Noncommutative SU(N) theories, the axial anomaly, Fujikawa's method and the Atiyah-Singer index

Fujikawa'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, $\theta^{\mu\nu}$, 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.

It 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 note 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 θ µν and for the most general Seiberg-Witten map which is local at each order in θ µν and commutes with hermiticity and complex conjugation.The case of noncommutive gauge theories with Dirac fermions and with a nonsemisiple gauge group is also analysed when the theory is defined by means of the symmetric Seiberg-Witten map [6,7].
Let a µ 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 A µ -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 A µ , that this map renders A µ 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 θ µν the Seiberg-Witten map is local, i.e., that it is a polynomial of the fields and their derivatives with dimensionless coefficients other than θ µν .Note that if, barring θ µν , 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 θ µν and the theory would not be local at each order in θ µν .It is not difficult to show that at first order in θ µν the most general Seiberg-Witten map that fulfils the previous requirements reads (1) Hermiticity of A µ demands κ i , i = 1, ..., 4 , to be real numbers.That the Seiberg-Witten map commutes with complex conjugation -i.e., Ψ[ψ, a µ , θ µν ] = Ψ[ψ * , −a * µ , −θ µν ] , see refs.[7,8]leads to z 2 = z 3 = z 4 = z 5 = 0 and restricts z 1 to be a real number.Notice that the terms in the Seiberg-Witten map that go with κ 4 and z 1 correspond, respectively, to field redefinitions of a µ 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 (non-physical) Green functions of the fields we shall compute.
The action of the noncommutative SU(N) theory we shall study is given by Tr denotes the trace operation on the matrix representation of SU(N) carried by ψ .In the previous equation, and A µ and Ψ are given by the Seiberg-Witten map above.⋆ stands for the Moyal product of functions: . 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 θ µν to be of "magnetic" type: θ 0i = 0 -, we obtain a theory whose action, S E , at first order in θ µν reads: S Y M 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 The operator iR / is gauge covariant and formally self-adjoint and, in K , it should be understood as a perturbation of the ordinary Dirac operator iD / .Note that this perturbation does not destroy the pairing between positive and negative eigenvalues that occurs in the spectrum of iD / .
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: a µ (x) → ig(x)∂ µ g −1 (x) + O(1/|x| 2 ) as x → ∞ .In keeping with the philosophy adopted in this paper, we shall take for granted that the eigenvalues and eigenfunctions of K can be computed by employing standard perturbation theory, using iR / 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 , in terms of the of a orthonormal set of eigenfunctions of K , say {ϕ n (x)} n .Recall that a n and bn are Grassmann variables.Then, the fermionic measure is defined as follows dψd ψ = n da n d bn .
The generating functional, Z[J aµ , ω, ω] , of the complete Green functions of our theory is defined by the following path integral where S E is defined by eqs.( 2) and ( 3), and the path integral measure dµ is equal to [da a µ ] n da n d bn .[da a µ ] is the measure over the space of gauge fields and contains the Faddeev-Popov factor.In the massless limit S E in eq. ( 2) is invariant under the following infinitesimal The current j µ 5 (x) is the U(1) A current, which is classically conserved and is given by The measure of the path integral above also changes under the previous local chiral transformations: 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 where 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 λ n 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 iD / the following paring property of the nonvanishing eigenvalues: for each nonvanishing eigenvalue λ n with eigenfunction, say, ϕ n (x) , there exists an eigenvalue −λ n with eigenfunction γ 5 ϕ n (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 + γ 5 )/2 .We shall come back to this issue at the end of this paper.
By going over to a plane wave basis, expanding the exponential e − K 2 Λ 2 , dropping all contributions with more that one θ µν and ignoring terms that yield traces of the type tr γ 5 = tr γ 5 γ µ γ ν = 0 , one obtains the following expression for the far r.h.s of eq. ( 7): ) II denotes the identity function on IR 4 .Notice that Tr also denotes trace over γ matrices, when there occur such matrices in the expression affected by Tr .The symbols D / (Λq) , D / 2 (Λq) and R / (Λq) are defined, respectively, by the following equalities: R / are given in eq.(3).
A ordinary (x) gives, of course, the abelian anomaly in ordinary 4-dimensional Euclidean space: Let us show next that the terms with k such that k ≥ 5 yield a vanishing contribution to A θ (x) in eq. ( 8).Let us consider a term coming from the expansion of which contains a , b and c factors of type D 2 , γ µ γ ν f µν and 2iΛq • D , respectively.Since {D / (Λq), R / (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 ≤ c max = k − 1 − b .Hence, the highest power of Λ that occurs in the term that we are analysing is Next, this term is to be multiplied by Λ 2(2−k) , so, for k > 2 , it will not survive in the large . This inequality and the constraint b ≥ 1 , leads to k > 4 .
We thus conclude that where T 1 , T 2 and T 3 correspond, respectively, to the contributions to A θ (x) -see eq. ( 8)with k = 2 , 3 and 4 : To carry out the computation of T 1 , T 2 and T 3 , we shall need the expansion of {D / (Λq), R / (Λq)} in powers of Λ : Let us work out T 1 in eq.(11).Using the fact that tr γ 5 = tr γ 5 γ µ γ ν = 0 , one concludes that Substituting in the previous equation the results in eq. ( 12), one shows that the contribution coming from S µν | Λ 1 vanishes upon integration over q and that S µν | Λ 2 yields a vanishing contribution since S µν | Λ 2 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 where II is the unit function on IR 4 and To calculate T 2 in eq. ( 11) will be shall express it as the sum of two terms, say, T (6γ) 2 and T (4γ) 2 , which involve the computation of the trace over six and four γ matrices, respectively: In the limit Λ → ∞ , only the piece S µν | Λ 2 of S µν -see eq. ( 12)-contributes to T (6γ) 2 . The computation of the corresponding integrals and some algebra yields Since tr γ 5 γ ρ γ σ γ µ γ ν ∼ ǫ ρσµν , one concludes that only the antisymmetric part of S µν in eq. ( 12) is relevant to the computation of T (4γ) 2 in eq. ( 14).Then, in the large Λ limit we have (12).By substituting in the previous equation the necessary integrals, and after some algebra, one obtains the following result: From the definition of T 3 in eq. ( 11), one readily learns that there are contributions to it involving 8 , 6 and 4 γ µ matrices.The contributions with 8 and 6 γ µ matrices vanish in the large Λ limit as Λ −2 and Λ −1 , respectively.The contributions with 4 γ µ matrices also go away as Λ → ∞ , since in this limit they are proportional 1/Λ 2 ǫ ρσµν S µν | Λ 2 and S µν | Λ 2 is symmetric in its indices µ and ν .In conclusion: Substituting the previous equation and eq.( 16), ( 15) and ( 13) in eq. ( 10), one obtains the following result: = 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 θ µν .Notice that one can always set κ 4 = 0 and that even in the event that one insisted in having a nonvanishing κ 4 , the contribution to A θ can be absorbed by performing the following finite and gauge invariant renormalization of the current j µ 5 in eq. ( 5): 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 θ µν is a highly nontrivial result.Indeed, A θ (x) being proportional to a truly anomalous term like Tr θ αβ ǫ µ 1 µ 2 µ 3 µ 4 [f αβ f µ 1 µ 2 f µ 3 µ 4 ] 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 + γ 5 )/2 , K being defined in eq. ( 3).But first let us exhibit some properties of 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 ∂ µ X µ , X µ 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 X µ , it would be a polynomial on the field a a µ and its derivatives such that s 0 X µ | aaa = 0 .s 0 is the free BRS operator -s 0 a a µ = ∂ µ c a -and X µ | aaa is the contribution to X µ which has 3 fields a a µ and 2 partial derivatives.Now, it has been shown in ref. [12] that the cohomology of s 0 over the space of polynomials of a a µ and its derivatives is constituted by polynomials of f a 0 µν = ∂ µ a a ν − ∂ ν a a µ and its derivatives.Hence, X µ | aaa = 0 , for it cannot expressed as a polynomial of f a 0 µν and its derivatives: we are one derivative short in X µ | aaa .X µ | aaa = 0 implies that X µ does not exist.The third property is that d 4 x P(x) does not necessarily vanish for fields with well-defined Pontrjagin number.For instance, in the SU(3) case, a a µ = a Let us now establish the connection between the abelian axial anomaly and the index of K(1 + γ 5 )/2 .Using eq. ( 7), one readily shows that n + and n − are respectively the number of positive and negative chirality zero modes of K in eq. ( 3).Of course, n + = dim Ker K(1 + γ 5 )/2 and n − = dim Ker K † (1 − γ 5 )/2 .Hence, the abelian anomaly is given by the index of K(1 + γ 5 )/2 : Now, we have assumed that the operator K differs from the Dirac operator iD / in a "infinitesimally small" -otherwise our expansions in θ µν would not make much sense-operator iR / that is hermitian and such that γ 5 K = −Kγ 5 .Then, one would hope [13] A by-product of our calculations is that the previous equation indeed holds as far as we have computed.Notice that if A θ in eq. ( 8) had received a contribution like P(x) = Tr θ αβ ǫ µ 1 µ 2 µ 3 µ 4 [f αβ f µ 1 µ 2 f µ 3 µ 4 (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 θ µν 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 a µ is given by a µ = s k=1 g k (a k µ ) a (T k ) a + N l=s+1 g l a l µ T l and the spinor ψ denotes a hypermultiplet carrying a given representation of the nonsemisimple gauge group.a k µ , g k and a l µ , g l 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 a µ is defined as in the previous equation.
It is a very interesting and open question to obtain the results presented in this paper by using the heat kernel expansion [16] due to its relevance in the mathematically rigorous proof of index theorems.