UTMS 2001–9

April 16, 2001

Exact Tachyon condensation
on noncommutative torus
byHiroshige Kajiura, Yutaka Matsuo
and Tadashi Takayanagi

T

UNIVERSITY OF TOKYOGRADUATE SCHOOL OF MATHEMATICAL SCIENCES
KOMABA, TOKYO, JAPANUTMS 2001-9
UT-930

hep-th/0104143
April, 2001

Exact Tachyon Condensation
on Noncommutative Torus

Hiroshige Kajiuraa , Yutaka Matsuob and Tadashi Takayanagiba) Graduate School of Mathematical Sciences, University of Tokyo
Komaba 3-8-1, Meguro-ku, Tokyo 153-8914, Japanb) Department of Physics, Faculty of Science, University of Tokyo
Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0034, Japan

Abstract
We construct the exact noncommutative solutions on tori. This gives an exact description of tachyon condensation on bosonic D-branes, non-BPS D-branes
and brane-antibrane systems. We obtain various bound states of D-branes after
the tachyon condensation. Our results show that these solutions can be generated
by applying the gauge Morita equivalence between the constant curvature projective modules. We argue that there is a general framework of the noncommutative
geometry based on the notion of Morita equivalence which underlies this speciﬁc
example.

1

Introduction

D-branes have been playing the most prominent role in recent developments of string
theory. This is not only because they are the non-perturbative objects but also because
they possess many interesting characteristics which require mathematically distinguisheddescriptions. It is very interesting to try to prove that these various viewpoints aremutually consistent. Such an eﬀort often gives profound relationships between physics
and mathematics.
One of such examples is the quantum ﬁeld theory on the noncommutative space arisingfrom open string theory in B ﬁeld background [1, 2]. In the commutative case, the spaceof functions gives the basis of the quantum ﬁeld theory, for example, through the mode
expansion. In the noncommutative space they are replaced by the noncommutative C ∗ algebra A which is at the heart of the principle of the noncommutative geometry (see forexample [3]). In this paper we are mainly interested in the description of D-branes withthe direct use of such principle.
In the commutative approach, the soliton charges of the D-branes are derived fromthe topological K-group [4, 5]. It is based on the observation that there are always themassless gauge particles which deﬁne the vector bundle on the D-branes. We need to takethe formal diﬀerence of vector bundles to describe the brane-antibrane pair annihilation
process [6, 7].
In noncommutative geometry, the topological information of the ‘manifold’ is given by
the operator algebra version of the K-group K(A). For example in Connes’ index theorem[3], the topological index is give by the pairing of the element of K-group with that of thecyclic cohomology group of A. We note that K0 (A) is deﬁned by an equivalence class ofprojection operators in Mat∞ (A), the inﬁnite dimensional matrix algebra with elements
in A.
From physical viewpoints, it is natural to conjecture that topological properties of
D-branes are described by the operator algebra K-group. It should be described throughthe solitonic conﬁgurations which are proportional to projection operators. Remarkably
such conﬁgurations indeed appear as classical solutions (GMS soliton) in the scalar ﬁeld
theory on the noncommutative plane (Moyal plane) in the large noncommutativity limit[8]. This idea was immediately applied to the string theory in the tachyon condensation
process [7, 6], e.g. bosonic D-branes, non-BPS D-branes and brane-antibrane systems[9, 10]. If we use nontrivial GMS soliton, the lower dimensional D-branes are generated.
As pointed out in [11, 12] it can be seen as the noncommutative generalization [13] of thecorrespondence between D-brane charges and K-theory [4, 5].
In this way the unstable systems of D-branes seem to give a good example for theapplication of the geometrical methods of the noncommutative geometry to the string
theory. There are two directions to proceed. One is to challenge noncommutative spaces

1with more complicated and richer structures. The simplest nontrivial example is the noncommutative torus. This example is interesting from physical side since we expect to haveanalogue of T-duality symmetry in the form of Morita equivalence. It was investigated in[14, 15] by using Powers-Rieﬀel projection. Unlike the Moyal plane, we observed a sortof instability [14]. It comes from the fact that we may construct the noncommutative
soliton with arbitrary small size. Mathematically it is related to the fact that K-groupof noncommutative tori is not quantized in Z but takes its value in R. It remained as a
puzzle whether it is natural to interpret the continuous value as the D-brane charge. Latera diﬀerent construction of the soliton conﬁguration on the torus and on the orbifold was
discussed in [16]. Among other things, a remarkable suggestion is to use Morita equivalence bimodule in the construction of the noncommutative soliton. Similar constructionof the projection operator on tori was also discussed in [17] and [18]. For fuzzy sphere,
GMS-like solitons were discussed in [19].
The other direction is to take the gauge ﬁeld into account and to construct the exact
solution without taking the large B limit. It was pioneered in [20] when the base space is
Moyal plane. Certain constraint on the coupling of ﬁeld strength and tachyon ﬁeld should
be satisﬁed in order to have such property.
In the present paper we continue to study the noncommutative soliton on the twotorus. We have mentioned two motivations, (1) how to resolve instability of the spectrumand (2) the construction of exact solution. The use of Morita equivalence initiated in [16]gives another motivation. For the nontrivial examples such as tori, we can not directly
construct the analogue of the shift operator. In this sense, we can not escape from using
more abstract Morita equivalence bimodule directly to construct the noncommutative
soliton. (A nice review of Morita equivalence for noncommutative torus is given by [21]).
Once we know how to use it, one may apply the method to other examples as well, namely
in the generic examples of the open string systems interpolating D-branes [2, 13]. We argue
that the Morita equivalence gives a natural generalization of the notion of the braneantibrane systems and leads to the description which is similar to the superconnection [5,22, 23, 24]. Inspired by this fact we propose an equation which deﬁnes the noncommutative
solitons on brane-antibrane systems in the generalized sense.
As we will see, we can obtain the exact solutions of the tachyon condensation on
the two-torus by employing the constant curvature connections. In the noncommutative
torus, the constant curvature connection parameterizes the equivalence class of the wholeprojective modules (analogue of vector bundle). As a result we obtain various D2-D0
bound states after the tachyon condensation on a non-BPS D2-brane. We also ﬁnd thatthe gauge Morita equivalence [25, 21] (bimodule between constant curvature connection)
plays the crucial role of generating solutions. In fact one can construct exact solutions for
any C ∗ -algebras if there exist the gauge Morita equivalence bimodules. Furthermore thisexact analysis explicitly shows that for ﬁnite B/g the above mentioned instability does2

not occur.
The paper is organized as follows. In section 2, we discuss tachyon condensation on
generic noncommutative spaces. After we review the Morita equivalence, we construct
the projection operator of the brane-antibrane systems in the generalized sense. We
see that the structure of superconnection [26] naturally appears as the linking algebrain the framework of C ∗ -algebra. In section 3 we discuss the tachyon condensation ona noncommutative two-torus. We construct the exact solutions for bosonic D-branes,
non-BPS branes and brane-antibrane systems in terms of ﬂat curvature connection. We
also discuss the solution generating rule for this examples by using the gauge Morita
equivalence. This section also includes a review of some mathematical results on the
projective modules. In section 4 we summarize the conclusions. In the appendix A wegive a review of the explicit example of projections in noncommutative tori and we also
show the calculations of their topological charges.

2

Morita equivalence and noncommutative soliton:
A General Strategy

We start from discussing relatively formal viewpoint which will be useful in the later
sections. While our main result is restricted to the noncommutative solitons on noncommutative tori in section 3, we think that our method can be basically applied to the otheropen string systems such as Dp − Dp as well.
Let us ﬁrst recall the deﬁnition of the noncommutative solitons [8]. They are thesolutions to the equation of motion,∂V ( φ)
= 0.
∂φ

(1)

Here is the (noncommutative) product of the given C ∗ -algebra A(0) which deﬁnes thenoncommutative geometry on the single D-brane. It is solved in the following form,
φ(x) =λi Pi , λi ∈ R, Pi ∈ A(0) ,(2)iwhere λi s are the solutions to the equation ∂V (λ)/∂λ = 0 and Pi ’s are the mutuallyorthogonal projections Pi · Pj = δij Pi , Pi∗ = Pi . In this sense, the construction of noncommutative solitons is reduced to ﬁnd the projection operators 1 .
In the mathematical context, the classiﬁcation of the projection operator is directly
related to the deﬁnition of the K0 -group of the operator algebra K-theory. This is related
1

If the number of the D-branes is greater than one (say N ), we need to consider the projector in thematrix algebra M atN (A).

3

to the fact that the noncommutative analogue of the vector bundle is described by the
projective module of given C ∗ -algebra A. Let us brieﬂy illustrate the correspondence byusing the relationship with the commutative theory.
Let M be a smooth manifold (base space) and E be a vector bundle over M. In thecontext of the topological K-theory, it is known that the isomorphism class of E is anelement of K 0 -group2 . By Swan’s theorem, for any E there exists vector bundle F suchthat E ⊕ F is a trivial bundle over M. Let C ∞ (M) be the smooth function on M. The
trivial bundle can be written as (C ∞ (M))N with some N ∈ N. Therefore, any vector
bundle over M can be obtained by acting P on (C ∞ (M))N , where P is a projection in
MatN (C ∞ (M))3 . The module which is constructed by applying the projection operator
to the free module is called projective module.
This characterization of the vector bundle can be generalized to noncommutative theory. A noncommutative algebra A(0) replaces C ∞ (M) for ‘noncommutative space’. The
free module (A(0))N corresponds to the rank N trivial bundle on commutative space. Projective module E is deﬁned as A(0) -module such that there exists the other A(0) -module Fwith E ⊕ F = (A(0) )N . Thus the noncommutative analogue of a vector bundle is the projective module E = P (A(0))N which is deﬁned by a projection operator P ∈ MatN (A(0) ).
In this sense, D-branes on noncommutative spaces are described by the projective modules, and the operator algebra K0 -group classiﬁes the D-branes on the noncommutative
space [13, 11, 12].
Let us come back to the issue of the construction of the projection operator. We would
like to use the Morita equivalence bimodule as the abstract building block to constructnoncommutative soliton.
Morita equivalence is one of the central idea of the classiﬁcation of C ∗ -algebra. Fromthe mathematical viewpoint, it is essential to determine when two C ∗ -algebras A, B deﬁne
the same type of noncommutative geometry. In the noncommutative geometry, the ideaof points is replaced by the set of ideals of C ∗ -algebra. It is then known that two C ∗ algebras A, B have the same set of ideals if there are A-B Morita equivalence bimodule
A XB (for example see [27]). It is deﬁned as a bimodule on which A (resp. B) acts fromthe left (resp. right) with two types of inner products  , A ,  , B of A XB with value in
A and B, respectively with the following conditions
ax, yA = ax, yA ,x, y∗A = y, xA

,

a∈A,

x, ybB = x, yB b ,x, y∗B = y, xB

,

b∈B,

xb, yA = x, yb∗ A ,
2
0

K .

3

ax, yB = x, a∗ yB .

(3)
(4)

In general, the formal diﬀerence of the isomorphism class of vector bundles over M is the element of
We deﬁne the algebra M atN (A) as the N times N matrix algebra with elements in A.

4

The most important property which should be satisﬁed by them is the associativity
x, yA z = xy, zB ,x, y, z ∈ A XB .

(5)

Two C ∗ -algebras A and B which have such a Morita equivalence bimodule A XB is called
Morita equivalent.
In the string theory, there is a natural interpretation of such equivalence relation.
It is well-known that the noncommutativity arises in the string theory on the D-branes
connected by the open string in the presence of B ﬁeld. On the two ends of open string, wehave two D-branes and generally two diﬀerent types of noncommutative geometry deﬁned
on them. Suppose they are deﬁned by the C ∗ -algebras A, B 4 . Under such circumstances,
it is natural to conjecture (for example, see [2, 13]),• The bimodule naturally interpreted as the open string ﬁeld Ψ where A (resp. B)
acts from the left (resp. right).
• Two inner products are identiﬁed the product of open string ﬁelds. We have two
type of inner product because we have two choices (which side of the open string)for the contraction.
• The associativity of the product corresponds to that of the product of the open
strings.
Although the actual justiﬁcation of these statements is far from being obvious at thisstage, it gives a nice intuition to the otherwise abstract nature of Morita equivalence.
In the following, we use the Morita equivalence bimodule to deﬁne the projection
operator (= noncommutative soliton). In an abstract language, it can be described asfollows [28, 29, 27, 16]. In the very deﬁnition of the Morita equivalence, we actuallyneed to impose that the inner product  , A,B maps A XB into a dense set of the A, B
respectively. Namely for any operator a ∈ A, there should be a ﬁnite set xi , yi ∈ A XB suchthat a = i xi , yi A . Suppose A has identity as its element and take 1 = N
i=1 xi , yi A .
From xi , yi one may deﬁne the projection operator in MatN (B) as P ≡ yi, xj B sinceyi , xj B yj , xk B =yi, xj yj , xk B B = yi,
xj , yj A xk B = yi, xk B .
(6)
j

j

j

Unlike the original GMS soliton where the projection operator deﬁnes the lower dimensional D-branes, it seems rather hard to identify the nature of the projected space. However, from the mathematical side, A can be embedded into MatN (B), asA ∼ P · MatN (B) · P .
4(7)More precisely, let A(0) be the algebra corresponding to the noncommutative base space, and let Eα(resp. Eβ ) be the projective A(0) -module (D-brane) related to A (resp. B), then A = EndA(0) Eα andB = EndA(0) Eβ . Let a ∈ A(0) , A ∈ EndA(0) E and ξ ∈ E. EndA(0) E means (a · ξ)A = a(ξ · A). This is
the natural noncommutative generalization of the deﬁnition of endomorphisms for vector bundles.

5

Suppose A (resp. B) describes the noncommutative geometry on D-brane α (resp. β).
Then a physical interpretation of above identity is that α can appear as the noncommutative soliton on the N copies of the D-branes β through the tachyon condensation process
in a generalized sense.
In order that this type of interpretation is possible, we need the analogue of D D̄ pair
for this generalized setting. In the study of D D̄ pair, the tachyon condensation process is
described by the combination of gauge ﬁelds and tachyon ﬁelds (so called ‘superconnection’[26]) as argued in [5, 22, 23, 24],

T
d + A1(8)T̄
d + A2where each entry represents the various sectors of the open string.
There exists an analogue of superconnection in C ∗ -algebra which is called the linking
algebra C (see for example, [27]). It is deﬁned in such way as containing the algebras A, B
as its complementary components. Namely there exists a projection P ∈ C such thatA = P · C · P and B = (1 − P ) · C · (1 − P ). It is known that the linking algebra C exists
if and only if two C ∗ -algebras A, B are Morita equivalent. Namely if the bimodule A XB
exists, one may deﬁne the linking algebra by two by two matrices,


a x
(9)
a ∈ A, b ∈ B, x, y ∈ A XB .
ȳ b
One may easily check that the matrix multiplication of such 2 × 2 matrices is well-deﬁned
1 0
and the obvious projection operator P =gives the projection into A. By com0 0
paring (8) and (9) one sees that the Morita equivalence bimodule plays the analogous rôleas that of the tachyon ﬁeld and that the two gauge ﬁelds correspond to the endomorphism
algebras A = EndA(0) Eα and B = EndA(0) Eβ .
This analogy was implicitly used in the explicit calculation of the tachyon condensationof D D̄ system in the Moyal plane [30, 20]. We summarize our conventions for Moyal planein the appendix A. We recall that the noncommutative soliton in this case was constructed
out of the partial isometry described by the tachyon ﬁelds T and T̄ which satisfy,
T T̄ T = T,

T̄ T T̄ = T̄ .

(10)

With these relations, T T̄ and T̄ T becomes projection operators and deﬁnes the noncommutative soliton. Explicit form of T can be constructed by using the shift operator
S = n |nn + 1|, which satisﬁes S S̄ = 1 and S̄S = 1 − |00| and so on [20]. The
tachyon ﬁeld is given by T = S m (m ∈ N) with this solution generating rule. In this case,T gives the isometry between the subsets of the identical algebra A and A acts on T fromthe both side.
6

In our more general situation, the rôle of T resembles that of the Moyal plane case but
the algebra acting on T from left is in general diﬀerent from that acting from right. Thus
T should be an element of an equivalence bimodule. If we replace the partial isometryby the Morita equivalence bimodule, the statement which corresponds to (10) is that one
may choose an element T of the bimodule which satisﬁes
1. P = T, T A and Q = T, T B are the projectors of A, B.
2. It satisﬁes the analog of the partial isometry relation,
T, T AT = T T, T B = T .

(11)

While it is a nontrivial question whether we may ﬁnd such T , the requirement of these two
conditions are actually equivalent [28]. Suppose that the partial isometry-like equation(11) holds. P = T, T A and Q = T, T B are clearly self-adjoint, and direct calculationshows that P 2 = P and Q2 = Q by using the properties of the two inner products(3)(4)(5). Conversely, if we start from the condition P 2 = P , the partial isometry-likeequations follows from vanishing of the norm,
T, T AT − T, T, T A T − T A = 0.

(12)

One can also see from this proof that P 2 = P implies Q2 = Q.
We would like to propose that the equation (11) deﬁnes the noncommutative soliton
on the brane-antibrane systems in the generalized sense. We will see the explicit examples
on noncommutative tori in section 3.4.

3

Noncommutative Torus and D-branes

Because our discussions so far are given in the abstract language, it is desirable toinvestigate explicit examples in order to illuminate the idea. The simplest example is Dbranes on noncommutative plane (Moyal plane) but it is too simple to use the machinery
we would like to examine since it reduces to the (inﬁnite dimensional) matrix algebra.
Thus in this section we consider D-branes on noncommutative tori, where Morita equivalence is interpreted as T-duality [1, 25, 31, 32] and has a rich structure.
The algebra of noncommutative two-torus is generated by unitary elements U1 and U2
with the relation,
U1 U2 = U2 U1 e2πiθ ,(13)where the real number θ ∈ [0, 1] is the parameter of the algebra. We write this algebraby Aθ . The generators can be written in terms of noncommutative coordinates (x1 , x2 )with [x1 , x2 ] = −2πiα θ as follows
1√

U1 = eix / α ,

2

√



U2 = eix / α .
7(14)

A generic element a ∈ Aθ can be expanded by Ui as5a=amn U1m U2n .

(15)

m,n∈Z

While we can not realize this algebra as the matrix algebra for the irrational θ, one may
formally deﬁne the trace for Aθ as follows
Tr a = a00 ,(16)

by using the above expansion. This is equal to the integration over (x1 , x2 ). It is obviousthat it satisﬁes the fundamental relation Tr(ab) = Tr(ba).
As we saw in section 2, the gauge bundle on the D-brane is described by the projectiveAθ -module. It is known that the isomorphic class of the projective modules on a noncommutative torus is classiﬁed by their Chern characters [29] which speciﬁes the element of
K0 (Aθ ). The relation of these mathematical facts to the RR-couplings of brane-antibrane
systems will be discussed later in the subsection 3.4.
In this section, after explaining some mathematical backgrounds on the projective
module on the noncommutative tori, we discuss mass spectrum of BPS D-branes andtheir T-duality transformation. Finally we investigate tachyon condensation on non-BPS
D-branes and brane-antibrane systems.

3.1

Projective modules, Constant Curvature Connection and
Morita equivalence on Noncommutative Torus

In this subsection, we explain some mathematical results on Aθ , especially the projections, the projective modules, the constant curvature connections and Morita equivalence6
which will be essential for later arguments. The readers who is familiar with the subject
may skip this subsection.
We start from deﬁning the projection operators on Aθ . In this algebra the unitaryequivalence class7 of projections is characterized by the value of the trace Tr [28, 33] asfollows
Tr(Pn−mθ ) = n − mθ,(0 ≤ n − mθ ≤ 1),(17)

where for each pair of integers n,m which satisfy 0 ≤ n−mθ ≤ 1 there exists a equivalenceclass of projections and we wrote an element in this class as Pn−mθ . In other words K0 group K0 (Aθ ) is given by
K0 (Aθ ) = Z + θZ ∈ R .
5(18)

To be exact we should assume amn tend to zero faster than any powers of |n| + |m| as |n| + |m| → ∞.
See [21] for an more pedagogical review.
7
A projection P is said to be unitary equivalent to another projection Q if an unitary element u ∈ Aexists such that P = u∗ Qu.
6

8

The explicit construction of projections was given in [28] and is called Powers-Rieﬀel
projection. Some details of this construction are reviewed in the appendix A.
Let us turn to projective modules on the noncommutative torus Aθ . Among various
projective modules, we are interested in those which has the constant curvature connection. The explicit construction can be found in [34, 29, 1, 21] and we review this below.
Starting from a set of the rapidly decreasing functions on R, φj (x), j ∈ Zm . We deﬁne
the right action of Ui (i = 1, 2) as follows,n− θ)m(φU2 )j (x) = φj (x)e2πi(x+jn/m) .
(φU1 )j (x) = φj−1(x +(19)

One may easily conﬁrm that the operators U1 , U2 thus deﬁned satisﬁes U1 U2 = e2πiθ U2 U1 .
It shows that the Schwartz space is right Aθ module. It contains two integer parametersn ∈ Z, m ∈ Z ≥ 0 and we denote it as En,m . It has the following properties,
1. The natural ‘covariant derivative’ for this module is given by∇1 = −

2πimx,n − mθ

∇2 =

∂
,
∂x

(20)

which satisﬁes ∇i Uj = 2πiδij Uj . It has the constant curvature,
F12 = −F21 =2πim.
n − mθ

(21)b−aθ). This acts on En,m from
2. Its endomorphism EndAθ En,m is given by Aθ̃ (θ̃ = − n−mθthe left. The integers a, b are determined from n, m by the condition an − bm = 1.
The generators of Aθ̃ act on them as(Z1 φ)j (x) = φj−a(x + 1/m)
xj

(Z2 φ)j (x) = φj (x)e2πi( n−mθ + m ) .

(22)

One may easily conﬁrm that the action of Zi are compatible with the action of Ui
since [Zi , Uj ] = 0. Note that if the module is free (n, m) = (1, 0), then θ̃ = θ.
Therefore the Schwartz space deﬁnes the Morita equivalence bimodule between Aθ and Aθ̃ .
More generally, if the integers (n, m) is not coprime, then one can construct the reducible
projective module En,m = En/d,m/d ⊕ En/d,m/d ⊕ · · · ⊕ En/d,m/d , where we have deﬁned
d = g.c.d(n, m). From this one can conclude that Aθ̃ is Morita equivalent to Matd (Aθ )andif there exist coprime integers (n0 , m0 ) ≡ (n/d, m/d) and (a, b) such that θ̃ = n0b−aθ−m0 θ
an0 − bm0 = −1 [28].
It will be useful to deﬁne the explicit form of A = Aθ̃ and B = Aθ inner productfor the Schwartz space. This problem is solved in much more general sense by Rieﬀel
9[29]. However we write down some of the explicit forms for our speciﬁc example 8 . Forφ, ψ ∈ S(R × Zm ), we deﬁne,1(Z −n Z −m φ, ψ)A Z1m Z2nn − mθ m,n 2 1

=(φ, ψU2−n U1−m )B U1m U2n

φ, ψA =φ, ψBm,n ∞(φ, ψ)B = (ψ, φ)A =

dx
−∞



φi (x)ψi (x).

(23)

(24)

i∈Zm

The basic properties of inner product (3) and (4) follows from the identities (aφ, ψ)A,B =(φ, a∗ ψ)A,B , (φ b, ψ)A,B = (φ, ψ b∗ )A,B and so on. The derivation of the associativity (5) ismuch more nontrivial. We need to use the Poisson resummation formula,


 ∞
∞
∞

1  ˜ 2πm˜f (αn) =eimx f (x)dx.
(25)ff(m) =
α
α
−∞
n=−∞
m=−∞
After some calculations, one may conﬁrm,(φ, ψAχ)i (x) = (φψ, χB )i (x) =(φU1 )ri (x)(Z1s ψU1r )i (x)(Z1s χ)i (x).

(26)

r,s∈Z

Among all connections on a given projective module, the constant curvature connectionis the most useful in the physical application. One reason for this is that such a special
connection appears as the solution of the BPS equation [25, 21]. Another reason will begiven later in the arguments of the exact solution for the tachyon condensation.
Finally we would like to add a comment on the topological invariants. From theconstant curvature connection, it is rather easy to evaluate the Chern character [29, 25]
and it is known that each component becomes integer after the modiﬁcation similar toMyers term [35]9 . It is of some interest to conﬁrm this fact by using other form of
the connection/curvature. In general a connection on projective module (“Levi-Civita
connection”) is constructed in the form (see for example [34, 21, 11]) ∇i = Pn−mθ ·δi ·Pn−mθ .
The trace of the curvature (ﬁrst Chern class) is the cyclic 2-cocycle τ2 for Pn−mθ . In
appendix A we evaluate it by using the Powers-Rieﬀel projection as Pn−mθ and deriveτ2 = m. This is of course consistent with the computation from the constant curvatureconnection (21).
8General formula for the generic Heisenberg module is not diﬃcult to write down. One outline issketched in Appendix A of [16].
9This modiﬁcation can also be regarded as ‘quantum eﬀect’ in terms of the quantum twisted bundle[36]. This procedure shows the relation between a bundle on the commutative torus and that on thenoncommutative torus geometrically, and agrees with so-called Seiberg-Witten map[2]. We will use thismap in eq.(37).

10

3.2

Noncommutative Description of BPS D-branes and T-duality

Here we discuss the spectrum and the T-duality transformation rule of the BPS Dbranes on a two dimensional torus with a B-ﬁeld ﬂux. As is well-known, there are two
viewpoints for this system. One is the conventional description (commutative description) using the closed string variables. The other is by the open string variables with
the noncommutative geometry [1, 2]. The results given here will be useful in the later
discussions. Some general arguments can be found in [37, 21].
We investigate the dynamics of D-branes on the noncommutative two-torus Aθ . Since
we restrict our interest to the two dimensional case, we discuss D2-D0 bound states on
the torus below. If the D2 charge and D0 charge are given by (n, −m), the mass of D2-D0
bound state is determined as follows|n| Sdet (g + 2πα(B + F )),(27)
MBP(n,m) = √ 
α gs
where we have deﬁned the gauge ﬁeld strength
J m
F =
,
2πα n


J≡

0 −1
1 0


.

(28)

In this formula we have taken the eﬀect of −m D0-branes into account as the shift ofgauge ﬁeld strength. Note that the value of n or −m can be negative integer because of
T-duality. For a review of T-duality on general tori see [38]. In the following we examine
the transformations from a D2-brane to various D2-D0 bound states.
We deﬁne E = g + 2πα B ∈ Mat2 (R). A single D2-brane has the mass,S
MBP(1,0) =1 √
det(E) .
gs α 

(29)

The T-duality group on T2 is given by SO(2, 2|Z) and this acts on g, B and gs as follows[38]Ẽ = T (E) = (AE + B) (CE + D)−1
g˜s = T (gs ) = gs det (CE + D)


A B
T =
∈ SO(2, 2|Z).
C D

− 21

(30)
(31)
(32)

S
The mass MBP(1,0) transforms into the following formS
T (MBP(1,0) )

1√
det T (E)T (gs ) α1 
= √det(AE + B) .
gs α 
=11(33)

When the dimension of the torus is two, the T-duality group can be decomposed asSO(2, 2|Z)  SL(2, Z) × SL(2, Z). One of two SL(2, Z) groups
is the modular transfor
2
mation of the target space T . Because it preserves gs and det(E), we will not consider
this part. The other SL(2, Z) is related to Morita equivalence for noncommutative T2[1, 25, 31, 32] and we concentrate on this part. This SL(2, Z) transformation can beembedded into SO(2, 2|Z) as follows
SL(2, Z) 2→ SO(2, 2|Z)




n m
n1 mJ
→
.
−b −a
bJ −a1(34)

Applying this transformation eq.(33) can be rewritten asS
T (MBP(1,0) ) =1 √
det(n(g + 2παB) + mJ) .
gs α 

(35)

This is the same value as (27) and conﬁrms that the D2-brane mass in the backgroundẼ = T (E) is equal to the mass of (n, −m) D2-D0 bound state. Note also that the massfor (n, −m) is equal to that for (−n, m). This implies that these two conﬁgurations should
be an identical state and we can restrict the integers (n, −m) to n − mθ ≥ 0.
We translate these results into the noncommutative description. For simplicity, we ﬁx
the choice of the parameter Φ [25, 32, 2] as B = −Φ. The relation between the variablesin the open and closed string theories is given [2] as follows
B = −Φ = −1J,2πα θ

1
G = −(2πα )2 B B,g1

Gs = gs det(2πα Bg −1 ) 2 .

(36)

This map is deﬁned so that the mass of the single D2-brane coincides in both (open/
closed) descriptions. The transformed backgrounds G and Gs are called open string
metric and open string coupling respectively.
When the ﬁeld strength F is constant, the ﬁeld strength F̂ in the open string description10 is given by [2]
F̂ =

F
.
1 + 2πα θJF(37)

If we apply this to the D0-D2 bound states, the ﬂux (28) is transformed into
F̂ =

1
m
J.

2πα n − mθ(38)This is the same as the constant curvature connection on the projective module En,mreviewed in the previous subsection.
10Note that the curvature in mathematical conventions in section 3.1 is related to the ﬁeld strength F̂
here such that Fij = −4π 2 iα F̂ij .

12

Furthermore, the mass of D0-D2 bound states can also be written in terms of the open
string variables as follows
|n| 
S
BP S
√
)
=
M
=
det (g + 2πα(B + F ))T (MBP
1,0(n,m)
α  gs



1 m
|n|
det g − ( − )J= √θnα  gs1 ndet(2παBg −1 )− 2√
det −(2πα )2 Bg −1 B(n − θm) + J
=

θ
α gs

n − mθ
det(G + 2πα(F + Φ)).
(39)
= √α  Gs
Note that the rank |n| of the gauge ﬁeld in the commutative description is replaced with
the non-integer ‘rank’ n − mθ ≥ 0. In noncommutative geometry such an appearance of
non-integers is not surprising but very natural. Indeed this is equal to the dimension of
the projective module dim(En,m ) = n − mθ. As we will show later, it appears naturally
in the processes of tachyon condensation. We also comment that the above formula iscorrect for any choice of Φ.
Finally we present an interpretation of the above results from the viewpoints of Tduality T on the noncommutative side. It is derived by rewriting the action T in (34) interms of the open string variables in (36) [25, 32, 21],θ̃ =b − aθ
, G̃µν = (n − mθ)2 Gµν ,n − mθ

G̃s = (n − mθ)Gs , 2πα Φ̃ = (n − mθ)2 (2παΦ +m
J).
n − mθ

(40)

The transformation for θ is exactly the same as the Morita equivalence (22).
By the deﬁnition of the transformation (40), the map from the closed string variable
to the open string variable (36) and the action of the T-duality group on both (openS/closed ) sides (32) and (40) are compatible. Therefore the mass of a bound state MBP(n,m)
S
in the last line of (39) can also be obtained by acting the transformation (40) on MBP(1,0)
in the open string variables
SMBP(n,m) = √1α G̃sn − mθ
det(G̃ + 2πα Φ̃) = √α  Gsdet(G + 2πα (F̂ + Φ)) .

(41)

This shows that a (n, −m) D-brane on the noncommutative torus Aθ is a single D2-brane
on Aθ̃ . This result is consistent with the previous arguments in the commutative (closed
string) side. Notice that the curvature F̂ on the noncommutative torus Aθ vanishes on
the corresponding single D2-brane on Aθ̃ due to the shift of Φ in eq.(40).
The above arguments of T-duality can also be applied to non-BPS D-branes andbrane-antibrane systems in the same way. We will see later that this T-duality on the13noncommutative side is more directly related to Morita equivalence in the arguments oftachyon condensation.

3.3

Tachyon condensation on non-BPS D-branes

Let us discuss the tachyon condensation on non-BPS D-branes (see for example [6])on the noncommutative torus Aθ . The same arguments can be applied to the bosonicstring. Because any D2-D0 bound state of non-BPS D-branes can be transformed into a
D2-brane, we can begin with a non-BPS D2-brane. The relation between the variables inopen and closed string theories is the same as (36) and we continue to choose the value ofΦ as Φ = −B to obtain the simplest expression. The solutions, however, do hold without
any modiﬁcation for general values of Φ with somewhat lengthy calculations.
On any non-BPS D-brane there exists11 a (real scalar) tachyon ﬁeld T and a gauge
ﬁeld Aµ . As argued in [39] the eﬀective action of a non-BPS D2-brane can be written as√
2
dtTr V (T ) det(G + 2πα (F̂ + Φ)) + O([∇, T ], [∇, F̂ ]),(42)
S=√
α  Gswhere [∇, T ], [∇, F̂ ] denote the covariant derivative of the tachyon ﬁeld T and the gauge
ﬁeld strength F ; the symbol O([∇, T ], [∇, F̂ ]) means those terms which include one ormore derivatives of T and F̂ . As we will see below our exact arguments of tachyon
condensation do not depend on the detailed form of the derivative terms. The factor V (T )in front of the Born-Infeld term represents the tachyon potential. We normalized the valueof the tachyon potential such that its value before and after the tachyon condensation into
the vacuum are given by V (1) = 1 and V (0) = 0 following from Sen’ conjecture [7, 6].
We use here the open string variable and therefore all the ﬁelds on the brane are
regarded as the operators on the noncommutative torus Aθ . On the non-BPS D-brane,
the tachyon and the gauge ﬁeld belongs to the adjoint representation of the gauge group.
In the language of the noncommutative geometry they are expressed as elements in theendomorphism EndA(0) E of the projective module E. Before the tachyon condensation,
the projective module should represent the original D2-brane itself E1,0 = Aθ . After thetachyon condensation, it should be projected into a nontrivial projective module of Aθ
which we investigate below.
We would like to solve the equation of motion of the tachyon ﬁeld by imposing several
assumptions. Suﬃcient conditions are∂V (T )= 0,∂T[∇, T ] = 0.

(43)

The ﬁrst equation is equivalent to the equation of motion if we take the large B limit
as discussed in [8, 9, 10]. The solutions to this equation are given by the projections in
11

In this paper we discuss the cases where the transverse scalars do not have expectation values.

14the noncommutative torus algebra Aθ . The projections in Aθ are classiﬁed by the valuesof trace as in (17) and we write T = Pn−mθ . Next we examine the second condition[∇, T ] = 0. This is satisﬁed12 if we use the connection ∇i = Pn−mθ (δi + Ai )Pn−mθ . It
deﬁnes the projective module E = Pn−mθ E1,0 and the endomorphism EndAθ E is given byPn−mθ Aθ Pn−mθ . The projective module E is twisted and its ﬁrst Chern class (or cyclic
2-cocycle) is given by τ2 (Pn−mθ ) = m as we explain in the appendix A.
Now let us turn to the equation of motion for the gauge ﬁeld [∇i , F̂ij ] = 0. It issatisﬁed if the ﬁeld strength is proportional to Pn−mθ . In a sense, Pn−mθ can be regarded
as the identity in the algebra EndAθ E. The ﬁeld strength which is proportional to theprojector should be regarded as constant curvature reviewed in the section 3.2.
One may prove that for any projection of type Pn−mθ the projective module of theform E = Pn−mθ E1,0 = Pn−mθ Aθ has a constant curvature connection. As we saw inthe subsection 3.1, for every n, m one can construct a constant curvature connection
as the Heisenberg projective module. Because it is projective, it should be written asẼ = P̃n−mθ ANθ for a certain projection P̃n−mθ in MatN (Aθ ). Because any projection in
MatN (Aθ ) for a given value of trace belongs to the same K-theory class [28], we can changeP̃n−mθ into any Pn−mθ ∈ Aθ via an unitary transformation. The transformed projective
module E = Pn−mθ Aθ also possesses the induced constant curvature connection.
In this way we have found exact solutions of the equation of motion derived from (42).
T = Pn−mθ ,

F̂ =

m
1
Pn−mθ J.
2πα n − mθ

(44)

Here we represents the ﬁelds as elements in the ‘large’ algebra Aθ . In the small algebra
EndAθ E, both T and Ai are proportional to the identity. Unlike the Moyal plane case,the small algebra is Morita equivalent to Aθ and can be rewritten as
EndAθ E = Pn−mθ Aθ Pn−mθ = Matd (Aθ̃ ),(θ̃ =b − aθ
s.t. an0 − bm0 = −1), (45)n0 − m0 θ

where we have deﬁned d = g.c.d.(n, m) and (n, m) = d(n0 , m0 ).
We proceed to discuss what will be generated via the tachyon condensation (44). The
mass of this excitation can be evaluated by neglecting the derivative terms because of[∇, T ] = [∇, F ] = 0,√2
Tr Pn−mθ det(G + 2πα(F̂ + Φ))M(n,m) = √α  Gs√2|n| 
= √det (g + 2πα (B + F )),(46)
α  gs12

One can also satisfy [∇, T ] = 0 if the connection acts only on the projective module E  = (1 −Pn−mθ )E1,0 . However a little analysis shows that this is not consistent with the descent relation [40]and does not have a expected tension. Therefore we believe that this does not correspond to physicalsolutions and neglect these in this paper.

15√
where the ﬂux F is given by (28). The factor 2 is peculiar to non-BPS D-branes. The
above calculation can be done in the same way as in (39). The dimension of the projective
module naturally appears here as the trace of the projection. It is interesting that thetachyon condensation on unstable D-brane systems gives an explicit physical realizationof the mathematically fundamental relation between projective modules and projections.
If we assume gij = R2 δij to make discussion clearer, the mass reduces to√2  2 2n R + (n/θ − m)2 .
(47)
M(n,m) = √α  gsIt explicitly shows that the resulting state is a bound state of n non-BPS D2-branes and(−m) non-BPS D0-branes. We comment that our arguments of tachyon condensation
naturally derive the fact that −m or n can be negative which is consistent with the resultin the previous subsection13 . In this way we obtain all kinds of D2-D0 bound states via
tachyon condensation and thus our results are consistent with T-duality.
If we take the large B/g limit, the mass spectrum is proportional to dim(E) = n − mθas can be seen from (47) and it is dense in R. This means that there exists a excitation ofwhich energy is arbitrary small. In [14] we investigated the tachyon condensation in thislimit and suggested that it leads to the instability. It means that any projection Pn−mθ
can be divided into inﬁnite numbers of mutually orthogonal smaller projections while
the total value of the trace is preserved. This was proved by investigating an explicit
representation of projections.
The argument can be simpliﬁed as follows. We start from a projection Pn−mθ in Aθ .
It can be regarded as the identity in the small algebra Matd (Aθ̃ ) = Pn−mθ Aθ Pn−mθ . One
can ﬁnd another projection Q in the small algebra. The original projection is decomposed
into two mutually orthogonal projections,Pn−mθ = (1 − Q)Pn−mθ + QPn−mθ .

(48)

One may continue this operation repeatedly to give the inﬁnitely small mutually orthogonal projector.
We would like to claim that such instability does not appear for ﬁnite B/g. As
can easily be seen from the mass formula (47), the bound state can be divided intod = g.c.d (n, m) pieces but is not divided further. It means the instability cannot appear.
We may interpret it from our exact tachyon solution. The key point is the requirement of
the constant ﬁeld strength in (44) which is absent in large B/g limit. It is permitted tobe divided into only d mutually orthogonal parts, even though the tachyon ﬁeld in (44)13

One may ask the physical interpretation for the negative n. Assume that n is negative and thus −m
is positive. Since D0-branes with B-ﬁeld on a torus generate (non-BPS) D2-brane, we can say that thenegative n means the annihilation of these induced non-BPS D2-branes with the |n| non-BPS D2-branes.

16can be divided into inﬁnitely many pieces. We conclude that for ﬁnite B/g the boundstates are all stable if g.c.d (n, m) = 1.
The corresponding projective module En,m = Pn−mθ E1,0 can be written as a direct sumof d projective modules of the same type as we saw in section 3.1. As shown in [41] (see also[21]), the moduli space of constant curvature connection is given by the symmetric productof d copies of a two-torus (T2 )d /Sd . This is actually the same as the physical moduli spaceof the solutions of (42) up to gauge transformation. The freedom of unitary transformationof the tachyon ﬁeld is absorbed in the gauge transformation T → UT U ∗ , U ∈ Aθ andwhat remains is only the moduli space of constant curvature connection for the projective
module En,m = Pn−mθ E1,0 . We note that this is consistent with the physical intuition. The
moduli space of a bound state for coprime (n, m) parameterizes the transverse coordinate
for a D0-brane, namely T2 . It is then obvious that the moduli space of the bound statesof d D0-branes should be (T2 )d /Sd .
The ﬂuctuations of the gauge and tachyon ﬁelds around the solution (44) belong to
Matd (Aθ̃ ). This is physically interpreted as the gauge group on the brane is U(d).
Up to now we concentrate on the tachyon condensation from a single D2-brane. We
may, of course, start from plural or even inﬁnitely many D2-branes. In such situation, we
can obtain arbitrary projective modules En,m from the projection in Mat∞ (Aθ ).
It is also important to note the relation between tachyon condensation on the above
noncommutative torus and that on the non-compact ﬂat plane (Moyal plane). The exact
solution for the latter [20] can be rewritten in our convention as ﬂows,
T = Pl ,

F̂ =

1
Pl J,
Θ

(l ∈ Z ≥ 0)(49)

where the noncommutativity Θ is deﬁned as [x1 , x2 ] = iΘ. The level-l projection Pl in
the Moyal plane algebra is given by l−1
k=0 |kk| and it corresponds to the generation ofl D0-branes. To obtain such situation from the torus, we need to take the large radius
limit or equivalently the small θ limit. In such a situation, the value of n is restricted to
0 or 1 which is consistent with the result (49). We note that in this limit one cannot takeB/g → ∞ limit and thus the instability does not occur.
Solution generating technique and Morita equivalence
We have seen the exact description of tachyon condensation is characterized by theconstant curvature connection. It is interesting to ask what is the solution generating
method which relates various solutions. In the Moyal plane, the exact solutions for tachyon
condensation were constructed in [20] by using the shift operator S = ∞n=0 |n + 1n|.
We would like to ﬁnd the analogous transformation on the two-torus Aθ .
In this case we have to be careful since the corresponding operator in general interplotes
the diﬀerent C ∗ -algebras. Namely after the tachyon condensation the original free module
E1,0 = Aθ (a D2-brane) is changed into the twisted projective module En,m (a D2-D0
17

bound state). As we have seen in section 2, the transformation between these two solutions
should be identiﬁed with the Morita equivalence bimodule S. It depends on the integers
n, m and satisﬁes
En,m = S ⊗Aθ E1,0 .

(50)This maps the endomorphism EndAθ E1,0 = Aθ into EndAθ En,m = Matd (Aθ̃ ). Physicallythis induces the transformation of the world-volume ﬁeld theories and this gives an explicit
realization of the descent relation [40].
In order to describe the exact solution, the projective module En,m should have theconstant curvature. In other words, we have to impose on the Morita equivalence Aθ̃ -Aθ
bimodule that S should keep this additional constraint. Actually it just ﬁts the deﬁnitionof the gauge Morita equivalence bimodule discussed in [21, 25]. We claim that this is theanalogue of the shift operator on the noncommutative torus.
We mention that the T-duality transformation can also be represented by the gauge
Morita equivalence bimodules as argued in [25, 32, 21]. On a noncommutative torus Aθ
there is a one-to-one correspondence between Aθ -modules and the solutions found in (44).
They correspond to the projective module En,m and take their values in EndAθ En,m . If we
perform T-duality so that the (n,−m) brane is transformed into d D2-branes (or equally
applying the Morita equivalence Aθ ∼ Aθ̃ ), the projective module En,m is changed into
the free module Ẽd,0 in the algebra Aθ̃ . Such T-duality transformation can be constructedby Aθ -Aθ̃ gauge Morita equivalence bimodule X
Ẽd,0 = En,m ⊗Aθ X .

(51)

The endomorphism EndAθ En,m = EndAθ̃ Ẽd,0 acts on Ẽd,0 from the left in (51). Especially,
the solution T = 1 ∈ EndAθ En,m is the identity in EndAθ̃ Ẽd,0 . On the other hand,
1mF̂ = 2πα n−mθ J ∈ EndAθ En,m is translated to 0 ∈ EndAθ̃ Ẽd,0 , because Ẽd,0 is the freemodule over Aθ̃ . The diﬀerence between the values of the constant curvatures comesfrom the constant curvature of X. This is equivalent to the shift of the ﬁeld Φ by Tduality as in (40). These imply that for general noncommutative algebras the exact
solution for tachyon condensation can be generated in the same way if there exist gauge
Morita equivalence bimodules.

3.4

Tachyon condensation on brane-antibrane systems

Brane-antibrane systems are more complicated and intriguing than non-BPS D-branes
from the viewpoint not only of string theory but also of the noncommutative geometry.
The crucial diﬀerence from non-BPS D-branes is that the tachyon ﬁeld becomes complex
and belongs to a Morita equivalence A-B bimodule (we write this as X), where A and
B are the algebras of the brane and the antibrane as we discussed in section 2. If we18

deﬁne E and F as the projective modules which represent the brane and the antibrane,
respectively and deﬁne A(0) as the noncommutative base space, the algebras A, B is given
by A = EndA(0) E, B = EndA(0) F . Remember that a Morita equivalence bimodule X isdeﬁned by the bimodule which possesses two types of inner product  , A ,  , B andsatisﬁes the conditions (3,4,5).
As in the previous subsection we use the example of noncommutative two-tori andconsider only D2-D0 bound states. We assume the original brane-antibrane system ismade of a (n1 , −m1 ) brane and a (n2 , −m2 ) antibrane. For simplicity we consider only
the case where the pairs of integers (n1 , −m1 ) and (n2 , −m2 ) are coprime. A pair of
integers (n, −m) denote the indices of D2-D0 bound state or equivalently of those thecorresponding projective module En,m . Note that if one speciﬁes (n, m) such that n−mθ ≥0, then there are two types of D-branes, that is, branes and antibrane. In our examplesof the noncommutative torus the fundamental algebra A(0) is given by A(0) = Aθ , whereθ is represented in terms of closed string variables as in (36). The algebras A and B aregiven byb1 − a1 θ,
n1 − m1 θ
b2 − a2 θ
B = Aθ2 , θ2 =,n2 − m2 θ

A = Aθ1 , θ1 =(a1 n1 − b1 m1 = −1)(a2 n2 − b2 m2 = −1).

(52)

The tachyon ﬁeld T belongs to a Aθ1 –Aθ2 bimodule. There are also the gauge ﬁelds on
the brane and the antibrane. We denote these as A(1) and A(2) . These ﬁelds belong to
A = Aθ1 and B = Aθ2 , respectively. The covariant derivative of the tachyon ﬁeld is givenby the connection for a bimodule X (see for example [25, 21]) speciﬁed by the requirement∇X (aT ) = δA (a)T + a∇X T(∀a ∈ A),∇X (T b) = (∇X T )b + T δB (b) (∀b ∈ B),(53)

where δA and δB denote the derivation in A and B, respectively. We also use the covariant
derivative of ﬁeld strengths F̂ (1) and F̂ (2) . Each of them is given by the commutator witha connection for the algebra A or B as in the previous subsection.
Now we have prepared to discuss the tachyon condensation on noncommutative tori.
The eﬀective action for a D2−D2 with two gauge ﬂuxes was already computed in [24]using the boundary string ﬁeld theory [42, 43, 44]. Applying this to our system on anoncommutative torus the result is given by

1
S = √
dtTrA V (T, T A ) det(G + 2πα(F̂ (1) + Φ))
α  Gs1
dtTrB V (T, T B ) det(G + 2πα(F̂ (2) + Φ))+ √α  Gs+ O(∇X T, [∇A , F̂ (1) ], [∇B , F̂ (2)]),(54)
19

where O(∇X T, · · ·) denotes the derivative terms. We deﬁned the traces for A and B byembedding these algebras in MatN (A(0) ) for a suﬃcient large integer N. These satisfy
the following relation [28, 29]
TrA T1 , T2  = TrB T2 , T1 ,(55)

and are normalized by TrM atN (A(0) ) 1 = N.
We would like to solve the equation of motion for the action (54). Below we give
solutions by imposing ansatz similar to the previous subsection. We assume the existenceof the partial isometry-like equation [30, 20, 19] for the tachyon ﬁeld (11)T, T A T = T,

T T, T B = T.

(56)

Note that these two equations are equivalent thanks to the relation (5). The solutions to this equation give the stationary points of the tachyon potential V (T, T A )and V (T, T B ). To make exact solutions for ﬁnite B/g, we should take account of thegauge ﬁelds. It is easy to see the equation of motions for the tachyon T and gauge ﬁelds(A(1) , A(2) ) are satisﬁed if we require (56) and∇X T = 0,(57)[∇A , F̂ (1) ] = 0, [∇B , F̂ (2)] = 0,(58)

are satisﬁed. Since (56) is equivalent to the statement that T, T A ∈ A and T, T B ∈ B
are both projections as explained in section 2, we can write these as follows
T, T A = 1 − Pα+βθ1 (≡ 1 − P1 ),T, T B = 1 − Pγ+δθ2 (≡ 1 − P2 ).

(59)

Using the relation (55) we obtain the constraint,
n1 (1 − α) − b1 β = n2 (1 − γ) − b2 δ,

m1 (1 − α) − a1 β = m2 (1 − γ) − a2 δ.

(60)

They determine γ and δ in terms of α and β. The tachyon ﬁeld which condensates as in(59) belongs to the bimodule (1 − P1 ) · X · (1 − P2 ). For this tachyon ﬁeld the potential
is evaluated as
V (T, T A ) = Pα+βθ1 ,V (T, T B ) = Pγ+δθ2 .

(61)

This is because we use the convention that the original brane-antibrane system corresponds to T = 0 and that the vanishing of brane and antibrane corresponds to T, T A = 1
and T, T B = 1, respectively14 .
14

Notice that this is the opposite convention to the tachyon potential of non-BPS D-branes in section

3.3.

20

Let us turn to the next equation (57). This is satisﬁed if the gauge ﬁelds belong toA(1) ∈ P1 AP1 ,A(2) ∈ P2 BP2 .

(62)

The last equation (58) is solved if we assume that both of the gauge ﬁelds have constantcurvatures1
βa1 + m1 α
1 ∈ P1 AP1 ,2πα n1 α + b1 β − (βa1 + m1 α)θ
δa2 + m2 γ
1=
1 ∈ P2 BP2 .

2πα n2 γ + b2 δ − (δa2 + m2 γ)θ

F̂ (1) =F̂ (2)(63)

It ﬁnishes our calculation of the physical solutions of the tachyon condensation. Our
result does not depend on the detailed form of the derivative term because of (57,58).
We evaluate mass spectrum of these classical solutions,
1
TrA Pα+βθ1 det(G + 2πα (F̂ (1) + Φ))
α  Gs1
TrB Pγ+δθ2 det(G + 2πα (F̂ (2) + Φ))+ √α  Gs
|N1 |
|N2 |
= √
det(g + 2πα (B + F (1) )) + √det(g + 2πα (B + F (2) )). (64)α gs
α  gsM = √We have deﬁned the ﬂuxes as follows
F (1) =J M1,2πα N1

F (2) =J M2,2πα N2(65)

where integers M1 , N1 , M2 and N2 are given by
N1 = αn1 + βb1 ,M1 = βa1 + αm1

N2 = γn2 + δb2 ,

M2 = δa2 + γm2 .

(66)

We ﬁnd that the products of the tachyon condensation are identiﬁed with a (N1 , −M1 )brane and a (N2 , −M2 ) anti-brane. If these integers are not coprime, each bound state
can be divided into several parts as before. Note that what are produced after the tachyon
condensation depend on the projections (61). The original brane-antibrane system corresponds to α = γ = 1, β = δ = 0. If one assumes that the dimension of the projective
module E is larger than that of F , then the tachyon ﬁeld γ = 0, δ = 0 gives the maximal
condensation and this will produce a (n1 − n2 , −m1 + m2 ) brane. In the opposite case thetachyon ﬁeld α = 0, β = 0 will generate a (n2 − n1 , −m2 + m1 ) anti-brane.
For general decay modes the diﬀerences of the D2-brane charge (N1 − N2 ) and theD0-brane charge (M2 − M1 ) are preserved as follows
N1 − N2 = n1 − n2 ,M2 − M1 = m2 − m1 .
21

(67)

The charge conservation can also be discussed in the framework of operator algebra
K0 -group K0 (Aθ ). If one would like to consider the K0 -group of noncommutative torus,
the Chern character [34] gives enough information [29]. In the brane-antibrane system
which corresponds to the pair of projective modules (E, F ) ∈ K0 (Aθ ) the K-theory charge
is given by the diﬀerencech(E) − ch(F ) = TrA exp(2παF̂ (1) ) − TrB exp(2πα F̂ (2) )= (n1 − m1 θ − n2 + m2 θ) + (m1 − m2 )dx1 dx2 ,(68)

where dx1 dx2 is the two form along the two-torus. It is known that the RR-couplings ona brane-antibrane system can be written by using K 0 -type superconnection [22, 23, 24].
Applying this idea to our examples we obtain the following RR-couplings
SRR ∼ CRR ∧ TrA V (T, T A ) exp(2παF̂ (1) ) − TrB V (T, T B ) exp(2πα F̂ (2) ) . (69)Note that in our noncommutative description the derivative of tachyon ﬁeld is always
zero and does not contribute. Also notice that the potential in the above can be regarded
as the identities in the algebras of gauge ﬁelds EndAθ E and EndAθ F because of (61).
Thus the conservation of RR-charge (67) is equal to that of operator algebra K-theory
charge and therefore this gives a further support to the relation between D-brane charge
and K-theory [4, 5, 13]. The intriguing characteristic that the above ‘noncommutative
Chern character’ depends on θ will correspond to the physical fact that D0-branes in the
B-ﬁeld background generate D2-brane charge [45]. Therefore it will also be interesting toclarify the relation between the RR-couplings for non-abelian transverse scalars [35, 24]
and the above RR-couplings on various noncommutative tori [45] (see also [46] for the
Moyal plane).

4

Conclusions

We ﬁrst discussed the open string theory in general noncommutative background. We
considered a general framework to handle open strings and D-branes in a uniﬁed way byutilizing the Morita equivalence. In particular we proposed the equation which deﬁnes
noncommutative solitons on general brane-antibrane systems.
From this viewpoint we have examined the exact solutions on noncommutative toriin tachyonic systems. For non-BPS branes the solutions are given by the tachyon ﬁeld
which is proportional to the projection and the gauge ﬁeld with a constant curvature. This
respects the one-to-one correspondence between a projection and a projective module. We
have also shown that these solutions can be generated by employing the (gauge) Morita
equivalence. Our exact description of tachyon condensation including the gauge ﬁeld
solves for ﬁnite B/g the previously observed instability problem.
22

More complicated and thus more intriguing examples are brane-antibrane systems.
In this case the tachyon ﬁeld belongs to the Morita equivalence bimodule and we can
impose the partial isometry-like relation instead of the equation of motion. We canconstruct the exact solutions and determine the decay products. We ﬁnd the RR-chargesof brane-antibrane systems can be represented by the superconnection-like extension of
the Connes’s Chern character and check that these charges conserved in the process oftachyon condensation. This also veriﬁes the fact that the D-brane charge is classiﬁed by
the operator algebra K-theory.
Note added: After completing our calculations, we noticed the preprint [47] on the net
which has some overlaps with our results in section 3.3.

Acknowledgments
We are grateful to M. Hamanaka, K. Ichikawa, A. Kato, K. Ohmori, Y. Sugawara, S.
Terashima and T. Uesugi for useful discussions. We are also happy to thank H. Moriyoshi
for his clear lecture on noncommutative geometry and useful comments. H.K. and T.T.
are supported by JSPS Research Fellowships for Young Scientists. Y.M. is supported
in part by Grant-in-Aid (# 707) from the Ministry of Education, Science, Sports andCulture of Japan.

A

Powers-Rieﬀel Projections and Cyclic Cocyles

Here we give a review of Powers-Rieﬀel projections [28] on two dimensional noncommutative tori and of the calculations of their topological charges [34, 3]. We also mention
some other projections constructed in [14].
First let us consider projections in any C ∗ -algebra A and assume that there exist a
trace Tr : A → C and derivations δi : A → A such that δi (ab) = δi (a)b + aδi (b). We also
normalize the trace as Tr(1) = 1. The index i corresponds to the basis of the derivations15 .
A projection p ∈ A is deﬁned to be a self-adjoint idempotent p∗ = p = p2 . The Connes’s
Chern Character is deﬁned as the exponential of the curvature F ∈ EndA E of projective
module E [34]:∞ F  1
ch(E) = Tr exp=
τ2k (F, F, · · ·, F ).
2πi
k!
k=0

(1)

The each term τ2k (F, F, · · ·, F ) of the above expansion represents the contribution which
is proportional to F k . Note that τ0 is equal to the trace of identity and it gives thedimension dim(E) of the projective module.
15

More precisely, the derivation is deﬁned as an action of a Lie group G on the algebra A [34]. Thenthe basis of the derivation can be said as those of the Lie algebra.

23

Since any projective module E is described as P ·AN using a projection P in MatN (A)for a suﬃciently large integer N, one can rewrite the curvature in terms of the projection.
More explicitly one can choose a connection of E as ∇i = P · δi · P [34, 21]. Note thatthe topological quantity such as the Chern character does not depend on the choice ofthe connection. Then the curvature F ∈ EndA E is expressed as
Fij = [P δi P, P δj P ] = P (δi P )(δj P ) − P (δi P )(δj P ) + P (δi δj − δj δi )P.

(2)

Because the trace in EndA E = P · MatN (A) · P is naturally induced from the trace in
MatN (A) normalized as Tr(1) = N, one can always calculate the Chern character (1) if
the projection P is given. Thus we write the 2k-th part by τ2k (P ). This can be regarded
as the cyclic 2k-cocycle where the projections are substituted.
For example, the cyclic 0-cocycle is given by
τ0 (P ) = Tr(P ) = dim(E).

(3)

Now let us investigate the explicit examples of cyclic cocycles. First we consider aﬂat two dimensional plane (Moyal plane) algebra. We employ the operator representationand deﬁne the noncommutative coordinate (x1 , x2 ) as [x1 , x2 ] = iΘ. Further we deﬁne
the creation and annihilation operator a† = √12Θ (x1 − ix2 ) and a = √12Θ (x1 + ix2 ) suchthat they satisfy [a, a† ] = 1. Using this one can deﬁne the basis of the Hilbert spaceas the familiar n-number state |n = √1n! (a† )n |0. Then the algebra is expanded by|nm| (n, m ≥ 0). Let us consider the projection Pn = n−1
k=0 |kk| for a ﬁnite integer
n. The trace of this operator is given by Tr(Pn ) = 2πΘ · n, where we have normalized thetrace so that it is equal to the integration dx1 dx2 in the c-function representation. Now
it is straightforward to calculate τ2 (Pn ) since the derivations δ1 , δ2 along the coordinate
x1 , x2 are given by
δ1 =

i 2
[x , ],
Θi
δ2 = − [x1 , ].
Θ

(4)Thus we obtain the result as follows

1
Tr Pn [a† , Pn ][a, Pn ] − Pn [a, Pn ][a† , Pn ]
2πΘ


1
=
Tr Pn + Pn a† Pn aPn − Pn aPn a† Pn
2πΘ
= n.

τ2 (Pn ) =(5)

Therefore the value of τ2 (Pn ) is quantized and is positive. Note that this value is equal tothe ﬁrst Chern class of the projective module and therefore should be quantized. Then
one may ask if one can obtain negative integers ? The answer is yes and one can construct
the corresponding projection as 1 − Pn . In this case we obtain the value τ2 (1 − Pn ) = −n.
24After this elementary example, let us turn to the two dimensional noncommutative
torus Aθ . We assume θ is irrational because for rational θ there are ﬁnite dimensional
representations of the algebra and the calculations are simpliﬁed (see also [17, 14, 16, 18]).
As we have mentioned in section 3.1, the projections in Aθ are generally characterized
by their values of trace as in eq.(17). The explicit construction of projections (PowersRieﬀel projection) was given in [28] and let us review this below.
Because U1 and U2 do not commute (U1 U2 = U2 U1 e2πiθ ), we can diagonalize only U2
2
and deﬁne c-number x2 as U2 = e2πix . Thus we obtain the following (inﬁnite dimensional)
representation of Aθ
U1 |x2  = |x2 + θ,
2πix2

2

U2 |x  = e

(6)

2

|x .

(7)

Then the trace of an element a ∈ Aθ is given by 1
Tr(a) =
dx2 x2 |a|x2 .

(8)0

In order to ﬁnd explicit projections P we assume the following form
P = U1∗ (g(U2 ))∗ + f (U2 ) + g(U2 )U1 .

(9)

As we will see below, one can construct a projection for each n, m even under this restriction. Acting on the position space |x2 , we require P 2 |x2  = P |x2 . This deﬁnes a
projection in Aθ if and only if f and g satisfy the following relations22g(e2πix )g(e2πi(x +θ) ) = 0 ,
2

2

2

2

2

g(e2πix )[1 − f (e2πix ) − f (e2πi(x +θ) )] = 0 ,22

f (e2πix )[1 − f (e2πix )] = |g(e2πix )|2 + |g(e2πi(x −θ) )|2 .

(10)

Explicit forms of f, g which satisfy these relations are given as follows. Choose any
2small 8 > 0 such that 8 < θ and θ + 8 < 1, and let F (x2 ) ≡ f (e2πix ) for one period begiven in the range x2 ∈ [0, 1] by


x2 /8
x2 ∈ [0, 8]
1x2 ∈ [8, θ]F (x2 ) =,(11)

1 − (x2 − θ)/8x2 ∈ [θ, θ + 8]
 0x2 ∈ [θ + 8, 1]
Then deﬁne g for one period by

2πix2

g(e

)=

F (x2 )(1 − F (x2 ))x2 ∈ [0, 8],x2 ∈ [8, 1] .

0
25

(12)

It is easy to see that the functions f and g, deﬁned as the periodic extensions of theabove, satisfy the relation (10). It can be easily shown that
 1
 1
2
2
2
dx < x |P |x >=dx2 F (x2 ) = θ .
(13)
Tr P =
00Thus the projection P now constructed corresponds to Pθ .
Now how about more general projections Pn−mθ ? Such general projections can be constructed by slightly modifying the above constructed Pθ as follows. The general projectionPn−mθ can be regarded as the projection Pθ in the algebra Aθ if we deﬁne θ = n − mθ.
It is easy to see that the algebra Aθ can be embedded in Aθ by replacing (U1 , U2 ) with(i) (U1 , U2m ) or (ii) (U1m , U2 ). Since one can construct the projection Pθ in the previous
way, we obtain the projection Pn−mθ in Aθ as desired.
In the ﬁrst choice (i), the projection is described by functions f and g with period
1/|m| and the width of each lump of the function f is given by (n − mθ)/|m|. This
preserves the form (9) and is called Powers-Rieﬀel projection [28]. On the other hand, inthe second choice (ii) the requirement for being a projection is given by the equation (10)with θ replaced by mθ and this is not included in the form (9). Then the width of thelump is enlarged to n − mθ. This construction was given in [14] and used in the proof
that any projection can be divided into inﬁnite numbers of mutually orthogonal smaller
projections, where the total value of trace is preserved. Note also that in either case, thetotal area occupied by the lump is 0 ≤ n − mθ ≤ 1.
Next we turn to the calculation of cyclic 2-cocyle τ2 (P ). Deﬁne the derivations δ1 , δ2along the two directions of the two dimensional torus as follows
δj Uk = 2πiδjk Uk ,(14)

where δij is the ordinary Kronecker’s delta. Equivalently, one can express the derivations
1
as follows by using the noncommutative coordinate (x1 , x2 ) deﬁned by U1 = e2πix , U2 =2e2πix :
δ1 = −i2π 2[x , ],
θ

δ2 = i2π 1[x , ].
θ(15)

From these we can see that δ1 and δ2 do commute. Then the cyclic 2-cocyle is deﬁnedbyτ2 (a, b, c) =1
Tr [a δ1 (b)δ2 (c) − a δ2 (b)δ1 (c)] .
2πi

26

(16)

If we substitute a = b = c = Pθ , then we get [34]
τ2 (Pθ )
= −(4πi)Tr f  (U2 )(g(U2 ))2 − f (U2 )g  (U2 )g(U2 )U2+U1 f (U2 )U1∗ g  (U2 )g(U2 )U2 − (g(U2))2 U1 f  (U2 )U2 U1∗
 1
 1
d
d
2
2
2
2 2
dx (f (x + θ) − f (x )) 2 (g(x ) ) + 2
dx2 2 (f (x2 + θ) − f (x2 ))(g(x2)2 )
=−
dx
dx
0
0 1
2
df (x )
= −6
dx2
(g(x2 )2 )
2
dx
0
 1
= −6
dt (t − t2 ) = −1.
(17)0

It is also possible to generalize the result for the projections Pn−mθ and we obtain
τ2 (Pn−mθ ) = m.

(18)

In order to see this one has only to note that for the description (i) the evaluation of τ2
is equal to counting of the number of lumps with sign and also that for the description(ii) the factor m is due to the derivation of U1m . Indeed this value of τ2 is the same asthat computed from the previously discussed projective modules En,m (0 ≤ n − mθ ≤ 1)
2πim
1 ∈ EndAθ En,m as follows
which possess the constant curvature F = n−mθ
τ2 (F ) =1
Tr(F ) = m.
2πi

(19)

Finally let us discuss the relation between the results in the Moyal plane and thosein noncommutative torus. Since the radius of the torus is scaled in proportion to √1θ , thenoncommutative torus will approaches the Moyal plane in the limit θ → 0. In this limit
the value of integers n, m which satisfy 0 ≤ n − mθ ≤ 1 is restricted to n = 0 or n = 1.
Thus we obtain the projection Pmθ and P1−mθ . This is consistent with the previous resultthat in the Moyal plane algebra the projection is given by Pm or 1 − Pm up to unitaryequivalence.

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30

UTMS 2001-9
UT-930

hep-th/0104143
April, 2001

Exact Tachyon Condensation
on Noncommutative Torus

Hiroshige Kajiuraa , Yutaka Matsuob and Tadashi Takayanagiba) Graduate School of Mathematical Sciences, University of Tokyo
Komaba 3-8-1, Meguro-ku, Tokyo 153-8914, Japanb) Department of Physics, Faculty of Science, University of Tokyo
Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0034, Japan

Abstract
We construct the exact noncommutative solutions on tori. This gives an exact description of tachyon condensation on bosonic D-branes, non-BPS D-branes
and brane-antibrane systems. We obtain various bound states of D-branes after
the tachyon condensation. Our results show that these solutions can be generated
by applying the gauge Morita equivalence between the constant curvature projective modules. We argue that there is a general framework of the noncommutative
geometry based on the notion of Morita equivalence which underlies this speciﬁc
example.

1

Introduction

D-branes have been playing the most prominent role in recent developments of string
theory. This is not only because they are the non-perturbative objects but also because
they possess many interesting characteristics which require mathematically distinguisheddescriptions. It is very interesting to try to prove that these various viewpoints aremutually consistent. Such an eﬀort often gives profound relationships between physics
and mathematics.
One of such examples is the quantum ﬁeld theory on the noncommutative space arisingfrom open string theory in B ﬁeld background [1, 2]. In the commutative case, the spaceof functions gives the basis of the quantum ﬁeld theory, for example, through the mode
expansion. In the noncommutative space they are replaced by the noncommutative C ∗ algebra A which is at the heart of the principle of the noncommutative geometry (see forexample [3]). In this paper we are mainly interested in the description of D-branes withthe direct use of such principle.
In the commutative approach, the soliton charges of the D-branes are derived fromthe topological K-group [4, 5]. It is based on the observation that there are always themassless gauge particles which deﬁne the vector bundle on the D-branes. We need to takethe formal diﬀerence of vector bundles to describe the brane-antibrane pair annihilation
process [6, 7].
In noncommutative geometry, the topological information of the ‘manifold’ is given by
the operator algebra version of the K-group K(A). For example in Connes’ index theorem[3], the topological index is give by the pairing of the element of K-group with that of thecyclic cohomology group of A. We note that K0 (A) is deﬁned by an equivalence class ofprojection operators in Mat∞ (A), the inﬁnite dimensional matrix algebra with elements
in A.
From physical viewpoints, it is natural to conjecture that topological properties of
D-branes are described by the operator algebra K-group. It should be described throughthe solitonic conﬁgurations which are proportional to projection operators. Remarkably
such conﬁgurations indeed appear as classical solutions (GMS soliton) in the scalar ﬁeld
theory on the noncommutative plane (Moyal plane) in the large noncommutativity limit[8]. This idea was immediately applied to the string theory in the tachyon condensation
process [7, 6], e.g. bosonic D-branes, non-BPS D-branes and brane-antibrane systems[9, 10]. If we use nontrivial GMS soliton, the lower dimensional D-branes are generated.
As pointed out in [11, 12] it can be seen as the noncommutative generalization [13] of thecorrespondence between D-brane charges and K-theory [4, 5].
In this way the unstable systems of D-branes seem to give a good example for theapplication of the geometrical methods of the noncommutative geometry to the string
theory. There are two directions to proceed. One is to challenge noncommutative spaces

1with more complicated and richer structures. The simplest nontrivial example is the noncommutative torus. This example is interesting from physical side since we expect to haveanalogue of T-duality symmetry in the form of Morita equivalence. It was investigated in[14, 15] by using Powers-Rieﬀel projection. Unlike the Moyal plane, we observed a sortof instability [14]. It comes from the fact that we may construct the noncommutative
soliton with arbitrary small size. Mathematically it is related to the fact that K-groupof noncommutative tori is not quantized in Z but takes its value in R. It remained as a
puzzle whether it is natural to interpret the continuous value as the D-brane charge. Latera diﬀerent construction of the soliton conﬁguration on the torus and on the orbifold was
discussed in [16]. Among other things, a remarkable suggestion is to use Morita equivalence bimodule in the construction of the noncommutative soliton. Similar constructionof the projection operator on tori was also discussed in [17] and [18]. For fuzzy sphere,
GMS-like solitons were discussed in [19].
The other direction is to take the gauge ﬁeld into account and to construct the exact
solution without taking the large B limit. It was pioneered in [20] when the base space is
Moyal plane. Certain constraint on the coupling of ﬁeld strength and tachyon ﬁeld should
be satisﬁed in order to have such property.
In the present paper we continue to study the noncommutative soliton on the twotorus. We have mentioned two motivations, (1) how to resolve instability of the spectrumand (2) the construction of exact solution. The use of Morita equivalence initiated in [16]gives another motivation. For the nontrivial examples such as tori, we can not directly
construct the analogue of the shift operator. In this sense, we can not escape from using
more abstract Morita equivalence bimodule directly to construct the noncommutative
soliton. (A nice review of Morita equivalence for noncommutative torus is given by [21]).
Once we know how to use it, one may apply the method to other examples as well, namely
in the generic examples of the open string systems interpolating D-branes [2, 13]. We argue
that the Morita equivalence gives a natural generalization of the notion of the braneantibrane systems and leads to the description which is similar to the superconnection [5,22, 23, 24]. Inspired by this fact we propose an equation which deﬁnes the noncommutative
solitons on brane-antibrane systems in the generalized sense.
As we will see, we can obtain the exact solutions of the tachyon condensation on
the two-torus by employing the constant curvature connections. In the noncommutative
torus, the constant curvature connection parameterizes the equivalence class of the wholeprojective modules (analogue of vector bundle). As a result we obtain various D2-D0
bound states after the tachyon condensation on a non-BPS D2-brane. We also ﬁnd thatthe gauge Morita equivalence [25, 21] (bimodule between constant curvature connection)
plays the crucial role of generating solutions. In fact one can construct exact solutions for
any C ∗ -algebras if there exist the gauge Morita equivalence bimodules. Furthermore thisexact analysis explicitly shows that for ﬁnite B/g the above mentioned instability does2

not occur.
The paper is organized as follows. In section 2, we discuss tachyon condensation on
generic noncommutative spaces. After we review the Morita equivalence, we construct
the projection operator of the brane-antibrane systems in the generalized sense. We
see that the structure of superconnection [26] naturally appears as the linking algebrain the framework of C ∗ -algebra. In section 3 we discuss the tachyon condensation ona noncommutative two-torus. We construct the exact solutions for bosonic D-branes,
non-BPS branes and brane-antibrane systems in terms of ﬂat curvature connection. We
also discuss the solution generating rule for this examples by using the gauge Morita
equivalence. This section also includes a review of some mathematical results on the
projective modules. In section 4 we summarize the conclusions. In the appendix A wegive a review of the explicit example of projections in noncommutative tori and we also
show the calculations of their topological charges.

2

Morita equivalence and noncommutative soliton:
A General Strategy

We start from discussing relatively formal viewpoint which will be useful in the later
sections. While our main result is restricted to the noncommutative solitons on noncommutative tori in section 3, we think that our method can be basically applied to the otheropen string systems such as Dp − Dp as well.
Let us ﬁrst recall the deﬁnition of the noncommutative solitons [8]. They are thesolutions to the equation of motion,∂V ( φ)
= 0.
∂φ

(1)

Here is the (noncommutative) product of the given C ∗ -algebra A(0) which deﬁnes thenoncommutative geometry on the single D-brane. It is solved in the following form,
φ(x) =λi Pi , λi ∈ R, Pi ∈ A(0) ,(2)iwhere λi s are the solutions to the equation ∂V (λ)/∂λ = 0 and Pi ’s are the mutuallyorthogonal projections Pi · Pj = δij Pi , Pi∗ = Pi . In this sense, the construction of noncommutative solitons is reduced to ﬁnd the projection operators 1 .
In the mathematical context, the classiﬁcation of the projection operator is directly
related to the deﬁnition of the K0 -group of the operator algebra K-theory. This is related
1

If the number of the D-branes is greater than one (say N ), we need to consider the projector in thematrix algebra M atN (A).

3

to the fact that the noncommutative analogue of the vector bundle is described by the
projective module of given C ∗ -algebra A. Let us brieﬂy illustrate the correspondence byusing the relationship with the commutative theory.
Let M be a smooth manifold (base space) and E be a vector bundle over M. In thecontext of the topological K-theory, it is known that the isomorphism class of E is anelement of K 0 -group2 . By Swan’s theorem, for any E there exists vector bundle F suchthat E ⊕ F is a trivial bundle over M. Let C ∞ (M) be the smooth function on M. The
trivial bundle can be written as (C ∞ (M))N with some N ∈ N. Therefore, any vector
bundle over M can be obtained by acting P on (C ∞ (M))N , where P is a projection in
MatN (C ∞ (M))3 . The module which is constructed by applying the projection operator
to the free module is called projective module.
This characterization of the vector bundle can be generalized to noncommutative theory. A noncommutative algebra A(0) replaces C ∞ (M) for ‘noncommutative space’. The
free module (A(0))N corresponds to the rank N trivial bundle on commutative space. Projective module E is deﬁned as A(0) -module such that there exists the other A(0) -module Fwith E ⊕ F = (A(0) )N . Thus the noncommutative analogue of a vector bundle is the projective module E = P (A(0))N which is deﬁned by a projection operator P ∈ MatN (A(0) ).
In this sense, D-branes on noncommutative spaces are described by the projective modules, and the operator algebra K0 -group classiﬁes the D-branes on the noncommutative
space [13, 11, 12].
Let us come back to the issue of the construction of the projection operator. We would
like to use the Morita equivalence bimodule as the abstract building block to constructnoncommutative soliton.
Morita equivalence is one of the central idea of the classiﬁcation of C ∗ -algebra. Fromthe mathematical viewpoint, it is essential to determine when two C ∗ -algebras A, B deﬁne
the same type of noncommutative geometry. In the noncommutative geometry, the ideaof points is replaced by the set of ideals of C ∗ -algebra. It is then known that two C ∗ algebras A, B have the same set of ideals if there are A-B Morita equivalence bimodule
A XB (for example see [27]). It is deﬁned as a bimodule on which A (resp. B) acts fromthe left (resp. right) with two types of inner products  , A ,  , B of A XB with value in
A and B, respectively with the following conditions
ax, yA = ax, yA ,x, y∗A = y, xA

,

a∈A,

x, ybB = x, yB b ,x, y∗B = y, xB

,

b∈B,

xb, yA = x, yb∗ A ,
2
0

K .

3

ax, yB = x, a∗ yB .

(3)
(4)

In general, the formal diﬀerence of the isomorphism class of vector bundles over M is the element of
We deﬁne the algebra M atN (A) as the N times N matrix algebra with elements in A.

4

The most important property which should be satisﬁed by them is the associativity
x, yA z = xy, zB ,x, y, z ∈ A XB .

(5)

Two C ∗ -algebras A and B which have such a Morita equivalence bimodule A XB is called
Morita equivalent.
In the string theory, there is a natural interpretation of such equivalence relation.
It is well-known that the noncommutativity arises in the string theory on the D-branes
connected by the open string in the presence of B ﬁeld. On the two ends of open string, wehave two D-branes and generally two diﬀerent types of noncommutative geometry deﬁned
on them. Suppose they are deﬁned by the C ∗ -algebras A, B 4 . Under such circumstances,
it is natural to conjecture (for example, see [2, 13]),• The bimodule naturally interpreted as the open string ﬁeld Ψ where A (resp. B)
acts from the left (resp. right).
• Two inner products are identiﬁed the product of open string ﬁelds. We have two
type of inner product because we have two choices (which side of the open string)for the contraction.
• The associativity of the product corresponds to that of the product of the open
strings.
Although the actual justiﬁcation of these statements is far from being obvious at thisstage, it gives a nice intuition to the otherwise abstract nature of Morita equivalence.
In the following, we use the Morita equivalence bimodule to deﬁne the projection
operator (= noncommutative soliton). In an abstract language, it can be described asfollows [28, 29, 27, 16]. In the very deﬁnition of the Morita equivalence, we actuallyneed to impose that the inner product  , A,B maps A XB into a dense set of the A, B
respectively. Namely for any operator a ∈ A, there should be a ﬁnite set xi , yi ∈ A XB suchthat a = i xi , yi A . Suppose A has identity as its element and take 1 = N
i=1 xi , yi A .
From xi , yi one may deﬁne the projection operator in MatN (B) as P ≡ yi, xj B sinceyi , xj B yj , xk B =yi, xj yj , xk B B = yi,
xj , yj A xk B = yi, xk B .
(6)
j

j

j

Unlike the original GMS soliton where the projection operator deﬁnes the lower dimensional D-branes, it seems rather hard to identify the nature of the projected space. However, from the mathematical side, A can be embedded into MatN (B), asA ∼ P · MatN (B) · P .
4(7)More precisely, let A(0) be the algebra corresponding to the noncommutative base space, and let Eα(resp. Eβ ) be the projective A(0) -module (D-brane) related to A (resp. B), then A = EndA(0) Eα andB = EndA(0) Eβ . Let a ∈ A(0) , A ∈ EndA(0) E and ξ ∈ E. EndA(0) E means (a · ξ)A = a(ξ · A). This is
the natural noncommutative generalization of the deﬁnition of endomorphisms for vector bundles.

5

Suppose A (resp. B) describes the noncommutative geometry on D-brane α (resp. β).
Then a physical interpretation of above identity is that α can appear as the noncommutative soliton on the N copies of the D-branes β through the tachyon condensation process
in a generalized sense.
In order that this type of interpretation is possible, we need the analogue of D D̄ pair
for this generalized setting. In the study of D D̄ pair, the tachyon condensation process is
described by the combination of gauge ﬁelds and tachyon ﬁelds (so called ‘superconnection’[26]) as argued in [5, 22, 23, 24],

T
d + A1(8)T̄
d + A2where each entry represents the various sectors of the open string.
There exists an analogue of superconnection in C ∗ -algebra which is called the linking
algebra C (see for example, [27]). It is deﬁned in such way as containing the algebras A, B
as its complementary components. Namely there exists a projection P ∈ C such thatA = P · C · P and B = (1 − P ) · C · (1 − P ). It is known that the linking algebra C exists
if and only if two C ∗ -algebras A, B are Morita equivalent. Namely if the bimodule A XB
exists, one may deﬁne the linking algebra by two by two matrices,


a x
(9)
a ∈ A, b ∈ B, x, y ∈ A XB .
ȳ b
One may easily check that the matrix multiplication of such 2 × 2 matrices is well-deﬁned
1 0
and the obvious projection operator P =gives the projection into A. By com0 0
paring (8) and (9) one sees that the Morita equivalence bimodule plays the analogous rôleas that of the tachyon ﬁeld and that the two gauge ﬁelds correspond to the endomorphism
algebras A = EndA(0) Eα and B = EndA(0) Eβ .
This analogy was implicitly used in the explicit calculation of the tachyon condensationof D D̄ system in the Moyal plane [30, 20]. We summarize our conventions for Moyal planein the appendix A. We recall that the noncommutative soliton in this case was constructed
out of the partial isometry described by the tachyon ﬁelds T and T̄ which satisfy,
T T̄ T = T,

T̄ T T̄ = T̄ .

(10)

With these relations, T T̄ and T̄ T becomes projection operators and deﬁnes the noncommutative soliton. Explicit form of T can be constructed by using the shift operator
S = n |nn + 1|, which satisﬁes S S̄ = 1 and S̄S = 1 − |00| and so on [20]. The
tachyon ﬁeld is given by T = S m (m ∈ N) with this solution generating rule. In this case,T gives the isometry between the subsets of the identical algebra A and A acts on T fromthe both side.
6

In our more general situation, the rôle of T resembles that of the Moyal plane case but
the algebra acting on T from left is in general diﬀerent from that acting from right. Thus
T should be an element of an equivalence bimodule. If we replace the partial isometryby the Morita equivalence bimodule, the statement which corresponds to (10) is that one
may choose an element T of the bimodule which satisﬁes
1. P = T, T A and Q = T, T B are the projectors of A, B.
2. It satisﬁes the analog of the partial isometry relation,
T, T AT = T T, T B = T .

(11)

While it is a nontrivial question whether we may ﬁnd such T , the requirement of these two
conditions are actually equivalent [28]. Suppose that the partial isometry-like equation(11) holds. P = T, T A and Q = T, T B are clearly self-adjoint, and direct calculationshows that P 2 = P and Q2 = Q by using the properties of the two inner products(3)(4)(5). Conversely, if we start from the condition P 2 = P , the partial isometry-likeequations follows from vanishing of the norm,
T, T AT − T, T, T A T − T A = 0.

(12)

One can also see from this proof that P 2 = P implies Q2 = Q.
We would like to propose that the equation (11) deﬁnes the noncommutative soliton
on the brane-antibrane systems in the generalized sense. We will see the explicit examples
on noncommutative tori in section 3.4.

3

Noncommutative Torus and D-branes

Because our discussions so far are given in the abstract language, it is desirable toinvestigate explicit examples in order to illuminate the idea. The simplest example is Dbranes on noncommutative plane (Moyal plane) but it is too simple to use the machinery
we would like to examine since it reduces to the (inﬁnite dimensional) matrix algebra.
Thus in this section we consider D-branes on noncommutative tori, where Morita equivalence is interpreted as T-duality [1, 25, 31, 32] and has a rich structure.
The algebra of noncommutative two-torus is generated by unitary elements U1 and U2
with the relation,
U1 U2 = U2 U1 e2πiθ ,(13)where the real number θ ∈ [0, 1] is the parameter of the algebra. We write this algebraby Aθ . The generators can be written in terms of noncommutative coordinates (x1 , x2 )with [x1 , x2 ] = −2πiα θ as follows
1√

U1 = eix / α ,

2

√



U2 = eix / α .
7(14)

A generic element a ∈ Aθ can be expanded by Ui as5a=amn U1m U2n .

(15)

m,n∈Z

While we can not realize this algebra as the matrix algebra for the irrational θ, one may
formally deﬁne the trace for Aθ as follows
Tr a = a00 ,(16)

by using the above expansion. This is equal to the integration over (x1 , x2 ). It is obviousthat it satisﬁes the fundamental relation Tr(ab) = Tr(ba).
As we saw in section 2, the gauge bundle on the D-brane is described by the projectiveAθ -module. It is known that the isomorphic class of the projective modules on a noncommutative torus is classiﬁed by their Chern characters [29] which speciﬁes the element of
K0 (Aθ ). The relation of these mathematical facts to the RR-couplings of brane-antibrane
systems will be discussed later in the subsection 3.4.
In this section, after explaining some mathematical backgrounds on the projective
module on the noncommutative tori, we discuss mass spectrum of BPS D-branes andtheir T-duality transformation. Finally we investigate tachyon condensation on non-BPS
D-branes and brane-antibrane systems.

3.1

Projective modules, Constant Curvature Connection and
Morita equivalence on Noncommutative Torus

In this subsection, we explain some mathematical results on Aθ , especially the projections, the projective modules, the constant curvature connections and Morita equivalence6
which will be essential for later arguments. The readers who is familiar with the subject
may skip this subsection.
We start from deﬁning the projection operators on Aθ . In this algebra the unitaryequivalence class7 of projections is characterized by the value of the trace Tr [28, 33] asfollows
Tr(Pn−mθ ) = n − mθ,(0 ≤ n − mθ ≤ 1),(17)

where for each pair of integers n,m which satisfy 0 ≤ n−mθ ≤ 1 there exists a equivalenceclass of projections and we wrote an element in this class as Pn−mθ . In other words K0 group K0 (Aθ ) is given by
K0 (Aθ ) = Z + θZ ∈ R .
5(18)

To be exact we should assume amn tend to zero faster than any powers of |n| + |m| as |n| + |m| → ∞.
See [21] for an more pedagogical review.
7
A projection P is said to be unitary equivalent to another projection Q if an unitary element u ∈ Aexists such that P = u∗ Qu.
6

8

The explicit construction of projections was given in [28] and is called Powers-Rieﬀel
projection. Some details of this construction are reviewed in the appendix A.
Let us turn to projective modules on the noncommutative torus Aθ . Among various
projective modules, we are interested in those which has the constant curvature connection. The explicit construction can be found in [34, 29, 1, 21] and we review this below.
Starting from a set of the rapidly decreasing functions on R, φj (x), j ∈ Zm . We deﬁne
the right action of Ui (i = 1, 2) as follows,n− θ)m(φU2 )j (x) = φj (x)e2πi(x+jn/m) .
(φU1 )j (x) = φj−1(x +(19)

One may easily conﬁrm that the operators U1 , U2 thus deﬁned satisﬁes U1 U2 = e2πiθ U2 U1 .
It shows that the Schwartz space is right Aθ module. It contains two integer parametersn ∈ Z, m ∈ Z ≥ 0 and we denote it as En,m . It has the following properties,
1. The natural ‘covariant derivative’ for this module is given by∇1 = −

2πimx,n − mθ

∇2 =

∂
,
∂x

(20)

which satisﬁes ∇i Uj = 2πiδij Uj . It has the constant curvature,
F12 = −F21 =2πim.
n − mθ

(21)b−aθ). This acts on En,m from
2. Its endomorphism EndAθ En,m is given by Aθ̃ (θ̃ = − n−mθthe left. The integers a, b are determined from n, m by the condition an − bm = 1.
The generators of Aθ̃ act on them as(Z1 φ)j (x) = φj−a(x + 1/m)
xj

(Z2 φ)j (x) = φj (x)e2πi( n−mθ + m ) .

(22)

One may easily conﬁrm that the action of Zi are compatible with the action of Ui
since [Zi , Uj ] = 0. Note that if the module is free (n, m) = (1, 0), then θ̃ = θ.
Therefore the Schwartz space deﬁnes the Morita equivalence bimodule between Aθ and Aθ̃ .
More generally, if the integers (n, m) is not coprime, then one can construct the reducible
projective module En,m = En/d,m/d ⊕ En/d,m/d ⊕ · · · ⊕ En/d,m/d , where we have deﬁned
d = g.c.d(n, m). From this one can conclude that Aθ̃ is Morita equivalent to Matd (Aθ )andif there exist coprime integers (n0 , m0 ) ≡ (n/d, m/d) and (a, b) such that θ̃ = n0b−aθ−m0 θ
an0 − bm0 = −1 [28].
It will be useful to deﬁne the explicit form of A = Aθ̃ and B = Aθ inner productfor the Schwartz space. This problem is solved in much more general sense by Rieﬀel
9[29]. However we write down some of the explicit forms for our speciﬁc example 8 . Forφ, ψ ∈ S(R × Zm ), we deﬁne,1(Z −n Z −m φ, ψ)A Z1m Z2nn − mθ m,n 2 1

=(φ, ψU2−n U1−m )B U1m U2n

φ, ψA =φ, ψBm,n ∞(φ, ψ)B = (ψ, φ)A =

dx
−∞



φi (x)ψi (x).

(23)

(24)

i∈Zm

The basic properties of inner product (3) and (4) follows from the identities (aφ, ψ)A,B =(φ, a∗ ψ)A,B , (φ b, ψ)A,B = (φ, ψ b∗ )A,B and so on. The derivation of the associativity (5) ismuch more nontrivial. We need to use the Poisson resummation formula,


 ∞
∞
∞

1  ˜ 2πm˜f (αn) =eimx f (x)dx.
(25)ff(m) =
α
α
−∞
n=−∞
m=−∞
After some calculations, one may conﬁrm,(φ, ψAχ)i (x) = (φψ, χB )i (x) =(φU1 )ri (x)(Z1s ψU1r )i (x)(Z1s χ)i (x).

(26)

r,s∈Z

Among all connections on a given projective module, the constant curvature connectionis the most useful in the physical application. One reason for this is that such a special
connection appears as the solution of the BPS equation [25, 21]. Another reason will begiven later in the arguments of the exact solution for the tachyon condensation.
Finally we would like to add a comment on the topological invariants. From theconstant curvature connection, it is rather easy to evaluate the Chern character [29, 25]
and it is known that each component becomes integer after the modiﬁcation similar toMyers term [35]9 . It is of some interest to conﬁrm this fact by using other form of
the connection/curvature. In general a connection on projective module (“Levi-Civita
connection”) is constructed in the form (see for example [34, 21, 11]) ∇i = Pn−mθ ·δi ·Pn−mθ .
The trace of the curvature (ﬁrst Chern class) is the cyclic 2-cocycle τ2 for Pn−mθ . In
appendix A we evaluate it by using the Powers-Rieﬀel projection as Pn−mθ and deriveτ2 = m. This is of course consistent with the computation from the constant curvatureconnection (21).
8General formula for the generic Heisenberg module is not diﬃcult to write down. One outline issketched in Appendix A of [16].
9This modiﬁcation can also be regarded as ‘quantum eﬀect’ in terms of the quantum twisted bundle[36]. This procedure shows the relation between a bundle on the commutative torus and that on thenoncommutative torus geometrically, and agrees with so-called Seiberg-Witten map[2]. We will use thismap in eq.(37).

10

3.2

Noncommutative Description of BPS D-branes and T-duality

Here we discuss the spectrum and the T-duality transformation rule of the BPS Dbranes on a two dimensional torus with a B-ﬁeld ﬂux. As is well-known, there are two
viewpoints for this system. One is the conventional description (commutative description) using the closed string variables. The other is by the open string variables with
the noncommutative geometry [1, 2]. The results given here will be useful in the later
discussions. Some general arguments can be found in [37, 21].
We investigate the dynamics of D-branes on the noncommutative two-torus Aθ . Since
we restrict our interest to the two dimensional case, we discuss D2-D0 bound states on
the torus below. If the D2 charge and D0 charge are given by (n, −m), the mass of D2-D0
bound state is determined as follows|n| Sdet (g + 2πα(B + F )),(27)
MBP(n,m) = √ 
α gs
where we have deﬁned the gauge ﬁeld strength
J m
F =
,
2πα n


J≡

0 −1
1 0


.

(28)

In this formula we have taken the eﬀect of −m D0-branes into account as the shift ofgauge ﬁeld strength. Note that the value of n or −m can be negative integer because of
T-duality. For a review of T-duality on general tori see [38]. In the following we examine
the transformations from a D2-brane to various D2-D0 bound states.
We deﬁne E = g + 2πα B ∈ Mat2 (R). A single D2-brane has the mass,S
MBP(1,0) =1 √
det(E) .
gs α 

(29)

The T-duality group on T2 is given by SO(2, 2|Z) and this acts on g, B and gs as follows[38]Ẽ = T (E) = (AE + B) (CE + D)−1
g˜s = T (gs ) = gs det (CE + D)


A B
T =
∈ SO(2, 2|Z).
C D

− 21

(30)
(31)
(32)

S
The mass MBP(1,0) transforms into the following formS
T (MBP(1,0) )

1√
det T (E)T (gs ) α1 
= √det(AE + B) .
gs α 
=11(33)

When the dimension of the torus is two, the T-duality group can be decomposed asSO(2, 2|Z)  SL(2, Z) × SL(2, Z). One of two SL(2, Z) groups
is the modular transfor
2
mation of the target space T . Because it preserves gs and det(E), we will not consider
this part. The other SL(2, Z) is related to Morita equivalence for noncommutative T2[1, 25, 31, 32] and we concentrate on this part. This SL(2, Z) transformation can beembedded into SO(2, 2|Z) as follows
SL(2, Z) 2→ SO(2, 2|Z)




n m
n1 mJ
→
.
−b −a
bJ −a1(34)

Applying this transformation eq.(33) can be rewritten asS
T (MBP(1,0) ) =1 √
det(n(g + 2παB) + mJ) .
gs α 

(35)

This is the same value as (27) and conﬁrms that the D2-brane mass in the backgroundẼ = T (E) is equal to the mass of (n, −m) D2-D0 bound state. Note also that the massfor (n, −m) is equal to that for (−n, m). This implies that these two conﬁgurations should
be an identical state and we can restrict the integers (n, −m) to n − mθ ≥ 0.
We translate these results into the noncommutative description. For simplicity, we ﬁx
the choice of the parameter Φ [25, 32, 2] as B = −Φ. The relation between the variablesin the open and closed string theories is given [2] as follows
B = −Φ = −1J,2πα θ

1
G = −(2πα )2 B B,g1

Gs = gs det(2πα Bg −1 ) 2 .

(36)

This map is deﬁned so that the mass of the single D2-brane coincides in both (open/
closed) descriptions. The transformed backgrounds G and Gs are called open string
metric and open string coupling respectively.
When the ﬁeld strength F is constant, the ﬁeld strength F̂ in the open string description10 is given by [2]
F̂ =

F
.
1 + 2πα θJF(37)

If we apply this to the D0-D2 bound states, the ﬂux (28) is transformed into
F̂ =

1
m
J.

2πα n − mθ(38)This is the same as the constant curvature connection on the projective module En,mreviewed in the previous subsection.
10Note that the curvature in mathematical conventions in section 3.1 is related to the ﬁeld strength F̂
here such that Fij = −4π 2 iα F̂ij .

12

Furthermore, the mass of D0-D2 bound states can also be written in terms of the open
string variables as follows
|n| 
S
BP S
√
)
=
M
=
det (g + 2πα(B + F ))T (MBP
1,0(n,m)
α  gs



1 m
|n|
det g − ( − )J= √θnα  gs1 ndet(2παBg −1 )− 2√
det −(2πα )2 Bg −1 B(n − θm) + J
=

θ
α gs

n − mθ
det(G + 2πα(F + Φ)).
(39)
= √α  Gs
Note that the rank |n| of the gauge ﬁeld in the commutative description is replaced with
the non-integer ‘rank’ n − mθ ≥ 0. In noncommutative geometry such an appearance of
non-integers is not surprising but very natural. Indeed this is equal to the dimension of
the projective module dim(En,m ) = n − mθ. As we will show later, it appears naturally
in the processes of tachyon condensation. We also comment that the above formula iscorrect for any choice of Φ.
Finally we present an interpretation of the above results from the viewpoints of Tduality T on the noncommutative side. It is derived by rewriting the action T in (34) interms of the open string variables in (36) [25, 32, 21],θ̃ =b − aθ
, G̃µν = (n − mθ)2 Gµν ,n − mθ

G̃s = (n − mθ)Gs , 2πα Φ̃ = (n − mθ)2 (2παΦ +m
J).
n − mθ

(40)

The transformation for θ is exactly the same as the Morita equivalence (22).
By the deﬁnition of the transformation (40), the map from the closed string variable
to the open string variable (36) and the action of the T-duality group on both (openS/closed ) sides (32) and (40) are compatible. Therefore the mass of a bound state MBP(n,m)
S
in the last line of (39) can also be obtained by acting the transformation (40) on MBP(1,0)
in the open string variables
SMBP(n,m) = √1α G̃sn − mθ
det(G̃ + 2πα Φ̃) = √α  Gsdet(G + 2πα (F̂ + Φ)) .

(41)

This shows that a (n, −m) D-brane on the noncommutative torus Aθ is a single D2-brane
on Aθ̃ . This result is consistent with the previous arguments in the commutative (closed
string) side. Notice that the curvature F̂ on the noncommutative torus Aθ vanishes on
the corresponding single D2-brane on Aθ̃ due to the shift of Φ in eq.(40).
The above arguments of T-duality can also be applied to non-BPS D-branes andbrane-antibrane systems in the same way. We will see later that this T-duality on the13noncommutative side is more directly related to Morita equivalence in the arguments oftachyon condensation.

3.3

Tachyon condensation on non-BPS D-branes

Let us discuss the tachyon condensation on non-BPS D-branes (see for example [6])on the noncommutative torus Aθ . The same arguments can be applied to the bosonicstring. Because any D2-D0 bound state of non-BPS D-branes can be transformed into a
D2-brane, we can begin with a non-BPS D2-brane. The relation between the variables inopen and closed string theories is the same as (36) and we continue to choose the value ofΦ as Φ = −B to obtain the simplest expression. The solutions, however, do hold without
any modiﬁcation for general values of Φ with somewhat lengthy calculations.
On any non-BPS D-brane there exists11 a (real scalar) tachyon ﬁeld T and a gauge
ﬁeld Aµ . As argued in [39] the eﬀective action of a non-BPS D2-brane can be written as√
2
dtTr V (T ) det(G + 2πα (F̂ + Φ)) + O([∇, T ], [∇, F̂ ]),(42)
S=√
α  Gswhere [∇, T ], [∇, F̂ ] denote the covariant derivative of the tachyon ﬁeld T and the gauge
ﬁeld strength F ; the symbol O([∇, T ], [∇, F̂ ]) means those terms which include one ormore derivatives of T and F̂ . As we will see below our exact arguments of tachyon
condensation do not depend on the detailed form of the derivative terms. The factor V (T )in front of the Born-Infeld term represents the tachyon potential. We normalized the valueof the tachyon potential such that its value before and after the tachyon condensation into
the vacuum are given by V (1) = 1 and V (0) = 0 following from Sen’ conjecture [7, 6].
We use here the open string variable and therefore all the ﬁelds on the brane are
regarded as the operators on the noncommutative torus Aθ . On the non-BPS D-brane,
the tachyon and the gauge ﬁeld belongs to the adjoint representation of the gauge group.
In the language of the noncommutative geometry they are expressed as elements in theendomorphism EndA(0) E of the projective module E. Before the tachyon condensation,
the projective module should represent the original D2-brane itself E1,0 = Aθ . After thetachyon condensation, it should be projected into a nontrivial projective module of Aθ
which we investigate below.
We would like to solve the equation of motion of the tachyon ﬁeld by imposing several
assumptions. Suﬃcient conditions are∂V (T )= 0,∂T[∇, T ] = 0.

(43)

The ﬁrst equation is equivalent to the equation of motion if we take the large B limit
as discussed in [8, 9, 10]. The solutions to this equation are given by the projections in
11

In this paper we discuss the cases where the transverse scalars do not have expectation values.

14the noncommutative torus algebra Aθ . The projections in Aθ are classiﬁed by the valuesof trace as in (17) and we write T = Pn−mθ . Next we examine the second condition[∇, T ] = 0. This is satisﬁed12 if we use the connection ∇i = Pn−mθ (δi + Ai )Pn−mθ . It
deﬁnes the projective module E = Pn−mθ E1,0 and the endomorphism EndAθ E is given byPn−mθ Aθ Pn−mθ . The projective module E is twisted and its ﬁrst Chern class (or cyclic
2-cocycle) is given by τ2 (Pn−mθ ) = m as we explain in the appendix A.
Now let us turn to the equation of motion for the gauge ﬁeld [∇i , F̂ij ] = 0. It issatisﬁed if the ﬁeld strength is proportional to Pn−mθ . In a sense, Pn−mθ can be regarded
as the identity in the algebra EndAθ E. The ﬁeld strength which is proportional to theprojector should be regarded as constant curvature reviewed in the section 3.2.
One may prove that for any projection of type Pn−mθ the projective module of theform E = Pn−mθ E1,0 = Pn−mθ Aθ has a constant curvature connection. As we saw inthe subsection 3.1, for every n, m one can construct a constant curvature connection
as the Heisenberg projective module. Because it is projective, it should be written asẼ = P̃n−mθ ANθ for a certain projection P̃n−mθ in MatN (Aθ ). Because any projection in
MatN (Aθ ) for a given value of trace belongs to the same K-theory class [28], we can changeP̃n−mθ into any Pn−mθ ∈ Aθ via an unitary transformation. The transformed projective
module E = Pn−mθ Aθ also possesses the induced constant curvature connection.
In this way we have found exact solutions of the equation of motion derived from (42).
T = Pn−mθ ,

F̂ =

m
1
Pn−mθ J.
2πα n − mθ

(44)

Here we represents the ﬁelds as elements in the ‘large’ algebra Aθ . In the small algebra
EndAθ E, both T and Ai are proportional to the identity. Unlike the Moyal plane case,the small algebra is Morita equivalent to Aθ and can be rewritten as
EndAθ E = Pn−mθ Aθ Pn−mθ = Matd (Aθ̃ ),(θ̃ =b − aθ
s.t. an0 − bm0 = −1), (45)n0 − m0 θ

where we have deﬁned d = g.c.d.(n, m) and (n, m) = d(n0 , m0 ).
We proceed to discuss what will be generated via the tachyon condensation (44). The
mass of this excitation can be evaluated by neglecting the derivative terms because of[∇, T ] = [∇, F ] = 0,√2
Tr Pn−mθ det(G + 2πα(F̂ + Φ))M(n,m) = √α  Gs√2|n| 
= √det (g + 2πα (B + F )),(46)
α  gs12

One can also satisfy [∇, T ] = 0 if the connection acts only on the projective module E  = (1 −Pn−mθ )E1,0 . However a little analysis shows that this is not consistent with the descent relation [40]and does not have a expected tension. Therefore we believe that this does not correspond to physicalsolutions and neglect these in this paper.

15√
where the ﬂux F is given by (28). The factor 2 is peculiar to non-BPS D-branes. The
above calculation can be done in the same way as in (39). The dimension of the projective
module naturally appears here as the trace of the projection. It is interesting that thetachyon condensation on unstable D-brane systems gives an explicit physical realizationof the mathematically fundamental relation between projective modules and projections.
If we assume gij = R2 δij to make discussion clearer, the mass reduces to√2  2 2n R + (n/θ − m)2 .
(47)
M(n,m) = √α  gsIt explicitly shows that the resulting state is a bound state of n non-BPS D2-branes and(−m) non-BPS D0-branes. We comment that our arguments of tachyon condensation
naturally derive the fact that −m or n can be negative which is consistent with the resultin the previous subsection13 . In this way we obtain all kinds of D2-D0 bound states via
tachyon condensation and thus our results are consistent with T-duality.
If we take the large B/g limit, the mass spectrum is proportional to dim(E) = n − mθas can be seen from (47) and it is dense in R. This means that there exists a excitation ofwhich energy is arbitrary small. In [14] we investigated the tachyon condensation in thislimit and suggested that it leads to the instability. It means that any projection Pn−mθ
can be divided into inﬁnite numbers of mutually orthogonal smaller projections while
the total value of the trace is preserved. This was proved by investigating an explicit
representation of projections.
The argument can be simpliﬁed as follows. We start from a projection Pn−mθ in Aθ .
It can be regarded as the identity in the small algebra Matd (Aθ̃ ) = Pn−mθ Aθ Pn−mθ . One
can ﬁnd another projection Q in the small algebra. The original projection is decomposed
into two mutually orthogonal projections,Pn−mθ = (1 − Q)Pn−mθ + QPn−mθ .

(48)

One may continue this operation repeatedly to give the inﬁnitely small mutually orthogonal projector.
We would like to claim that such instability does not appear for ﬁnite B/g. As
can easily be seen from the mass formula (47), the bound state can be divided intod = g.c.d (n, m) pieces but is not divided further. It means the instability cannot appear.
We may interpret it from our exact tachyon solution. The key point is the requirement of
the constant ﬁeld strength in (44) which is absent in large B/g limit. It is permitted tobe divided into only d mutually orthogonal parts, even though the tachyon ﬁeld in (44)13

One may ask the physical interpretation for the negative n. Assume that n is negative and thus −m
is positive. Since D0-branes with B-ﬁeld on a torus generate (non-BPS) D2-brane, we can say that thenegative n means the annihilation of these induced non-BPS D2-branes with the |n| non-BPS D2-branes.

16can be divided into inﬁnitely many pieces. We conclude that for ﬁnite B/g the boundstates are all stable if g.c.d (n, m) = 1.
The corresponding projective module En,m = Pn−mθ E1,0 can be written as a direct sumof d projective modules of the same type as we saw in section 3.1. As shown in [41] (see also[21]), the moduli space of constant curvature connection is given by the symmetric productof d copies of a two-torus (T2 )d /Sd . This is actually the same as the physical moduli spaceof the solutions of (42) up to gauge transformation. The freedom of unitary transformationof the tachyon ﬁeld is absorbed in the gauge transformation T → UT U ∗ , U ∈ Aθ andwhat remains is only the moduli space of constant curvature connection for the projective
module En,m = Pn−mθ E1,0 . We note that this is consistent with the physical intuition. The
moduli space of a bound state for coprime (n, m) parameterizes the transverse coordinate
for a D0-brane, namely T2 . It is then obvious that the moduli space of the bound statesof d D0-branes should be (T2 )d /Sd .
The ﬂuctuations of the gauge and tachyon ﬁelds around the solution (44) belong to
Matd (Aθ̃ ). This is physically interpreted as the gauge group on the brane is U(d).
Up to now we concentrate on the tachyon condensation from a single D2-brane. We
may, of course, start from plural or even inﬁnitely many D2-branes. In such situation, we
can obtain arbitrary projective modules En,m from the projection in Mat∞ (Aθ ).
It is also important to note the relation between tachyon condensation on the above
noncommutative torus and that on the non-compact ﬂat plane (Moyal plane). The exact
solution for the latter [20] can be rewritten in our convention as ﬂows,
T = Pl ,

F̂ =

1
Pl J,
Θ

(l ∈ Z ≥ 0)(49)

where the noncommutativity Θ is deﬁned as [x1 , x2 ] = iΘ. The level-l projection Pl in
the Moyal plane algebra is given by l−1
k=0 |kk| and it corresponds to the generation ofl D0-branes. To obtain such situation from the torus, we need to take the large radius
limit or equivalently the small θ limit. In such a situation, the value of n is restricted to
0 or 1 which is consistent with the result (49). We note that in this limit one cannot takeB/g → ∞ limit and thus the instability does not occur.
Solution generating technique and Morita equivalence
We have seen the exact description of tachyon condensation is characterized by theconstant curvature connection. It is interesting to ask what is the solution generating
method which relates various solutions. In the Moyal plane, the exact solutions for tachyon
condensation were constructed in [20] by using the shift operator S = ∞n=0 |n + 1n|.
We would like to ﬁnd the analogous transformation on the two-torus Aθ .
In this case we have to be careful since the corresponding operator in general interplotes
the diﬀerent C ∗ -algebras. Namely after the tachyon condensation the original free module
E1,0 = Aθ (a D2-brane) is changed into the twisted projective module En,m (a D2-D0
17

bound state). As we have seen in section 2, the transformation between these two solutions
should be identiﬁed with the Morita equivalence bimodule S. It depends on the integers
n, m and satisﬁes
En,m = S ⊗Aθ E1,0 .

(50)This maps the endomorphism EndAθ E1,0 = Aθ into EndAθ En,m = Matd (Aθ̃ ). Physicallythis induces the transformation of the world-volume ﬁeld theories and this gives an explicit
realization of the descent relation [40].
In order to describe the exact solution, the projective module En,m should have theconstant curvature. In other words, we have to impose on the Morita equivalence Aθ̃ -Aθ
bimodule that S should keep this additional constraint. Actually it just ﬁts the deﬁnitionof the gauge Morita equivalence bimodule discussed in [21, 25]. We claim that this is theanalogue of the shift operator on the noncommutative torus.
We mention that the T-duality transformation can also be represented by the gauge
Morita equivalence bimodules as argued in [25, 32, 21]. On a noncommutative torus Aθ
there is a one-to-one correspondence between Aθ -modules and the solutions found in (44).
They correspond to the projective module En,m and take their values in EndAθ En,m . If we
perform T-duality so that the (n,−m) brane is transformed into d D2-branes (or equally
applying the Morita equivalence Aθ ∼ Aθ̃ ), the projective module En,m is changed into
the free module Ẽd,0 in the algebra Aθ̃ . Such T-duality transformation can be constructedby Aθ -Aθ̃ gauge Morita equivalence bimodule X
Ẽd,0 = En,m ⊗Aθ X .

(51)

The endomorphism EndAθ En,m = EndAθ̃ Ẽd,0 acts on Ẽd,0 from the left in (51). Especially,
the solution T = 1 ∈ EndAθ En,m is the identity in EndAθ̃ Ẽd,0 . On the other hand,
1mF̂ = 2πα n−mθ J ∈ EndAθ En,m is translated to 0 ∈ EndAθ̃ Ẽd,0 , because Ẽd,0 is the freemodule over Aθ̃ . The diﬀerence between the values of the constant curvatures comesfrom the constant curvature of X. This is equivalent to the shift of the ﬁeld Φ by Tduality as in (40). These imply that for general noncommutative algebras the exact
solution for tachyon condensation can be generated in the same way if there exist gauge
Morita equivalence bimodules.

3.4

Tachyon condensation on brane-antibrane systems

Brane-antibrane systems are more complicated and intriguing than non-BPS D-branes
from the viewpoint not only of string theory but also of the noncommutative geometry.
The crucial diﬀerence from non-BPS D-branes is that the tachyon ﬁeld becomes complex
and belongs to a Morita equivalence A-B bimodule (we write this as X), where A and
B are the algebras of the brane and the antibrane as we discussed in section 2. If we18

deﬁne E and F as the projective modules which represent the brane and the antibrane,
respectively and deﬁne A(0) as the noncommutative base space, the algebras A, B is given
by A = EndA(0) E, B = EndA(0) F . Remember that a Morita equivalence bimodule X isdeﬁned by the bimodule which possesses two types of inner product  , A ,  , B andsatisﬁes the conditions (3,4,5).
As in the previous subsection we use the example of noncommutative two-tori andconsider only D2-D0 bound states. We assume the original brane-antibrane system ismade of a (n1 , −m1 ) brane and a (n2 , −m2 ) antibrane. For simplicity we consider only
the case where the pairs of integers (n1 , −m1 ) and (n2 , −m2 ) are coprime. A pair of
integers (n, −m) denote the indices of D2-D0 bound state or equivalently of those thecorresponding projective module En,m . Note that if one speciﬁes (n, m) such that n−mθ ≥0, then there are two types of D-branes, that is, branes and antibrane. In our examplesof the noncommutative torus the fundamental algebra A(0) is given by A(0) = Aθ , whereθ is represented in terms of closed string variables as in (36). The algebras A and B aregiven byb1 − a1 θ,
n1 − m1 θ
b2 − a2 θ
B = Aθ2 , θ2 =,n2 − m2 θ

A = Aθ1 , θ1 =(a1 n1 − b1 m1 = −1)(a2 n2 − b2 m2 = −1).

(52)

The tachyon ﬁeld T belongs to a Aθ1 –Aθ2 bimodule. There are also the gauge ﬁelds on
the brane and the antibrane. We denote these as A(1) and A(2) . These ﬁelds belong to
A = Aθ1 and B = Aθ2 , respectively. The covariant derivative of the tachyon ﬁeld is givenby the connection for a bimodule X (see for example [25, 21]) speciﬁed by the requirement∇X (aT ) = δA (a)T + a∇X T(∀a ∈ A),∇X (T b) = (∇X T )b + T δB (b) (∀b ∈ B),(53)

where δA and δB denote the derivation in A and B, respectively. We also use the covariant
derivative of ﬁeld strengths F̂ (1) and F̂ (2) . Each of them is given by the commutator witha connection for the algebra A or B as in the previous subsection.
Now we have prepared to discuss the tachyon condensation on noncommutative tori.
The eﬀective action for a D2−D2 with two gauge ﬂuxes was already computed in [24]using the boundary string ﬁeld theory [42, 43, 44]. Applying this to our system on anoncommutative torus the result is given by

1
S = √
dtTrA V (T, T A ) det(G + 2πα(F̂ (1) + Φ))
α  Gs1
dtTrB V (T, T B ) det(G + 2πα(F̂ (2) + Φ))+ √α  Gs+ O(∇X T, [∇A , F̂ (1) ], [∇B , F̂ (2)]),(54)
19

where O(∇X T, · · ·) denotes the derivative terms. We deﬁned the traces for A and B byembedding these algebras in MatN (A(0) ) for a suﬃcient large integer N. These satisfy
the following relation [28, 29]
TrA T1 , T2  = TrB T2 , T1 ,(55)

and are normalized by TrM atN (A(0) ) 1 = N.
We would like to solve the equation of motion for the action (54). Below we give
solutions by imposing ansatz similar to the previous subsection. We assume the existenceof the partial isometry-like equation [30, 20, 19] for the tachyon ﬁeld (11)T, T A T = T,

T T, T B = T.

(56)

Note that these two equations are equivalent thanks to the relation (5). The solutions to this equation give the stationary points of the tachyon potential V (T, T A )and V (T, T B ). To make exact solutions for ﬁnite B/g, we should take account of thegauge ﬁelds. It is easy to see the equation of motions for the tachyon T and gauge ﬁelds(A(1) , A(2) ) are satisﬁed if we require (56) and∇X T = 0,(57)[∇A , F̂ (1) ] = 0, [∇B , F̂ (2)] = 0,(58)

are satisﬁed. Since (56) is equivalent to the statement that T, T A ∈ A and T, T B ∈ B
are both projections as explained in section 2, we can write these as follows
T, T A = 1 − Pα+βθ1 (≡ 1 − P1 ),T, T B = 1 − Pγ+δθ2 (≡ 1 − P2 ).

(59)

Using the relation (55) we obtain the constraint,
n1 (1 − α) − b1 β = n2 (1 − γ) − b2 δ,

m1 (1 − α) − a1 β = m2 (1 − γ) − a2 δ.

(60)

They determine γ and δ in terms of α and β. The tachyon ﬁeld which condensates as in(59) belongs to the bimodule (1 − P1 ) · X · (1 − P2 ). For this tachyon ﬁeld the potential
is evaluated as
V (T, T A ) = Pα+βθ1 ,V (T, T B ) = Pγ+δθ2 .

(61)

This is because we use the convention that the original brane-antibrane system corresponds to T = 0 and that the vanishing of brane and antibrane corresponds to T, T A = 1
and T, T B = 1, respectively14 .
14

Notice that this is the opposite convention to the tachyon potential of non-BPS D-branes in section

3.3.

20

Let us turn to the next equation (57). This is satisﬁed if the gauge ﬁelds belong toA(1) ∈ P1 AP1 ,A(2) ∈ P2 BP2 .

(62)

The last equation (58) is solved if we assume that both of the gauge ﬁelds have constantcurvatures1
βa1 + m1 α
1 ∈ P1 AP1 ,2πα n1 α + b1 β − (βa1 + m1 α)θ
δa2 + m2 γ
1=
1 ∈ P2 BP2 .

2πα n2 γ + b2 δ − (δa2 + m2 γ)θ

F̂ (1) =F̂ (2)(63)

It ﬁnishes our calculation of the physical solutions of the tachyon condensation. Our
result does not depend on the detailed form of the derivative term because of (57,58).
We evaluate mass spectrum of these classical solutions,
1
TrA Pα+βθ1 det(G + 2πα (F̂ (1) + Φ))
α  Gs1
TrB Pγ+δθ2 det(G + 2πα (F̂ (2) + Φ))+ √α  Gs
|N1 |
|N2 |
= √
det(g + 2πα (B + F (1) )) + √det(g + 2πα (B + F (2) )). (64)α gs
α  gsM = √We have deﬁned the ﬂuxes as follows
F (1) =J M1,2πα N1

F (2) =J M2,2πα N2(65)

where integers M1 , N1 , M2 and N2 are given by
N1 = αn1 + βb1 ,M1 = βa1 + αm1

N2 = γn2 + δb2 ,

M2 = δa2 + γm2 .

(66)

We ﬁnd that the products of the tachyon condensation are identiﬁed with a (N1 , −M1 )brane and a (N2 , −M2 ) anti-brane. If these integers are not coprime, each bound state
can be divided into several parts as before. Note that what are produced after the tachyon
condensation depend on the projections (61). The original brane-antibrane system corresponds to α = γ = 1, β = δ = 0. If one assumes that the dimension of the projective
module E is larger than that of F , then the tachyon ﬁeld γ = 0, δ = 0 gives the maximal
condensation and this will produce a (n1 − n2 , −m1 + m2 ) brane. In the opposite case thetachyon ﬁeld α = 0, β = 0 will generate a (n2 − n1 , −m2 + m1 ) anti-brane.
For general decay modes the diﬀerences of the D2-brane charge (N1 − N2 ) and theD0-brane charge (M2 − M1 ) are preserved as follows
N1 − N2 = n1 − n2 ,M2 − M1 = m2 − m1 .
21

(67)

The charge conservation can also be discussed in the framework of operator algebra
K0 -group K0 (Aθ ). If one would like to consider the K0 -group of noncommutative torus,
the Chern character [34] gives enough information [29]. In the brane-antibrane system
which corresponds to the pair of projective modules (E, F ) ∈ K0 (Aθ ) the K-theory charge
is given by the diﬀerencech(E) − ch(F ) = TrA exp(2παF̂ (1) ) − TrB exp(2πα F̂ (2) )= (n1 − m1 θ − n2 + m2 θ) + (m1 − m2 )dx1 dx2 ,(68)

where dx1 dx2 is the two form along the two-torus. It is known that the RR-couplings ona brane-antibrane system can be written by using K 0 -type superconnection [22, 23, 24].
Applying this idea to our examples we obtain the following RR-couplings
SRR ∼ CRR ∧ TrA V (T, T A ) exp(2παF̂ (1) ) − TrB V (T, T B ) exp(2πα F̂ (2) ) . (69)Note that in our noncommutative description the derivative of tachyon ﬁeld is always
zero and does not contribute. Also notice that the potential in the above can be regarded
as the identities in the algebras of gauge ﬁelds EndAθ E and EndAθ F because of (61).
Thus the conservation of RR-charge (67) is equal to that of operator algebra K-theory
charge and therefore this gives a further support to the relation between D-brane charge
and K-theory [4, 5, 13]. The intriguing characteristic that the above ‘noncommutative
Chern character’ depends on θ will correspond to the physical fact that D0-branes in the
B-ﬁeld background generate D2-brane charge [45]. Therefore it will also be interesting toclarify the relation between the RR-couplings for non-abelian transverse scalars [35, 24]
and the above RR-couplings on various noncommutative tori [45] (see also [46] for the
Moyal plane).

4

Conclusions

We ﬁrst discussed the open string theory in general noncommutative background. We
considered a general framework to handle open strings and D-branes in a uniﬁed way byutilizing the Morita equivalence. In particular we proposed the equation which deﬁnes
noncommutative solitons on general brane-antibrane systems.
From this viewpoint we have examined the exact solutions on noncommutative toriin tachyonic systems. For non-BPS branes the solutions are given by the tachyon ﬁeld
which is proportional to the projection and the gauge ﬁeld with a constant curvature. This
respects the one-to-one correspondence between a projection and a projective module. We
have also shown that these solutions can be generated by employing the (gauge) Morita
equivalence. Our exact description of tachyon condensation including the gauge ﬁeld
solves for ﬁnite B/g the previously observed instability problem.
22

More complicated and thus more intriguing examples are brane-antibrane systems.
In this case the tachyon ﬁeld belongs to the Morita equivalence bimodule and we can
impose the partial isometry-like relation instead of the equation of motion. We canconstruct the exact solutions and determine the decay products. We ﬁnd the RR-chargesof brane-antibrane systems can be represented by the superconnection-like extension of
the Connes’s Chern character and check that these charges conserved in the process oftachyon condensation. This also veriﬁes the fact that the D-brane charge is classiﬁed by
the operator algebra K-theory.
Note added: After completing our calculations, we noticed the preprint [47] on the net
which has some overlaps with our results in section 3.3.

Acknowledgments
We are grateful to M. Hamanaka, K. Ichikawa, A. Kato, K. Ohmori, Y. Sugawara, S.
Terashima and T. Uesugi for useful discussions. We are also happy to thank H. Moriyoshi
for his clear lecture on noncommutative geometry and useful comments. H.K. and T.T.
are supported by JSPS Research Fellowships for Young Scientists. Y.M. is supported
in part by Grant-in-Aid (# 707) from the Ministry of Education, Science, Sports andCulture of Japan.

A

Powers-Rieﬀel Projections and Cyclic Cocyles

Here we give a review of Powers-Rieﬀel projections [28] on two dimensional noncommutative tori and of the calculations of their topological charges [34, 3]. We also mention
some other projections constructed in [14].
First let us consider projections in any C ∗ -algebra A and assume that there exist a
trace Tr : A → C and derivations δi : A → A such that δi (ab) = δi (a)b + aδi (b). We also
normalize the trace as Tr(1) = 1. The index i corresponds to the basis of the derivations15 .
A projection p ∈ A is deﬁned to be a self-adjoint idempotent p∗ = p = p2 . The Connes’s
Chern Character is deﬁned as the exponential of the curvature F ∈ EndA E of projective
module E [34]:∞ F  1
ch(E) = Tr exp=
τ2k (F, F, · · ·, F ).
2πi
k!
k=0

(1)

The each term τ2k (F, F, · · ·, F ) of the above expansion represents the contribution which
is proportional to F k . Note that τ0 is equal to the trace of identity and it gives thedimension dim(E) of the projective module.
15

More precisely, the derivation is deﬁned as an action of a Lie group G on the algebra A [34]. Thenthe basis of the derivation can be said as those of the Lie algebra.

23

Since any projective module E is described as P ·AN using a projection P in MatN (A)for a suﬃciently large integer N, one can rewrite the curvature in terms of the projection.
More explicitly one can choose a connection of E as ∇i = P · δi · P [34, 21]. Note thatthe topological quantity such as the Chern character does not depend on the choice ofthe connection. Then the curvature F ∈ EndA E is expressed as
Fij = [P δi P, P δj P ] = P (δi P )(δj P ) − P (δi P )(δj P ) + P (δi δj − δj δi )P.

(2)

Because the trace in EndA E = P · MatN (A) · P is naturally induced from the trace in
MatN (A) normalized as Tr(1) = N, one can always calculate the Chern character (1) if
the projection P is given. Thus we write the 2k-th part by τ2k (P ). This can be regarded
as the cyclic 2k-cocycle where the projections are substituted.
For example, the cyclic 0-cocycle is given by
τ0 (P ) = Tr(P ) = dim(E).

(3)

Now let us investigate the explicit examples of cyclic cocycles. First we consider aﬂat two dimensional plane (Moyal plane) algebra. We employ the operator representationand deﬁne the noncommutative coordinate (x1 , x2 ) as [x1 , x2 ] = iΘ. Further we deﬁne
the creation and annihilation operator a† = √12Θ (x1 − ix2 ) and a = √12Θ (x1 + ix2 ) suchthat they satisfy [a, a† ] = 1. Using this one can deﬁne the basis of the Hilbert spaceas the familiar n-number state |n = √1n! (a† )n |0. Then the algebra is expanded by|nm| (n, m ≥ 0). Let us consider the projection Pn = n−1
k=0 |kk| for a ﬁnite integer
n. The trace of this operator is given by Tr(Pn ) = 2πΘ · n, where we have normalized thetrace so that it is equal to the integration dx1 dx2 in the c-function representation. Now
it is straightforward to calculate τ2 (Pn ) since the derivations δ1 , δ2 along the coordinate
x1 , x2 are given by
δ1 =

i 2
[x , ],
Θi
δ2 = − [x1 , ].
Θ

(4)Thus we obtain the result as follows

1
Tr Pn [a† , Pn ][a, Pn ] − Pn [a, Pn ][a† , Pn ]
2πΘ


1
=
Tr Pn + Pn a† Pn aPn − Pn aPn a† Pn
2πΘ
= n.

τ2 (Pn ) =(5)

Therefore the value of τ2 (Pn ) is quantized and is positive. Note that this value is equal tothe ﬁrst Chern class of the projective module and therefore should be quantized. Then
one may ask if one can obtain negative integers ? The answer is yes and one can construct
the corresponding projection as 1 − Pn . In this case we obtain the value τ2 (1 − Pn ) = −n.
24After this elementary example, let us turn to the two dimensional noncommutative
torus Aθ . We assume θ is irrational because for rational θ there are ﬁnite dimensional
representations of the algebra and the calculations are simpliﬁed (see also [17, 14, 16, 18]).
As we have mentioned in section 3.1, the projections in Aθ are generally characterized
by their values of trace as in eq.(17). The explicit construction of projections (PowersRieﬀel projection) was given in [28] and let us review this below.
Because U1 and U2 do not commute (U1 U2 = U2 U1 e2πiθ ), we can diagonalize only U2
2
and deﬁne c-number x2 as U2 = e2πix . Thus we obtain the following (inﬁnite dimensional)
representation of Aθ
U1 |x2  = |x2 + θ,
2πix2

2

U2 |x  = e

(6)

2

|x .

(7)

Then the trace of an element a ∈ Aθ is given by 1
Tr(a) =
dx2 x2 |a|x2 .

(8)0

In order to ﬁnd explicit projections P we assume the following form
P = U1∗ (g(U2 ))∗ + f (U2 ) + g(U2 )U1 .

(9)

As we will see below, one can construct a projection for each n, m even under this restriction. Acting on the position space |x2 , we require P 2 |x2  = P |x2 . This deﬁnes a
projection in Aθ if and only if f and g satisfy the following relations22g(e2πix )g(e2πi(x +θ) ) = 0 ,
2

2

2

2

2

g(e2πix )[1 − f (e2πix ) − f (e2πi(x +θ) )] = 0 ,22

f (e2πix )[1 − f (e2πix )] = |g(e2πix )|2 + |g(e2πi(x −θ) )|2 .

(10)

Explicit forms of f, g which satisfy these relations are given as follows. Choose any
2small 8 > 0 such that 8 < θ and θ + 8 < 1, and let F (x2 ) ≡ f (e2πix ) for one period begiven in the range x2 ∈ [0, 1] by


x2 /8
x2 ∈ [0, 8]
1x2 ∈ [8, θ]F (x2 ) =,(11)

1 − (x2 − θ)/8x2 ∈ [θ, θ + 8]
 0x2 ∈ [θ + 8, 1]
Then deﬁne g for one period by

2πix2

g(e

)=

F (x2 )(1 − F (x2 ))x2 ∈ [0, 8],x2 ∈ [8, 1] .

0
25

(12)

It is easy to see that the functions f and g, deﬁned as the periodic extensions of theabove, satisfy the relation (10). It can be easily shown that
 1
 1
2
2
2
dx < x |P |x >=dx2 F (x2 ) = θ .
(13)
Tr P =
00Thus the projection P now constructed corresponds to Pθ .
Now how about more general projections Pn−mθ ? Such general projections can be constructed by slightly modifying the above constructed Pθ as follows. The general projectionPn−mθ can be regarded as the projection Pθ in the algebra Aθ if we deﬁne θ = n − mθ.
It is easy to see that the algebra Aθ can be embedded in Aθ by replacing (U1 , U2 ) with(i) (U1 , U2m ) or (ii) (U1m , U2 ). Since one can construct the projection Pθ in the previous
way, we obtain the projection Pn−mθ in Aθ as desired.
In the ﬁrst choice (i), the projection is described by functions f and g with period
1/|m| and the width of each lump of the function f is given by (n − mθ)/|m|. This
preserves the form (9) and is called Powers-Rieﬀel projection [28]. On the other hand, inthe second choice (ii) the requirement for being a projection is given by the equation (10)with θ replaced by mθ and this is not included in the form (9). Then the width of thelump is enlarged to n − mθ. This construction was given in [14] and used in the proof
that any projection can be divided into inﬁnite numbers of mutually orthogonal smaller
projections, where the total value of trace is preserved. Note also that in either case, thetotal area occupied by the lump is 0 ≤ n − mθ ≤ 1.
Next we turn to the calculation of cyclic 2-cocyle τ2 (P ). Deﬁne the derivations δ1 , δ2along the two directions of the two dimensional torus as follows
δj Uk = 2πiδjk Uk ,(14)

where δij is the ordinary Kronecker’s delta. Equivalently, one can express the derivations
1
as follows by using the noncommutative coordinate (x1 , x2 ) deﬁned by U1 = e2πix , U2 =2e2πix :
δ1 = −i2π 2[x , ],
θ

δ2 = i2π 1[x , ].
θ(15)

From these we can see that δ1 and δ2 do commute. Then the cyclic 2-cocyle is deﬁnedbyτ2 (a, b, c) =1
Tr [a δ1 (b)δ2 (c) − a δ2 (b)δ1 (c)] .
2πi

26

(16)

If we substitute a = b = c = Pθ , then we get [34]
τ2 (Pθ )
= −(4πi)Tr f  (U2 )(g(U2 ))2 − f (U2 )g  (U2 )g(U2 )U2+U1 f (U2 )U1∗ g  (U2 )g(U2 )U2 − (g(U2))2 U1 f  (U2 )U2 U1∗
 1
 1
d
d
2
2
2
2 2
dx (f (x + θ) − f (x )) 2 (g(x ) ) + 2
dx2 2 (f (x2 + θ) − f (x2 ))(g(x2)2 )
=−
dx
dx
0
0 1
2
df (x )
= −6
dx2
(g(x2 )2 )
2
dx
0
 1
= −6
dt (t − t2 ) = −1.
(17)0

It is also possible to generalize the result for the projections Pn−mθ and we obtain
τ2 (Pn−mθ ) = m.

(18)

In order to see this one has only to note that for the description (i) the evaluation of τ2
is equal to counting of the number of lumps with sign and also that for the description(ii) the factor m is due to the derivation of U1m . Indeed this value of τ2 is the same asthat computed from the previously discussed projective modules En,m (0 ≤ n − mθ ≤ 1)
2πim
1 ∈ EndAθ En,m as follows
which possess the constant curvature F = n−mθ
τ2 (F ) =1
Tr(F ) = m.
2πi

(19)

Finally let us discuss the relation between the results in the Moyal plane and thosein noncommutative torus. Since the radius of the torus is scaled in proportion to √1θ , thenoncommutative torus will approaches the Moyal plane in the limit θ → 0. In this limit
the value of integers n, m which satisfy 0 ≤ n − mθ ≤ 1 is restricted to n = 0 or n = 1.
Thus we obtain the projection Pmθ and P1−mθ . This is consistent with the previous resultthat in the Moyal plane algebra the projection is given by Pm or 1 − Pm up to unitaryequivalence.

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30

Preprint Series, Graduate School of Mathematical Sciences, The University of Tokyo
UTMS
2000–65 Eiji Ogasa: n-dimensional links, their components, and their band-sums.
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