aa r X i v : . [ h e p - t h ] O c t Conformal anomalies of CFT’s with boundaries
D.V. Fursaev
Dubna State UniversityUniversitetskaya str. 19141 980, Dubna, Moscow Region, Russiaandthe Bogoliubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear ResearchDubna, Russia
Abstract
The trace anomaly of conformal field theories in four dimensions is characterizedby ’ a ’ and ’ c ’-functions. The scaling properties of the effective action of a CFT inthe 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 fordifferent theories and different boundary conditions. One of the boundary chargesdepends on the bulk c charge. Anomaly of the effective action
Conformal anomalies are well-known. In a conformal field theory (CFT) in four dimen-sions the expectation value of the trace of stress energy tensor can be written in thefollowing universal form [1]: h T µµ i = − a E − c I − c π ∇ R , (1.1) E = 132 π h R − R µν R µν + R µνλρ R µνλρ i , (1.2) I = − π C µνλρ C µνλρ , (1.3)where E is the volume density of the Euler characteristics of the underlying backgroundmanifold M , and C µνλρ is the Weyl tensor of M . One can also define the integral anomalyby the variation of the effective action W , A ≡ ∂ σ W [ e σ g µν ] σ =0 = Z M √ gd x h T µµ i , (1.4)under scaling with a constant factor σ . The right hand side of (1.4) relates A to the traceanomaly (1.1) and holds on a closed manifold.By analogy with two-dimensional CFT’s constants a and c are called charges, or a -function and c -function, when they are allowed to run under renormalization. There arearguments [2] that a -function changes monotonically when going from a critical point toa critical point.The aim of this work is to study integral anomaly of the effective action (1.4) when M has a boundary. In this case the right hand side of (1.4) cannot be written solely asthe integral of the trace anomaly since new boundary terms may appear. An experiencewith 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 therenormalization group [3], [4].We study different CFT’s in four dimensions with the boundary conditions whichdo not break the conformal invariance on the classical level and show that the integralanomaly has the following universal structure: A = − a χ − c i + q j + q j . (1.5)Quantities χ , i , j and j are scale invariant functionals, χ is the Euler characteristicsof the background manifold, i is the integral of I over M , see (1.3). The boundariesresult in two new terms with constants q and q (the boundary charges). The boundaryfunctionals in (1.5) are j = 116 π Z ∂ M √ Hd x C µνλρ N ν N ρ ˆ K µλ ≡ π Z ∂ M √ Hd xG , (1.6) j = 116 π Z ∂ M √ Hd x Tr( ˆ K ) ≡ π Z ∂ M √ Hd xG . (1.7)We use the following notations: g µν is the metric of the background manifold M , themetric induced on the boundary ∂ M of M is H µν = g µν − N µ N ν , N ν is a unit outward2able 1: Charges in the anomaly of the effective action Theory a c q q boundary conditionreal scalar Dirichletreal scalar
RobinDirac spinor mixedgauge Boson absolutegauge Boson relative pointing normal vector to ∂ M , K µν = H λµ H ρν N λ ; ρ is the extrinsic curvature tensor of ∂ M ,ˆ K µν = K µν − H µν K/ K µν . Conformal invariance of j , j follows fromthe fact that ˆ K µν transforms homogeneously under conformal transformations. Definitionsof the Riemann tensor, the Ricci tensor, the scalar curvature, respectively, are R λµνρ = − Γ λµν,ρ + ... , R µν = R λµλν , R = R µµ . Our definitions of R and K µν coincide with those usedby Dowker and Schofield in [5] and by Vassilevich in [6]. The only difference with thoseworks is in the sign of R λµνρ and direction of N µ . In our paper, in [5], and in [6] the scalarcurvature on S n is positive. The curvatures constructed with the metric of ∂ M will bedenoted as ˆ R λµνρ , ˆ R µν , ˆ R .Our results for the boundary charges in (1.5) are listed in Table 1. q can depend onthe type of the boundary conditions.It should be noted that (1.5) for a scalar field with the Dirichlet boundary conditionfollows from results by Dowker and Schofield [5]. Table 1 agrees with [5] for this case.Anomalous rescalings of the effective action of Dirac fields and gauge bosons have beenalso studied by Moss and Poletti [7] for Einstein spaces with boundaries.To our knowledge the model independent nature of Eq. (1.5) has not been emphasizedso far.In Sec. 2 we relate the anomaly to known computations of boundary terms in thecorresponding heat kernel coefficients. Anomalies in the models presented in Table 1 arediscussed in Sec. 3. Concluding remarks are given in in Sec. 4. We point out a universalrelation between c and q (valid at least for all above models). It hints that q is not anindependent new charge. We use the relation between the anomaly and the heat coefficient of a Laplace operator∆ = −∇ + X for the corresponding conformal theory A = ηA , (2.1)3here η = +1 for Bosons and η = −
1. We ignore in (3.4) a possible contribution of zeromodes of ∆. The heat coefficients for the asymptotic expansion of the heat kernel of ∆are defined as K (∆; t ) = Tr e − t ∆ ≃ X p =0 A p (∆) t ( p − / , t → . (2.2)If the classical theory is scale invariant the heat coefficient A is a conformal invariant, seee.g. [8]. Therefore A can be represented as a linear combination of conformal invariantsconstructed of the geometrical characteristics of M , ∂ M and embedding of ∂ M in M .On dimensional grounds, these invariants (in four dimensions) are χ , i , j and j . Thebulk part of A is determined by χ and i , and it is well known.We are interested in boundary terms in A . To find them one should take into accountthe boundary term in χ . The definition of the Euler number for a four-dimensionalmanifold with a boundary is as follows: χ = B [ M ] + S [ ∂ M ] , (2.3) B [ M ] = Z M √ gd x E , (2.4) S [ ∂ M ] = 132 π Z ∂ M √ Hd xQ , (2.5) Q = − h det K µν + ˆ G µν K µν i , ˆ G µν = ˆ R µν − H µν ˆ R . (2.6)Derivation of these formulae can be found in [5]. Some values of χ are: χ = 1, if M isa domain in R with the spherical boundary ∂ M = S , χ = 0, if M is a domain of atorus S × R with a boundary ∂ M = S × S .The boundary part of A , therefore, is A bd = η ( q j + q j − aS ) . (2.7)It is the aim of computations to check that the coefficient at S does equal − ηa , where a is the same constant which appears in the trace anomaly (1.1).Appearance of S among counter terms in a one-loop effective action dates back toworks in 1980’s, see [10]. If M is a domain of a flat Euclidean background and ∂ M = S one easily finds for (1.5) that A = − a . In this case the anomaly is solely determinedby S . Mode-by-mode computations of the anomaly for the spinor and gauge fields havebeen done in [11],[12].Our starting point is formula (5.33) from Vassilevich’s review [6] for A for mixedboundary conditions. We put there f = 1, n = 4. The boundary conditions are( ∇ N − S )Π + φ = 0 , Π − φ = 0 , (2.8)where ∇ N = N µ ∇ µ , Π ± are corresponding projectors, Π + + Π − = 1, definition of S coincides with [6].After converting total derivatives in the bulk into surface terms and some algebra theboundary part A bd of A can be written as A bd = 1(4 π ) Z ∂ M √ Hd x Tr C , (2.9)4 = Π + C +4 + Π − C − + C + − , (2.10) C +4 = − Q + 115 G + 245 G − (cid:18) X − R (cid:19) K + 12 ∇ N (cid:18) X − R (cid:19) +43 (cid:18) S Π + + 13 K (cid:19) − (cid:18) X − R (cid:19) S + (cid:18) S + 13 K (cid:19) (cid:18)
215 Tr K − K (cid:19) , (2.11) C − = − Q + 115 G + 235 G − (cid:18) X − R (cid:19) K − ∇ N (cid:18) X − R (cid:19) , (2.12) C + − = −
13 (Π + − Π − )Π +: a Ω aµ N µ −
215 Π +: a Π +: a K −
415 Π +: a Π +: b K ab −
43 Π +: a Π +: a Π + S . (2.13)Here we use ’flat’ indices a, b in the tangent space to the boundary, see [6]. Ω µν is the fieldstrength of the connection defined by (2.10) in [6]. The notations are Tr K m = K µ µ ...K µ p µ , K = Tr K .In deriving (2.10)-(2.13) we used Gauss-Codazzi identities and the relations: G = R µνλρ K µλ N ν N ρ − R µν ( N µ N ν K + K µν ) + 16 KR , (2.14) G = Tr K − K Tr K + 29 K , (2.15) Q = 8 R µνλρ K µλ N ν N ρ − R µν ( N µ N ν K + K µν ) + 4 KR +83 K + 163 Tr K − K Tr K , (2.16)which can be easily obtained from definitions (1.6),(1.7),(2.6).Representation of the boundary terms in form (2.10)-(2.13) follows Moss and Poletti[7]. Calculations of A in case of boundaries have been done by several authors. The keypaper is by Branson and Gilkey [9]. A complete list of references can be found in [6]. In this case X = 1 /
6. For the Dirichlet condition Π + = 0, C + − = 0, and the boundarycharges are determined by C − , (2.12).Conformally invariant Robin condition requires S = − K/
3, Π + = 1. Then again C + − = 0, and boundary charges follow from (2.11). In case of a massless Dirac field ψ the operator is ∆ (1 / = ( iγ µ ∇ µ ) . The boundaryconditions are mixed ones,Π − ψ | ∂ M = 0 , ( ∇ N + K/ + ψ | ∂ M = 0 , (3.1)5here Π ± = (1 ± iγ ∗ N µ γ µ ), and γ ∗ is a chirality gamma matrix. Therefore, X = R/ S = − Π + K/ µν = 14 R µνσρ γ σ γ ρ . The rest computation is straightforward. One findsTr(Π + C +4 + Π − C − ) = r (cid:18) − Q + 115 G + 1315 G + 172 KR + 11620 K − K Tr( K ) (cid:19) , (3.2)Tr( C + − ) = r (cid:18) − R µνλρ K µλ N ν N ρ −
115 Tr( K ) + 120 K Tr( K ) (cid:19) , (3.3)where r = 4 is the number of components of the Dirac spinor in four dimensions. Incomputing C + − one uses the relationΠ +: a = i γ b γ ∗ K ba . The data of Table 1 follow from the sum of (3.2) and (3.3) and relations (2.14)-(2.16).
By following [6] we consider the quantization of an Abelian gauge field V µ in the Lorentzgauge ∇ V = 0. The results of Table 1 are valid for a gauge invariant combination A = A (∆ (1) ) − A (∆ ( gh ) ) , (3.4)where (∆ (1) ) νµ = −∇ δ νµ + R νµ is the vector Laplacian, and ∆ ( gh ) = −∇ is the Laplacianfor ghosts.We study two sorts of boundary conditions. The absolute (or electric [7]) boundarycondition: N µ F µν | ∂ M = 0 , (3.5)and relative (or magnetic [7]) boundary condition: N µ ˜ F µν | ∂ M = 0 , (3.6)where F µν = ∇ µ V ν − ∇ ν V µ , and ˜ F µν is the Hodge dual of F µν .The both conditions are manifestly gauge and conformally invariant.Condition (3.5) requires that components of an electric field normal to ∂ M and com-ponents of the magnetic field which are tangential to ∂ M vanish on the boundary. Thismeans that the boundary is a perfect conductor. In condition (3.6) the roles of electricand magnetic fields are interchanged.In the Lorentz gauge the absolute boundary condition is reduced to the followingconditions on the vector field and ghosts [6]:( δ νµ ∇ N + K νµ ) V + ν | ∂ M = 0 , V − µ | ∂ M = 0 , (3.7)6here V ± = Π ± V , and (Π + ) νµ = δ νµ − N µ N ν , (Π − ) νµ = N µ N ν . (3.8)The corresponding boundary condition for a ghost field c is ∂ N c | ∂ M = 0 . (3.9)The relative condition in the same gauge is( ∇ N + K ) V + µ | ∂ M = 0 , V − µ | ∂ M = 0 , (3.10)(Π + ) νµ = N µ N ν , (Π − ) νµ = δ νµ − N µ N ν . (3.11) c | ∂ M = 0 . (3.12)By taking into account that X µν = R µν , (Ω λρ ) νµ = R νµλρ , for the vector component withthe absolute condition (3.7), (3.8) ( S µν = − K µν ) we findTr(Π + C +4 + Π − C − ) =( N µ R µa ) : a − R ,N − Q + 415 G − G − KR + R µν ( N µ N ν K + K µν ) , (3.13)Tr( C + − ) = 23 R µνλρ K µλ N ν N ρ + 45 Tr( K ) − K Tr( K ) . (3.14)For the ghost part with condition (3.9) S = 0 and C = C +4 = − R ,N − Q + 115 G + 245 G + 118 KR + 14405 K + 245 K Tr( K ) , (3.15)For the vector component with relative conditions (3.10), (3.11) ( S µν = − Kδ µν )Tr(Π + C +4 + Π − C − ) = − ( N µ R µa ) : a + 16 R ,N − Q + 415 G + 845 G − KR + R µν ( N µ N ν K + K µν ) − K − K Tr( K ) , (3.16)Tr( C + − ) = − R µνλρ K µλ N ν N ρ −
815 Tr( K ) + 1615 K Tr( K ) . (3.17)For the ghost part with condition (3.12) S = 0 and C = C − = 112 R ,N − Q + 115 G + 235 G + 118 KR . (3.18)In deriving (3.13), (3.16) we used the identity12 R ,N = ( N µ R µa ) : a + KR µν N µ N ν − R µν K µν + N µ N ν N λ R µν ; λ . The first term in the right hand side of this identity is the total derivative on ∂ M . Let usemphasize that the two sets of boundary conditions for the gauge field result in the sameboundary terms in the anomalous scaling of the effective action.7 Concluding remarks
The aim of this paper was to demonstrate a model independent form of the integralanomaly (1.5) in the presence of boundaries and obtain specific values of the boundarycharges for some typical CFT’s. It would be important now to study evolution of theboundary charges under the renormalization group.Computations of boundary charges for other models can be continued along the lines ofthe 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 moreboundary charges. A principal challenge here is the knowledge of boundary terms in theheat coefficient A .An interesting issue is a possible relation between bulk charges a , c and boundarycharges q , q . A conjecture of [13] is that q and c may be related. Indeed, all modelspresented in Table 1 satisfy a universal relation q = 8 c . (4.1)We leave a general proof and implications of (4.1) for a future analysis.It should be noted that formula (1.5) was also discussed in a recent work [14] whichappeared several days earlier of the present publication. Acknowledgement
The author is grateful to I.L. Buchbinder and D.V. Vassilevich for helpful discussions,A.A. Tseytlin, G. Esposito and K. Jensen for useful comments. Many thanks go to S.N.Solodukhin for a remark on the connection of q with c . The remark also helped tofix misprints in Table 1 for the spinor field. This work was supported by RFBR grant13-02-00950. 8 eferences [1] M.J. Duff, Observations on Conformal Anomalies , Nucl. Phys.
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