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JHEP10(2008)033 June 17, 2008 October 2, 2008 October 7, 2008 stantons, n the boundary September 26, 2008 Received: Accepted: Published: Revised: xplicitly take the T-dual of ons is derived from the tachyon Published by IOP Publishing for SISSA Tachyon Condensation, D-branes, Solitons Monopoles and In We ﬁnd an exact solution, with a nonzero net D-brane charge, i [email protected] Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502,E-mail: Japan SISSA 2008 c

String Field Theory. Seiji Terashima Abstract: string field theory ofthis brane-anti-brane configuration. pairs The on Nahm-transformation of acondensation. the torus. instant We e Keywords: Tachyon condensation on torus and T-duality JHEP10(2008)033 8 1 3 4 10 D-brane − oliton on a D-brane pairs a D olding of the − ng modes and the D0 pairs 5 ex soliton of the tachyon − ve been intensively investi- [1]. On the other hand, we ns of the tachyon condensa- al bundle. Since the torus is utions, for example, the kink sional unstable D-branes, like yon condensation on torus will with codimension one and two, ranes from unstable D-branes by use of the charge conservation. In this [12 – 14], which were known as the achyon condensation of a D r ld theory for the bosonic string was found d via the boundary state [10, 11]. [3 – 7] or in the boundary state [8] on on” is massless). This solution corresponds R 1 – 1 – . This soliton will represent a D0-brane. Since the torus is 2 D0-brane pairs T − D-brane systems. By the marginal deformation, we always has − , it seems easy to construct such soliton, however, the orbif 2 However, it will be very difficult to find an exact solution of a s R 3 is topologically trivial and non-compact, there is no windi r R D0 pairs on torus D2-brane pair on − − Those solutions include the topologically non-trivial sol Since 3.1 Nahm transformation3.2 and tachyon condensation T-dual on of D0 the D0 2 D-brane pair on a torus. To be explicit, let us consider a vort Recently, the boundary string field theory was reconstructe On a torus with the self dual radius, an exact solution of the t Recently, the exact solution in the Witten’s cubic string fie . 2 3 1 r − D solutions in the boundarysimplest string non field trivial theory compact can manifold,be the have interesting. study a of trivi the tach ascent relations. R in [9]. it may represent a closed string vacuum. or the vortex, which representrespectively. lower dimensional This D-branes, constructions of lowerthe dimensional tachyon D-b condensation were knowncan construct as higher the dimensional decent D-branesthe relations from matrix lower models, dimen by the tachyon condensation on gated in the past decadetion, [1, were 2]. found Among in the them, boundary the exact string solutio field theory 1. Introduction The open sting tachyon condensation on unstable D-branes ha 2. D0-branes on torus and3. T-dual D0 Contents 1. Introduction or non BPS D-branespaper which we do will not study have the net soliton D-brane with charges a beca net D-brane charge. of a D2 was given in [1,to 15] the by lower the dimensional marginal D deformations (the “tachy 4. Concluding remarks an orbifold of JHEP10(2008)033 4 (1.2) (1.1) are the -branes, D-brane 5 p − (2) µ + 4)-branes A )-branes and p q D( + 0) with non-zero nsformation was and p D4-brane with the N ∼ (1) µ D-brane pairs which N T µ A − , ) D he boundary string field 2 D-brane pairs on a torus change the dimensions of yon condensation [23, 24]. ,x − e Nahm-transformation [16, 2 up without a probe D-brane )-branes and the D x y. This is the exact solution ( q N) with the instanton number es infinitely many D instanton on the T-dual torus using the D( , G , ) + aive extensions of the boundary state, undle on the anti-D2-brane, a general 2) ! N ng the ADHM(N) cases [19, 20]. , p solution ( n case and we will see that the Nahm 2 2 ce (plus the T-dual transformation). ξ L 4 , therefore, the i ′ →∞ = 1 e T α u µ « because it is a compact space. It will be because of the non periodicity of (1.1). ( ( 1 is much bigger than string scale. ences when away from on-shell background fields. 2 2 the D0-brane solution is represented by a 1 , 4 ξ T is the tachyon and πL Z ) 2 2 2 / T ) gauge instanton. If we introduce a probe , R 2 T iξ + limit. Obviously, it is difficult to extend this N R – 2 – n + = 0 „ = 1 ξ 2 H (2) µ ( Z → ∞ T D4-branes with the u A . ∈ X µ n u k ξ = =

. In string theory, the bound state of u 4 T (1) µ to an anti-self dual configuration of U(k) with the instanton ˜ T 4 A )= 2 T ,ξ 1 is given by the U( ξ ( 4 T T to a solution on 2 is equivalent to )-brane is the simplest one. R 4 p T -branes. This method can be applied to ) is a periodic function of 2 p on on a dual torus -branes on ,ξ 1 p N ξ D-brane pairs and we will see that the soliton on the D0 ( 4)-brane and consider the low energy limit on it, the Nahm tra D − G k − As an application of the soliton solution, we can consider th In this paper, we consider the tachyon condensation of D . If we take the bundle on the D2-brane as (2.7) and the trivial b We assume we can employ off-shell boundary states, which are n ) , which has a sub-stringy size if the size of 4 p 5 i instanton on on four dimensional torus µ ( 4 ˜ T k following non-periodic configuration: this method gives the equivalence between the descriptions transformation is naturally interpreted as this equivalen It is worth noticing that this equivalence is exact in solution is not straightforward. Actually, on interpreted as the T-dualRecently, the transformation ADHM [18], transformation was generalizi given inand a nor D-brane a set low energyFor limit a [21, bound 22] state by the of method two using different the D-branes, tach say the D( 17] which maps an anti-self dualk gauge field (instanton) of U( number and using the D represents a D(2 and find exact solutions,theory or which the have boundary a state formalism.pairs net instead Our D-brane construction of charge, us a inthe pair. t D By the T-dual transformation, we can solution (1.1) on Moreover, the gauge fields can not be trivial on interesting, but, highly non-trivial to construct a vortex in the boundary string field theory. Here D( as in [21, 22]. They have possibility of suffering from diverg where gauge fields on theof D2-brane the and equations anti-D2-brane, of respectivel motion in the A tachyon field would be written as JHEP10(2008)033 , ′ ) e ν µ ξ ion U ( µ (2.3) (2.4) (2.1) (2.2) πR ˜ is the A 2 µ ˜ < A D-branes D-branes. µ = 0 gauge. x 0 N and A ≤ ×∞ ν N πL matrices. Here we 2 N < . This is actually a gauge ν w to obtain the T-dual ν × of the D0-branes in the U ξ , es [25]. 1 N h translation operators ≤ is not summed over except − ν ′ section 4. s on torus. We take the T- ν nstantons is derived from the . ype II superstring theory on a )Ω ion 3, we give an exact solution N N 1 and 0 1 ) ν , then ). ν ξ r ′ ν ξ ( ˜ ( T U µ πR N 1 πR ˜ 2 A × 2 − ν l configurations although finding those are not ′ unitary matrices: ions on torus is expected to be on-shell for all µν N retation in consistency with the boundary string + . µν δ U δ ′ ν N µ + α =Ω R ∂ + , whose periodic coordinates 0 × ) which commutes with µ ∂ξ r ν µ i ∂ = ∂ξ Z U X N – 3 – X / ν , ξ, ′ ( r Note that this implies that the gauge field = ( ν L ν ′ ν ξ L R N 1 6 πα i . × − e r = ν N ˜ =Ω T U r although there are many fields on the D0-branes. In U = = 2 D0-branes should not change the (2.2). Thus the transformat 1 µ T ν µ − N ′ X ν U ν X D-branes on tours will be equivalent to the will be operator valued Hermite U U µ µ ,...,r N X . We will define matrix and X ′ ν ν unitary matrix, i.e. a gauge transformation of the U -branes on = 1 is always fixed, and thus abbreviated below. If we consider th -branes, i.e. a unitary matrix N P r r , the unitary matrix t N r , it can be considered as the static configuration in the µ × D D µ Z N × / N N matrix as an operator. N X r × N R N = r ), where ) is an t is an ξ ∞×∞ T ( as discussed in [21, 22]. ( which should be operator valued µ ′ ′ ν µ is the periodic coordinate of the T-dual torus ν α ˜ whose coordinates X A ν x ξ r This paper is organized as follows. In section 2, we review ho Since We consider only the scalars corresponding to the locations By the orbifolding, we need the following identification wit The gauge transformation of the R 6 is generated by a order in discussed in this paper. Actually, the instanton configurat However, the off-shell boundary states havefield a theories. natural interp Furthermore, our main concern is the on-shel where picture of the D0-branes onin torus, according the to boundary [25]. stringdual In sect of field this theory configuration. oftachyon The condensation. brane-anti-brane Nahm-transformation of We pair the conclude i with some discussions in 2. D0-branes on torus and T-dual In this section, we will(rectangular) review torus how as to an describe orbifold D0-branes in t time-independent transformation of the this paper, the time Here according to [25, 26] and how to take thetorus, T-dual of the D0-bran on regard an gauge field of the explicitly indicated by along Throughout this paper, we take a convention that an index where Ω A representation of (2.2) is JHEP10(2008)033 − 1 ) is . If ξ − 1 ξ (2.6) (2.5) (2.7) (3.1) ( . ˜ Ω µ . Note r µ ˜ . ˜ A T ! ˜ , p A 2 µ 1 1 ˆ − x T ˜ Ω iral spinors. p 2 ⊗ × )-branes in the 1 0 )= M p − 2 p × 2 ,ξ 1 M is the Dirac gamma D(2 1 nction (or the gauge 1 . The solution which − -branes on the torus µ p -branes is non-trivial, r 2 ⊗ 1 πL r M D − p ) is trivial, i.e. R p 2 ξ 2 . Then, from the T-dual + 2 ( × ) is m calars with the same v.e.v eans that a solution of the tion including all order in × s µ p 0 1 where and Γ 1 x g e on the dual torus 2 ˜ ξ ( A − 1 ( p µ D0-branes on the T-dual torus lent to µ − 2 2 µ,ν A = 1 is the tachyon which acts on the ˜ 1 ˜ Ω A iδ m ) µ T

2 of the bound state of the D0-brane . ˜ Ω ]= = µ , µ , ν D0-brane pairs on a torus 1 ∂ p N ˆ X p = 1 ( 2 v − , , X i ˜ ” Γ 2 µ Fξ , )) ν ˜ x − Ω ν 2 n x D0-brane pairs on = · · · L 1 (ˆ ν L ξ − µ 2 2 − 2 , 1 “ ˜ 2 Γ ˜ m 2 A A Ω -brane within ξ – 4 – 1 L =1 µ πL with gauge field r i , r ν Γ − ˜ 2 e A p satisfy [ˆ p with the gauge field -vector. If the bundle of D P µ 2 2 − i ˆ p = µ ˜ i r = = 0 Ω N ˆ − R p ˜ 1 T e , ⊗ 1 ˜ ≡ ˜ F Ω ˜ µ A )= x M 2 × M πL (1 )-branes . p r + 2 , whose component is not necessary a periodic function of ⊗ T r 2 -branes on µ ˜ r T = 2 case as an example. The bound state of the D2-brane and the is a base of a D(2 ,ξ Γ D 1 r u ξ N M ( v N acts on the anti-D0-branes, which correspond to the anti-ch µ will be given by ˜ † A →∞ ) which satisfies Γ u 2 T p and ˜ T -branes on = lim r Z and D ! 1 is the transverse scalars of D0-branes and the ∈ is an integer which is the D0-brane charge. The transition fu − 1 D0 pairs on torus 0 T µ m ν ˜ Ω expansions, but leading order in the string coupling † − n ) m X 0 1 ′ T ˜ Finally, let us consider a bound state of a D0-brane and First, we consider the infinitely many D0 α Ω

µ . First, we sketch how to construct a D . We will consider ∂ r r ( ˜ boundary string field theory of infinitely many D0 periodic, the base of the Hilbert space is spanned by transformation) between the different patches is given by In this section, we will construct a solution which is equiva where i the bundle of the is a connection on the T the base of the Hilbert space will be the sections of the bundl where Note that this isequations the of exact solution. motionsthe of Here an the exact D2-brane solution (string m field theory) ac map (2.3), we can read the D0-brane configuration and the is equivalent to the matrix of SO(2 3. D0 where Here we have setas that the the D0-branes. anti-D0-branes The has operators the ˆ transverse s D0-branes on D0-branes and T JHEP10(2008)033 ) is x by ( µ µ back (3.6) (3.2) (3.3) (3.4) (3.5) T A A . Here we p 2 is the bound epresentation R pairs le D0-branes. In 0 which is equivalent = 0 or, for example, rus. Thus the D p µ 2 , − 1 A T 0 )-branes. Thus the (3.1) − ν p rator (3.5) is a consistent Ω ρ ation of D0-branes which is . ∂ ion on the torus and see that ctions. D(2 coordinates. Thus we require µ ν , x state, (3.1) with the orbifolding f we set Ω )-branes with gauge field N i M of the orbifolding if we take p 1 = ν − µ 1 orus in the boundary state formalism. , by the orbifolding of D(2 . − ν πR µ ν 2 . The constraint for the tachyon , R p ρ )Ω M 2 ν πR µν x U ˆ p δ 2 ( 5) is a symmetry. Note that Z µ ). Since the orbifolding will consistently . ρ πi / x + U 2 ≤ p ( A e 2 u D/, X µ ν µ µ = ν R x A X T, – 5 – D0-branes given by the configuration (3.1) uni- µ →∞ D0-brane pairs on a torus = ≤ − U 7 u = = =Ω ρ )=Ω − p 1 1 2 ν U ν = lim − − ν ν U T = 2. ! U πR p µ 2 to 0 T U T ν X p † µν 2 U ν 0 δ T U R +

and the unit shift ( 3 µ in (2.1) for the D0-branes and the anti-D0-branes. Then the x p ( ν 2 ρ R which is extended from the gauge field on the torus, i.e. a pull A p 2 ) satisfies D(2p)-branes with a nontrivial gauge bundle on the torus, it R x , which is spanned by 0 ( p act on the Dirac spinors which transformed as a fundamental r 2 µ M µ T A D(2p)-branes and the lower dimensional D-branes, for examp )-branes with the gauge field ) gauge symmetry on the manifold spanned by X p is a transition function (or gauge transformation) on the to M M ν and D(2 T Therefore, the configuration (3.1) with the orbifolding ope Using this configuration we can construct the It is desirable and interesting to study the solution on the t M 7 which is from the fundamental property of the transitionconfiguration fun of the infinitely many D0 can be written as that state of formly distributed in If we consider this case, following [23]equivalent (see to the also bound [27]) state.the we We Nahm can will transformation find apply naturally this a appears. to configur the solut on the torus of the U( to by the generator (3.5) will be an exact solution on the torus i truncate the equations of motion or the (on-shell) boundary assume that where Ω where we take same Ω the orbifolding may be samethat as the constraint for transverse a gauge field on configuration (3.1) is consistent with the constraint (3.4) This is obvious if we notice that the D0 3.1 Nahm transformation and tachyon condensation on D connection of the map from an anti self-dual configuration for JHEP10(2008)033 . µ (3.8) (3.7) πL (3.13) (3.11) (3.14) (3.10) (3.12) (3.16) 2 D(2p)- < M µ ξ into a spinor e tachyon, , p ≤ ) 2 of the boundary x , ( ) R ξ x ( ψ ξ ξ and 0 ψ D/ µ p modes remain after the µ 2 x x µ T µ ξ ero modes only: ξ ′ ′ onalized by the gauge trans- 1 πα . 1 πα 2 2 i i is given by just a truncation of e ′ e p . p 2 µ ) (3.9) ) (3.15) µ : , ) 2 ξ πα ν x x ¯ i x X dξ ( ( 2 ( dξ U , j i ξ ξ ′ ξ ) p | ′ u D/, ψ ψ p 2 , x πα 2 Ψ µ µ µ R πα ( 2 )+ R x x 2 ν ξ D0 pairs which represents the 0 ′ x X µ µ R 0 →∞ α ψ (ˆ ξ ξ Z | − ν Z ′ ′ u ξ µ i ) = 0 i 1 1 h πα πα A x e · · · = lim ) = 0, is given by 2 2 ( · · · )= i i 2 i x ξ – 6 – = − 2 x e e limit. Then we see that the remaining D0-branes ! ( ψ j dξ µ )= ξ ξ Ψ( dξ 0 ˆ T p ′ x i ψ ′ . Now we decompose a spinor on 2 D/ ( D/ )= )= † ν 2 p πα ξ R 0 πα 2 µ x R x 2 U µ T →∞ 2 ( ( Ψ 0 ¯ ξ i ξ 0 R X ν Z u

, i.e. it satisfies Z Ψ Ψ U 1 ξ =Γ 1 ξ D/ dξ can be written as can be written as (3.9). Using this, we have dξ ′ D/ p ′ ν 1 2 1 πα R U πα 2 R 2 R 0 0 Z ) is a section of the spinor bundle on Z x ( ) on ξ x ) = )= ψ x x Ψ( D(2p)-branes with a nontrivial gauge bundle on the torus, th Ψ( D/ M ) which is a spinor on is the eigen state of the unitary operator ) is ) is a zero mode of x is a zero mode of the tachyon. This gives the D0-brane picture x x ξ ( ( i i ξ ξ i ψ ψ | For the In [23], we first take a configuration of D0 on torus and a plain wave like the Bloch wave function: state. where acts on Ψ( Thus any spinor Ψ( branes by the tachyon condensation.formation Then the and tachyon then is diag tachyon only condensation, namely D0-branes the which corresponds to zero where where Then, the zero modes of the tachyon, where which means that the Chan-Patton-Hilbert space to those composed by the the z because any eigen state of Indeed, Ψ have the transverse scalars or the matrix coordinate JHEP10(2008)033 ), x ( )= i ξ x ψ ( (3.18) (3.19) (3.21) (3.22) (3.23) (3.17) (3.20) ′ j ξ ψ ) ) = † x x ) ) ( ( ′ x ) invariant.) x i ξ j ξ parameterize ( ( ter the tachyon p ψ ′ i ξ ψ j ξ µ ψ † ∂ ψ ) ∂ξ µ x ′ ( ∂ i ξ ∂ξ ψ , µ | x ′ µ . ξ ) , ) ′ ′ µ ξ, j ′ ij x 1 Ψ is the SO(2 πα ξ δ h ( 2 † ) From the index theorem [28], j ξ ′ i − ξ ψ 8 ) which implies j , , ie . ′ µ i ) µ ξ ′ ij − x ξ ∂ δ ξ ( ) ξ D/ ∂ξ i )( ξ ′ , )+ is the instanton number. The zero ( ξ ) µ − † ψ ( x δ l ) x 1 ( µ )= ′ πα ξ µ ′ ( m ′ ( ˜ − ν j ξ 2 x i ξ A x δ ( )= ( ) Ψ i Ψ ξ πR j ξ x x even for cases with zero modes of both chiralities, D0-branes corresponding to the remaining µ ν ν ψ ′ ( j ( + ψ ξ L ′ ′ ′ = 2, ∂ , i i j ξ j ξ † )=Ω + 2 m ) x µ ∂ξ ) e ′ p p x ′ ) Ψ i Ψ µ ∂ ξ 2 ( ′ of x ξ † ∂ξ i ξ µ d − x ( ( – 7 – ) − µ ( i x ξ ψ ij p )= µ x i 2 ¯ in this basis. Moreover, from † ξ ˜ ( ψ x X A iδ orbifolding. ) ( T † i ξ ( ′ . For ) i ξ x Z ξ x p Ψ ( x j 2 ′ i πδ p Ψ i ξ ( i + 2 x ν i ξ Z exp )2 d p = Ψ ′ πα µ 2 U ) Ψ p Z x j x ∂ ξ d 2 ( i ∈ x ( p ∂ξ ′ T p p 2 p j µ ξ = 2 2 2 Z 2 ij d ) ˜ ,l ψ R A d µ ξ ) p X 0 and the all zero modes have positive chirality, i.e. Γ . We will see that the discrete eigen values of iδ † Z ( 2 ··· p i ¯ x ) , X 2 µ ′ ( 1 R D0-branes are remained and all anti-D0-branes disappear af π | ′ l ˜ ′ + 1 R x A Z j ξ 2 and we have used ′ ( Z µ m i ξ and Ψ m > x . (Because of the Euclidean nature, Ψ ′ π ν ∂ πα ξ, i 1 l p ν ψ is the number of the zero modes of † 2 ∂ξ α ξ h 2 ν ) ′ d ij x x m = 2 = = πR ( p p iδ πR i ξ 2 2 2 does not depend on ′ d T Ψ ′ j,ξ Z ≤ p +2 m 2 x ν ) and i,ξ ). p ′ ν π T ′ 1 ′ πα 2 2 x Z x x d µ ( ,m ′ p ¯ = = = j 2 = 2 X ξ ≤ ν R ψ µ · · · x Z † , ¯ ) X ′ We normalize the zero modes as Now we can evaluate the coordinate π 1 Here we assume that x 2 zero modes: 8 ( = 1 i ξ i where 0 where This means that ( ψ thus which is equivalent to mirror images of D0-branes by the modes are labeled by This can been seen from which means that onlycondensation. the However, this assumptions isas not discussed essential, in [21, 22]. where m we know that JHEP10(2008)033 ) p 2 ξ N ( can T ) = µ x ˜ p A (3.28) (3.29) (3.24) (3.27) (3.25) (3.26) ( 2 . Here ξ R )-branes ψ with the µ ) p ν n p ′ U pace of the 2 µ ξ,n , ) on T D(2 πα ( ) R is a constant. ′ 2 x ψ x µ m µ ( ζ + x j ξ as a combination ) ” µ ν µ ξ ( n )-branes on the with the coordinates ′ ψ p ′ µ † p πα R ). ) 1 2 2 ′ , πα x ) 2 x + T ( D(2 branes is ), where with the gauge field ( x i quivalence between the ′ µ ξ i ξ ( ξ p = ψ ξ “ 2 ψ N , ′ formula given in [29] for the πα ′ µ )-branes on the ψ ˜ T π ˜ 1 F p p x 2 πα (2 µ ψ, same 2 p e x / µ i 2 and any spinor Ψ( µ µ x e d , ξ D(2 p ζ ” ) Tr ′ p 2 µ Z 2 , x p 1 n πα ( ν 2 ∈ T R N ′ ′ 2 ˜ µ T j ξ Z µ i X L ) = n πα e Z R ψ x 2 . p p ) which is periodic, namely, µν ( ν 2 2 µ + ′ ). Note that = x µ )= ¯ µ α ν ξ πδ X dξ dξ ξ x A ( L ξ,n ( “ ′ ′ ( µ ′ e p p ′ j ξ ˜ 2 2 + 2 ψ A πα πα 1 – 8 – R R πα , N 2 2 µ Ψ µ 2 x 0 0 π i † ξ )= F 2 µ D0-branes on the ) e Z Z x n e x ( = ′ ( j ξ µ m ) is given by i ξ · · · · · · ′ ) µ πα Tr R )= ξ ν 2 ξ 2 2 Ψ ( ( p x ′ 2 )-brane solution on the torus, (3.1). Now we assume that µ ( )-branes on the dual torus ψ x dξ dξ 1 p T ˜ πα p p A ′ ′ ) is anti-self dual, the formula (3.21) is indeed the Nahm 2 Z 2 ξ,n 2 2 i x d D0-branes with πα πα e R R ( Ψ 2 2 = p D(2 -brane pairs µ 2 0 0 0 m Z A Z Z R ) and the m m ∈ ) = 0, thus a linear combinations of D Z 1 1 x µ x ( )-brane is trivial and − n ( µ dξ dξ p = j ξ 0 with the gauge field ) , we find an equivalence between the ′ ′ A P j ν 1 1 p ν i ( 2 πα πα R R R and the 2 2 ˜ ψ ν T ) 0 0 ˆ p = 2 and µ ξ ξ ) is a constant spinor. Then, is just a translation operator. As we have seen, the Hilbert s ) = Z Z ) and the ( πi p A x D/ x ν 2 ν ( ( e = ˜ ξ ξ,n Ω U µ ( ν ) = ψ A ψ x Ψ( = Ω ν ) and Therefore, we find an equivalence between the If we take The transition function for (3.20). From the relation (2.3), the T-dual of the latter D0- U x ( µ ξ ¯ be written as In this case, Chan-Paton index is spanned by the spinors on the 3.2 T-dual of the D for D4-brane with Let us take the T-dual of the D(2 which is just a Fourier transformation with the momentum we defined the bundle on the D(2 are followed from the index theorem. on the dual torus which satisfies ψ with the gauge field gauge field X given by (3.21) using the Dirac zero modes (3.16). transformation of [16, 17], which is a generalization of the ADHM case. Weof note the that two different the equivalences: Nahm (1) transformation the can T-dual be and viewed (2) the e where and JHEP10(2008)033 9 )- p (3.34) (3.31) (3.32) (3.35) (3.36) (3.33) )-brane and . Therefore, p p . Thus the 2 ′ R ξ,n . ) , e tachyon in the above to Ψ ) x ( , the transition function x ) stisfying ( x µν ξ,n . We interpreted that the ξ,n ( δ ′ ξ,n Ψ . The tachyon condensation e orbifold of ψ + p u ξ,n πα , 2 s although it is not periodic. µ 2 ψ ˜ ) ξ,n T n µ D2-branes, on the other hand, )-branes in this T-dual picture, µ , ζ p ) (3.30) y many pairs of D(2 E ) = − n = x x , µ − ( u ( ′ µ µ , )= D0 pairs is more convenient than its n ξ,n πL ξ,n n x u D/ ′ − µ,ν ( ψ Ψ δ µ µ ν ξ,n + 2 . This means that the gauge fields of the ξ + x and ξ,n πα and the spinor index of a constant spinor R µ ′ µ µ ν →∞ , µ 2 ν ψ ξ ) pairs, especially, for a configuration with α ∂ n R u ζ ν } ′ ξ ∂ξ p i u E L . Then, the tachyon configuration of D(2 → U i + 1 e πα ′ − µ 2 µν µ D/ ), from a spinor n ξ,n i µ – 9 – ξ x { )= e ξ πα U πδ = lim ( )= , D/ ( ξ , on the dual torus µ ) ( x µ = p µ is the location of the a solitonic D0-branes on the T and ( ˜ x 2 A T +2 )= Γ ( Γ ν µ ν µ ′ Z x u ξ,n ˜ )-anti-D(2 ξ ζ )-branes and anti-D(2 Ω ( U ξ,n p p 1 Ψ πα ψ = ν ξ,n 2 = 0. Here we defined ˜ }∈ ) = ˜ U ′ µ µ ξ Ψ µ ξ ( n )= ˜ ˜ A { T )= x )-branes in the T-dual picture vanish. ( x p ( ξ,n and is implicitly given by ξ,n ) is an eigen state of ψ is represented as 2 , where p x µ ν ′ Ψ 2 ( ˜ is the unitary operator which maps Ψ µ n U T x ξ,n D/ ξ,n ). For a non-trivial = ˆ µ,ν x δ ( µ =Ψ + µ . µ X p ) is an eigen state of ,n A n ′ 2 x ξ ˜ → ( T µ )-brane, labeled by ) pairs on n ξ,n p p U are the Chan-Paton indices. parameterize the world volume of the pairs of D2 )-branes and anti-D(2 Now we expect that the T-dual of the tachyon will be given by th Since Ψ We note that the configuration (3.8) of the D0 p } We thank Koji Hashimoto for suggesting this solution. } 9 µ µ . We note that Ψ ξ n a non-trivial T-dual configuration (3.32) of D(2 a basis is given by { ψ anti-D(2 in the T-dual picture, this system is equivalent to infinitel is a basis of the Hilbert space labeled by { D(2 In this basis, basis of the Chan-Paton bundle since we regard the torus as th is given by which is diagonal in where Ψ and also an eigen state of the tachyon dual torus in this basis. anti-D(2 of the infinitely many pairs of D(2 is given by where tachyon (3.32) is a consistent configuration on the dual toru JHEP10(2008)033 r T (3.37) elation tion for = 0 will remain µ n alytic) K-homology tionsfor the decent rus. If we take the -brane charge, of the D0-brane pairs on e KK-theory discussed s by the K-theory. The − mments and discussions. xplicitly) assumed in this malism though we believe mapped to the boudnary -theory to the analytic K- cussed in [33 – 35] are also this tachyon condensation. . It would be interesting to method is not restricted to s, like ALE spaces, are also n [13] although the winding Culture, Sports, Science and n the analytic K-homology by ane-anti-brane pairs on torus. 6].) It would be desired to do e decent relation. On the other e K-theory (using the infinitely h other. )-brane with p , ) µ ζ − µ ξ ( µ – 10 – uγ ) = ˜ ξ )-brane and anti-D(2 ( p ˜ T , the original infinitely many D0 → ∞ µ R and the solution (3.1) is same as the solution for the ascent r = 2) case, it will be interesting to study the Nahm transforma r p R D9 pairs [30, 31]). On the other hand, (3.1) represnts the (an − , then only the pair of D(2 Finally, we will comment on the classification of the D-brane Let us take the large radius limit of the torus or the T-dual to There are several interesting future directions. Since our In this paper, we do not explicitly use the boundary state for →∞ µ found in [12,relation 13]. (3.37) and Thus the ascent we relation can (3.1) say are T-dual thatconfiguration eac (3.37) on for the themany torus decent D9 relation the is two related solu to th which is just the Atiyah-Bott-Shapirohand, solution [5 if – 7] we for take th the and the configuration (3.32) becomes L become those on class in [13]. Therefore,homology. we However, the expext winding that modes are theassuming neglected T-dual to the maps obtai size the of K modes the compactified are manifold important isin for very [32] large the will i be T-dual importantinvestigate picture. the to study relation to the The role it of duality further. the of widing th modes 4. Concluding remarks In this paper,tachyon we condesation in found the an boundaryThe string exact field Nahm-transformation theory solution, of of the with br We instantons also a was found nonzero derived the from net T-dual configutration D of this. the instanton (i.e. D0-D8 or D0-D6 cases.paper. Morevoer, the Therefore, BPS properties thecovered are non-BPS in not cases, thie (e for paper.interesting. exampole To models extend di our result to other orbifold the exact solutions instate. the (The boundary marginal deformation stringit case field explicitly. [15] theory was can studied in be [3 Acknowledgments S. T. isS. grateful T. to is K. partly supported HashimotoTechnology. by and the S. Japan Ministry Sugimoto of for Education, useful co JHEP10(2008)033 , , D 63 , JHEP JHEP , , Adv. A 14 , ]. Phys. Rev. nal non-BPS , (1999) 281 (2005) 5513 , Commun. Math. ndary state , 3 (1998) 012 eld theory ri 08 A 20 Commun. Math. Phys. , (1999) 027 Int. J. Mod. Phys. 12 JHEP , , hep-th/0305006 ]. JHEP Exact description of D-branes in , ]; ]. ]. ]. Int. J. Mod. Phys. ]. (2004) 93 [ , Adv. Theor. Math. 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