Commutative Geometry for Non-commutative D-branes by Tachyon Condensation

There is a difficulty in defining the positions of the D-branes when the scalar fields on them are non-abelian. We show that we can use tachyon condensation to determine the position or the shape of D0-branes uniquely as a commutative region in spacetime together with non-trivial gauge flux on it, even if the scalar fields are non-abelian. We use the idea of the so-called coherent state method developed in the field of matrix models in the context of the tachyon condensation. We investigate configurations of noncommutative D2-brane made out of D0-branes as examples. In particular, we examine a Moyal plane and a fuzzy sphere in detail, and show that whose shapes are commutative $\mathbb{R}^2$ and $S^2$, respectively, equipped with uniform magnetic flux on them. We study the physical meaning of this commutative geometry made out of matrices, and propose an interpretation in terms of K-homology.


Introduction
D-branes in superstring theory are dynamical hypersurfaces in spacetime on which gauge fields and transverse scalar fields live. On a single D-brane, the transverse scalar fields represent the displacement of the worldvolume in spacetime. However, this interpretation cannot be applied naively for a stack of N D-branes, since the scalar fields take values in N × N hermitian matrices, which are not mutually diagonalizable in general. Soon after the discovery of D-branes, the idea that such non-commuting scalar fields represent noncommutative (NC) geometry [1,2] came out. It is most readily seen by the matrix quantum mechanics for multiple D0-branes [3] or the matrix model [4], which is (at least formally) seen as a model for D-instantons. According to this matrix geometry picture, various NC configurations of scalar fields, representing NC spaces such as the Moyal plane [5] and the fuzzy sphere [6] are considered. In these examples, the non-abelian scalar fields on lower dimensional D-branes make the system couple to the RR 3-form potential due to the Myers term and the effective theories on them become NC gauge theories. Here there appears a puzzle: Such a NC worldvolume lives in usual commutative spacetime while a NC space cannot be embedded into commutative spacetime in a usual sense in general. Therefore, the position or the shape of a NC D-brane in the commutative spacetime is far from obvious in particular.
This problem has been discussed from various viewpoints. In [7], the position of NC D-brane systems is estimated as the distribution of D-brane charges by using the D-brane charge density formula given in [8,9]. The original charge density formula is improved by assuming that fuzzy sphere configurations have single spherical shell structures, which gives a consistent improvement of the formula. This suggests that the worldvolume of NC D-brane system has a definite shape in the spacetime.
The way to determine the shape of D-brane system is not unique. The original interpretation that the diagonal elements of the scalar fields express the position of the worldvolume in spacetime has been generalized in [10], where the authors discuss that the position of the worldvolume should be identified by taking the "almost diagonal gauge" [11] of the scalar fields.
In [12], another interesting method of defining the shape of NC D-branes was proposed. In this method, in addition to NC D-branes, one introduces a probe D0-brane and considers open strings connecting the probe brane and the NC D-branes. The point is that the lowest energy of an open string is always proportional to the length of the string. Then, moving the position of the probe brane, one can find massless modes of open string only when the probe brane hits the NC branes so that the length of one of the open strings becomes zero. Thus, the set of all possible positions of the probe brane such that the open strings have massless modes can be interpreted as the shape of the NC branes. The energy of the open string can be measured by using a Dirac operator on the open strings and thus the shape of NC branes is defined as loci of zeros of the Dirac operator. See also [13,14] for analysis of this method.
The relation between NC and commutative geometries has been further developed as a mathematical correspondence between commutative geometry and matrices. In [15], a systematic way to extract a commutative space from a given configuration of matrices has been developed. In this approach, a Hamiltonian operator plays an important role, which is assumed to include matrices accompanied by coordinates of a Euclidean space R n as parameters. The commutative manifold living in R n is identified as loci of zero eigenstates of the Hamiltonian and some geometrical quantities such as Poisson structures and Riemannian metrics can also be extracted by the coherent states [15] (See also [16]). Although the large N limit of the matrices has been considered in [15], it has been pointed out in [17] that this idea works even at finite N with the use of the quasi coherent states, and it has been discussed that a Dirac-like operator can play the same role as the Hamiltonian. Interestingly enough, the obtained formulation is deeply related to that developed in the context of the superstring theory discussed in [12]. We thus refer to the method developed in [12,15,17] collectively as "the coherent state method" hereafter.
In this paper, we point out that the coherent state method also plays important roles in the context of the tachyon condensation in the superstring theory [18]. The basic idea is to identify the Dirac-like operator in the coherent state method with a tachyon profile on a system of unstable D-branes. With this identification, the coherent state method can be interpreted as the tachyon condensation, and the resultant commutative manifolds can be regarded as D-branes living in the commutative spacetime. The advantage of this interpretation is two-fold: First, the parameter space R n in the coherent state method can be interpreted as a worldvolume of this unstable D-branes. Second, it gives a clear reason why the ground state should be chosen to extract the commutative worldvolume.
Technically our analysis in this paper is an application of the technique developed so far [19]. This method has been applied to realize the Nahm construction of monopoles and the ADHM construction of instantons [20,21] and/or to realize a spherical D-brane [22]. In the latter case, a system of 2 non-BPS D3-branes is considered, where a tachyon profile T representing a D0-brane is deformed by a constant shift. By diagonalizing the tachyon T , the system is shown to condensate to a spherical D2-brane with a gauge flux of the unit monopole charge. Since the diagonalization is just the change of basis, the original deformed D0-brane and the spherical D2-brane with flux are unitary equivalent. This construction is similar but different from the well-known Myers dielectric D2-brane. In the former case, a D2-brane is made out of a single D0-brane and its worldvolume is a commutative S 2 , while in the latter case a D2-brane is made out of multiple D0-branes and its worldvolume is a fuzzy sphere. In this paper, we apply the tachyon condensation to the latter case and show that the fuzzy sphere has an equivalent expression to a system on a commutative sphere. For the latest result of the related topic, see [23], which has some overlap with the present paper and has been appeared on arXiv at the same time with the present paper.
The organization of this paper is as follows. In the section 2, we consider the system of k D0-branes with matrix-valued scalar fields on them, in terms of tachyon field of 2k non-BPS D3-branes. By using the idea of the coherent state method in this setting, we claim that the shape of D0-branes is a commutative region M of spacetime, and is determined uniquely by the zeros of the tachyon field. We also explain a general mechanism of appearing a gauge flux on M. In the section 3, we apply the method to NC D2-branes on the Moyal plane and the fuzzy sphere, that are made of D0-branes with the Myers term. We identify the shapes of these systems as commutative R 2 and S 2 , respectively. In the section 4, we discuss the meaning of the shapes more closely and propose an interpretation in terms of K-homology. The section 5 is devoted to conclusion and discussion.
2 Geometry from Matrices by Tachyon Condensation 2.1 Multiple D0-branes in non-BPS D-branes Consider a system of N non-BPS D3-branes whose worldvolume is R×R 3 in 10-dimensional Minkowski spacetime. The effective action is a U(N) gauge theory coupled with 6 transverse scalar fields and a tachyon field. In this paper, we focus on static configurations of the tachyon field only. We also restrict the gauge connection to be trivial. In this setting, D3-branes are rigid and its spatial worldvolume is identified with a part of spacetime. We set the spatial coordinates x = (x 1 , x 2 , x 3 ). Moreover, the Chan-Paton bundle, a complex vector bundle over R 3 with the fiber C N , is trivial R 3 × C N because of no gauge field. The tachyon field T (x) is a hermitian N × N-matrix valued scalar field.
Our argument below does not rely on the explicit form of the action, but for definiteness, we assume the tachyon potential has the form V (T ) = e −T 2 (i.e., we assume BSFT-type theory [24,25]). Because it is unstable around the (false) vacuum T = 0, the tachyon condensation occurs. At the true vacuum T = u1 N (u → ∞), non-BPS D3-branes disappear. In addition, lower dimensional D-branes can be realized as solitonic configurations [18].
Among them, let us consider k D0-brane configuration with fluctuations. We take N = 2k and set the tachyon profile as is a collection of transverse scalar fields on k D0-branes, that are k × k hermitian matrices. In the second expression, we used complex notation with z = x 1 + ix 2 and Φ = Φ 1 + iΦ 2 . Note that x i should be understood as x i ⊗ 1 k more precisely. The tachyon profile (2.1) without fluctuation, Φ = 0, represents indeed k D0-branes sitting at the origin x = 0 in the limit u → ∞, which is known as the ABS construction [26,27]. This is essentially seen by the tachyon potential which is proportional to the delta function δ(x) in the u → ∞ limit. Thus, under the tachyon condensation, the spatial worldvolume R 3 reduces to the origin, leaving the pointlike defect. This fact is most rigorously shown by using boundary states (see for example [28]): The boundary state for non-BPS D3-branes with this tachyon profile added on as a boundary interaction reduces to the boundary state for k D0-branes in the limit u → ∞, with the correct tension and the RR-charge. Even adding fluctuations, the profile (2.1) reduces to k D0-branes with transverse scalars, where scalar fields appear as a boundary interaction. The resulting effective action S D0 [Φ] for k D0-branes is given by the DBI action and the Chern-Simons term, which in particular includes the Myers term [6]. In the opposite way, a matrix model S D0 [Φ] can be embedded into the theory of non-BPS D3branes. This explains why the tachyon field appears in considering the shape of D-branes with non-commuting scalar fields. Note that, in this treatment, the condensation itself is obtained without matrix scalar fields Φ, and Φ are turned on afterwards as perturbation. Equivalent but more direct way is to consider the condensation of the profile (2.1) with Φ. The resulting defect should be the deformation of the point-like defect by matrices Φ. Indeed, as shown in [22], a deformation of the single D0-brane (k = 1) profile drastically changes the condensation defect to a spherical D2-brane 1 . Our claim in this paper is that the position or the shape of D-branes is determined by diagonalizing the tachyon field T , not the scalar fields Φ themselves.
In the following, we will consider such configurations of matrix scalar fields Φ that represent noncommutative D2-branes as typical examples. In particular, we investigate a NC plane and a fuzzy sphere in detail. By embedding D0-branes into non-BPS D3-branes, we will see that the spatial worldvolume R 3 shrinks to a commutative 2-dimensional space after the tachyon condensation. Moreover, this process induces a non-trivial gauge fields inevitably, whose field strength carries the D0-brane charge k.

Tachyon condensation and gauge flux production
Before treating explicit examples, we describe the schematic structure of the tachyon condensation for the configuration (2.1). Technically, the analysis is the same as the coherent state method mentioned in the introduction. We also explain how a non-trivial U(1) gauge flux is induced from the tachyon condensation. In order to consider the case of not only finite but also infinite N, we formulate the problem in terms of Hilbert spaces and projective modules.
Tachyon condensation The Chan-Paton bundle for N non-BPS D3-brane in our setting is a trivial complex vector bundle E = R 3 × C N over R 3 , whose typical fiber C N is a Hilbert space. Then, the space of sections of the Chan-Paton bundle is a free module A N of rank N, with A = C ∞ (R 3 ). Denote an orthonormal basis (ONB) for C N as | a (a = 0, 1, 2, . . . , N − 1). Then, a generic section is written as (2. 3) The tachyon field T (x) is an operator-valued function on R 3 . It is an element of the endomorphism End(E) and is written as According to (2.1), we assume that each matrix element T a b is at order u, and the limit u → ∞ will be taken. Note that in this profile (2.1), the matrices Φ act at each x (not only at the origin).
In order to extract the condensation defect, we need to diagonalize the potential V (T ) = e −T 2 , or the tachyon field T (x) itself at each point x on R 3 . Any hermitian matrix can be diagonalized by a unitary matrix. In an infinite dimensional Hilbert space and operators acting on it, the corresponding notion is the spectral decomposition. Assume the spectral decomposition at each point x, we find the eigenstates for the tachyon field as Here an eigenstate | ψ a (x) , which is a section of the Chan-Paton bundle, and the unitary operator U(x) are position dependent. In more familiar term, (2.4) is a gauge transformation of the tachyon field. In general, eigen-functions t a (x) are u-dependent but U(x) is u-independent (see the examples below). Then, the tachyon potential is written as This shows that, at each x, the component with t a (x) = 0 tends to 0 in the limit u → ∞. That is, the tachyon potential picks up the tachyon zero modes at each point. For example 2 , if only one component is the zero mode t 0 (x) = 0 for any x, then In this case, all the excited state | a (a = 0) are annihilated under the tachyon condensation. Note that P 0 = | 0 0 | is a rank 1 projection operator acting on the typical fiber, and the P (x) is unitary equivalent to P 0 . It means that the tachyon condensation picks up a 1-dimensional subspace U(x) | 0 from the N-dimensional fiber at each point x. More generally, it may happen that t 0 (x) = 0 for some region M ⊂ R 3 , but t 0 (x) = 0 otherwise. In this case, the tachyon potential also projects out the region M. Schematically, where δ(M) denotes a delta function distribution with its support on M. The original information of a choice of matrices Φ in (2.1) is transferred to two kinds of information, δ(M) and P (x). Note that this procedure is completely point-wise, and in general the unitary operator U(x) is not globally defined as a smooth function on the whole R 3 . For such cases, we may apply the procedure by considering patch-wise. That is, choose an open covering {U I } of In the latter case, the patch-wise condensation is needed. In these cases, a defect after the condensation is interpreted as a D2-brane on M. What kind of M appears depends of course on the choice of the matrices Φ.
In summary, the tachyon condensation just picks up the zeros of the eigenfunction and as a result a defect remains on a region M. This is technically the same as the coherent state method mentioned in the introduction. In fact, the tachyon profile T (2.1) is exactly the same as the Dirac-like operator in the literature [12,13,14,17] and T 2 corresponds to the Hamiltonian in [15,16]. Our claim in this paper is that the tachyon condensation gives a new physical interpretation of this prescription, based on the dynamics of the non-BPS D-branes. Although we are working with static configurations, the condensation is essentially a dynamical process and the zero modes survive as a result of the dynamics. This is in contrast with the previously proposed interpretations in [12,15,17] of the coherent state method, which are based on statics.
Gauge flux production Tachyon potential of the form (2.7) or (2.8) induces a U(1)flux. We here briefly describe the mechanism of this effect. For more detail we refer the reader to [22].
On the Chan-Paton bundle, the tachyon potential (2.7) plays the role of a projection operator P (x), which picks up a subspace U(x) | 0 at each fiber. This defines a projective module P A N , which is identified as the space of sections of a line bundle on R 3 . Since U(x) is a unitary operator, U(x) | a forms an orthonormal basis at each fiber. We may then write a generic element of the free module A N in this new basis as (2.9) An element of the projective module P A N is then given by Since P (x) depends on x, the exterior derivative d does not preserve the module P A N in general. This leads to the notion of connections. A natural connection on P A N , called the Grassmannian connection, is defined by ∇ = P • d, which acts as In components, we obtain ψ 0 → dψ 0 + iAψ 0 the covariant exterior derivative on the line bundle with a U(1) gauge potential A. In the case that the tachyon potential has the form (2.8), this gauge field is also confined to the region M ⊂ R 3 because of the delta-function distribution. In this case, (2.12) has components only along M (for the proof, see [22]). If U(x) is not globally defined and there are two different diagonalizations {U I (x)} at a point, then two gauge potentials of the form (2.12) are related by U(1) transition function. This gauge potential should possess a non-trivial U(1)-flux in our setting. Note that the D0-brane charge k for the original k D0-brane system described by Φ should be maintained as a magnetic flux of charge k in the D2-brane on M. We will see this explicitly in the next section. Note that the induced gauge potential A in (2.12) can also be seen as a Berry connection, if we regard the base space R 3 as a parameter space of the single Chan-Paton space C N . This viewpoint is appeared in [20,21] in the context of the tachyon condensation, and in [15] in the coherent state method.

Examples for NC D2-branes
The k D0-brane solution with fluctuation is given by (2.1) where z = x 1 + ix 2 and Φ = Φ 1 + iΦ 2 . Note that x i should be understood as x i ⊗ 1 k more precisely. Φ i (i = 1, 2, 3) are transverse scalar fields on k D0-branes and are k × k hermitian matrices. In this section, we consider examples that matrices Φ i represent NC D2-branes, on a Moyal plane and a fuzzy sphere. The shape of these branes are a commutative R 2 and S 2 , respectively.

Moyal plane
A NC D2-brane on the Moyal plane can be made out of k D0-branes, if the scalar field has the profile wherex 1 andx 2 are coordinates on a Moyal plane satisfying [x 1 ,x 2 ] = iθ. By defining the creation/annihilation operators bŷ the scalar fields (3.2) are rewritten in complex notation as In order to realize them, it is necessary to take k → ∞ and replace matrices Φ i with operators acting on the Hilbert space ℓ 2 (N). By inserting (3.2) into (3.1), the tachyon profile becomes It acts on the Chan-Paton bundle with typical fiber to be the Hilbert space H = ℓ 2 (N)⊗C 2 . Let {| n, ǫ |n = 0, 1, 2, · · · , ǫ = ±} be its ONB, where n and ǫ denote the eigenstate for the number operatorN =â †â and the eigenvalues of the Pauli matrix σ 3 , respectively. That is, two component vectors give the basis of H, where | n is the ONB of ℓ 2 (N).

Condensation
We will now study the tachyon condensation of this profile (3.5). To this end, we use the displacement operator for α ∈ C, which is a unitary operator and defines a coherent state | α = D(α) | 0 [29]. The basic properties are By using these properties, z-dependence in (3.5) is extracted as where the unitary operator U(z) is given by Under the tachyon condensation u → ∞, the surviving mode under the condensation is zero eigenstates of T 2 (x), It exists only for x 3 = 0. SinceN has the spectrum {n = 0, 1, 2, . . .}, T 2 (x) has a zero mode of the form at each point with arbitrary z = x 1 + ix 2 and x 3 = 0. The tachyon potential reduces to the projection operator onto this zero mode: Here we used the fact e −u 2 (x 3 ) 2 → u √ π δ(x 3 ) in the limit u → ∞. From the delta function, we see that the remnant of this condensation is the real 2-dimensional surface M = R 2 = C, which is considered as a spatial worldvolume of a D2-brane. We emphasize that the obtained worldvolume parameterized by z andz is commutative, although we start with a Moyal plane configuration. On the other hand, the projection operator P (z) of the Chan-Paton bundle picks up a coherent state D(α) | 0 at each point z on M. Because it is 1-dimensional subspace at each fiber, the Chan-Paton bundle reduces to a line bundle on M. It means a single D2-brane with the gauge group U(1). Moreover, because the fiber D(α) | 0 smoothly depends on the base space (recall (3.10)), this line bundle is non-trivial. This information is encoded in the unitary operator U(z), and we see a further consequence on the gauge flux in the following.
It is worth emphasizing that this result is completely different from perturbative picture of multiple D0-branes, where a D0-brane is sitting at the origin but is fluctuating around the origin to the "directions" of the non-commuting scalar fields Φ, that is, a single Moyal plane. In our picture, matrices Φ originally give a family of Moyal planes on R 3 as a Chan-Paton bundle of the non-BPS D3-branes, which however reduces to a line bundle on M = R 2 by the tachyon condensation. The schematic picture is given in Figure 1. Eigenstates For completeness, we here diagonalize (3.5). We give the solution for the eigenvalue problem (3.14) Note that states of the form D(α) | n, ǫ give an ONB, since the displacement operator is a unitary operator. Acting T on these states, it is easy to recognize that D(α) | 0, + is already an eigenstate T D(α) | 0, + = ux 3 D(α) | 0, + . Thus we write it as | ψ 0,+ = D(α) | 0, + with t 0,+ (x) = ux 3 . Next, for a fixed n (n ≥ 1), two states D(α) | n, + and On this doublet, T is effectively a matrix for each n, which is easily diagonalized by the unitary matrix The eigenvalues are t n,ǫ (x) = uǫ|T (n) | and the corresponding eigenstates are We can express all the eigenstates as | ψ n,ǫ = W D(α) | n, ǫ , by defining W as 1 on | 0, + and W (n) on the doublet at n as above. These states are orthonormal ψ n,ǫ | ψ n ′ ,ǫ ′ = δ nn ′ δ ǫǫ ′ , since W † W = 1.
In summary, the set of eigenstates consists of a ground state (singlet) | ψ 0,+ = D(α) | 0, + with its eigenvalue t 0,+ (x) = ux 3 , and the family of doublets | ψ n,ǫ (n ≥ 1) with eigenvalues t n,ǫ (x) = uǫ (x 3 ) 2 + 2θn. Under the tachyon condensation, all the doublets are annihilated, because t n,ǫ (x) = 0 for all x, while the singlet survives on the plane x 3 = 0 as states in (3.12). Note that the mixing between states | n and | ± is inevitable. This structure cannot be seen by the part | n only (i.e, Chan-Paton space for D0-branes).
Gauge flux The tachyon potential (3.13) defines the projective module or equivalently a complex line bundle over M = R 2 . The corresponding U(1) gauge connection is given by the Grassmannian connection according to (2.12). In the present case, the U(1) gauge field on M = R 2 is given by After some calculations, we find The corresponding field strength on M is given by A uniform magnetic flux on a D2-brane is interpreted as the D0-brane charge density, and its presence indicates that the resulting system is a bound state of D2 and D0-branes, where D0-branes are dissolved into a D2-brane. In fact, in the Chern-Simons term for a D2-brane, the coupling to the RR 1-form is It says that there is a dissolved D0-brane per unit volume 2πθ. Therefore, the original information on a Moyal plane is converted to a commutative plane with a uniform magnetic flux. This equivalence between commutative and noncommutative descriptions of the D2-D0 bound states is first shown in [30], in terms of boundary states. We here reproduce the same result within the effective theory on non-BPS D3-branes, but the equivalence is realized in a more direct way. That is, once the D2-D0 bound states is represented in the tachyon profile, the equivalence is realized by the unitary transformation that diagonalizes the tachyon profile.

Fuzzy sphere
A NC D2-brane on a fuzzy sphere can be made out of k D0-branes, if the scalar field has the profile where ρ is a real parameter and L i (i = 1, 2, 3) are su(2) generators in the spin-ℓ irreducible representation [31]. Thus it is possible for k ≥ 2. We denote corresponding k = 2ℓ + 1 a naive guess of the radius of this fuzzy sphere is ρ k 2 −1 4 . We will compare it with the radius of S 2 obtained from the tachyon condensation below.
Condensation Here we study the tachyon condensation by diagonalizing T in (3.24).
To this end, we examine the two terms in (3.24) separately in detail.
a) The term σ · x in (3.24) is independent of the choice of Φ, and it can be diagonalized only patch-wise [22].
First at the origin x = 0 in R 3 , this term does not contribute to T and is already diagonal. We then divide R 3 except x = 0 into two regions: and in the second expression the standard polar coordinates are used. Thus, U N is R 3 except for the negative x 3 -axis, while U S is R 3 except for the positive x 3 -axis. In each region U N and U S , σ · x is diagonalized as by the corresponding unitary matrix-valued function on R 3 They depend only on the angular coordinates Ω = (θ, ϕ) and are written in the polar coordinates as The expression R N in (3.29) is familiar in quantum mechanics with the diagonalization of a spin with respect to the directionx = x |x| , if S i = σ i 2 is considered as the spin-1 2 representation. But note that the diagonalization by R N is ill-defined at the south pole θ = π. 3 In order to cover all the direction, we need another open set U S .
b) The term σ · L in (3.24) or more properly, the term S · L is similar to the spin-orbit interaction in quantum mechanics. Thus, under the total spin J = L+S, the tensor product representation , the operator σ · L has the eigenvalue ℓ in all states in [ℓ + 1 2 ], while the eigenvalue −(ℓ + 1) in all states in [ℓ − 1 2 ]. This shows that two kinds of states should be mixed, in order to diagonalize σ · L. The ONB (2ℓ + 2 states) of [ℓ + 1 2 ] are given by eigenstates of J 3 as where m = −ℓ − 1, −ℓ, . . . , ℓ, and Note that two particular states In order to diagonalize T in (3.24), we have to consider both aspects of a) and b) simultaneously, that is, we have to consider the total spin b) in a patch-wise way a).
Condensation in U N First, we consider points in the open set U N . As stated, R N in a) appears in the spin along the axis through x. In general, for an angular momentum operator J , the term x · J determines the new "north pole" direction through x. Then eigenvalues of J ′ 3 =x · J can also be used to label the ONB. Here J i and J ′ 3 are related by an SO(3) rotation Λ i j that sends the unit vector through the point Ω = (θ, ϕ) on the unit sphere to that pointing to the north pole x = (0, 0, 1). This rotation is generated by the unitary operator, The previous R N in (3.29) is the spin 1/2 case of (3.34). This implies the spin-j analogue of (3.27), In particular, the transformed state R N (Ω) | j of the highest weight state | j is called the Bloch (spin) coherent state [29]. In our case, consider (3.34) for the total spin J = L + S. We can then split it as (3.34) for S and L, respectively. For the tachyon profile (3.24), it is obvious that this operator still diagonalizes x · σ = 2x · S. On the other hand, it keeps σ · L = 2S · L invariant, since it is an SO(3) scalar operator. Therefore, the tachyon profile is written as Note that the Ω=(θ, ϕ)-dependence is absorbed into R N (Ω). It is then reasonable to use the orthonormal basis of the form R N (Ω) | m, ǫ to find the eigenstates of T . According to b), the second term σ · L in (3.38) is diagonalized by the states of the form , but we should also take into account the first term. It turns out that two particular states R N (Ω) | ℓ, + and R N (Ω) | −ℓ, − are already the eigenstates of T . By using 40) which is zero at a point in U N with |x| = ρℓ. Thus, a sphere with radius ρℓ survives under the tachyon condensation. Similarly, we have which is always negative (no zero locus), thus, this state is completely annihilated under the tachyon condensation.
For the remaining eigenstates, consider a two dimensional subspace of the form for a fixed m with m = −ℓ, −ℓ + 1, . . . , ℓ − 1 and with arbitrary coefficients a m (x) and b m (x). The point is that T (x) is closed within this subspace: This matrix is diagonalized in a standard way (see Appendix A.3 for more detail) and the eigenvalues at each point (i.e., functions) are found to be λ (m) are always non-zero at any point x ∈ U N . Therefore, the tachyon condensation annihilates the corresponding eigenstates.
In summary, the eigenvalues of T (x) and the corresponding eigenstates are given by where the explicit form of the matrix W (m) (|x|) is given in the Appendix A.3. The first state becomes a zero mode at a point x ∈ U N with the radius |x| = ρℓ. The other states always vanish under the tachyon condensation.
Condensation in U S In U S , the eigenvalues of the tachyon profile are the same as in (3.47) in U N , but another unitary operator R S is needed to diagonalize T and a different state survives under the tachyon condensation. To see this, consider a point on the negative x 3 -axis, x = (0, 0, −|x|). Because of σ · x = −|x|σ 3 and by using (3.39), we find It shows that the surviving state is the lowest weight state | −ℓ, − around the south pole, if |x| = ρℓ. This is extended to any point x = (|x|, θ, ϕ) ∈ U S (θ = 0) by sending it to the south pole generated by the unitary operator Then, the Bloch coherent stateR S (Ω) | −ℓ, − with respect to the south pole is shown to be an eigenstate of T (x): which is the zero mode if |x| = ρℓ. Finding the other eigenstates is similar to perform. Although this treatment is sufficient when we are interested only in U S , it should actually be consistent with the result in U N in the overlapping region U N S = U N ∩ U S . The remaining 1-dimensional fibers of both construction should be identified with each other, that is, the difference should be at most a U(1) phase. To this end, we insert an extra rotation Θ = e −iπJ 2 on the state in U S to diagonalize T (x). For J = S, it is the Wigner time reversal operator, which flips | + and | − . In general, due to the relations the state Θ | m, ǫ is the eigenstate with alternating the sign: By the same relations, we also have (3.54) We now define 4 Then the tachyon profile in U S is written as Since the term inside the bracket is the same as in U N , it clearly shows that the eigenvalues of T are the same as U N as required, and the corresponding eigenstates are given by The first state becomes a zero mode at any point x ∈ U S with the radius |x| = ρℓ. The other states always vanish under the tachyon condensation.
Gluing in the overlap U N S The zero mode in U N and U S are now written respectively as R N (Ω) | ℓ, + and R S (Ω) | ℓ, + . In the overlapping region U N S , they are identified up to a U(1) gauge transformation (transition function, more properly). To see this, it is worth to rewrite R S (Ω) in (3.55) as (valid for θ = 0, π) By acting this on the state | m, ǫ , it implies This shows that two states R S (Ω) | m, ǫ and R N (Ω) | m, ǫ are related by a U(1) phase for fixed (m, ǫ). Hence, the transition function is U(1) 2k -valued: (3.60) 4 There is an constant phase ambiguity to define Θ. It can be shown that the spin 1/2 part R (S) S coincides with R S in (3.28) so that our choice of Θ is to be consistent with [22].
Structure of the tachyon potential Having found the eigenvalues of T (x), it is easy to write the tachyon potential in the spectral decomposition. Then, it has the form (2.8) as Here we used the fact e −u 2 (|x|−ρℓ) 2 → u √ π δ(|x| − ρℓ) in the limit u → ∞ on the radial delta function [22].
Matrices Φ originally give a family of fuzzy spheres on R 3 as a Chan-Paton bundle of the non-BPS D3-branes. This potential controls the reduction of both the worldvolume and the Chan-Paton space. The worldvolume R 3 of the non-BPS D3-branes reduces to the sphere M = S 2 defined by |x| = ρℓ, that is considered as a spherical D2-brane. This sphere is commutative and embedded in R 3 (thus in the spacetime as well). At each point on the sphere specified by Ω = (θ, ϕ), the original Chan-Paton space reduces to a 1-dimensional subspace R N (Ω) | ℓ, + on U N or R S (Ω) | ℓ, + on U S , that are related by a U(1) transition function on U N S . It is essentially the Bloch coherent state. The schematic picture is given in Figure 2. The projection operators P N (Ω) and P S (Ω) also define an induced U(1)-gauge connection as we will see.
It is interesting that the obtained radius ρℓ of S 2 is different from the expected "radius" of the fuzzy sphere ρ ℓ(ℓ + 1). Our result should also be compared with the radius given by the charge density formula [7] 5 . The radius can be any value at this stage, because of an arbitrary constant ρ. It would be determined by the dynamics of D2-brane, since ρ is regarded as a constant mode of a transverse scalar field on the spherical D2-brane, as analyzed by [6] [22]. 5 The original charge density is supported on the family of spherical shells at finite k, but the author of [7] argued that the commutator corrections improve the formula to give a single sphere with a physical radius ρ ℓ(ℓ + 1). Our result supports this improvement but the radius does not coincide with each other. Induced gauge connection The tachyon potential (3.62) defines the projective module or equivalently a complex line bundle over M = S 2 . The corresponding U(1) gauge connection (2.12) is given patch-wise, i.e., a gauge potentials A N on U N and A S on U S are given respectively by After some algebra, we obtain where k = 2ℓ + 1. On the overlap U N S , they are related by the U(1) transition function e −ikϕ : The U(1) field strength is defined patch-wise by F | U N = dA N and F | U S = dA S , but in fact it is globally defined: This configuration is nothing but the Wu-Yang k-monopole [32]. The RR-charge originally carried by k D0-branes is maintained by this U(1)-flux on a spherical D2-brane, where D0branes are dissolved into a D2-brane. In fact, in the Chern-Simons term for a D2-brane, the coupling to the RR 1-form is This result is independent of ρ.

Myers term and K-homology
We are considering the problem of mutually non-commuting matrix scalar fields Φ on multiple D0-branes. The system of k D0-branes with matrix scalar fields Φ reduces by the tachyon condensation to the region M in the (R 3 part of) spacetime, equipped with a Chan-Paton bundle E over M with k-magnetic flux. Thus, we call the region M, or a pair (M, E) as the shape of D0-branes. In our simple example above, both M = R 2 and M = S 2 are regarded as the worldvolume of a D2-brane and the resulting system (M, E) is identified as D2-D0 bound state, where k D0-branes are resolved into the D2-brane. This is consistent with that the Myers term produces D2-brane charge density. However, it is not evident if the region M can always be identified as D-branes in a more general scalar fields Φ. In this section, we discuss the technical result in the previous section from conceptual grounds, and propose a possible physical interpretation of the shape of multiple D0-branes 6 . The point of our notion of the shape is that it is completely independent of the coordinate interpretation for Φ and of the large N. We only use the fact that the zero locus of the tachyon profile gives a defect made out of D0-branes. The underlying belief is that all the D-brane systems are described as solitons by the tachyon condensation and that K-theory classifies all of them [18,27]. Therefore, it is natural to understand the meaning of the shape along this belief, instead of the coordinate interpretation. In the following, we elaborate on the structure of solitons and then propose that the shape fits nicely to the classification by the K-homology, that is, the Poincaré dual to K-theory. This says that the shape is classified as a D-brane system. In particular, the Myers term can be incorporated in K-homology.

Structure of the solitons
The original ABS construction (T (x) in (2.1) with Φ = 0) represents a (k-tuple of) codimension 3 soliton sitting at the origin. It winds the field space SU(2) once around S 2 at the asymptotic infinity |x| → ∞ in R 3 . Let us first discuss to what extent the addition of the matrix scalar fields Φ on D0-branes changes the structure of the soliton from the original ABS construction of D0-branes.
One may think that adding scalar fields Φ in (2.1) to the tachyon profile does not change this asymptotic structure since it is just a continuous deformation of the ABS solution. It is true for finite u and for finite k. However, as we will soon see below, the asymptotic behavior itself can be changed by adding proper Φ with k = ∞. Furthermore, even if k is finite, it may affect the structure of the soliton in the limit of u → ∞. Note that the scalar fields Φ change the tachyon profile T (x) at each point x, not just at the origin.
To see this more explicitly, we first recall the Moyal case. After the change of basis as in (3.9), only the zero eigen-function t 0,+ (x) = ux 3 contributes to the remaining defect. This zero mode has the form of a codimension 1 kink along the x 3 -direction. The asymptotic behavior is t 0,+ (x 3 = ±∞) = ±∞, which is evidently different from the ABS construction before adding Φ. The kink charge is shown to be related to the D2-brane charge 1 uvol(R) dx 3 ∂ 3 t 0,+ (x) = 1. This drastic change of the asymptotic region is due to the k → ∞ effect.
Next we move to the fuzzy S 2 case. In this case, the asymptotic region is unchanged, but the structure at the origin is deformed. After the change of basis, as seen for example in (3.47), the zero eigenfunction is t ℓ,+ (x) = u(|x| − ρℓ). This satisfies the boundary condition t ℓ,+ (|x| = 0) = −uρℓ → −∞ and t ℓ,+ (|x| = ∞) = ∞, that relates two different vacua. The limit u → ∞ is important in this situation. Then, it behaves as a kink along the radial direction. This is the same behavior with the spherical D2-brane studied in [22].
In both the cases, since the structure of the soliton is changed, it is no longer a system made of only D0-branes. The appearance of a kink after the deformation is a sign of that the defect is actually a D2-D0 bound state. For more general scalar fields Φ, there may appear defects with all possible codimensions 0, 1, 2, 3. Of course, if we start not with D3-brane but with non-BPS D9-branes, all the nine transverse scalar fields can also be considered as a deformation of codimension nine ABS construction.

More on the shape of D0-branes
Although we have considered only two examples, the Moyal plane and the fuzzy sphere, the analysis itself can be applied for more general cases. In general, the shape M of D0-branes is just the zero locus of the tachyon profile for given matrices Φ. Here an important fact is the tachyon field can always be diagonalized for any Φ. Thus, the zero locus M is always determined uniquely.
When all k × k matrices Φ are diagonal, then the zero locus M consists of k different points in R 3 . This is still true for k = ∞. For example, let Φ 2 =x 1 and Φ 2 =x 2 with commutative [x 1 ,x 2 ] = 0 (i.e., θ = 0 in the Moyal case), then the shape is given by the point set M = R 2 . We know that this M does not mean a D2-brane (Neumann boundary state along R 2 ) but infinitely many D0-branes aligned on R 2 (a family of Dirichlet boundary states). Thus, M itself does not see this difference. On the other hand, in the Moyal case, we know that the shape M = R 2 is a D2-brane worldvolume of a D2-D0 bound state, that is, a smooth submanifold in R 3 . It is seen by noticing that a point (z,z) in M = R 2 and the origin is connected by the displacement operator D(α) of coherent states. That is, the existence of differential structure is guaranteed by the unitary operator U(z,z). This is also consistent with the fact that a coordinate operatorx 1 of the Moyal plane is simultaneously a differential operator iθ∂ 2 = [x 1 , ·]. Thus, it is important to include the information on the connection into the shape (M, E), in order to distinguish these cases.
If several zero modes appear, M consists of several pieces, each of which may have different dimension in general. In a very particular case, if there are n zero modes and are degenerate on the same region M, then E becomes a U(n) bundle over M. This is in contrast to the conventional description of D-branes. The difference is apparent when considering fluctuations Φ ′ = Φ + δΦ further. In the conventional description, δΦ is identified as a matrix scalar field on M, but in our treatment, we seek zero locus again, and obtain another shape (M ′ , E ′ ). In this sense, M is always commutative and no matrix scalar fields appear on M.
Before going to K-homology, we make a brief comment on the boundary state description of D-branes. A system of coincident D-branes is most rigorously defined by a boundary state equipped with a boundary interaction representing fields on D-branes. In this description, D-branes have a definite position defined by a Dirichlet boundary condition, and matrix scalar fields are treated as boundary perturbations. A bound state of n D2-branes and k D0-branes can be described by either (1) D2-brane picture: D2-brane boundary state with a U(n) gauge field A carrying k D0-brane charge, or (2) D0-brane picture: D0-brane boundary state with U(k) scalar fields Φ carrying n D2-brane charge. Schematically, the equivalence of two pictures is given by (4.1) In the Moyal case, the equivalence of two pictures are shown in [30]. Both the pictures represent the same mixed boundary condition from different viewpoints: The picture (1) represents it as a deformation of the Neumann boundary state by a boundary interaction (constant U(1) gauge flux) while (2) does as a deformation of the Dirichlet boundary state. In terms of tachyon condensation, the system can also be realized by the boundary state of non-BPS D3-branes with a tachyon field T . The advantage of this realization is that both pictures are two different choices of basis for the tachyon profile and thus they are manifestly unitary equivalent. The D0-brane picture (2) corresponds to the basis that diagonalizes the ABS construction, and the D2-brane picture (1) corresponds to the basis that diagonalizes the full tachyon profile including Φ 7 . We stress here, however, that the concept of the shape is independent of the choice of the pictures (1) and (2). Although the shape in the Moyal case happens to be well described in the picture (1), it is just by chance. For generic Φ, although the boundary state (4.2) can be still defined and we can read off the shape and/or the boundary condition from this expression, there is no guarantee that the obtained shape is always well described in a specific picture like (1).

K-homology
The shape of D0-branes described so far fits nicely to the classification of D-branes by the K-homology group as announced. In particular, we emphasize that the Myers term can be incorporated in this classification. Let us recall the definition of the K-homology [33]. A K-cycle for a topological space X is a triple (M, E, φ), where M is a compact spin c manifold without boundary, E → M is a complex vector bundle and φ : M → X is a continuous map. The (topological or geometric) K-homology group is defined by K * (X) = {(M, E, φ)}/ ∼, where the equivalence relation is generated by (a) bordism, (b) direct sum and (c) vector bundle modification defined by the relation which will be appeared in (4.3). Here * = 0 (1) corresponds to M with even (odd) dimension, respectively.
Since the K-homology group is a Poincaré dual to the K-theory group, it is natural to conjecture that the K-homology classifies D-branes. This is first described in the concrete form in [34] (see also previous discussions [35,36] and subsequent development [37,38,39]). A K-cycle is conjectured to be a D-brane itself, where M is a worldvolume of a BPS Dp-brane, 8 E is a Chan-Paton bundle on M, and φ is a embedding of M to the spacetime X. The equivalence relations have been also interpreted as physical equivalences: (a) is a continuous deformation of a D-brane, (b) is a gauge symmetry enhancement of coincident D-branes, and (c) is a dielectric effect [34,37,38]. In the following, however, we discuss that we should modify the physical interpretation of the equivalence relation (c).
We start with pointing out that there are several subtleties in the above interpretation. First, since φ is not necessarily an embedding but just a continuous map, there can be such an M whose dimension is larger than that of X in principle. Therefore, precisely speaking, the physical interpretation described above can be applied only when we implicitly regard φ as an embedding [40]. Next, there is no room for a matrix-valued scalar fields in K-cycles, since only a single scalar field (U(1) part) is implicitly assumed when we consider φ to be the embedding. As a result, we cannot incorporate the Myers term in this interpretation in particular. The Myers term Tre i Φ i Φ C in the RR-coupling (Chern-Simons term) for D0-branes is originally obtained by applying T-duality to the RR-coupling for D9-branes which includes the Chern character C ∧ Tre F . In the latter case, a non-trivial gauge flux F = 0 is topologically distinct from F = 0, indicating RR-coupling to higher rank RRpotentials, known as branes within branes [41]. This information is already incorporated as the Chern character for a K-cycle [33] as shown in [34]. Similarly, since non-commuting scalar fields Φ produce a RR-coupling to higher rank RR-potentials through the Myers term, T-duality requires that such configuration is distinguished from commuting one. If the K-homology classifies all possible D-branes, it should be able to take into account the matrix scalar fields. Now, let us turn to the situation in this paper. In the coherent state method, the shape of k D0-branes is a region M in the spatial part X = R 3 of the spacetime, which can naturally be identified with φ(M) in the K-cycle, with the canonical inclusion map φ. There is also a U(1) Chan-Paton bundle with k-magnetic charge. As stated, if zero locus are degenerate, it is extended to a non-abelian Chan-Paton bundle φ * E. Therefore, our shape naturally corresponds to a K-cycle (M, E, φ), even if matrix scalar fields Φ are non-commuting. In particular, the Myers term is implicitly incorporated in this new interpretation.
To see the effect of the Myers term more explicitly, we recall the equivalence (c), the vector bundle modification [33], (4. 3) The r.h.s. is obtained from the l.h.s. through the clutching construction: π :M → M is a sphere bundle over M whose fiber is an even dimensional sphere S 2n .Ĥ →M is a complex vector bundle overM whose fiber is a Bott generator on S 2n . Because of the appearance of the sphere,M has been interpreted as the worldvolume of a spherical Dbrane [34,37,38]. However, as seen by following [33] carefully, we should rather interpret M as a worldvolume of DD-system andĤ as an ABS construction representing M as a codimension 2n soliton in R 2n (whose one point compactification is S 2n above). In other words, the equivalence (4.3) should be just a physical equivalence between a D-brane and the same D-brane constructed by the tachyon condensation. This is consistent with that the image of both maps φ(M) = φ • π(M) represent the same region in X and the fiber S 2n does not seen in X.
In our situation, the K-cycle (M 0 , E 0 , φ 0 ) in the l.h.s. of (4.3) corresponds to k D0branes without scalar fields; Φ = 0. That is, M 0 is a point, φ 0 (M 0 ) = 0 in X = R 3 and E 0 = C k is a Chan-Paton space. It is equivalent to the r.h.s. of (4.3), whereM 0 = S 4 (the one-point compactification of R 4 ) andĤ 0 is an ABS construction of the codimension 4 soliton on the D4D4-system. Note that our non-BPS D3-branes are considered as a part of this system given by a kink solution along x 4 -direction. Thus,Ĥ 0 here is essentially given by (2.1) with Φ = 0.
Let us turn to the case of adding matrix scalar fields Φ. Under the present interpretation, the deformation of (M 0 , E 0 , φ 0 ) by Φ (with a non-zero Myers term) is naturally realized as the deformation of the r.h.s with the tachyon profile (2.1) with Φ. The obtained K-cycle can be non-equivalent to the point-like K-cycle from the above argument.
In that case, it should rather be equivalent to another K-cycle that is given by the shape of D0-branes with Φ. In our examples, we obtain a triple (M 1 , E 1 , φ 1 ) where M 1 = S 2 (in the Moyal case, one point compactification of R 2 ), E 1 is a U(1) Chan-Paton bundle with magnetic flux k, and φ 1 : S 2 → X. Although the non-equivalence between (M 0 , E 0 , φ 0 ) and (M 1 , E 1 , φ 1 ) should be proven mathematically, it should be consistent with the structure of solitons and the RR-coupling described above.
In summary, we propose that the shape of D0-branes with scalar fields corresponds to a K-cycle. We claim that the Myers term (non-commuting scalar fields) is incorporated as a non-equivalent deformation of K-cycles rather than the vector bundle modification.

Conclusion and discussion
We considered D-brane systems with non-commuting scalar fields Φ via tachyon condensation and gave a novel prescription to read off the shape of the noncommutative D-brane system as a commutative region in spacetime, by rearranging the idea of the method proposed in [12,15,17] (the coherent state method) as the tachyon condensation. In this interpretation, the shape of D-brane is defined as a set of zeros of the tachyon field together with a gauge flux on it. As typical examples, we closely investigated the Moyal plane and the fuzzy sphere but the generalization to other systems is straightforward. The point is that diagonalizing a tachyon profile is always possible for any matrix valued scalar fields Φ. We also argued that the shapes fit well to the classification of D-branes by the K-homology group. This shows that the D-branes made through the Myers term are incorporated in this classification.
Since we focused mainly on the topological aspects of the shape M corresponding to the K-homology, there are several issues that we did not touch upon. In this section, we briefly discuss two other aspects of the shape.
Metric on the shape From the point of view of the coherent state method, it is natural to define a metric of the shape only from the matrices Φ [15,16]. In the present context, it is suitable to be defined on the zero mode of the Chan-Paton bundle. There are several notions of metrics defined on a family of Hilbert spaces, such as the quantum Fisher metric, the Fubini-Study metric and the fidelity susceptibility. Here we adopt the definition of [42,43].
Let | ψ(q) be a state depending on external parameters denoted by q i . In the information theoretic geometry, the metric on the parameter space is defined by where the quantities C ij and A i are In our case, a tachyon zero mode has the form | ψ 0 (x) = U(x) | 0 , where x ∈ R 3 is considered to be parameters and U(x) is determined by Φ. Then, the above quantities are written as The latter is nothing but the induced gauge potential (2.12). This metric gives a length between zero mode states | ψ 0 (x) and | ψ 0 (x + dx) at two nearby points 9 essentially through the overlap ψ 0 (x + dx) | ψ 0 (x) . In other words, this length is defined along the fiber direction of the Chan-Paton bundle.
Applying it to the Moyal case, we obtain (see Appendix A.1 for a proof) (5.5) and the metric becomes

4)
This is a flat metric on R 2 but different from the induced metric of the flat Euclidean background by a Weyl factor. For the fuzzy sphere case, the quantities in (5.3) should be evaluated patch-wise with respect to the state R N (Ω) | ℓ, + on U N and R S (Ω) | ℓ, + on U S , respectively. It turns out, however, that the line element is the same in both of U N and U S (see Appendix A.4).
This is the round metric on S 2 with a Weyl factor. This metric should be compared with two alternative metrics: the induced metric of the sphere with radius |x| = ρℓ, ds 2 = ρ 2 ℓ 2 d 2 Ω, and the metric of the fuzzy sphere with radius ρ ℓ(ℓ + 1), ds 2 = ρ 2 ℓ(ℓ + 1)d 2 Ω. Again, the information metric (5.7) is different from both of them by a Weyl factor.
Although the difference between the information metric and the induced metric of the flat target space is only the Weyl factors in these examples, this is not the case in general. This can be most easily checked by adding perturbations to Φ in both examples. This difference can also be intuitively understood as follows. In the above examples, the Weyl factors are given by the inverse of the noncommutative parameters, which are also related to the densities of the D0-branes. The induced metric just depends on the shape in the target space, while the information metric picks up information of the density distribution of D0-branes. At least in the large-N limit, there exist two configurations of D0-branes such that they have a common shape in the target space but have different density distributions. Such two configurations will share the same induced metric but have different information metrics. This implies the inequivalence of the two metrics.
The appearance of the noncommutative parameters in the information metric also suggests that the information metric is the Kähler metric associated with the symplectic structure given by the gauge flux. In the above examples, the information metrics are indeed the Kähler metrics. In [16], it is also shown for a wide class of matrix configurations that the information metric is indeed reduced to the Kähler metric in the large-N limit.
Effective theory on the shape We close this paper with rather speculative discussion. We come back to the example of the fuzzy sphere. The shape (M, E) in this case is given by M = S 2 and E is a complex line bundle over M equipped with the k-monopole connection. This connection comes from displacements of the Bloch coherent states and this suggests that a smooth structure is guaranteed to exist. However, the situation is different from the Moyal case, because the fuzzy sphere is made of finite matrices. This is easily seen by considering algebra of functions on a fuzzy sphere and a commutative sphere. The algebra of functions on a fuzzy sphere is called the fuzzy spherical harmonics, which corresponds to the ordinary spherical harmonics with a restriction in the maximal angular momentum in order to match the degrees of freedom 10 . In order to close the latter algebra by restricted harmonics, it is needed to deform the product to a noncommutative one ( * -product). Although our shape S 2 is a commutative region in spacetime, when considering functions on it, this suggests that it behaves as a noncommutative space. This is not a contradiction because the function algebra is needed only if we consider an effective field theory on the shape. Of course, we do not need to consider a fluctuation as transverse scalar fields on the shape as stated before. When the fuzzy sphere configuration Φ corresponds to the shape (M = S 2 , E), then adding fluctuations Φ ′ = Φ + δΦ gives another shape (M ′ , E ′ ). However, it would also be convenient to find the effective theory description, as in the conventional D2-brane picture. That is, the shape is kept as (M, E) but δΦ is treated as a field on the shape.
To find a new shape caused by a small fluctuation, the standard perturbation theory in quantum mechanics can be applied. The perturbed tachyon profile T [Φ ′ ] can be considered as a Dirac-like operator with an interaction term uσ · δΦ. Then, the new zero mode of T [Φ ′ ] will be given by a linear combination of the ONB for unperturbed T [Φ]. For this purpose, the complete set of ONB found in this paper can be used.
In the language of boundary state, this procedure is understood as follows. The boundary state of the D0-brane picture is e −S b [Φ ′ ] | D0 with scalar fields Φ ′ = Φ + δΦ. By realizing it as the D3-brane boundary state (4.2), it would be rewritten as the form e −S b [δΦ] | M, E . Here the state | M, E corresponds to the shape M = S 2 for a fuzzy sphere. It would not be the conventional Neumann boundary state along S 2 direction, but will be the variant of the mixed boundary state, if the shape behaves as a noncommutative space. The fields on the shape S 2 is extracted by the boundary interaction e −S b [δΦ] . It is interesting to see whether the effective theory for δΦ is given by a noncommutative field theory on M = S 2 with a * -product. This problem is closely related to the situation of the Seiberg-Witten map [44]. It is interesting to study fluctuations around the Moyal plane and the fuzzy sphere and investigate the relation to the Seiberg-Witten map. We hope to report on this issue in the near future.

Acknowledgements
We would like to thank S. Terashima for useful discussions and comments. The work of G. I. was supported, in part, by Program to Disseminate Tenure Tracking System, MEXT, Japan and by KAKENHI (16K17679). The work of S. M. was supported in part by Grant-in-Aid for Scientific Research (C) 15K05060.
and then calculate a gauge potential and a metric. To this end, we will use an identity By using (A.1), we obtain and thus Next we calculate the metric. In (5.3), A i is given by (A.10) and C ij is obtained as By using these, the components in the metric are and thus the line element becomes (A.14)

A.2 Rotation
Let Λ be the rotation matrix that sends x = (x 1 , x 2 , x 3 ) to x = (0, 0, r), and let R be the corresponding unitary operator such that We have two possibilities: (a) Rotation about an axis n = (− sin ϕ, cos ϕ, 0) with an angle −θ.
We will see (b) first. The operation (1) is generated by R 1 = e i(−ϕ)J 3 = e −iϕJ 3 . In fact, which means The operation (2) is generated by R 2 = e i(−θ)J 2 = e −iθJ 2 . In fact, which means The operation (3) is generated by R 3 = e iϕJ 3 . In fact, which means Then the sequence of (1) to (3) is generated by R = R 1 R 2 R 3 and On the other hand, R is rewritten as which says that R generates (a). By using J ± = J 1 ± iJ 2 , R is also written as The case of spin 1 2 The unitary operator R in this case is given by (A.25)