Axial Anomaly in the $SU(N)$ Gauge Matrix Model

The $SU(N)$ Yang-Mills matrix model admits self-dual and anti-self-dual instantons. When coupled to $N_f$ flavors of massless quarks, the Euclidean Dirac equation in an instanton background has $n_+$ positive and $n_-$ negative chirality zero modes. We show that the index $(n_+ - n_-)$ is equal to a suitably defined instanton charge. Further, we show that the path integral measure is not invariant under a chiral rotation, and relate the non-invariance of the measure to the index of the Dirac operator. Axial symmetry is broken anomalously, with the residual symmetry being a finite group. For $N_f$ fundamental fermions, this residual symmetry is $\mathbb{Z}_{2N_f}$, whereas for adjoint quarks it is $\mathbb{Z}_{4N_f}$.


Introduction
An anomaly occurs when a symmetry of a classical theory fails to survive in the corresponding quantum theory. The earliest example is the axial anomaly [1,2]. In massless QCD the classical action has symmetries corresponding to the vector and axial rotations on a Dirac spinor Ψ, defined as U (1) V : Ψ → e iα Ψ, U (1) A : Ψ → e iαγ 5 Ψ (1. 1) respectively. In the quantum theory, the former survives while the latter is anomalously broken. The axial anomaly leads to the nonconservation of the axial current j µ A =Ψγ µ γ 5 Ψ where F µν is the gauge curvature. In this article we will demonstrate a surprising result: the quantum SU (N ) gauge matrix model coupled to N f massless quarks also exhibits an anomalous breaking of axial symmetry. We show this by solving for the zero modes of the Euclidean Dirac operator in the background of instanton gauge configurations, and evaluating its index. The index is related to a suitably defined instanton charge of the background configuration, giving us the matrix model version of the Atiyah-Singer index theorem. Finally, following Fujikawa [3,4], we relate the index to the non-invariance of the path-integral measure, which gives the anomaly. For the fermions in the fundamental representation of SU (N ) (i.e. quarks), there remains a residual Z 2N f symmetry, while for adjoint fermions the residual symmetry is Z 4N f . We emphasise that our result holds for any finite N ≥ 2.
The surprise here is that the quantum SU (N ) gauge matrix model, at least at first sight, is vastly different from SU (N ) gauge field theory. Even the structure of instanton configurations is very different. In the gauge field theory, distinct self-dual instantons are labelled by the instanton charge π 3 (SU (N )) = Z, whereas in the corresponding matrix model, they are labelled by a finite set of integers. We find that even though the number of distinct self-dual instantons is finite, it is enough to disturb the balance between left-and right-handed fermion zero modes, leading to the anomalous breaking of axial symmetry.
That anomalies can also occur in quantum mechanical situations has been known for some time now [5][6][7] and can be traced to domain problems. The Hamiltonian H, being an unbounded operator, can only be defined on a dense domain D(H) of the Hilbert space. A symmetry is broken anomalously if the generator J of the symmetry transformation does not preserve D(H). More recently, anomalies in quantum mechanics have been shown to play an important role in the possible phase structure of θ-vacua in QCD [8,9]. In this work, we show the anomaly using Euclidean methods rather than domain issues.
The matrix model discussed here was first presented in [10][11][12], and has been shown to be an excellent candidate for an effective low-energy approximation of SU (N ) Yang-Mills theory on S 3 × R. In particular, a numerical investigation of its spectrum gave remarkably accurate predictions for the light glueball and hadron masses [13,14]. To serve as a correct low-energy approximation of Yang-Mills theory, however, this quantum-mechanical model must also exhibit the axial anomaly. Our present work provides this important conceptual support.
The article is organized as follows. In Section 2, we present a brief derivation of the gauge matrix model coupled to massless quarks. In Section 3, we discuss the instantons in the pure gauge matrix model, and show how a suitably defined instanton charge can be assigned to elementary self-dual instanton configurations. In Section 4, we solve for the zero modes of the Dirac operator in the instanton background, and compute the index of the Dirac operator using Callias' Index Theorem. In Section 5, we show that in the presence of the instanton, the fermion path integral measure is not invariant under axial transformations, and thus the axial symmetry is broken anomalously. In Section 6, we couple adjoint fermions to the matrix model and show that the axial symmetry is broken in this situation as well. In Section 7 we discuss the implications of our result.

The Matrix Model
We briefly recall the construction of the Yang-Mills matrix model, first discussed in [10,11]. The spatial R 3 is compactified to S 3 , which is then isomorphically mapped to an SU (2) subgroup of the gauge group SU (N ). We consider the general left-invariant Maurer-Cartan one-form Ω on SU (N ), where M is a (N 2 − 1)-dimensional real matrix, g ∈ SU (N ) and T a are the generators of SU (N ) in the fundamental representation, satisfying Tr T a T b = 1 2 δ ab . Under the mapping S 3 → SU (2) ⊂ SU (N ), the orthonormal basis of vector fields on S 3 can be identified with iX i , where X i is the left-invariant vector field on SU (2). The gauge fields A i ≡ M ia T a (i = 1, 2, 3) are the pullback of the Maurer-Cartan form Ω under this map on S 3 . Here M ia is a 3×(N 2 −1)-dimensional real matrix depending on time only. Under spatial rotations R ∈ SO(3) and gauge transformations h ∈ SU (N ), it transforms as The matrix model degrees of freedom are elements of M N , the set of all 3 × (N 2 − 1)-dimensional real matrices M ia , and gauge transformations act as M → M S(h) T , h ∈ SU (N ). The configuration space of the model is the base space of principle bundle Ad SU (N ) → M N → M N /Ad SU (N ). This fibre bundle has been previously studied and found to be twisted, not admitting global sections [15,16], and this fact lies at the heart of the Gribov problem in Yang-Mills theory. The curvature F ij is obtained by the pullback of the structure equation dΩ + Ω ∧ Ω to S 3 : For dynamics, we need a gauge-invariant Lagrangian. The obvious choice for the kinetic term 1 2Ṁ iaṀia = Tr(Ȧ iȦi ) is not gauge-invariant because the time-derivative of A i does not transform covariantly under time-dependent gauge transformations. Rather, To remedy this, we introduce a set of time-dependent real functions, conveniently named as M 0a , and the matrix A 0 ≡ M 0a T a . We then define the gauge-covariant time derivative of A i as A 0 is therefore the parallel-transporter needed to define the covariant derivative along the temporal direction. The electric field E i ≡ D t A i and the magnetic field B i ≡ 1 2 ijk F jk are given by We choose the dynamics of the gauge fields to be governed the Yang-Mills Lagrangian where the potential V (A) ≡ 1 g 2 Tr B i B i consists of quadratic, cubic and quartic interaction terms: We have set the radius R of S 3 to 1, it can easily be put back when necessary. The model described above has been used to make variational estimates of the glueball spectrum [13], and has reproduced the predictions of lattice gauge theory with surprising accuracy.
Quarks are easy to introduce in the matrix model [12,14]. These are spinors Ψ αA , where A and α denote the color and spin indices respectively. Ψ αA are Grassmann-valued matrices that depend only on time, and transform in the fundamental representation of color SU (N ) and in the spin-1 2 representation of spatial rotations: respectively, where u(h) is the fundamental representation of h and D 1 2 (R) is the spin-1 2 representation of R.
The time derivativeΨ does not transform covariantly under time-dependent gauge transformations, but D t Ψ = (∂ t − iA 0 )Ψ does. Thus the Lagrangian with minimally coupled massless quarks is given by where the fermionic part is given by the gauge-covariant Dirac Lagrangian on S 3 [12]: In the chiral basis, the γ-matrices take the form where 1 is the identity matrix in two dimensions and σ i are the Pauli matrices.
It is now straightforward to perform a Wick rotation t → −iτ and A 0 → iA 0 in the Lagrangian L [18,19]. Integrating L over Euclidean time τ gives us the Euclidean action where D τ = ∂ ∂τ − A 0 and the Euclidean Dirac operator / D is given by The Euclidean action S E is invariant under U (1) A axial rotation Ψ → e iαγ 5 Ψ. However, as we shall see, quantum effects break the axial U (1) A symmetry.

Instantons in SU (N ) matrix model
Classical vacuum configurations of the pure Yang-Mills matrix model are given by those A i for which V (A) = 0. From (2.9), it is easy to see that the vacuum configurations satisfy This has solutions A i = 0, or A i = L i , where L i is a generator of the Lie algebra of SU (2). Thus classically, the matrix model has multiple degenerate vacua, which correspond to the matrices A i forming a general N -dimensional representation of the generators of SU (2). The existence of degenerate classical minima naturally opens the possibility of quantum mechanical tunnelling between them. The tunnelling behaviour is captured by instantons, the finite action solutions of equations of motion of the action (2.15). Defining the Euclidean chromoelectric field as we can rewrite the Euclidean action as This action is extremized when The solutions to (3.4) with the plus (minus) sign give the (anti-)self-dual instantons. The quantity is the integral of a total derivative, is insensitive to localized τ fluctuations and hence an invariant. We make use of our gauge freedom in (2.2) and (2.6) to transform (3.4) to the temporal gauge, with A 0 = 0 [17]. Then the instanton equation (3.4) becomes into We can reverse this process and start with any solution of equation (3.6). A solution with A 0 = 0 is obtained by considering all possible curves h(τ ) in SU (N ) that takes finite values at the boundaries τ = ±∞. For each such curve h, the instanton solutions are The equations (3.4,3.6) and their solutions have been studied by [20], and also [21] in a different context. Substituting the ansatz i 's are generators of SU (2) (in an arbitrary N -dimensional representation), we obtain which have solutions φ a (τ ) The subscripts s and a denote self-dual and anti-self-dual solutions respectively. Their profiles are plotted in Fig. 1).
The self-dual instanton tunnels from the classical minimum at i . If we transform the solutions by the curve h(τ ) to obtain A h µ , we obtain instanton solutions that go between L for the self-dual and h b = h(∞) for the anti-self-dual solution. This does not affect any of the subsequent arguments, since quantities of interest like the instanton number and charge defined below are gauge invariant.
Up to a normalization factor c, the instanton number T is given by For the instanton (3.8), (3.14) . We can fix c = 4 by requiring that T takes integer values for all cases. Thus We could define T to be the instanton charge; however, the form (3.16) is not very enlightening as there is no simple relation between the {j (α) r } and N . Luckily, for the (anti-)self-dual instanton, we can construct a charge that is expressible only in terms of r (α) 0 , the number of irreps at τ = ±∞. We define E i as The charge T new is still an integral over a total τ -derivative and hence a topological invariant. It depends only on (r 0 ), rather than the Casimirs or other labels of each individual irrep. As we will demonstrate in the next section, T new for a (anti-)self-dual instanton is equal to the index of the Dirac operator in that instanton background.

Index of / D in Instanton Background
The Dirac operator (2.17) is Hermitian and obeys { / D, γ 5 } = 0. So for every eigenfunction ψ n of / D with a non-zero eigenvalue λ n , there is an eigenfunction γ 5 ψ n with eigenvalue −λ n . This is not necessarily true for the eigenfunctions with zero eigenvalue (the zero modes). Since the Dirac operator is Hermitian, for non-zero eigenvalues, ψ n and γ 5 ψ n are orthogonal.
The zero modes can also be arranged to be eigenfunctions of γ 5 : In the Weyl basis, the Euclidean Dirac operator (2.17) is The zero modes (in the temporal gauge) are fundamental Dirac spinors of the form Ψ Aα . When A h 0 = − dh dτ h −1 , the corresponding zero mode is obtained via the gauge transformation Ψ h = u(h)Ψ. In any case, the number of zero modes and hence the index do not change under the gauge transformation. So we can make our entire argument in temporal gauge.
Let us study the simple case when A i = φ s/a (τ )L i , i.e. L = L i . Then the matrix σ i A i is a 2N -dimensional Hermitian matrix whose τ -dependence is via φ s/a (τ ) only. Its eigenvectors are τ -independent, so L can be diagonalized and / D brought to the form Here, ξ i (τ )'s are the eigenvalues of φ s/a (τ )σ i ⊗ L i + 3 2 1. Positive and negative chirality zero modes are of the form The index can therefore be expressed as We can find the ψ ± easily from (4.4): corresponding to the eigenvalue ξ n (τ ) of Σ, we obtain the zero mode where C ± n is a constant column vector with a non-zero entry only in the n th row. Let us count the normalizable solutions for a given L i . Say L i consists of r 0 irreducible blocks (if L i is the N -dimensional irrep, r 0 = 1), each block of dimension N r ≡ (2j r + 1). The eigenvalues are then straightforward to calculate using addition of angular momenta: in each irreducible block, φ s/a (τ )σ i ⊗L i has two distinct eigenvalues −φ s,a (τ ) Nr+1 2 and φ s,a (τ ) Nr−1 2 with multiplicities N r − 1 and N r + 1 respectively. In total, Σ has (N r − 1) degenerate eigenvalues λ 1,s/a ≡ 1 2 [3 − φ s/a (τ )(N r + 1)], and (N r + 1) degenerate eigenvalues λ 2,s/a ≡ 1 2 [3 + φ s/a (τ )(N r − 1)]. For either of these eigenvalues, the integral in (4.8) evaluates to   To determine the index, we count the number of normalizable solutions for each chirality and compute (n + − n − ). For example with N r = 3, the explicit plots of the solutions are shown in Fig.2a and Fig.2b. In general there are N r − 1 normalizable solutions of with positive and negative chirality for self-dual and anti-self-dual instantons respectively.
We could, of course, analyze the time-dependence and normalizability of each of these solutions case by case; however, for a Dirac operator of the form (4.3), there is a much neater way to determine the index using Callias' index theorem [22]. The index of / D is given by (4.14) The τ -dependence of ξ i is entirely contained in the function φ s/a , which takes values 0 or 1 at ±∞. Then As 0 ≤ φ s/a (τ ) ≤ 1, the term in the second bracket of (4.15) always evaluates to 0, and the eigenvalues with degeneracy (N r + 1) do not contribute to the index at all. The φ s/a at τ = ±∞ is given in (3.11) and (3.12). Therefore, we have where the subscripts s and a represent self-dual and anti-self-dual instanton backgrounds respectively. For N r > 2, the term in the square brackets in (4.16) evaluates to 2. Corresponding to each block of size N r ≥ 3, there exist N r − 1 normalizable zero modes of / D with one chirality (+1 for the self-dual instanton background and −1 for anti-self-dual one) and none with the other. This can be verified through examination of (4.12) and (4.13) at the boundaries.
When N r = 1, the index evaluates to 0. Consequently there are no normalizable zero modes corresponding to the blocks of size N r = 1, i.e., the trivial representation.  For the blocks with N r = 2, it seems at first sight that the index (4.16) is undefined. This is the case for which one eigenvalue of Σ vanishes at either τ = −∞ or +∞, depending on whether background is self-dual or anti-self-dual. Thus for large |τ |, the Dirac operator resembles that of a free particle. The corresponding zero mode is not strictly normalizable (it is delta-function normalizable); it is a zero energy resonance, or a threshold state. The resonance states are plotted in Fig.3c and Fig.3d.
To summarize, if L i 's consist of r 0 irreducible blocks with r 1 one-dimensional and r 2 2-dimensional blocks, there are r 2 zero-energy resonances of one chirality and for each of the remaining r 0 − r 1 − r 2 blocks with dimension N r ≥ 3, there are N r − 1 normalizable zero modes of one chirality. The chirality is +1 for the self-dual background and −1 for the anti-self-dual background. As we shall see in the next section, both zero modes and zero energy resonance states have non-zero contribution to the axial anomaly. Therefore to correctly take into account all contributions, we must extend the definition of ind / D to include zero modes as well as zero energy resonances with positive and negative chirality, as in [23]. Then we can write down the index as For the more general (anti-)self-dual instanton (3.8), the index is This is the same as the new topological charge of the instanton background, obtained in (3.18). This is the matrix model version of the Atiyah-Singer index theorem: We emphasize that T new is a quantity computed from the pure gauge sector, while the index of the Dirac operator counts the difference between the number of zero modes of opposite chiralities. It is remarkable that there exists such a simple relation between the two: a priori, there is no reason to expect this equality. Moreover, as we will show in Section 6, this relation between the charge and the index can be suitably adapted for adjoint fermions as well, hinting towards its universal nature in the context of the matrix model. However, despite the demonstration of the equivalence, there is no simple explanation of its origin and as it stands, the relation T new = ind / D is fortuitous.

Non-invariance of fermion measure
We adapt Fujikawa's method to demonstrate the axial anomaly, and relate it with the index of the Dirac operator. Specifically we show that under a U (1) A transformation, the measure of the fermionic path integral is not invariant. The Jacobian of the transformation gives the integrated anomaly. The Euclidean fermionic path integral is given by We expand the fermionic field Ψ in the basis of the eigenfunctions of / D in a given background field configuration as Ψ = a n Φ n ,Ψ = n b n Φ n , where Φ n 's include eigenfunctions of / D with non-zero eigenvalues, zero modes as well as zero-energy resonances. In this basis, the fermionic path integral measure is given by Under a U (1) A rotation with an infinitesimal α, the transformed Ψ can be expanded as where the coefficients transform linearly: a n = C mn a m , b n = C mn b m . The anomaly function A(τ ) is defined as As a result, the path integral measure transforms as Evaluating A(τ ) allows us to compute the change in the path integral measure and hence the anomaly. Now (5.8) is a summation over an infinite number of modes, and is formally divergent. We introduce a regulator e −β / D 2 and formally take the limit β → 0 at the end: We already found that for non-zero eigenvalues, Φ n and γ 5 Φ n are orthogonal, and so eigenfunctions of / D with non-zero eigenvalues do not contribute to (5.10). However, the zero modes can have non-zero contribution to (5.10). Furthermore, the zero-energy scattering states, although not normalizable, can be delta-function normalized. Hence, they also have a well-defined contribution to (5.10).
Setting α = constant, the integrated anomaly from (5.9) is where χ ± k include the zero modes as well as the zero-energy resonances. As / Dχ ± k = 0, the summations are independent of β, and we can evaluate them at β = 0. Therefore As χ ± k are also eigenvectors of γ 5 , the above implies Thus in a background gauge configuration where the Dirac operator has a nonzero index, the fermion measure is not invariant under axial transformations, and axial symmetry is anomalously broken. Specifically, in the SU (N ) instanton background (3.8), ind / D is given by (4.18), and the integrated anomaly is given by (5.14) Because (r 0 ) is always an integer, e ∓2iα dτ A(τ ) = e ∓2iα(r (2) 0 ) = 1 when α = nπ for any N (here n ∈ Z). This means that under axial rotations with α = nπ, the anomaly vanishes. Thus, the U (1) A is anomalously broken to a discrete Z 2 residual symmetry group.
The generalization to N f flavours of fermions is straightforward. The fermionic action is given by The Dirac operator is diagonal in flavour, and we simply obtain N f copies of the spectrum of / D for a single flavour. The axial symmetry is now broken to Z 2N f .

Adjoint Weyl Fermion
We consider here the case of adjoint Weyl fermions, which are relevant for supersymmetric gauge matrix models [24]. We show that in this case too, there is a non-trivial index of Dirac operator in the instanton background, and hence, an anomaly.
The Euclidean fermionic action with an adjoint Weyl fermion can be written as where F i = M ia G a and G a = −if abc are the SU (N ) generators in the adjoint representation: As before, we choose the temporal gauge.

Embedding the Instanton in (N 2 − 1) Dimensions
The matrix F i is an (N 2 − 1)-dimensional matrix, and is just the embedding of the background gauge field i , F i takes the form are the representations of SU (2) obtained by embedding L (α) i in (N 2 − 1) dimensions. To understand this embedding, we recall that adjoint representation of colour is derived from the tensor product of the fundamental and the anti-fundamental representation: Since L i and J i are embeddings of the SU (2) Lie algebra into the N -dimensional and the (N 2 − 1)dimensional representations of SU (N ) respectively, (6.4) induces a map between the embeddings as well.
If L i is the N -dimensional irreducible representation with spin-j = N −1 2 , the induced map gives the angular-momentum algebra: On the other hand, if L i is a reducible representation made of r blocks with spin j r = Nr−1 2 in each block, we obtain (⊕ r j r ) ⊗ (⊕ q j q ) = r,q (j r + j q ) ⊕ (j r + j q − 1) ⊕ · · · ⊕ (|j r − j q |) . (6.6) In the above direct sum, one spin-0 block arises from the singlet 1 of the RHS in (6.4), elimination of which gives us J i . So for L i in a representation r N r , J i is given by one singlet representation removed from the following direct sum: We can again define the spin in each block as j , and the electric and magnetic fields with respect to the embedding F i as which satisfy the instanton equation E i = ±B i . Then the quantity has the same form as (3.16).
To express T adj in a more convenient form like before, we define E i exactly as in (3.17) as i − e i . Then, the new instanton charge for the embedding is given by 0 . (6.12)

Index of the Dirac Operator
The index calculation is exactly as in Section 4 from (4.14) and (4.15), by replacing N Furthermore, we observe that all the even-dimensional blocks occurs an even number of times. This is understandable because such a representation can come only from the cross terms in (6.7) (for which r = q), which occur twice in the direct sum. So all distinct eigenvalues occur an even number of times, i.e., are at least doubly degenerate. This can also be seen as follows. Given σ i ⊗ F i + 3 2 1 ϕ = λϕ, (6.14) taking the complex conjugate and simplifying using σ * i = −σ 2 σ i σ 2 and F * i = F i gives σ i ⊗ F i + 3 2 1 (σ 2 ⊗ 1)ϕ * = λ(σ 2 ⊗ 1)ϕ * . (6.15) Thus corresponding to an eigenvector ϕ, we have a degenerate eigenvector (σ 2 ⊗ 1)ϕ * . The index is always an even integer, and so the anomaly e −2iα ind / D takes value 1 when α = nπ 2 , n ∈ Z. Therefore a single adjoint Weyl fermion breaks the U (1) A axial symmetry to a residual Z 4 subgroup.
For N f flavours, the residual symmetry is Z 4N f .

Discussion
As we mentioned in the introduction, the gauge matrix model is very different from the corresponding gauge field theory. Nevertheless, the matrix model does retain important non-perturbative features of the full field theory. As we have demonstrated here, the axial anomaly is one such feature. In the usual discussion of the axial anomaly in non-Abelian gauge field theories, only the irreducible connections are considered, and it is the instanton number of such connections that is related to the fermion zero modes. A priori, there is no reason that the residual axial symmetry should match with the corresponding result in the field theory, and it is surprising that it matches for the case of fundamental fermions. Whether this coincidence has a deeper reason requires further investigation. Our result on the anomaly provides a strong conceptual support to the numerical investigations of the matrix model [14]. In addition to reproducing the masses of light hadrons with excellent accuracy, the numerics also show that the pseudo-scalar mesons are much lighter than their scalar counterparts. Furthermore, it also finds the η -meson to be considerably heavier than the η-meson. The result on the axial anomaly presented in this article serves to strengthen the position of the matrix model as an effective low-energy approximation of QCD.
There is a plethora of matrix models that have been studied in the literature [25][26][27] that have a non-Abelian gauge symmetry and fermions, and which remain a subject of continuing interest [28][29][30][31][32][33]. Whether axial symmetry continues to hold at the quantum level in these models needs to be understood, in light of the results presented here.
Lastly, the axial anomaly is present for any SU (N ) gauge group, and there is no reason to expect that it is washed out in the large N limit.