application/xmlNoncommutative [formula omitted] theories, the axial anomaly, Fujikawa's method and the Atiyah–Singer indexC.P. MartínC. Tamarit[formula omitted] anomalySeiberg–Witten mapNoncommutative gauge theoriesPhysics Letters B 620 (2005) 187-194. doi:10.1016/j.physletb.2005.06.028journalPhysics Letters BCopyright © unknown. Published by Elsevier B.V.Elsevier B.V.0370-26936203-44 August 20052005-08-04187-19418719410.1016/j.physletb.2005.06.028http://dx.doi.org/10.1016/j.physletb.2005.06.028doi:10.1016/j.physletb.2005.06.028http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/JournalsS300.2PLB22103S0370-2693(05)00815-410.1016/j.physletb.2005.06.028TheoryNoncommutative SU(N) theories, the axial anomaly, Fujikawa's method and the Atiyah–Singer indexC.P.Martíncarmelo@elbereth.fis.ucm.esC.Tamaritctamarit@fis.ucm.esDepartamento de Física Teórica I, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, 28040 Madrid, SpainEditor: N. GloverAbstractFujikawa's method is employed to compute at first order in the noncommutative parameter the U(1)A anomaly for noncommutative SU(N). We consider the most general Seiberg–Witten map which commutes with hermiticity and complex conjugation and a noncommutative matrix parameter, θμν, which is of “magnetic” type. Our results for SU(N) can be readily generalized to cover the case of general nonsemisimple gauge groups when the symmetric Seiberg–Witten map is used. Connection with the Atiyah–Singer index theorem is also made.PACS11.15.-q11.30.Rd12.10.DmKeywordsU(1)A anomalySeiberg–Witten mapNoncommutative gauge theoriesIt is difficult to overstate the importance of the Abelian chiral anomaly in physics [1,2]. A most beautiful explanation of the existence of this anomaly was supplied by Fujikawa [3], who showed that it comes from the lack of invariance of the fermionic measure under chiral transformations. Fujikawa's method of computing anomalies also provides a way of easily exhibiting the relationship between the Abelian chiral anomaly and the Atiyah–Singer index theorem [3,4]. The method in question is called a nonperturbative method since no expansion in the coupling constant is carried out.The purpose of this Letter is to use Fujikawa's method to work out the Abelian chiral anomaly for noncommutative SU(N) gauge theories with Dirac fermions [5] up to first order in the noncommutative matrix parameter θμν and for the most general Seiberg–Witten map which is local at each order in θμν and commutes with hermiticity and complex conjugation. The case of noncommutative gauge theories with Dirac fermions and with a nonsemisimple gauge group is also analyzed when the theory is defined by means of the symmetric Seiberg–Witten map [6,7].Let aμ be an ordinary SU(N) gauge field. Let ψ denote an ordinary massive Dirac fermion carrying a given representation of SU(N). Following Ref. [5], we construct the noncommutative fields Aμ, the gauge field, and Ψ, the Dirac fermion, by applying the Seiberg–Witten map to their ordinary counterparts. As in Ref. [7,8], we shall assume that ψ does not enter the Seiberg–Witten map that yields Aμ, that this map renders Aμ hermitian and that it commutes, the Seiberg–Witten map, with complex conjugation when acting on fermion fields. We shall also assume that at each order in the noncommutative matrix parameter θμν the Seiberg–Witten map is local, i.e., that it is a polynomial of the fields and their derivatives with dimensionless coefficients other than θμν. Note that if, barring θμν, we would allow for dimensionful coefficients, such as masses, etc., then, the Seiberg–Witten map would have an infinite number of terms at each order in θμν and the theory would not be local at each order in θμν. It is not difficult to show that at first order in θμν the most general Seiberg–Witten map that fulfills the previous requirements reads (1)Aμ=aμ−14{aα,∂βaμ+fβμ}+i(κ2−κ12)θαβDμ[aα,aβ]+κ3θαβDμfαβ+κ4θμβDνfνβ,Ψ=ψ+[−12θαβaα∂β+i4θαβaαaβ+iκ3θαβfαβ−(κ2−κ12)θαβ[aα,aβ]+z1θαβfαβ+i2z2θαβ[γβ,γρ]DαDρ−z32θαβ[γα,γρ]fρβ+iz4θαβγαγβD2−z5θαβγαγβγμγνfμν]ψ. Hermiticity of Aμ demands κi, i=1,…,4, to be real numbers. That the Seiberg–Witten map commutes with complex conjugation—i.e., Ψ[ψ,aμ,θμν]=Ψ[ψ∗,−aμ∗,−θμν], see Refs. [7,8]—leads to z2=z3=z4=z5=0 and restricts z1 to be a real number. Notice that the terms in the Seiberg–Witten map that go with κ4 and z1 correspond, respectively, to field redefinitions of aμ and ψ, so that their actual values have no effect on physical quantities. However, we shall keep these parameters arbitrary and see whether they can be used to simplify the values of the (nonphysical) Green functions of the fields we shall compute.The action of the noncommutative SU(N) theory we shall study is given by S=∫d4x{−14g2TrFμν⋆Fμν+Ψ¯⋆(iD̸⋆−m)Ψ}. Tr denotes the trace operation on the matrix representation of SU(N) carried by ψ. In the previous equation, Fμν=∂μAν−∂νAμ−i[Aμ,Aν]⋆,D̸⋆=∂̸−iA̸⋆, and Aμ and Ψ are given by the Seiberg–Witten map above. ⋆ stands for the Moyal product of functions: (f⋆g)(x)=f(x)exp(i2θαβ∂α←∂β→)g(x). Since we shall use Fujikawa's method to compute the Abelian anomaly, we must define the theory for the Euclidean signature of space–time. Upon Wick rotation—we shall play it safe [9] and consider θμν to be of “magnetic” type: θ0i=0—we obtain a theory whose action, SE, at first order in θμν reads: (2)SE=SYM−∫d4xψ¯(K+iM(x))ψ.SYM is the contribution coming from the pure noncommutative Yang–Mills action, whose actual value will be irrelevant to us. The differential operator K and the function M(x) are given by (3)K=iD̸+iR̸,R̸=(−14+2z1)θαβfαβγμDμ−12θαβγρfραDβ+z1θαβγμDμfαβ−iκ4θμβDνfνβγμ,M(x)=m[1+(−14+2z1)θαβfαβ(x)]. The operator iR̸ is gauge covariant and formally self-adjoint and, in K, it should be understood as a perturbation of the ordinary Dirac operator iD̸. Note that this perturbation does not destroy the pairing between positive and negative eigenvalues that occurs in the spectrum of iD̸.We shall assume that the ordinary Dirac operator has a discrete spectrum. The latter is achieved by imposing on the fields boundary conditions that allow, by means of the stereographic projection, for the compactification of ordinary 4-dimensional Euclidean space to a 4-dimensional unit sphere [10,11]. In particular, we shall assume that the ordinary gauge fields satisfy the standard boundary condition: aμ(x)→ig(x)∂μg−1(x)+O(1/|x|2)as x→∞. In keeping with the philosophy adopted in this Letter, we shall take for granted that the eigenvalues and eigenfunctions of K can be computed by employing standard perturbation theory, using iR̸ as a perturbation. Thus, following Fujikawa [3], we shall use the eigenfunctions of K to define the fermionic measure of the path integral. One expands first the fermion fields ψ(x)=∑nanφn(x), ψ¯(x)=∑nb¯nφn†(x), in terms of the of a orthonormal set of eigenfunctions of K, say {φn(x)}n. Recall that an and b¯n are Grassmann variables. Then, the fermionic measure is defined as follows: dψdψ¯=∏ndandb¯n.The generating functional, Z[Jaμ,ω,ω¯], of the complete Green functions of our theory is defined by the following path integral (4)Z[Jμa,ω,ω¯]=1N∫dμexp{−SE+∫d4x[Jaμ(x)aμa(x)+ω¯(x)ψ(x)+ψ¯(x)ω(x)]}, where SE is defined by Eqs. (2) and (3), and the path integral measure dμ is equal to [daμa]∏ndandb¯n. [daμa] is the measure over the space of gauge fields and contains the Faddeev–Popov factor. In the massless limit SE in Eq. (2) is invariant under the following infinitesimal U(1) rigid chiral transformations δψ(x)=iαγ5ψ(x), δψ¯(x)=iαψ¯(x)γ5. Hence, under the infinitesimal local Abelian chiral transformations δψ(x)=iα(x)γ5ψ(x), δψ¯(x)=iα(x)ψ¯(x)γ5, the action SE undergoes the change δSE=−∫d4x[α(x)∂μj5μ(x)−2α(x)ψ¯(x)M(x)γ5ψ(x)]. The current j5μ(x) is the U(1)A current, which is classically conserved and is given by (5)j5μ(x)=ψ¯(x)[γμ−(14−2z1)θαβfαβγμ−12θαμγρfρα]γ5ψ(x). The measure of the path integral above also changes under the previous local chiral transformations: dμ→dμ[1−∫d4xα(x)A(x)]. The symbol A(x) denotes the following formal expression (6)A(x)=2i∑nφn†(x)γ5φn(x). These results and the fact that the path integral in Eq. (4) does not change under changes of ψ and ψ¯, leads to the following anomalous Ward identity 《[∂μj5μ(x)−2ψ¯(x)M(x)γ5ψ(x)+iω¯(x)γ5ψ(x)+iψ¯(x)γ5ω(x)]》=《A(x)》, where 《⋯》=1N∫dμ⋯exp{−SE+∫d4x[Jaμ(x)aμa(x)+ω¯(x)ψ(x)+ψ¯(x)ω(x)]}.As it stands in Eq. (6), A(x) is a formal object that is in demand of a proper definition. The latter is achieved as follows (7)A(x)=2ilimΛ→∞∑nφn†(x)γ5e−λn2/Λ2φn(x)=2ilimΛ→∞∑nφn†(x)γ5e−K2/Λ2φn(x).λn denotes a generic eigenvalue of K, K being defined in Eq. (3). The previous equation provides a gauge invariant definition of A(x) obtained by using the operator that gives the dynamics of fermions in the chiral limit. Besides, the spectrum of the operator K has in common with the spectrum of iD̸ the following paring property of the nonvanishing eigenvalues: for each nonvanishing eigenvalue λn with eigenfunction, say, φn(x), there exists an eigenvalue −λn with eigenfunction γ5φn(x). That this pairing property holds is necessary to establish a connection between of the value of A(x) and the index of the operator K(1+γ5)/2. We shall come back to this issue at the end of this Letter.By going over to a plane wave basis, expanding the exponential e−K2/Λ2, dropping all contributions with more that one θμν and ignoring terms that yield traces of the type trγ5=trγ5γμγν=0, one obtains the following expression for the far r.h.s. of Eq. (7): (8)A(x)=Aordinary(x)+Aθ(x)+O(θ2),Aordinary(x)=∑k=2∞limΛ→∞2i∫d4q(2π)4e−q2Λ2(2−k)k!Trγ5D̸2k(Λq)I,Aθ(x)=∑k=2∞∑l=0k−1limΛ→∞2i∫d4q(2π)4e−q2Λ2(2−k)k!Trγ5D̸2l(Λq){D̸(Λq),R̸(Λq)}D̸2(k−1−l)(Λq)I.I denotes the identity function on R4. Notice that Tr also denotes trace over γ matrices, when there occur such matrices in the expression affected by Tr. The symbols D̸(Λq), D̸2(Λq) and R̸(Λq) are defined, respectively, by the following equalities: D̸(Λq)=D̸+iΛq̸,D̸2(Λq)=D2+2iΛq⋅D−i2fμνγμγν,R̸(Λq)=R̸+i[(−14+2z1)θαβfαβΛq̸−12θαβγρfραΛqβ].D̸ and R̸ are given in Eq. (3).Aordinary(x) gives, of course, the Abelian anomaly in ordinary 4-dimensional Euclidean space: (9)Aordinary(x)=i(4π)2εμνρσTrfμνfρσ. Let us show next that the terms with k such that k⩾5 yield a vanishing contribution to Aθ(x) in Eq. (8). Let us consider a term coming from the expansion of D̸2l(Λq){D̸(Λq),R̸(Λq)}D̸2(k−1−l)(Λq)=[D2−i2γμγνfμν+2iΛq⋅D]l{D̸(Λq),R̸(Λq)}[D2−i2γμγνfμν+2iΛq⋅D]k−1−l, which contains a, b and c factors of type D2, γμγνfμν and 2iΛq⋅D, respectively. Since {D̸(Λq),R̸(Λq)} supplies two γ matrices to the term in question, we conclude that the trace over the Dirac matrices will vanish unless 2b+2⩾4, i.e., unless b⩾1. Now, notice that a+b+c=k−1, so that c is bounded from above as follows: c⩽cmax=k−1−b. Hence, the highest power of Λ that occurs in the term that we are analyzing is cmax+2=k−b+1. Next, this term is to be multiplied by Λ2(2−k), so, for k>2, it will not survive in the large Λ limit if k−b+1<2(k−2). This inequality and the constraint b⩾1, leads to k>4. We thus conclude that (10)Aθ=T1+T2+T3, where T1, T2 and T3 correspond, respectively, to the contributions to Aθ(x) (see Eq. (8)) with k=2, 3 and 4: (11)T1=∑l=01limΛ→∞2i∫d4q(2π)4e−q212Trγ5D̸2l(Λq){D̸(Λq),R̸(Λq)}D̸2(1−l)(Λq)I,T2=∑l=02limΛ→∞2i∫d4q(2π)4e−q213!Λ2Trγ5D̸2l(Λq){D̸(Λq),R̸(Λq)}D̸2(2−l)(Λq)I,T3=∑l=03limΛ→∞2i∫d4q(2π)4e−q214!Λ4Trγ5D̸2l(Λq){D̸(Λq),R̸(Λq)}D̸2(3−l)(Λq)I. To carry out the computation of T1, T2 and T3, we shall need the expansion of {D̸(Λq),R̸(Λq)} in powers of Λ: (12){D̸(Λq),R̸(Λq)}=γμγνSμν=γμγν(Sμν|Λ0+Sμν|Λ1+Sμν|Λ2),Sμν|Λ0=d1θαβ(DμfαβDν+2fαβDμDν)−12θαβ(DμfναDβ+fναDμDβ+fμαDβDν)+z1θαβ(DμDνfαβ+D{νfαβDμ}o)−iκ4(θνβDμDαfαβ+θ{νβDαfαβDμ}o),Sμν|Λ1=iΛ(q{μRν}o)+id1Λθαβ(qνDμfαβ+fαβq{μDν}o)−i2Λθαβqβ(Dμfνα+f{μαDν}o),Sμν|Λ2=−Λ2θαβq{μ(d1fαβqν}o−12fν}oαqβ).{}o indicates that only the indices μ and ν are symmetrized. d1=−1/4+z1.Let us work out T1 in Eq. (11). Using the fact that trγ5=trγ5γμγν=0, one concludes that T1=limΛ→∞2i∫d4q(2π)4e−q212Trγ5{−i2γμγνfμν{D̸(Λq),R̸(Λq)}I−i2{D̸(Λq),R̸(Λq)}γμγνfμν}. Substituting in the previous equation the results in Eq. (12), one shows that the contribution coming from Sμν|Λ1 vanishes upon integration over q and that Sμν|Λ2 yields a vanishing contribution since Sμν|Λ2 is symmetric in μ and ν. Then, the computation of the integral and traces on the r.h.s. of the previous equation leads to (13)T1=−2(4π)2εμ1μ2μ3μ4Tr(fμ1μ2Sμ3μ4|Λ0+Sμ1μ2|Λ0fμ3μ4)=i8π2θαβεμ1μ2μ3μ4Tr(2d1fαβfμ1μ2fμ3μ4−fμ2αfμ1βfμ3μ4)+116π2θαβεμ1μ2μ3μ4Tr[fμ1μ2(Dμ3fμ4αDβI−2d1Dμ3fαβDμ4I)+Dμ1fμ2αfμ3μ4DβI−2d1Dμ1fαβfμ3μ4Dμ2I]+iκ44π2θμ4βεμ1μ2μ3μ4TrDμ3(fμ1μ2Dνfνβ), where I is the unit function on R4 and d1=−1/4+2z1.To calculate T2 in Eq. (11) will be shall express it as the sum of two terms, say, T2(6γ) and T2(4γ), which involve the computation of the trace over six and four γ matrices, respectively: (14)T2=T2(6γ)+T2(4γ),T2(6γ)=∑l=02limΛ→∞(−i2)∫d4q(2π)4e−q21(3)!Λ2Trγ5[(γργσfρσ)lγμγνSμν(γκγτfκτ)(2−l)]I,T2(4γ)=limΛ→∞∫d4q(2π)4e−q21(3)!Λ2Trγ5[{D2γργσfρσ+γργσfρσD2,γμγνSμν}+D2γμγνSμνγργσfρσ+γργσfρσγμγνSμνD2]I+limΛ→∞∫d4q(2π)4e−q22i(3)!ΛTrγ5[{q⋅Dγργσfρσ+γργσfρσq⋅D,γμγνSμν}+q⋅DγμγνSμνγργσfρσ+γργσfρσγμγνSμνq⋅D]I. In the limit Λ→∞, only the piece Sμν|Λ2 of Sμν (see Eq. (12)) contributes to T2(6γ). The computation of the corresponding integrals and some algebra yields (15)T2(6γ)=i2(4π)2θαβ(−8d1−1)εμ1μ2μ3μ4Trfαβfμ1μ2fμ3μ4. Since trγ5γργσγμγν∼ερσμν, one concludes that only the antisymmetric part of Sμν in Eq. (12) is relevant to the computation of T2(4γ) in Eq. (14). Then, in the large Λ limit we have T2(4γ)=∫d4q(2π)4e−q2i(3)!Trγ5[{q⋅Dγργσfρσ+γργσfρσq⋅D,γμγνS[μν]|Λ1}+q⋅DγμγνSμν|Λ1γργσfρσ+γργσfρσγμγνS[μν]|Λ1q⋅D]I.S[μν]|Λ1 is the antisymmetric part of Sμν|Λ1 in Eq. (12). By substituting in the previous equation the necessary integrals, and after some algebra, one obtains the following result: (16)T2(4γ)=−116π2θαβεμ1μ2μ3μ4Tr[fμ1μ2Dμ3fμ4αDβI+Dμ1fμ2αfμ3μ4DβI−2d1(fμ1μ2Dμ3fαβDμ4I+Dμ1fαβfμ3μ4Dμ2I)].From the definition of T3 in Eq. (11), one readily learns that there are contributions to it involving 8, 6 and 4 γμ matrices. The contributions with 8 and 6 γμ matrices vanish in the large Λ limit as Λ−2 and Λ−1, respectively. The contributions with 4 γμ matrices also go away as Λ→∞, since in this limit they are proportional to derivatives of 1/Λ2ερσμνSμν|Λ2 and Sμν|Λ2 is symmetric in its indices μ and ν. In conclusion: T3=0.Substituting the previous equation and Eqs. (16), (15) and (13) in Eq. (10), one obtains the following result: (17)Aθ(x)=T1+T2(6γ)+T2(4γ)+T3=−i32π2θαβεμ1μ2μ3μ4Tr[fαβfμ1μ2fμ3μ4+4fαμ3fμ4βfμ1μ2]+iκ44π2θμ4βεμ1μ2μ3μ4TrDμ3(fμ1μ2Dνfνβ)=∂μ(iκ44π2θμ3βεμμ1μ2μ3Trfμ1μ2Dνfνβ). The identity Trθαβεμ1μ2μ3μ4[fαβfμ1μ2fμ3μ4+4fαμ3fμ4βfμ1μ2]=0 has been used to get the second equality in the previous expression.In view of Eq. (17), one concludes that there is no anomalous contribution at first order in θμν. Notice that one can always set κ4=0 and that even in the event that one insisted in having a nonvanishing κ4, the contribution to Aθ can be absorbed by performing the following finite and gauge invariant renormalization of the current j5μ in Eq. (5): j5renμ=j5μ−iκ44π2θμ3βεμμ1μ2μ3Trfμ1μ2Dνfνβ.Let us choose SU(N), with N>2, as our ordinary gauge group. That A(x) in Eq. (8) be equal to A(x)ordinary up to first order in θμν is a highly nontrivial result. Indeed, Aθ(x) being proportional to a truly anomalous term like Trθαβεμ1μ2μ3μ4[fαβfμ1μ2fμ3μ4] is consistent with power counting and gauge invariance. And yet, as shown in Eq. (17) all contributions of this type cancel each other. Why? One may answer this question by establishing the connection between the Abelian anomaly A(x) and the index of K(1+γ5)/2, K being defined in Eq. (3). But first let us exhibit some properties of P(x)=Trθαβεμ1μ2μ3μ4[fαβfμ1μ2fμ3μ4(x)].The first property we want to display is that for SU(2), P(x)=0. The second property is that for SU(N), with N>2, P(x) cannot be expressed as ∂μXμ, Xμ being a gauge invariant polynomial of the gauge field and its derivatives. This is why we called P(x) a truly anomalous contribution for SU(N), N>2. That P(x) possesses this property can be shown as follows. If there exist such an Xμ, it would be a polynomial on the field aμa and its derivatives such that s0Xμ|aaa=0. s0 is the free BRS operator s0aμa=∂μca, and Xμ|aaa is the contribution to Xμ which has 3 fields aμa and 2 partial derivatives. Now, it has been shown in Ref. [12] that the cohomology of s0 over the space of polynomials of aμa and its derivatives is constituted by polynomials of f0μνa=∂μaνa−∂νaμa and its derivatives. Hence, Xμ|aaa=0, for it cannot expressed as a polynomial of f0μνa and its derivatives: we are one derivative short in Xμ|aaa. Xμ|aaa=0 implies that Xμ does not exist. The third property is that ∫d4xP(x) does not necessarily vanish for fields with well-defined Pontryagin number. For instance, in the SU(3) case, aμa=aμ(BPST)a+δa8bμ, with aμ(BPST)a being standard embedding of the BPST SU(2) instanton into SU(3) and bμ being a 4-dimensional vector field with components bμ=ωμνxνρ2(n−1)(r2+ρ2)n,n>1,ωμν=−ωνμ,ωμν=Sign(ν−μ)(μ⋅ν)for μ≠ν, yields ∫d4xP(x)≠0 and has Pontryagin number equal to 1. To simplify the computation choose a θμν with θ12=−θ21 as its only nonvanishing components.Let us now establish the connection between the Abelian axial anomaly and the index of K(1+γ5)/2. Using Eq. (7), one readily shows that −i2∫d4xA(x)=∑n=1n+∫d4xφn+†(x)φn+(x)−∑n=1n−∫d4xφn−†(x)φn−(x)=n+−n−.n+ and n− are, respectively, the number of positive and negative chirality zero modes of K in Eq. (3). Of course, n+=dimKerK(1+γ5)/2 and n−=dimKerK†(1−γ5)/2. Hence, the Abelian anomaly is given by the index of K(1+γ5)/2: (18)−i2∫d4xA(x)=indexK(1+γ5)/2. Now, we have assumed that the operator K differs from the Dirac operator iD̸ in a “infinitesimally small”—otherwise our expansions in θμν would not make much sense—operator iR̸ that is hermitian and such that γ5K=−Kγ5. Then, one would hope [13] that (19)indexK(1+γ5)/2=indexiD̸(1+γ5)/2=132π2∫d4xεμνρσTrfμν(x)fρσ(x). A by-product of our calculations is that the previous equation indeed holds as far as we have computed. Notice that if Aθ in Eq. (8) had received a contribution like P(x)=Trθαβεμ1μ2μ3μ4[fαβfμ1μ2fμ3μ4(x)], then, in view of the discussion in the previous paragraph and Eq. (18), we would have concluded that the first equality in Eq. (19) would not be correct. Obviously, this analysis can be extended and conjecture that at any order in θ the Abelian anomaly for noncommutative SU(N) is saturated by the ordinary Abelian anomaly. This conjecture is further supported by the second order in θμν Feynman diagram calculations carried out in Ref. [14].Finally, our results can be readily extended to the case of noncommutative gauge theories with a nonsemisimple gauge group, when the noncommutative theory is constructed by using the symmetric form of the Seiberg–Witten map as defined in Ref. [7]. In this case the Seiberg–Witten map is the same as the map displayed in Eq. (1) by now aμ is given by aμ=∑k=1sgk(aμk)a(Tk)a+∑l=s+1NglaμlTl and the spinor ψ denotes a hypermultiplet carrying a given representation of the nonsemisimple gauge group. aμk, gk and aμl, gl are the ordinary gauge field and coupling constants associated, respectively, to each simple and U(1) factor of the nonsemisimple group. The reader is referred to Ref. [15] for further details on the notation. It is clear that Eqs. (9) and (17) will also be valid in the nonsemisimple case provided aμ is defined as in the previous equation.It is a very interesting and open question to obtain the results presented in this Letter by using the heat kernel expansion [16] due to its relevance in the mathematically rigorous proof of index theorems.AcknowledgementsThis work has been financially supported in part by MEC through grant BFM2002-00950. The work of C. Tamarit has also received financial support from MEC trough FPU grant AP2003-4034.References[1]J.FrohlichB.Pedrinihep-th/0002195[2]S.L.Adlerhep-th/0411038[3]K.FujikawaPhys. Rev. D2119802848K.FujikawaPhys. Rev. D2219801499Erratum[4]L. Alvarez-Gaumé, HUTP-85/A092, Lectures given at International School on Mathematical Physics, Erice, Italy, 1–14 July, 1985[5]B.JurcoL.MollerS.SchramlP.SchuppJ.WessEur. Phys. J. C212001383hep-th/0104153[6]X.CalmetB.JurcoP.SchuppJ.WessM.WohlgenanntEur. Phys. J. C232002363hep-ph/0111115[7]P.AschieriB.JurcoP.SchuppJ.WessNucl. Phys. 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