Matrix regularization for gauge theories

We consider how gauge theories can be described by matrix models. Conventional matrix regularization is deﬁned for scalar functions and is not applicable to gauge ﬁelds, which are connections of ﬁber bundles. We clarify how the degrees of freedom of gauge ﬁelds are related to the matrix degrees of freedom, by formulating the Seiberg-Witten map between them.


Introduction
The matrix models are expected to describe superstring theory and M-theory [1][2][3], and are thought to be able to describe various degrees of freedom of extended objects in those theories such as strings, D-branes and M2/M5-branes.In particular, for coincident D-branes, non-Abelian gauge fields arise as massless modes of open strings, and the matrix models should also be able to describe gauge theories.In fact, many concrete examples [4][5][6][7][8][9][10][11][12][13][14][15] have shown that this is indeed possible.Although matrices and gauge fields are mathematically quite different (The former is just finite sets of numbers with a specific algebraic structure, while the latter is connection 1-forms of fiber bundles), there are certainly several known ways to link them together.
When the space-time is flat, there is a known map called the Seiberg-Witten map [16], which relates the ordinary gauge field to a noncommutative gauge field which has a deformed gauge transformation law with the star product.Since the star product is induced from the operator product, the noncommutative gauge field can be regarded as a field that takes the value in operators, or in other words, matrices of infinite dimension.Thus, the Seiberg-Witten map provides an example of relating a gauge field to a matrix.
Such correspondences between gauge theories and matrix models are also known when the space is compact and the matrix size is finite.Theories on sphere and torus [4,6,7] are the best understood concrete examples.The gauge fields arise as tangential fluctuations of the corresponding noncommutative background of the matrix models.In these cases, by using the Fourier mode expansion (as found in [10] for example), one can directly relates the degrees of freedom in gauge theories and those in matrix models in the large-N limit.
Yet another method of formulating gauge fields from matrices is known in the context of tachyon condensation [17][18][19].This method is closely related to a quantization scheme called the Berezin-Toeplitz quantization [20], which is also discussed in this paper.A gauge field appears as the connection 1-form used in this quantization and this field is identified with the dynamical gauge field on stable D-branes arising from the tachyon condensation.
There is also another formulation called Eguchi-Kawai model [21], which was discovered earlier than the above examples.The model is defined in terms of unitary matrices and is shown to reproduce the planar sector of a corresponding gauge theory provided that some modifications called quanching or twisting [22][23][24][25] is implemented, which protects the model from the notorious problem of the spontenious U(1) D symmetry breaking.
Thus, there are indeed several different methods of relating gauge fields and matrices.However, these methods look quite different form each other and there seems to be a potential viewpoint that gives a unified understanding of them.In this paper, we reconsider the problem of relating gauge fields and matrices and find a new method based on the Berezin-Toeplitz quantizaiton.Our method also provides a relation between the above known methods and gives a unifying understanding for this problem.By using this method, we can formulate the Seiberg-Witten map for finite size matrices.
The Berezin-Toeplitz quantization was also used in the context of regularizing the worldvolume theory of a membrane [26] (The quantization is sometimes called matrix regularization in this context.).The quantization is a map from continuous functions on the spatial world volume M to a finite size matrices, such that the Poisson algebra of functions is semi-classically realized in the commutator algebra of matrices.More specifically, for any function f on M, its image T (f ) of the quantization map T is a finite size (say N × N) matrix and one of the main properties of the map is that [T (f ), T (g)] = − i N T ({f, g}) + O(1/N 2 ) for any functions f and g, where { , } is a Poisson bracket on M. For an embedding function X A : M → R D (A = 1, 2, • • • , D) of the membrane living in the flat space, the quantization maps X A to a matrix XA = T (X A ).The infinitesimal area-preserving diffeomorphism of the form δX A = {α, X A }, where α is a local parameter, is mapped in the above sense to the infinitesimal unitary similarity transformation of the matrix, δ XA = iN[ α, XA ].Thus, the regularized theory keeps the matrix version of the area-preserving diffeomorphism [26].
In this paper, we mainly focus on coincident D-branes in string theory and consider how they are described in terms of matrices.The data needed to describe the D-branes consist of the embedding function X, which defines the position of the D-branes in the target space, and the excitations of open strings.In the low energy limit, the bosonic excitatons consists of a non-Abelian gauge field B and adjoint scalar fields Φ1 .We construct a map from the data (X, B, Φ) to matrices X(X, B, Φ), such that the gauge symmetry for B and Φ is also realized in the commutator algebra of matrices.Schematically, the map we will construct has the form where δX denote fluctuations of the embedding function X.In the large-N limit, the fields B and Φ correspond to the tangential and vertical fluctuations of X, respectively.If we include the 1/N corrections, this relation becomes more complicated and we will argue that the relation is given by a generalization of the Seiberg-Witten map.This relation works for arbitrary symplectic manifold M and thus, gives a general way of relating gauge fields and matrices.For S 2 , we explicitly derive the Seiberg-Witten map up to the next leading order in 1/N.We then apply this method to a specific matrix model with a cubic interaction and relate the model with a gauge theory on S 2 .The same relation was studied in the large-N limit in [9,11] (See also [12,13,27]) in terms of the momentum expansion.Our method can also reveal the effect of 1/N corrections in this relation.We evaluate the 1/N correction of the classical action in this relation.
We also clarify the relation between our method and the other methods mentioned above.We first show that the gauge field in [19] can be identified in the large-N limit with the gauge field B in our setup.If we include 1/N corrections, they are related by a nonlinear redefinition of the fields.We then discuss a connection to Eguchi-Kawai model by applying the Berezin-Toeplitz quantization to Wilson line operators, not to the embedding function.The quantization maps Wilson lines to finite size matrices, and in the case of torus, we demonstrate that the plaquete action written in terms of the Wilson lines is related to Eguchi-Kawai model through the quantization.
This paper is organized as follows.In section 2, we review the Berezin-Toeplitz quantization for vector bundles, which is needed in the subsequent section.In section 3, we first review the Seiberg-Witten map and then generalize it for compact spaces based on the Berezin-Toeplitz quantization.Based on this generalization, we find the map (1.1).In section 4, we apply the Seiberg-Witten map to a cubic matrix model and see its relation to a massive BF theory on S2 .In section 5, we discuss the relation between our method and the other formulations.Section 6 is devoted to summary and discussion.

Berezin-Toeplitz quantization for vector bundles
In this section, we review the Berezin-Toeplitz quantization for vector bundles developed in [28][29][30][31][32].For simplicity, we consider 2-dimensional case, but higher dimensional case can also be treated in a similar manner [33].See also [34][35][36], where the quantization is discussed in terms of the Landau problem.
First, we describe the basic setup.Let M be a closed Riemann surface.We denote by g a Riemannian metric on M and by ω the volume form of g.Since ω is a closed nondegenerate 2-form, it gives a symplectic structure on M. We also introduce the so-called prequantum line bundle L such that it is a complex line bundle and its curvature is proportional to the symplectic form as where A (L) is the connection 1-form and V = M ω 2π is the volume of M. Note that by definition the bundle L has the unit monopole charge (the first Chern number) as We also introduce spinor fields on M to construct the quantization.Let S be the standard (2component) spinor bundle on M. We then consider twisted spinor fields which are smooth sections of S ⊗ L p ⊗ E, where p is a positive integer and E is an arbitrary finite-rank vector bundle with hermitian metric and hermitian connection.We define an inner product on Here, • is a hermitian inner product on the fiber (S ⊗ L p ⊗ E) x at each point x ∈ M. The Dirac operator on Γ(S ⊗ L p ⊗ E) is defined in terms of local coordinates as Here, γ α are the gamma matrices satisfying {γ α , γ β } = g αβ .The covariant derivative ∇ (E) α acting on Γ(S ⊗ L p ⊗ E) is written as where Ω α is the spin connection and is the connection of E. For sufficiently large p, the dimension of normalizable zero modes of the Dirac operator is given as dimKerD (E) = d (E) p + c (E) , where d (E) and c (E) are the rank and the first Chern number of E. This follows from the Atiyah-Singer index theorem and the vanishing theorem [28,29,31].We denote the projection operator onto the Dirac zero modes by Π (E) : Γ(S ⊗ L p ⊗ E) → KerD (E) .
Let E and E ′ be arbitrary finite-rank complex vector bundles over M with hermitian inner products and hermitian connections.Below, we focus on the homomorphism bundle Hom(E, E ′ ), which is a vector bundle whose fiber at a point x ∈ M is a vector space of linear maps from the fiber E x to E ′ x .The quantization we will construct below is a map from smooth sections of Hom(E, E ′ ) to finite-size matrices, that approximately preserves the algebraic structure of homomorphisms in the large matrix size limit.
Let us consider a field ϕ ∈ Γ(Hom(E, E ′ )).We can regard ϕ as a linear map from Γ(S ⊗L p ⊗E) to Γ(S ⊗ L p ⊗ E ′ ).We define the quantization map for ϕ ∈ Γ(Hom(E, E ′ )) by 3T p (ϕ) := Π (E ′ ) ϕΠ (E) . (2.6) The operator T p (ϕ) is called the Toeplitz operator of ϕ.From the above estimation of dimKerD (E) , one finds that T p (ϕ) can be represented as a (d From the definition, it satisfies where † on the left-hand side is the hermitian conjugate of the homomorphism, while that on the right-hand side is the hermitian conjugate of the rectangular matrix with respect to the Frobenius inner product.The quantization map (2.6) has some desired properties, which follow from an asymptotic expansion of the Toeplitz operator in the large-p limit.Let us consider two fields, ϕ ∈ Γ(Hom(E, E ′ )) and ϕ ′ ∈ Γ(Hom(E ′ , E ′′ )).For T p (ϕ) = Π (E ′ ) ϕΠ (E) and T p (ϕ ′ ) = Π (E ′′ ) ϕ ′ Π (E ′ ) , their product T p (ϕ ′ )T p (ϕ) satisfies the following asymptotic expansion in p = V /p [31]: (2.8) Here, the symbols C s on the right hand side are the bilinear differential operators C s : Γ(Hom(E ′ , E ′′ ))⊗ Γ(Hom(E, E ′ )) → Γ(Hom(E, E ′′ )).The first four symbols are explicitly given by Here, W αβ is the Poisson tensor induced from the symplectic form ω, F (E ′ ) is defined by αβ and R is the Ricci scalar for the metric g.In terms of the inverse of the vierbein, e α a (a = 1, 2) satisfying e α a e β a = g αβ , the Poisson tensor is explicitly given as W αβ = e α 1 e β 2 − e α 2 e β 1 .Similarly, we have ).For ϕ ∈ Γ(Hom(E, E ′ )) and ϕ ′ ∈ Γ(Hom(E ′ , E ′′ )), the covariant derivatives in (2.9) are defined as (2.10) If we use the local orthnormal basis defining , where ∇ a := e α a ∇ α (a = 1, 2), the symbols C s can be written as Note that when all of the vector bundles E, E ′ , E ′′ and so on are the trivial line bundles, the above quantization reduces to the conventional matrix regularization for functions on M.
For higher dimensional symplectic spaces, we can also define the quantization map satisfying an asymptotic relation, which takes the same form as (2.8) [33].In this case, the form of the bilinear differential operators C s are slightly modified from the 2-dimensional case except C 0 and C 1 , which remain to have the same form as above.

Seibeg-Witten map for finite size matrices
In this section, we first review the Seiberg-Witten map (SW map) for noncommutative plane [16] and then construct the map (1.1) by generalizing the SW map for closed noncommutative manifolds described by finite-size matrices.We also explicitly evaluate the SW map for S 2 up to the next leading order in 1/N.

Review of the Seiberg-Witten map on flat space
Let us consider the gauge group U(N) on a flat space.We can consider two kinds of U(N) gauge fields: one is the ordinary non-abelian gauge field B which transforms as and the other is the non-commutative gauge field B which has the deformed transformation law, Here, * is the non-commutative star product (Moyal product) satisfying for any functions f and ĝ, where corresponds to the non-commutative parameter and { , } is the standard Poisson bracket on plane.
The SW map is a map from the ordinary gauge field to the non-commutative gauge field, which is compatible with the above transformations [16].The compatibility of the transformations is expressed as Assuming that the map has a power series expansion with respect to as the condition (3.5) can be pertrubatively solved.The solution at the first order in is given as where

The Seiberg-Witten map for finite-size matrices
Let us consider the U(N) gauge group on a closed symplectic space M. Let B a and χ i be the gauge field and some adjoint scalar fields on M, which transform as Below, for a given isometric embedding we will define a map from the data (X A , B a , χ i ) to finite size matrices XA such that the above gauge transformation is compatible with the unitary similarity transformation of XA generated by The compatibility condition is expressed as which is very similar to (3.5).Thus, the map can be seen as a generalization of the SW map.
Our ansatz for the map is where, on the right-hand side, the Berezin-Toeplitz quantization is defined for an endmorphism bundle (namely, Hom(E, E ′ ) with E = E ′ ) on the N-dimensional vector bundle E with connection A (E) and 1 N is the identity matrix on the fiber of E. XA and α take values in Γ(Hom(E, E)) and XA can be considered as fluctuations X A .This ansatz is very natural from the viewpoint of D-branes, since the gauge field and scalar fields are known to arise as tangential and vertical fluctuations of D-branes, respectively.Taking this analogy further, we decompose the fluctuation to the tangential and vertical directions as Here, W is the Poisson tensor of M and a, b are indices of an orthonormal basis such that Note that when M is 2n-dimensional space, a, b run from 1 to 2n while i runs from 1 to D − 2n and the vectors {∇ a X, Y i } form an orthonormal basis of the D-dimensional vector space.The fields A a and φ i take values in Γ(Hom(E, E)) and hence they can be represented as N × N matrices.From the perspective of D-branes, A a and φ i seem to directly correspond to B a and χ i , respectively.As we will see below, this is indeed correct in the classical limit p → 0. However, when p corrections are taken into account, the compatibility condition (3.10) implies more complicated nonlinear relations between them, which are very similar to the SW map (3.7).
By substituting our ansatz (3.11) into the transformation law (3.9), and using the asymptotic expansion (2.8), we find that (3.9) implies the following transformation laws for A a and φ i5 , where, D s (ϕ, φ) = C s (ϕ, φ) − C s (φ, ϕ).In the classical limit p → 0, (3.13) reduces to These are the transformation laws of non-Abelian gauge field and adjoint scalar field, if we interpret A (E) as the background of A. Thus, we can assume the following form for the solution of the compatibility (3.10): The compatibility (3.10) is written in terms of A a and φ i as where we omitted the X dependence.We can determine the higher order terms in (3.15) by solving these conditions.We will demonstrate this for S 2 below.Note that though we do not see any local gauge symmetry on the matrix side (3.9), the above map relates such objects with the gauge field with local gauge symmetry.The emergence of the local gauge symmetry originates from a symmetry that keeps XA invariant.In fact, under the following transformation, the Toeplitz operator (3.11) is invariant.Thus, the gauge symmetry can be understood as the gauge redundancy of A (E) that constitutes the quantization scheme.The existence of the above symmetry is shown in Appendix A.

The Seiberg-Witten map for fuzzy S 2
Here, we consider fuzzy S 2 and we find a solution to (3.16) up to the next-leading order in p .
Let us consider S 2 with the standard metric and its isometric embedding into R 3 , satisfying In this case, the scalar curvature R is equal to 2 and the vector perpendicular to S 2 is given by Y A = X A .By using the notation 12) is written as For simplicity, we assume that A (E) = 0.Then, the transformation laws (3.13) for the fluctuations are explicitly given as In deriving above equations, we used an equation shown in Appendix C. By integrating these equations, we can find that χ 1 , B 1,− , B 1,+ and β 1 in (3.15) are given as A derivation of the above solution is shown in Appendix D.

Application to a cubic matrix model
In this section, we apply the SW map in section 3.3 to a cubic matrix model, and see how this model can be related to a gauge theory on S 2 .Such relation was also studied in earlier literatures [9,11] from different view points, and the above model is known to correspond to a massive BF theory on S 2 in the large-N limit.The SW map we constructed above enables us to see this relation including the 1/N corrections.Below, we assume that A (E) = 0 for simplicity, and the covariant derivatives shall act as for ϕ ∈ Hom(E, E) and ξ α ∈ Hom(E, E) ⊗ T * M. We write detailed computation in Appendix E and only present the result here.By using (3.21), the action (4.1) is rewritten in terms of the non-Abelian gauge field B and the adjoint scalar field χ as where we have defined the field strength of B and the gauge covariant derivative as By fine-tuning the parameter α as α = 2 3 − 1 3 p + • • • , one can always eliminate the linear term of χ.Then, for small p , we have This is indeed the action of the massive BF theory.If we integrating out χ, the above action reduces to the 2-dimensional pure Yang-Mills theory, which is known to be a topological field theory.It would be interesting to study how the p -corrections in (4.3) modify the topological nature of the Yang-Mills theory.A similar calculation is also possible for more general matrix models.For example, for a model with quartic and cubic interactions, which was also studied earlier in [37,38], we have (4.6)

Relations to other formulations
There are other formulations of gauge fields in terms of matrices.In this section, we will consider relations between our description and the others.

Gauge fields in tachyon condensation
It was shown in earlier work that the Berezin-Toeplitz quantization also plays important roles in the context of tachyon condensation [19].In this context, the connection 1-form A (E) in the Berezin-Toeplitz quantization is identified with the gauge field on D-branes.
In the classical limit p → 0, one can identify A (E) with B, since they have the common transformation laws (3.17) and A (E) in (3.15) can be eliminated by shifting B as B → B + A (E) .For finite p , A (E) in the higher order terms in (3.15) cannot be eliminated by such a simple shift.Instead, we can consider B → B + A (E) + O( p ) with the last term of O( p ) suitably fine-tuned to eliminate A (E) 's in (3.15).There may be a subtlety here when the bundle E has non-vanishing Chern numbers.In this case, since the matrix size of the quantization map depends on the Chern number of E, it would not be possible to eliminate A (E) 's keeping the same size of the matrices.Thus, above discussion should be modified such that only the fluctuations of A (E) that do not change the Chern numbers can actually be absorbed by shifting B a .

Matrix regularization for Wilson line operators
In this section, we describe the matrix regularization for Wilson line operators.We consider the 2-dimensional torus T 2 with the volume V = 2π for simplicity but it is easy to generalize it into the higher dimensional torus T 2n .
We introduce a new gauge field A for the bundle E that is independent of A (E) , and consider the problem of quantizing A. Below, we consider the Berezin-Toeplitz quantization with A (E) = 0 for simplicity, but A can still be nontrivial 7 .Since A is a connection and the gauge transformation law of A is different from that of local sections, the quantization in section 2 is not applicable to A. Instead, let us consider the following operators made of Wilson lines of A and acting on ψ ∈ Γ(S ⊗ L p ⊗ E): ψ(y 1 , y 2 ). (5.1) Here, l is a real parameter, γ is the shortest straight line 8 from y to x and P denotes the path ordering.
is what is called the Bergman kernel and is defined by where The above operators U l and V l gives linear maps on Γ(S ⊗ L p ⊗ E), and thus, we can consider their quantization just as described in section 2.
7 If we consider the field , where Φ α are adjoint scalars, A has the same transformation law with A (E) .Even when A (E) = 0, A is nontrivial by the degrees of freedom of Φ α .
8 On the torus, the shortest straight line is not unique for some long paths.However, since the Bergman kernel decays very quickly for such paths and hence it does not cause any problem.To define the operators more regorously, one can insert a function that is equal to 1 on a compact support around the point y and is vanishing elsewhere.
In order to study some properties of this quantization, the following asymptotic expansion of the Bergman kernel B (L) p (x|y) [39] is very useful: where P x is the projection onto the positive chirality and the functions P and J r are given as When p is sufficiently large (or equivalently when p is small), we can calculate as Thus, the operators U l and V l are essentially the Wilson lines along x 1 and x 2 directions, respectively in the large-p limit.We consider the quantization of U l and V l .As shown in appendix F, the Toeplitz operators of U l and V l satisfy and also Let us then consider the quantization of a single plaquette, ψ I (v) . (5.9) Here, we used the shorthand notation for the integration measure, (5.10) and the total path γ ′ (x|v) consists of the straight lines from v to w, w to z, z to y, and y to x.By using the relation (5.6) in the large-p limit, we can perform four integration.Furthermore, we can use the formula [39] for the diagonal asymptotic expansion of the Bergman kernel, given by where the coefficient b (5.12) By using (5.11), we obtain where F is the curvature of A and the function C is defined by with r = r 1 + r 2 +r 3 +r 4 +r 5

2
. Finally, by using (5.8), we find that in the large-p limit, where we ignored the field independent term on the right-hand side.The right-hand side is just the standard action for the pure Yang-Mills theory while the left-hand side is the action of the Eguchi-Kawai model.The only subtlety is the unitarity of the Toeplitz operators.However, because of relations such as U † l U l = 1 + O( p ) and the asymptotic properties of T p , they become unitary matrices in p → 0. Thus, we find that the model is obtained from the matrix regularization of Wilson line operators accompanied with the Bergman kernel.
The definition (5.1) may look awkward.It may be more useful to express the Bergman kernel in terms of the path integral [40] and define U l and V l as a path integral of Wilson lines.Such expression may be useful for generalizing above discussion to the case of curved nontrivial manifold.

Summary and discussion
In this paper, we studied how gauge fields can be described in matrix models.We considered the Berezin-Toeplitz quantization, which is a concrete realization of the matrix regularization, as a tool of connecting the matrices and continuous gauge fields.At first sight, since such quantization applies only to matter fields, there seems no room for gauge fields.However, when the quantization is applied to embedding functions of brane-like objects, tangential and transverse fluctuations of the embedding functions certainly contains the degrees of freedom for the gauge fields and scalar fields, respectively.We defined the Seiberg-Witten map that connects finite-size matrices and those fields and obtained an explicit form of the map (3.21) for S 2 up to the next leading order in 1/N.The emergence of the local gauge symmetry from matrices is quite nontrivial, and we argued that this originates from the gauge redundancy (3.17) in constructing the quantization map.
We also studied relationships with other formulations of gauge fields.The first formulation is the one discussed in the context of tachyon condensation [19].We argued that the gauge field defined in this paper is also related to the gauge field in tachyon condensation, through a redefinition of the field variables.The second formulation we discussed is Eguchi-Kawai model.We showed that variables in this model can be interpreted as quantized Wilson line operators in our setup.
It is important to consider the inverse problem of our problem to understand more deeply the relation between matrices and gauge fields.The inverse problem of fuzzy geometry (namely, finding a classical geometry from quantized geometry) has been discussed in [41][42][43][44][45] and it would be interesting to consider the dual description of our findings (A possible candidate for such description is provided by the Berry connection [44] in the matrix geometry.).Such description will be useful in understanding physics of tachyon condensation [46,47].
Let us consider the isometric embedding X A : S 2 → R 3 satisfying (3.18) and where g is the metric on S 2 .The tangential vectors ∂ α X A and the radial vector X A form a basis of the three dimensional vector space and other vectors such as ∇ α ∂ β X A should be able to be expanded in terms of the basis.Below, we will show First, note that the Christoffel symbol for the metric g is given in terms of X A as From this expression, we find that Thus, ∇ β ∂ γ X A is perpendicular to the sphere and thus should be proportional to X A .The factor of proportionality is deduced from where, we used X A ∂ α X A = 0 which follows from (3.18).This shows the equation (C.2).

D Derivation of SW map on S 2
In this Appendix, we derive the SW map (3.21) on S 2 .We first calculate χ 1 .By substituting the expansion φ = χ + p χ 1 + • • • into the transformation law (3.20), and picking up the term of O( p ), we find that where in the second equality, we used The last expression is integrable and we obtain the solution χ 1 and β 1 as in (3.21).We can calculate B 1 in the similar manner.We first consider the minus component B 1− .Its transformation law is derived from (3.20), and is calculated as where in the second equality, we substituted the expression of β 1 in (3.21) and in the third equality, we used . By integrating the above expression, we obtain the solution for B 1− as shown in (3.21).B 1+ can easily be found from (B 1− ) † = B 1+ .Note that the higher order terms of the SW map have ambiguities in adding gauge covariant quantities.We have considered the simplest case where such ambiguous terms are simply put zero.

E Calculation for the qubic matrix model
In this appendix, we show a derivation of (4.3) in detail.

E.1 Preliminary
Here, we show some relations that are needed for the derivation of (4.3) including the 1/p corrections.
We first calculate ωtrχ 2 .The gauge transformation law for χ 2 is obtained from (3.20) as By taking the trace and integration, we obtain Here, in obtaining the first equality, we used the equation, for any rank 2 tensor T .In obtaining the second equality, we substituted β 1 in (3.21).By integrating (E.2), we obtain where Q(B, χ) is a function of B and χ satisfying Note that χ 2 originally has an ambiguity in adding a gauge covariant quantity.The above function Q represents this ambiguity and we will consider the case of Q = 0 for simplicity.We then obtain From (3.21), we can easily find that ωtrχ 1 , ωtrχχ 1 and ωtr(iF 0 12 χ 1 ), are given as and (E.9) From (3.21), we also find that (E.10)

E.2 Relation between integration and trace
Here, we discuss the relation between the integral on S 2 and the matrix trace.Let us consider the bundle S ⊗ L p ⊗ E on S 2 with the connection for E vanishing.The Dirac zero modes {ψ I } I=1,2,••• ,p×dimE for such bundle are written as [30] ψ Here, we have set the radius of the sphere to be 1 and hence V = π 2π and p = V p = 1 2p .

E.3 Calculation of Tr XA XA
Here, we compute Tr XA XA including 1/p corrections.By using the (2.8) and (3.11), we have where we have ignored Tr T p (X A )T p (X A ) , since it is a constant and does not depend on A or φ.By substituting (3.12), we find that the first order term in p in (E.15) is simply given as Similarly, the second order terms in (E.15) are given as TrT p XA XA = TrT p (2A − A + + φ 2 ), (E.17) and the third order terms are given as By summing up all the contributions and using (E.14), we find up to total derivative terms that By using (3.12), we can evaluate the second order terms of p as Taking a sum of these, we find that the second order contribution in (E.20) is Next, the third order terms in p in (E.20) are calculated as Thus, we find that the third order contribution in (E.20) is Finally, let us consider the 4th order terms in p in (E.20).We first consider the linear terms in X.Most terms are actually vanishing, since we can ignore the total derivative terms and C 2 (X, X) = C 3 (X, X) = 0.The only surviving contributions are linear terms in φ, which take the following forms, The total coefficient for this type of contributions is given as We then compute the 4th order terms in p in (E.20) which are quadratic or qubic in X.By using (3.12), we find that each term is given as follows: Summing all the above terms, we obtain the 4th order contribution in p in (E.20) as Thus, we find that Trǫ ABC XA XB XC is given as follows.

F.2 Asymptotic expansion
Here, we show the equation (5.8).Below, to simplify the notation, we omit the E-dependence (for instance we denote D (E) simply by D).From the definition of the Toeplitz operator 2.6, we have are of O( 0 ), the second term in (F.10) is suppressed in the classical limit p → 0. We therefore obtain (5.8).