Parity-odd surface anomalies and correlation functions on conical defects
Maro Cvitan, Predrag Dominis Prester, Silvio Pallua, Ivica Smolić, Tamara Štemberga
aa r X i v : . [ h e p - t h ] D ec ZTF-EP-15-01
Parity-odd surface anomaliesandcorrelation functions on conical defects
M. Cvitan a , P. Dominis Prester b , S. Pallua a , I. Smoli´c a , T. ˇStemberga a a Theoretical Physics Division of Particles and Fields , Department of Physics, Faculty of Science, University of Zagreb , Bijeniˇcka cesta 32, 10000 Zagreb, Croatia b Department of Physics, University of Rijeka,Radmile Matejˇci´c 2, 51000 Rijeka, Croatia
Email: [email protected], [email protected], [email protected], [email protected], [email protected]
Abstract.
We analyse the parity-odd (“type P”) surface anomalies of the energy-momentum tensor correlators inconformal field theories, with an emphasis on d = 4 and d = 3 dimensional spacetimes. Using cohomologyanalysis we construct the expression for the most general P-type surface trace anomaly on a singular 2-dimensional surface in 4-dimensional bulk spacetimes. As an important example, we specialise to the casewhen the singular surface is a conical defect and show that the bulk P-type Pontryagin trace anomalyinduces such a surface trace anomaly. We show that this conical type P surface trace anomaly is givenpurely by the outer curvature tensor. In addition, we analyse parity-odd surface contact terms in energy-momentum tensor correlators in the flat spacetime induced by the conical defect by studying two specialcases in which the contact terms are induced by, (1) type P trace anomaly in d = 4 and, (2) gravitationalChern-Simons Lagrangian term in d = 3 spacetime dimensions. In both cases we show that the surfacecontact terms appear in correlators of the lower rank than the corresponding bulk surface terms.1 Introduction
It is known that when a quantum field theory is defined on the curved spacetime, an expectation valueof the energy-momentum tensor may receive quantum corrections in the form of local terms. Some ofthese terms break classical symmetries of the theory, the well-known example being the trace (also calledWeyl) anomaly in conformal field theories in even number 2 k of the spacetime dimensions [1–6]: h T µµ ( x ) i g = A ( x ) (1)Here T µν is the energy-momentum tensor of the theory, and A is the trace anomaly which is generallysome combination of monomials of k -th order in the Riemann tensor. The terms in the anomaly A are,therefore, local covariant expressions constructed from the background metric g µν , the Riemann tensorand covariant derivatives. A may come in three types: type A which is the k -th Euler invariant, type Bwhich consists of contracted tensor products of the Weyl tensor, and type P consisting of exterior productsof the Riemann curvature two-forms. The types A and B are parity-even, while type P is parity-odd and,although allowed by consistency conditions, was until recently usually neglected in the literature.Last decade witnessed a renewed interest in gravitational mechanisms of CP violation, see e.g., [7–15],including studying possibilities and consequences of the appearance of the type P anomalies [3, 5, 16–21].Indeed, in [20, 21] an old result of [22] was rederived and it was shown that in 4-dimensional quantumfield theories with chiral fermions in which the numbers of left and right chiralities are not the same, typeP anomalies are indeed present.It is of interest to study non-regular spacetimes containing singular submanifolds. Singular structuremay arise from the matter localised on the surface or from the topology, example of the former beingbranes and of the latter orbifolds. In some instances, possibly after analytical extensions, these singularsurfaces are equivalent to the conical defects. One notable example is provided by the replica method[23] for the calculation of the entanglement entropy (for reviews see [24, 25]), in which one effectively(by analytical continuation) introduces the conical singularity on the entangling surface (which is acodimension-2 submanifold) with a deficit parameter α and calculates the linear term in the expansionin (1 − α ).In the presence of a singular surface Σ the trace anomaly receives also surface contributions localisedon Σ, an example of which is given by the so called Graham-Witten anomalies [26]. In d = 4 dimensionsa generic cohomology analysis of the parity-even surface trace anomalies connected with codimension-2singular surfaces was performed in [27]. One purpose of this paper is to extend this analysis to theparity-odd sector and in this way complete the construction from [27]. We also calculate the parity-oddsurface trace anomaly in the important case of conical singularity in d = 4.Singular surfaces generally also induce surface contact terms in correlation functions of the energy-momentum tensor in the flat spacetime. In fact, as was observed in the case of the conical singularityin [28], in the flat spacetime surface contact terms appear in the correlation functions one point lower thanthe corresponding bulk contact terms. This is an interesting property which may have some importantuses in the future. We analyse and demonstrate this feature in the parity-odd sector on the two importantexamples containing a codimension-2 singular surface, (1) surface contact terms induced by the surfacetrace anomaly in d = 4, and (2) surface contact terms induced by the presence of the gravitational Chern-Simons terms in the quantum effective action in d = 3 when the singular surface is a conical singularity.We perform calculations using two methods, first developed in [29] and the second in [30]. A side resultof agreement of the two calculations is a non-trivial confirmation of the validity of both methods in theparity-odd sector. Our conventions are as follows. The Riemann tensor is R µανβ = ∂ ν Γ µαβ − . . . and the Ricci tensor is R µν = R µαµβ .The energy-momentum tensor is defined by T µν = 2 g − / δS/δg µν , where S is the action, and so h T µν i = 2 g − / δW/δg µν where the functional W [ g ] is defined by W [ g ] = − ln Z [ g ] = − ln R D φ exp( − S ). All calculations in this paper are performedin the Euclidean time. Bulk and surface trace anomalies
Let us assume that an otherwise regular (“bulk”) spacetime M with the metric g µν contains a singularsurface Σ, for which in this paper we shall assume to be codimension-2. A consequence is that some localquantities get contributions localised on Σ. Let us focus on a trace anomaly of a CFT defined on such aspacetime, A ω = 2 δ ω ln Z [ g ] = Z M ω h T µµ i (2)where Z [ g ] is the generating functional of the CFT on the spacetime with the metric g µν , and ω ( x ) isan infinitesimal local parameter of the Weyl transformation δg µν = ω g µν . In the presence of a singularsurface, the trace anomaly in general receives the bulk and the surface contributions, A ω = A ( b ) ω + A (Σ) ω , (3)which are local functionals defined as A ( b ) ω = Z M ω A ( b ) A (Σ) ω = Z Σ ω A (Σ) + Z Σ ∇ µ ω A (Σ) µ + · · · (4)The dots · · · above denote the terms which include the second or the higher derivatives of ω . Observethat in the case of the surface terms one cannot in general shift the derivatives acting on ω to the anomalydensity by using partial integration, as is always possible for the bulk term.In general, possible terms that can appear in the trace anomaly are restricted by the dimension, andby the consistency conditions. For the diffeomorphism covariant theories, in which the diff-anomaly isvanishing, consistency conditions reduce to δ ω A ω = 0 , (5)where ω is treated as an anticommuting variable satisfying δ ω ω = 0. The “true” anomaly consists of theterms which are not exact with respect to δ ω , i.e. one has to subtract all the terms which can be writtenas δ ω C , where C is some diff-covariant density.Cohomology analysis of the bulk anomalies has been thoroughly studied and it is known that possibleterms in A ( b ) fall into three-classes: type A, consisting of the Euler densities; type B, consisting of theWeyl-invariant terms; type P, consisting of the parity-odd terms. In d = 4, the most general form for thebulk trace anomaly is: A ( b ) = − a E + c
64 ( W µνρσ ) + p P (6)where a , c and p are constants depending on the CFT in question, E is the second Euler scalar (Gauss-Bonnet scalar), W µνρσ is the Weyl tensor, and P is the Pontryagin (pseudo)scalar P = 12 √ g ε µνρσ R αβµν R αβρσ . (7)Here, ε µνρσ is the Levi-Civita symbol (with components equal 0 or ± We assume Euclidean time, and so spacetimes will be Riemannian in this paper. .2 Codimension-2 surface trace anomalies in d = 4 Let us now focus our attention on a particular case of the codimension-2 surface anomalies in d = 4,where Σ is a 2-dimensional singular surface. The relevant geometric objects include the first fundamentalform (i.e. the induced metric) γ µν and the corresponding intrinsic Ricci curvature scalar ˆ R , the Weyltensor contracted with the induced metric W , the second fundamental form (i.e. the extrinsic curvature) K µνα and the outer curvature pseudoscalar Ω. From the dimensional analysis, it is easy to see thathigher order curvatures are irrelevant here.The cohomology of the parity-even sector was analysed and classified in [27] with the result that theterms with derivatives of ω are not present and A (Σ) is built out of the following densities:(type A) : ˆ E = ˆ R (8)(type B) : W = γ µρ γ νσ W µνρσ , ( C µνα ) = ( K µνα ) −
12 ( K α ) (9)So, the general form of the parity-even surface trace anomaly in this case is: (cid:0) A (Σ) ω (cid:1) even = Z Σ ω (cid:2) k ˆ E + k W + k ( C µνα ) (cid:3) (10)where k a are the coefficients that depend both on the theory and on the properties of the singular surface.We want to complete the analysis by finding the most general expression for the parity-odd (type P)surface trace anomaly. Now, it is not hard to see that there are just three linearly independent candidateterms which have the proper dimension: Z Σ ω Ω , Z Σ ω ǫ µναβ K µρα K νρβ and Z Σ ǫ αβ ∇ α ω K β . (11)Here ǫ αβ is the binormal on Σ and Ω is the outer curvature pseudoscalar, defined on Σ, which is obtainedfrom the outer curvature tensor Ω µναβ through Ω ≡ ǫ µναβ Ω µναβ (12)The first and the second term in (11) indeed satisfy (5), but the third term does not and it drops out.It may appear that there is one more potential candidate ǫ µνρσ γ αµ γ βν n λρ n κσ R αβλκ (13)where n µν = g µν − γ µν , which indeed satisfies the consistency condition, but it is not independent andso we leave it out. One can see this by using the Ricci equationΩ µνρσ = γ αµ γ βν n λρ n κσ R αβλκ + K µτρ K ντ σ − K µτσ K ντ ρ (14)which contracted with the Levi-Civita tensor gives ǫ µνρσ γ αµ γ βν n λρ n κσ R αβλκ = 2 Ω − ǫ µνρσ K µτρ K ντ σ (15)We see that the candidate (13) can be written as a linear combination of those in (11). The mathematical definitions and properties of these objects can be found in [31, 32]. In our notation the extrinsiccurvature is K µνρ = γ µα γ νβ ∇ α γ βρ . The outer curvature Ω µνρσ is defined using ( ˜ D µ ˜ D ν − ˜ D ν ˜ D µ ) X ρ = Ω µνρσ X σ , for X µ normal i.e. γ µν X ν = 0. Here, the derivative ˜ D is defined to act on normal vectors as ˜ D µ X ν = γ µα n νβ ∇ α X β , and ontangential-normal X βσ (where n µβ X βσ = 0 and γ ρσ X βσ = 0) as ˜ D µ X νρ = γ µα γ νβ n ρσ ∇ α X βσ . Note that Ω µναβ has only one independent component, so Ω contains the complete information about the outer curvaturetensor. ω , and A (Σ) is built out of the following two terms:(type P) : Ω , ǫ µναβ K µρα K νρβ (16)which means that the general form of the parity-odd surface trace anomaly in d = 4 on 2-dimensionalsingular surfaces is given by (cid:0) A (Σ) ω (cid:1) odd = Z Σ ω (cid:16) ˜ k Ω + ˜ k ǫ µναβ K µρα K νρβ (cid:17) . (17)Again, the coefficients ˜ k , in general depend both on the theory in hand and on the properties of thesingular surface Σ. In the next section we shall calculate them for the particular case when Σ is a conicaldefect surface. We want to present an explicit example where the type P surface trace anomalies are present, and forthis we take an important example of spacetimes with conical singularities.We assume that in otherwise regular d -dimensional spacetime M with metric g µν a conical defectwith an angle deficit 2 π (1 − α ) is introduced in a standard fashion such that there is a ( d − M α ,and the singular surface Σ by C . When necessary, we shall use the local coordinates x µ , µ = 1 , . . . , d inwhich Σ is defined with x = x = 0 and the conical defect is described by having an angle deficit in x - x plane.We shall be interested in the integrals over M α of local functions F of the curvature scalars constructedout of g µν , Levi-Civita tensor ǫ µ ··· µ d , Riemann tensor R µνρσ and covariant derivatives ∇ µ . It was shownin [33] that in the leading order in 2 π (1 − α ) such integrals can be split into the bulk part (integral over M ) and the surface part (integral over Σ) by using the following formula Z M α F = Z M F + 2 π (1 − α ) Z C ( ∂F∂R µρνσ ǫ µρ ǫ νσ − X r (cid:18) ∂ F∂R µ ρ ν σ ∂R µ ρ ν σ (cid:19) r K ρ σ λ K ρ σ λ q r + 1 × (cid:2) ( n µ µ n ν ν − ǫ µ µ ǫ ν ν ) n λ λ + ( n µ µ ǫ ν ν + ǫ µ µ n ν ν ) ǫ λ λ (cid:3) (cid:27) + O (cid:0) (1 − α ) (cid:1) (18)Summation over r and the definition of the parameter q r are explained in [33]. As in our examples F will be linear or quadratic in Riemann tensor, we can put q r = 0. We shall be interested only in thelowest-order correction in the expansion over (1 − α ).It is useful to introduce two orthonormal vector fields, n µ ( a ) , a = 1 ,
2, which constitute a basis in thesubspace of vectors normal to Σ. Then one can write n µν = X a =1 n µ ( a ) n ν ( a ) , ǫ µν = X a,b =1 n µ ( a ) n ν ( b ) ε ab (19)where ε ab is the two-dimensional Levi-Civita symbol. Using this it is easy to prove a useful relation ǫ µν ǫ ρσ = n µρ n νσ − n µσ n νρ (20)5 .2 Conical trace anomaly in d = 4 We now focus on the CFT defined on the four dimensional curved spacetime M α which contains theconical singularity located on the 2 dimensional surface Σ = C , as described in the previous section.As the trace anomaly (6) has a purely geometric description in a regular spacetime, we assume that itpreserves its form also in the presence of the conical singularity, in the sense of the regularisation of thespacetime already implicit in Dong’s formula (18). This means that the integrated anomaly is given by: A ω = Z M α ω (cid:16) − a E + c
64 ( W µνρσ ) + p P (cid:17) (21)We want to extract the surface contributions to the trace anomaly. In the cases of the type A and Bconical surface trace anomalies the results are known and can be found in [34,35]. Here we shall completethe analysis in d = 4 by calculating the type P conical surface trace anomalies. The easiest way toachieve this is by applying Dong’s formula (18) on (21). A tedious but straightforward calculation givesthe following result (cid:0) A ( C ) ω (cid:1) odd = 4 π p (1 − α ) Z C ω ǫ µνρσ (cid:0) γ αµ γ βν n λρ n κσ R αβλκ + 2 K µτρ K ντ σ (cid:1) (22)The first and the second term separately come from the first and the second term in (18), respectively.By using (15) we can write our result for the integrated type P conical surface anomaly in the morecompact and suggestive form using just the outer curvature tensor (cid:0) A ( C ) ω (cid:1) odd = 8 π p (1 − α ) Z C ω Ω (23)The pseudotensor Ω was defined in (12). Comparison with the generic formula (17) for the surface traceanomaly in d = 4 shows that for the conical anomaly one has ˜ k = 8 π p (1 − α ) and ˜ k = 0. We seethat in the case of the conical singularity, the parity-odd surface trace anomaly is a direct consequenceof a presence of the parity-odd bulk trace anomaly, and is completely determined by it. As alreadymentioned, there is now a strong evidence [20, 21] that CFT’s with chiral fermions indeed possess such abulk anomaly.It is interesting to note that the outer curvature scalar Ω can be written as the total 2-dimensionalgradient, and so is in some sense an outer analogue of the Euler term in d = 2 (which is an intrinsicRicci scalar) [31]. As the conical surface type A anomaly is purely given by the intrinsic Ricci scalar, itis maybe not surprising that the type P anomaly is given purely by Ω. Moreover, Penrose showed thatgenerally the spinor approach leads naturally to the construction of a single complex curvature invarianton 2-dimensional submanifolds whose real part is the Euler term (the intrinsic Ricci scalar) and imaginarypart is the outer curvature scalar Ω [37]. We believe that there is some interesting mathematics hereworthy of detail studying, but we leave such questions to our future research. As is well-known, the local terms in the 1-point energy-momentum correlation function on a curvedspacetime induce contact terms in higher-rank energy-momentum correlation functions on a flat space.Our definition of correlation functions in the flat space is such that h T µ ν ( x ) . . . T µ n ν n ( x n ) T µν ( x ) i = ( − n δδg µ ν ( x ) · · · δδg µ n ν n ( x n ) h T µν ( x ) i g (cid:12)(cid:12)(cid:12)(cid:12) g µν = δ µν (24)Of course, if h T µν ( x ) i contains both bulk and surface local terms there will be corresponding bulk andsurface contact terms. Eq. (24) is telling us that in the case of the surface contact terms generated by the Between the first and the second version of our paper a reference [36] appeared, with some results partially overlappingwith ours. Though the motivation and notation are different, the results there are in agreement with ours. d = 4 one should expect from (10) and (17) that they will appear already in the2pt energy-momentum correlation functions, while from (6) one expects that the bulk contact terms startat the 3pt functions. The fact that in general number of dimensions d, the surface contact terms start toappear in the correlation functions of one rank less than the bulk terms was noted in the reference [28]and then elaborated in [29] in the case of the contact terms connected with the parity even conical surfacetrace anomalies. This also follows from Dongs formula (18), which we will use for studying parity oddsurface contact terms.In this section we further demonstrate this phenomenon, of surface contact terms appearing “before”bulk surface terms in energy-momentum tensor correlation functions, on two examples. One example isthe contact term in d = 4 connected with the type P surface trace anomaly, and the other is the specificcontact term in d = 3 appearing when an effective action contains the gravitational Chern-Simons term.Let us emphasize that the correlators defined in (24) are, strictly speaking, not equal to standard T -ordered correlation functions. However, the difference is irrelevant for the calculation of the contactterms which we present in the rest of the section. d = 4 According to (24), the presence of the bulk trace anomalies in d = 4, given in (6), obviously induces bulkcontact terms in the energy-momentum tensor correlation functions of the type h T µ ν ( x ) · · · T µ n ν n ( x n ) T ρρ ( x ) i = ( − n δδg µ ν ( x ) · · · δδg µ n ν n ( x n ) h T ρρ ( x ) i g (cid:12)(cid:12)(cid:12)(cid:12) g µν = δ µν (25)when n ≥
2. In addition, when the singular surface Σ is present one expects also that the surface contactterms are present. In the parity-odd sector we showed that in this case the trace anomaly is given by (cid:0) h T ρρ ( x ) i g (cid:1) odd = p P ( x ) + (cid:16) ˜ k Ω( x ) + ˜ k ǫ µναβ K µρα K νρβ (cid:17) δ Σ (26)Here δ Σ is the covariant Dirac δ -function localised on the 2-surface Σ.We shall now show that the term proportional to ˜ k induces nonvanishing surface contact terms in(25) also for n = 1 correlation functions. To calculate this contribution we use the relation (15) and notethat the term proportional to ˜ k obviously does not contribute in (25) with n = 1, because K µρα = 0 inthe flat spacetime. Using the formula δR αβγκ ( x ′ ) δg µν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) g µν = δ µν = −
14 ( δ αµ δ κν ∂ γ ∂ β + δ βµ δ γν ∂ κ ∂ α ) δ (4) ( x − x ′ ) + ( µ ↔ ν ) − ( γ ↔ κ ) (27)and the fact that in the flat space R µνρσ = 0, we obtain (cid:0) h T µν ( x ) T ρρ ( x ′ ) i (cid:1) odd = − ˜ k ε µγρσ n ρν n σβ δ Σ ∂ β ∂ γ δ (4) ( x − x ′ ) + ( µ ↔ ν ) (28)Now we choose the coordinates such that Σ is defined by x = x = 0. Then, δ Σ = δ ( x ) δ ( x ) and wecan take n µ (1) = (1 , , ,
0) and n µ (2) = (0 , , , (cid:0) h T µν ( x ) T ρρ ( x ′ ) i (cid:1) odd = − ˜ k ˜ ε µa ˆ ε ν ˆ a δ ( x ) δ ( x ) ∂ a ∂ ˆ a δ (4) ( x − x ′ ) + ( µ ↔ ν ) + O ((1 − α ) ) (29)where ˆ a = 1 , a = 3 , ε and ˜ ε are 2-dimensional Levi-Civita symbols living on normal and tangential space, respectively (thebinormal and the volume form on Σ, respectively)ˆ ε = 1 , ˆ ε µa = 0 , ˜ ε = 1 , ˜ ε µ ˆ a = 0 (30)7n the special case of the conical surface trace anomaly we have shown that˜ k = 8 π (1 − α ) p (31)From the expression (29) it is obvious that the correlation function is nonvanishing only if one of theindices µ or ν is normal while the other one is tangential to Σ. One of the consequences is that the traceof (29) vanishes (cid:0) h T µµ ( x ) T νν ( x ′ ) i (cid:1) odd = 0 (32)As it was shown in [29] that the same is true for the type B anomaly, the only contribution to the traceof the n = 1 correlation function comes from the type A anomaly. As argued in [30], there is a correspondence between the correlation functions on the flat space with andwithout the conical defect. This can be used to check the result from the previous section for the case ofthe conical surface trace anomaly. The correspondence is given through the relation
PhO ( x ) . . . O N ( x N ) i M α = hO ( x ) . . . O N ( x N ) K i (33)where P = − lim α → ∂∂α (34)and the operator K is K = − π Z d d − y Z ∞ dz z T ( z , z = 0 , ~y ) (35)Here, the directions 1 and 2 are normal while the directions 3 , . . . , d are tangential to the conical defectsurface Σ, and ~y = ( z , . . . , z d ).Applying this to the particular case of the parity-odd contribution to the correlation functions ofthe energy-momentum tensor in d = 4 enables us to check the result from the last subsection by anindependent method. From (33) follows P (cid:0) h T ρρ ( x ) T µν ( y ) i M α (cid:1) odd = − π Z dz Z dz Z ∞ dz z h T ρρ ( x ) T µν ( y ) T ( z , z = 0 , z , z ) i (36)For clarity, we denote the correlation functions on the spacetime with the conical singularity with thesubscript C α . Now, from (6) it can be shown that in the regular flat spacetime one has h T ρρ ( x ) T µν ( y ) T ρσ ( z ) i odd = 2 p ε µραβ ( ∂ σ ∂ ν − δ σν ∂ )[ ∂ α δ (4) ( x − y ) ∂ β δ (4) ( x − z )]+ ( ρ ↔ σ ) + ( µ ↔ ν ) (37)where the differential operator inside the round brackets is explicitly given by ∂ σ ∂ ν − δ σν ∂ ≡ ∂∂y σ ∂∂z ν − δ σν ∂∂y κ ∂∂z κ (38)Plugging (37) into (36) we obtain P (cid:0) h T ρρ ( x ) T µν ( y ) i M α (cid:1) P − odd = − π p ε µ αβ Z ∞−∞ dz Z ∞−∞ dz Z ∞ dz z ×× ( ∂ ∂ ν − δ ν ∂ σ ∂ σ ) h ∂ α δ (4) ( x − y ) ∂ β δ (4) ( x − z ) i + ( µ ↔ ν ) (39)Let us concentrate on the first two lines in (39). Observe that β = 3 or 4 because of the integrations over z and z , and β = 2 due to the Levi-Civita symbol, so it must be that β = 1. From this follows that µ α must be tangential to the surface of the defect, i.e., µ = 3 and β = 4 or vice versa. Moreover, itcan be shown that ν = 3 or 4 because of the integration over z . Taking all this into account, and using Z ∞ dz z ∂ ∂z δ ( x − z ) = δ ( x ) (40)it is easy to show that (39) becomes P (cid:0) h T ρρ ( x ) T µν ( y ) i M α (cid:1) odd = − π p ˜ ε µa ˆ ε ν ˆ a δ ( x ) δ ( x ) ∂ ˆ a ∂ a δ (4) ( x − y ) + ( µ ↔ ν ) (41)where as before ˆ a = 1 , a = 3 ,
4, and the 2-dimensional Levi-Civita symbols are those defined in (30).From (39) and (34) and the fact that the bulk part of the 2-point correlation function vanishes in thelimit α → α gives in the lowest order in (1 − α ) the following result (cid:0) h T ρρ ( x ) T µν ( y ) i M α (cid:1) odd = − π (1 − α ) p ˜ ε µa ˆ ε ν ˆ a δ ( x ) δ ( x ) ∂ ˆ a ∂ a δ (4) ( x − y ) + ( µ ↔ ν ) (42)where we have used 2-dimensional Levi-Civita symbols defined in (30). As expected, the result is thesame as the corresponding one obtained by the Method 1 in the Sec. 4.1.1 (see Eq. (29)-(31)). d = 3 When 3-dimensional QFT’s are defined in a curved spacetime, expectation value of the energy-momentumtensor may develop a parity-odd contribution of the form h T µν ( x ) i odd = iw π ε αβ ( µ ∇ β R αν ) = − iw π C µν (43)where C µν is known as the Cotton-York tensor. As the integer part of the coefficient w can be removedby adding to the classical action a local counterterm, which is the well-known gravitational Chern-Simonsterm, it is sometimes stated that w is defined modulo 1 [17]. It is known that in regular spacetimes theCotton-York tensor is traceless and covariantly conserved C µµ = 0 , ∇ µ C µν = 0 (44)so as a consequence (43) does not contribute to the trace anomaly, which is expected from the generaltheorem stating that there are no trace anomalies in CFT’s defined in odd-dimensional spacetimes.Now we add into the spacetime a conical defect with the deficit angle 2 π (1 − α ) in the same manneras before. By assuming that (43) is valid also when the conical defect is present, and using( R µν ) C α = R µν + 2 π (1 − α ) n µν δ Σ + (terms containing the second fundamental form) (45)which follows from Dong’s formula (18), we obtain that for the flat metric g µν = δ µν ( h T µν ( x ) i M α ) odd = iw
24 (1 − α ) ε αβ ( µ ∂ β (cid:16) n αν ) δ Σ (cid:17) = iw
48 (1 − α ) ˆ ε ν ˆ a δ µ ∂ ˆ a [ δ ( x ) δ ( x )] + ( µ ↔ ν ) (46)where in the second equality we used (30) and also the vanishing of the second fundamental form. Again,this expression is nonvanishing only if one of the indices is in the normal direction (1 or 2) while the otherone is in the tangential direction (3). It is easy to see that (46) is traceless and covariantly conserved.This means that, as expected, it will not contribute to the trace anomaly. Arguments based on the path integral quantisation suggest that (i) the coupling constant of the purely gravitationalChern-Simons Lagrangian term in all odd spacetime dimensions is imaginary in the Euclidean regime, and (ii) the value ofthe coupling is quantised [10]. The parametrisation used in (43) is such that the contribution to w from the Lagrangiangravitational CS term must be an integer [38] if the only restriction on the spacetime is that it is a spin manifold. .2.2 Method 2 For a CFT defined in a regular flat spacetime a consequence of (43) is [5, 17] h T µν ( x ) T αβ ( y ) i odd = iw π ǫ µασ ( ∂ ν ∂ β − δ νβ ∂ ) ∂ σ δ (3) ( x − y ) + ( µ ↔ ν ) + ( α ↔ β ) (47)Now we can use the correspondence (33) to independently calculate the expectation value of the energymomentum tensor in a flat space with a conical singularity. In this way we obtain Ph T µν ( x ) i M α = h T µν ( x ) K i = − π Z ∞−∞ dy Z ∞ dy y h T µν ( x ) T ( y , y = 0 , y ) i (48)By using (47) we obtain h T µν ( x ) K i odd = − iw ε µ σ ( ∂ ν ∂ − δ ν ∂ ) Z ∞−∞ dy Z ∞ dy y ∂ σ δ (3) ( x − y ) (cid:12)(cid:12)(cid:12) y =0 + ( µ ↔ ν ) (49)Now, σ = 3 and ν = 3 because integration over y would be vanishing. From this follows that σ = 1 andone gets h T µν ( x ) K i odd = iw δ µ ( ∂ ν ∂ − δ ν ∂ a ∂ a ) (cid:20) δ ( x ) Z ∞ dy y ∂ δ ( x − y ) (cid:21) + ( µ ↔ ν )= iw δ µ ˆ ε ν ˆ a ∂ ˆ a [ δ ( x ) δ ( x )] + ( µ ↔ ν )= iw ε µαβ n αν ∂ β δ Σ + ( µ ↔ ν )where a = 1 , δ Σ = δ ( x ) δ ( x ) in the particular Cartesian coordinates we use here. Plugging thisinto (48), using (34) and integrating over α we obtain that in the leading order in (1 − α ) the final resultis ( h T µν ( x ) i M α ) odd = iw
48 (1 − α ) δ µ ˆ ǫ νa ∂ a [ δ ( x ) δ ( x )] + ( µ ↔ ν )= iw
48 (1 − α ) ǫ αβµ n αν ∂ β δ Σ + ( µ ↔ ν )= iw
24 (1 − α ) ǫ αβ ( µ ∂ β (cid:16) n αν ) δ Σ (cid:17) (50)We see that the final result is the same as the one obtained by the Method 1, which is (46). The expectation value of the energy-momentum tensor on a curved spacetime may receive quantumcorrections in the form of local terms. The well-know examples are the trace and diff- anomalies, whichbreak the classical symmetries of the theory. In some instances, e.g. in theories in four dimensions withunequal number of left and right chiral fermions, the parity breaking gravitational terms appear.We have studied some consequences of the presence of singular surfaces for quantum field theories incurved spacetimes, focusing on the parity violating sector. In particular, we have studied a parity-oddcontribution to the trace anomaly of the conformal field theories, which we dubbed the type P surfacetrace anomaly. By analysing the consistency condition we have found the most general form for the typeP surface trace anomaly in four dimensions. To have an example at hand, we have analysed the specialcase when the singular surface is due to conical singularities, which are important in its own right, andobtained the exact result for the type P conical surface trace anomaly. This surface anomaly appears only10hen the theory contains parity violating bulk trace anomaly and it turns out that it can be expressedfully by the outer curvature tensor, an interesting property which deserves further studying.In the second part of the paper we have studied, on the two examples, the influence of singular surfaceson the parity violating contact terms in energy-momentum correlation functions on the flat spacetime.One example consists of the surface contact terms in four dimensions connected with the type P surfacetrace anomalies. The second example consists of the surface contact terms generated by the presence ofthe gravitational Chern-Simons term in the effective action in the three dimensional flat spacetime. Inthe case of the conical singularity, we were able to perform calculations by using two methods, one purelygeometrical and the other using the connections between correlation functions in the flat spacetime withand without conical singularity. The agreement of the results gives a non-trivial confirmation of thevalidity of both methods in the parity-odd sector.Generalisations of our analyses to higher spacetime dimensions are possible, as type P trace anomaliesare expected to be present in 4 k dimensions, while gravitational Chern-Simons terms in effective actionsare expected to be present in (4 k −
1) dimensions [9, 12, 16]. However, computations become much morecomplicated so we left this to the future work.
Acknowledgements
The research has been supported by Croatian Science Foundation under the project No. 8946 and byUniversity of Rijeka under the research support No. 13.12.1.4.05. We thank Loriano Bonora on manylessons on anomalies and Bruno Lima de Souza for stimulating discussions.
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