Tsuguhiko Asakawa,

^{3}

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 a 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 non-commutative 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

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

This problem has been discussed from various viewpoints. In Ref. [

The way to determine the shape of the 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 Ref. [

In Ref. [

The relation between NC and commutative geometries has been further developed as a mathematical correspondence between commutative geometry and matrices. In Ref. [

In this paper, we point out that the coherent state method also plays important roles in the context of tachyon condensation in superstring theory [

Technically our analysis in this paper is an application of the technique developed so far [

The organization of this paper is as follows. In

Consider a system of

Our argument below does not rely on the explicit form of the action, but for definiteness, we assume the that tachyon potential has the form

Among them, let us consider

The tachyon profile (

Note that, in this treatment, the condensation itself is obtained without matrix scalar fields ^{1}

In the following, we will consider such configurations of matrix scalar fields

Before treating explicit examples, we describe the schematic structure of the tachyon condensation for the configuration (

The Chan–Paton bundle for

The tachyon field

In order to extract the condensation defect, we need to diagonalize the potential

Here an eigenstate

Then, the tachyon potential is written as

This shows that, at each ^{2}

In this case, all the excited states

Note that this procedure is completely point-wise, and in general the unitary operator

In summary, the tachyon condensation just picks up the zeros of the eigenfunction and as a result a defect remains on a region

A tachyon potential of the form (

On the Chan–Paton bundle, the tachyon potential (

An element of the projective module

Since

In components, we obtain

This gauge potential should possess a non-trivial

Note that the induced gauge potential

The

An NC D2-brane on the Moyal plane can be made out of

In order to realize them, it is necessary to take

By inserting Eq. (

It acts on the Chan–Paton bundle with typical fiber to be the Hilbert space

We will now study the tachyon condensation of this profile (

By using these properties, the

Under the tachyon condensation

It exists only for

Here we have used the fact

From the delta function, we see that the remnant of this condensation is the real 2D surface

It is worth emphasizing that this result is completely different from the perturbative picture of multiple D0-branes, where a D0-brane is sitting at the origin but fluctuates around the origin in the “directions” of the non-commuting scalar fields

The large plane represents the base space

For completeness, we here diagonalize Eq. (

Note that states of the form

On this doublet,

We can express all the eigenstates as

In summary, the set of eigenstates consists of a ground state (singlet)

The tachyon potential (

After some calculations, we find

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

This says that there is a dissolved D0-brane per unit volume

This equivalence between commutative and non-commutative descriptions of the D2–D0 bound states is first shown in Refs. [

An NC D2-brane on a fuzzy sphere can be made out of

By inserting Eq. (

Here, the ONB of the Chan–Paton Hilbert space

Here we study the tachyon condensation by diagonalizing

a) The term

First at the origin

They depend only on the angular coordinates

The expression ^{3}

b) The term

Note that two particular states,

In order to diagonalize

First, we consider points in the open set

In particular, the transformed state

In our case, consider Eq. (

Note that the

For the remaining eigenstates, consider a 2D subspace of the form

This implies

This matrix is diagonalized in a standard way (see

For all

In summary, the eigenvalues of

In

This shows that the surviving state is the lowest weight state

Then, the Bloch coherent state

Although this treatment is sufficient when we are interested only in

By the same relations, we also have

We now define^{4}

Then the tachyon profile in

Since the term inside the brackets is the same as in

The first state becomes a zero mode at any point

The zero modes in

To see this, it is worth rewriting

By acting this on the state

This shows that two states

In particular, we obtain

Having found the eigenvalues of

Here we have used the fact

Matrices

The large sphere represents the base space

It is interesting that the obtained radius ^{5}

The tachyon potential (

After some algebra, we obtain

The

This configuration is nothing but the Wu–Yang

This result is independent of

We are considering the problem of mutually non-commuting matrix scalar fields ^{6}

The point of our notion of the shape is that it is completely independent of the coordinate interpretation for

The original ABS construction (

One may think that adding scalar fields

To see this more explicitly, we first recall the Moyal case. After the change of basis as in Eq. (

Next we move to the fuzzy

In both 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 that the defect is actually a D2–D0 bound state. For more general scalar fields

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

When all

If several zero modes appear,

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

In the Moyal case, the equivalence of the two pictures is shown in Refs. [^{7}

The shape of D0-branes described so far fits nicely with the classification of D-branes by the K-homology group, as noted. In particular, we emphasize that the Myers term can be incorporated in this classification.

Let us recall the definition of the K-homology [

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 was first described in concrete form in Ref. [^{8}

We start by pointing out that there are several subtleties in the above interpretation. First, since

Now, let us turn to the situation in this paper. In the coherent state method, the shape of

To see the effect of the Myers term more explicitly, we recall the equivalence (c), the vector bundle modification [

The r.h.s. is obtained from the l.h.s. through the clutching construction:

In our situation, the K-cycle

Let us turn to the case of adding matrix scalar fields

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.

We considered D-brane systems with non-commuting scalar fields

We also argued that the shapes fit well with 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.

As typical examples, we closely investigated the Moyal plane and the fuzzy sphere but generalization to other systems is straightforward. The point is that we can always diagonalize a tachyon profile for any matrix-valued scalar fields

Since we focused mainly on the topological aspects of 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

Let

In our case, a tachyon zero mode has the form

The latter is nothing but the induced gauge potential (^{9}

Applying this to the Moyal case, we obtain (see

This is a flat metric on

For the fuzzy sphere case, the quantities in Eq. (

This is the round metric on

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

The appearance of the non-commutative 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 Ref. [

We close this paper with a rather speculative discussion. We come back to the example of the fuzzy sphere. The shape ^{10}

Although our shape

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

To find a new shape caused by a small fluctuation, the standard perturbation theory in quantum mechanics can be applied. The perturbed tachyon profile

In the language of the boundary state, this procedure is understood as follows. The boundary state of the D0-brane picture is

We would like to thank S. Terashima for useful discussions and comments. The work of G.I. was supported in part by the Program to Disseminate Tenure Tracking System, MEXT, Japan and by KAKENHI (16K17679). The work of S.M. was supported in part by a Grant-in-Aid for Scientific Research (C) 15K05060.

Open Access funding: SCOAP

For the displacement operator (

First we set

The higher-order terms

Similarly, by setting

The others in Eq. (

By using Eq. (

Next we calculate the metric. In Eq. (

By using these, the components in the metric are

Let

We have two possibilities:

(a) Rotation about an axis

(b) The sequence of (1) rotation about an axis

We will see (b) first. Operation (1) is generated by

Operation (2) is generated by

Operation (3) is generated by

Then the sequence of (1) to (3) is generated by

On the other hand,

The unitary operator

In general, a

The eigenstates

In our case,

Then, the eigenvalues

Note that it is also written as

Next, we will check whether

Since

Then two eigenvalues of

For later purposes, we define a unitary operator

This depends on

For Eq. (

Equations (

From the first line of Eq. (

Hence, the sum over even-order terms in Eq. (

Combining them, we obtain

The gauge potential in

For the metric, we need to evaluate

These are shown as follows:

By using these, the components in the metric

For the gauge potential

This expression is valid only for

For the metric, using Eq. (

Thus, in order to obtain

Combining these with

^{1} It is a deformation of Eq. (

^{2} This is just a working assumption. More general situations are discussed in

^{3} It is obvious in Eq. (

^{4} There is an constant phase ambiguity in defining

^{5} The original charge density is supported on the family of spherical shells at finite

^{6} The discussion in this section is mainly based on our answers to S.Terashima’s questions. We thank him for this private communication.

^{7} They are analogous to the interaction and the Heisenberg picture, respectively, in quantum mechanics.

^{8} More precisely, each connected component of

^{9} Of course, they should belong to the same open set in

^{10} In our case, because a monopole exists, it is better to think about (fuzzy) monopole harmonics.