matching conditions for the de Rham complex. 4 For the ghosts (0-forms), Thus, the ghosts are actually smooth across Σ. For 1-forms, the operator L is the Hodge-de Rham laplacian. We have and Gauge transformed vector fields satisfy matching conditions if the gauge parameter satisfies the matching condition of ghosts. The gauge invariance of these matching conditions follows by the construction and may be checked directly. The bulk gauge-field action is standard and the interface theory does not require a surface term.

Heat kernel coefficients and central charges
For any generalized Laplacian L, there is a small-t asymptotic expansion, where f is a smearing function which allows a computation of local heat kernel coefficients: Note that if there are boundaries or singular surfaces, a k (x, L) contains δ-functions and derivatives of δ-functions localized on the boundary or on the singular surface. The trace anomaly is given by where η = 1 for bosons and η = −1 for fermions. 4 See, for instance, [22] for a more general discussion.
Here we compute the heat kernel coefficients a d (f, L) (d = dim M) by using general expressions. Covariant derivatives are denoted by a semicolon. By a colon we shall denote covariant derivatives on Σ containing the Christoffel symbol corresponding to the induced metric h ij . 5 The heat kernel coefficients are local. This means that they are given by integrals of local polynomials. In a d (f, L), the integral over M contains invariants of the canonical dimension d, while the integral over Σ contains invariants of the dimension d − 1.
Before starting actual computations, let us list several consistency conditions [22,23] that must be satisfied: (i) The heat kernel coefficients have to be invariant with respect to exchanging the roles of the "+" and "−" sides of Σ. The coefficients need to be invariant with respect to which direction one looks at the system.
(ii) When the metric is smooth, we have K + ij = −K − ij , R + = R − , etc. In this case, the interface effectively disappears and only the bulk structure survives.
(iii) Assume that M is composed of two identical manifolds M + = M − glued together along their common boundary Σ. Let x ∈ M + and x * ∈ M − be corresponding points. By considering one can map the heat kernel on M with an interface Σ to the heat kernel of boundary value problems on M + . In our case, this mapping implies that the heat kernel coefficient a k for conformal scalar on M is a sum of the coefficients a k for a conformal scalar on M + with Dirichlet boundary conditions on Σ and for another scalar with conformal Robin boundary conditions. The heat kernel expansion for a spinor field on M has interface coefficients which are twice that for conformal spinor fields on M + . Similarly, the heat kernel for a d = 4 Maxwell field on M is a sum of the heat kernel for Maxwell fields on M + satisfying the so-called absolute and relative boundary conditions.
The computations in two and three dimensions are simple. For the conformally coupled scalar we have 5 We follow notation in [22]. Note the Riemann tensor in [22] has an overall minus sign as compared to our notation. The conventions for the Ricci tensor and scalar curvature are identical.
For the Dirac spinor, The expressions (24) -(27) could have been obtained by using only the conditions (i), (ii) and (iii). The d = 2 bulk integral is well known. The anomaly in d = 3 CFT is a surface term. On Σ, the expressions f (K + − K − ) in a 2 as well as f ( in a 3 are forbidden since they do not satisfy the condition (i). The expressions do not vanish on smooth geometries and thus are forbidden by the condition (ii). The remaining invariants are exactly the ones which appear in (24)-(27); they can be determined by comparing to BCFTs, as described in (iii) above.
Let us turn to four dimensions. The bulk contributions are standard. However, here we write them down explicitly with the total derivatives in T µ µ which are sometimes neglected. We have The parameter ξ = 1 6 corresponds to the conformal scalar, and ξ = 0 to the ghost. We have removed ghost contributions in the electromagnetic field case.
Next, we find the following interface contributions: where curvature structures I i and heat kernel coefficients γ i are given by These surface contributions are computed with the help of the basis considered in [22, Theorem 7.1] but we remark that [22] uses an overcomplete basis. Using Gauss-Codazzi equations, we can write I 13 = − 1 2 (I 1 + I 2 ) + I 3 and I 20 = 1 2 (

Interface trace anomaly for d = 4 ICFTs
By collecting everything together, we now obtain the trace anomaly We have dropped the anomaly 2R which will be cancelled by the conformal variation of a local counterterm; see (38) below. The central charges are given in the following

144
Via the standard folding trick, one can check that these central charge results are consistent with BCFT data obtained earlier in [18,19]. For instance, tr( K + + K − ) 3 becomes 8 tr K 3 after the folding; note this is to be compared to b 1 (Dirchlet)+b 1 (Robin) in BCFT. The relations read Note b ICFT 2 = 4c in free ICFTs while b BCFT 2 = 8c in free BCFTs. It was shown in [24] that such a relation can be violated by boundary marginal interactions in BCFT. While we do not include interactions here, we expect the interaction will correct the relation b ICFT 2 = 4c.
The integrated trace anomaly is locally conformally invariant. We have verified that all the derivative of the delta-function terms can be cancelled by the following local counterterms: where coefficients α i are given by spin s 360α 1 360α 2 360α 3 360α 4 s=0 Namely, the conformal transformation of (38), δ σ I ct , reproduces terms in a 4 with derivatives of f after replacing σ with f .
Interestingly, the requirement of vanishing derivatives of the smearing function f automatically removes the 2R anomaly in free theories (29), (30), (31) via the identity δ σ M R 2 = 122R. To our knowledge, such a connection to a vanishing 2R has not been mentioned before. Note central charges a, c, b 1 , b 2 do not renormalize while α-terms depend on normalization conditions and thus, in this sense, scheme-dependent. However, these α-coefficients are still meaningful as long as one stays in the heat-kernel scheme.
An interface theory with the N=4 super Yang-Mills multiplet has simple relations: The 2R anomaly has zero coefficient with the N=4 SYM multiplet. More generally, it would be interesting to search for bounds on these coefficients in interface CFT.

A conjecture
There are two structures, which satisfy the conditions (i) -(iii) given at the beginning of the Sec. 3 and give rise to conformal invariants after integration. However, these invariants never appear in T µ µ for the cases we studied here. In particular, they cannot be determined through the folding trick.
All invariants that appear in (35) in the final expression for T µ µ can be written as tr (A · B . . . C) with A, B, C tensors. These tensors are irreducible in the sense that they contain the curvatures in positive powers and cannot be written as products of tensors of a lower dimension. We observe that, for allowed invariants, all multiplets A, B, etc., are even with respect to the reflection + ↔ −. The invariants J 1 and J 2 instead contain odd factors. This observation leads us to conjecture the following new rule: The allowed interface anomaly must be factorizable in irreducible factors which are even under the reflection.
This rule would allow one to distinguish the invariants which appear in T µ µ from the ones which do not. However, we do not know why it should work generally. Note the counterterms (38) follow a similar pattern.
It will be interesting to test the conjecture by including interactions on the interface or in the bulk. The boundary trace anomaly of a graphene-like d = 4 interacting BCFT was recently discussed in [24]. (For supersymmetric generalizations, see [25].) By looking at an interface generalization of this theory, one could test the conjecture. While this graphenelike theory has interactions confined to the boundary, one could also study what happens with interactions in the bulk, for example by looking at maximally supersymetric SU (N ) Yang-Mills theory in the presence of an interface.
It would further be interesting to understand the implication of these results for displacement operator correlation functions in d ≥ 2 ICFTs. The displacement operator, D, is dual to the position of the interface and can be related to the difference between the normal-component of the stress tensors: In the boundary case, it is known that the coefficients of the two-and three-point functions of the displacement operator are proportional to the b 1 and b 2 central charges [24,26]. In the interface case, how are the boundary invariants related to the displacement operator? It seems natural to expect that the relation between b 1 , b 2 and the displacement operator continues to hold. But then it is not clear to what one should relate the coefficients of the additional invariants (40) and (41). Perhaps their absence correlates with the absence of corresponding operators on the interface.
It will be also nice to consider the trace anomalies and boundary/interface central charges in d = 5, 6 ICFTs to check if the above conjectured rule applies.