A Note on D-Branes of Odd Codimensions from Noncommutative Tachyons

a r X i v :h e p -t h /0008214v 2 19 S e p 2000CERN-TH/2000-256

SNUST-000801

TIFR/TH/00-48

hep-th/0008214

A Note on D-Branes of Odd Codimensions

From Noncommutative Tachyons 1Gautam Mandal a,b and Soo-Jong Rey a,c Theory Division,CERN,CH-1211,Genev′e ,Switzerland a Tata Institute for Fundamental Research Homi Bhabha Road,Mumbai 400005India b School of Physics &Center for Theoretical Physics Seoul National University,Seoul 151-747Korea c mandal@theory.tifr.res.in sjrey@gravity.snu.ac.kr abstract

On a noncommutative space of rank-1,we construct a codimension-one soliton explicitly and,in the context of noncommutative bosonic open string ?eld theory,identify it with the D24-brane.We compute the tension of the proposed D24-brane,yielding an exact value and show that it is related to the tension of the codimension-two D23-brane by the string T-duality.This resolves an apparent puzzle posed by the result of Harvey,Kraus,Larsen and Martinec and proves that the T-duality is a gauge symmetry;in particular,at strong noncommutativity,it is part of the U(∞)gauge symmetry on the worldvolume.We also apply the result to non-BPS D-branes in superstring theories and argue that the codimension-one soliton gives rise to new descent relations among the non-BPS D-branes in Type IIA and Type IIB string theories via T-duality.

In a series of recent important developments[1,2,3],it has been noted that noncommutative ?eld theories in various dimensions arise quite naturally at various corners of the moduli space of nonperturbative string theories,especially,when a nonzero value of the NS two-form potential B2is turned on,inducing noncommutativity on the open string dynamics.

Among the most interesting developments are noncommutative solitons,in particular,in the context of the string theories.At in?nite noncommutativity limit,Gopakumar,Minwalla and Strominger[4]have constructed explicit forms of static solitons by utilizing an isomorphism between?elds on noncommutative plane and operators on a single-particle Hilbert space,Hθ—viz.the Weyl-Moyal correspondence2.The method of[4]were adapted by Harvey,Kraus, Larsen and Martinec[6]and Dasgupta,Mukhi and Rajesh[7]to bosonic string and superstring theories,respectively,and have constructed a noncommutative version of D-branes,supporting the viewpoint that D-branes are built out of solitonic lumps of tachyon living on higher di-mensional D-branes[10].The result of[6,7]has an important implication for Sen’s conjecture [9]concerning universality of the tachyon potential in that,in the limit of in?nitely strong noncommutativity the level-zero truncation yields exactly Sen’s conjectural relation between the D-brane tension T D and the height of the tachyon potential V(T):

T D+ V(0)?V(T0) =0.

In Ref.[6],in the in?nite noncommutativity limit,bosonic D-branes were constructed explicitly only for those of even codimensions,viz.D23-,D21-,...D1-branes but the rest, viz.D24-,D22-,.....D0-branes,were left out.In the commutative limit,all D p-branes are on equal footing,as they are constructed out of the tachyon?eld as localized energy lumps of both odd and even codimensions.It is highly unlikely that,for some mysterious reasons, the D-branes of odd codimensions are suppressed relative to those of even codimensions as noncommutativity is turned on.How then do the D-branes of odd codimensions arise in the in?nite noncommutativity limit?In this Letter,we point out that,in both bosonic string and superstring theories,D-branes of odd codimensions can be constructed by applying a U(∞) symmetry transformation.It has already been emphasized in[6]that the U(∞)symmetry arises as a gauge symmetry on the worldvolume of noncommutative D-branes.Moreover,the D-branes we will?nd are T-dual to the ones found in[6].As such,our result may be interpreted as showing3that the T-duality,known to be a discrete(gauge)symmetry in compacti?ed bosonic and Type II superstring theories,is actually a part of the U(∞)gauge symmetry of the noncommutative D-branes.

We will begin with posing an issue concerning noncommutative D p-branes in bosonic open

string theory.In doing so,we will bring up the puzzle concerning missing D p-branes for p=even more explicit.At level-zero truncation,the bosonic open string?eld theory is described by the action of the tachyon?eld T:

S0=C

?G 1

2 T2+13

g st

=T25,(3) and G st,Gμνrefers to the open string coupling and metric,respectively.The?-subscript refers to the fact that?eld products are de?ned in terms of the noncommutative Moyal product:

T(y)?T(y)=exp i

θx, both in the quadratic gradient term and the cubic tachyon interaction term Eq.(2),the action Eq.(1)can be recast as4

S0=C

?G 1

2 ?a1?b2??b1?a2

T(x a)T(x b) x a,x b→x.(6)

The equation of motion is then given by

Gμν?μ?νT?V′?(T)=0, which,in the static and translationally invariant case,reduces to

V′?(T)=?T+

1

4Later,we will readdress the precise nature of theθ→∞limit.

In[4],a rotationally invariant static soliton satisfying Eq.(7)was found:

T(x)=T0·2exp ?r2

(2π)2 T(k)e i k·x,(8) Consider now operators x1, x2satisfying the Heisenberg algebra

x1, x2 =i.

Built out of the operator algebra is an auxiliary Hilbert space Hθof an auxiliary one-particle quantum mechanics.Given an operator T( x)acting on Hθ,one can de?ne its Fourier-mode function as:

T( x)= d2k

5Refs.[4,11,12]provide useful reviews of the formalism in the context of noncommutative?eld theories and gauge theories.

It then follows straightforwardly that

T

( x ) U (

x )←→T (x )?U (x )

Tr T ( x ) U ( x )···←→ d 2x √λ x 2 a ?=1

2 λ x 1?i 1

n |n +1 a ?|n =√

√

√√

6In the case of

Eq.(2),the critical points consist of {0,T 0}.

The conditions on C mn’s are imposed to ensure the projection operator properties: P· P= P and Tr P=1.As a simple choice with‘zero’angular momentum(that is,a diagonal projection operator),let us take

P n=|n n|(n=0,1,2,···).

From Eq.(9),one easily?nds the corresponding Fourier mode-function P n(k)as:

P n(k)=e?k2/4L n k2/2 ,k2=(k21+k22),

where L n denotes the n-th order Laguerre https://www.sodocs.net/doc/1812507515.html,ing(10)and inserting the Fourier-mode function into Eq.(8),one obtains the Weyl-Moyal(inverse)map P n(x)of the projection operator:

P n(x)=2(?1)n e??r2L n(2?r2)where?r2= λ2x21+1

√

2π|p p|=I.

The most general projection operator is given by

P= dpdp′2πC(p,q)C(q,p′)=C(p,p′)and dp

issues concerning in?nite noncommutativity and large deformation,which will be elaborated later.For de?niteness,we may de?ne| p0 operationally in terms of the following wave-packet:

| p0 = dpπexp ?(p? p0)2/σ2 |p .

To obtain the monochromatic plane wave state,we letσ→0in the end with a suitably de?ned limit procedure.

It is straightforward to compute,as before,the Weyl-Moyal(inverse)map P p

(x)of the

0 projection operator Eq.(13).The result is

P p0(x)=2exp ?(x2? p0)2

δ(x2? p0),(15)

2πR1

consisting of a delta-function in x2-direction while all dependence on x1-direction disappears. Here,R1is a suitably large radius of the x1-direction and the factor1/(2πR1)is to take care of the requisite normalization condition,where the‘box cut-o?’R1is related to the‘Gaussian cut-o?’σas R1～1/√

.

U(∞?1)×U(1)

For example,the various values ofλin Eq.(11)correspond to a one-parameter‘squeezed state’subspace in the above manifold.The corresponding tachyon solutions are related to one another by U(∞)gauge transformations

T(x)?→U(x)?T(x)?U?(x).

The U(∞)gauge transformations clearly respect the projection operator properties:(P?P)(x)=P(x)and d x/(2πθ)P(x)=1;they also preserve rank of the projection operator so that all points of a gauge orbit correspond to the same rank.Remarks similar to these and the possibility of squeezing deformation have been made in[4,6];we have found here an explicit application of these ideas to construct bosonic D24branes.

As a concrete illustration of the U(∞)gauge transformation,we now map the codimension-two pro?le Eq.(12)to the codimension-one pro?le Eq.(15).This is achieved by recalling that the parameterλis associated with the squeezing deformation,an aspect discussed already in [4].One?nds that theλ→0limit of Eq.(12)reduces exactly to Eq.(15)for the choice p0=0. In fact,in this limit,the squeezing deformation parameterλplays precisely the same role as the regularization parameterσin Eq.(14).The opposite squeezing limitλ→∞yields again the same state as Eq.(15)but with x1→x2,x2→?x1which amounts to a rotation in the noncommutative plane.The moduli space of the squeezing deformation is parametrized by λ∈[0,1].We have thus shown that the parameterλspans a one-parameter trajectory in M1associated with the squeezed state and the two extreme endpoint con?gurations represent geometrically codimension-one asλ→0and codimension-two asλ→1,respectively.

At this point,we would like to discuss an important subtlety7in applying the U(∞) gauge transformation and obtaining the codimension-one,kink con?guration.Recall that the action Eq.(5)was obtained by dropping terms involving derivatives along the noncommutative directions—both in the quadratic gradient term and in the cubic interaction term—by taking θ→∞limit.On the other hand,in applying the squeezing deformation over the Grassmannian M1and lettingλ→0,as described above,each derivative along the noncommutative directions is ampli?ed by a factor1/λ.Certainly,to retain the utility of the in?nite noncommutativity and construction of the noncommutative D-branes via the Moyal-Weyl correspondence,one ought to take a controlled limit keepingθe?≡λ2θ?xed,while lettingθ→∞andλ→0.In order to suppress all the terms involving derivatives along the noncommutative directions,it is necessary to takeθe??1.In fact,one easily?nds that,once the squeezing deformation is taken,the new expansion parameter in Eq.(5)is set byθe?instead ofθ.It is interesting to consider whether such subtleties involving large spatial derivatives arise in some other parts o f the Grassmannian M1as well.We expect that the standard large-θlimit can be taken only in the interior of the Grassmannian M1,but not along the boundary.As the boundary is approached,the large-θconstructions would in general require a suitably controlled scaling such as the one we have de?ned above.

Having obtained a codimension-one pro?le,we are naturally led to identify the soliton Eq.(15)with a noncommutative version of the bosonic D24-brane.To ascertain this identi?ca-

tion,we will now calculate the tension of the codimension-one soliton and compare it with the known string theory result.For a static solution,the action is simply the time interval times the mass,so the soliton mass can be equivalently calculated out of the action.We will calculate the action Eq.(5)for the tachyon?eld T sol(x,y)=T0P p0(x)where P p0(x)is given by Eq.(15). To be speci?c,we will take the closed string background as

ds2=θ(R21dx21+dx22)+dyμdyμ

eφ=g st

B2=B dx1∧dx2.(16) From the de?nition of the open string coupling G st and the open string metric Gμν[3],one

?nds(cf.[6])in the largeθlimit

G st=g st

√

2π?2st B

.(17)

Using the results that

V(T sol(x,y))=V(T0)P p0(x)

C

2πθP p0(x)=1 1/B=θ,

we?nd that the action Eq.(5)is evaluated to

S=T25(2π?st)2 dt dy3...dy25.(18) This agrees with the action of the codimension-two soliton evaluated in[6],re?ecting the U(∞) gauge invariance mentioned above, d x/(2πθ)U(x)?P p0(x)?U?(x)= d x/(2πθ)P p0(x)=1.

In[6],the action Eq.(18)and the corresponding tension were interpreted as those of a D23-brane extending along y3,...,y25.The symmetries of our codimension-one soliton are those of a24-brane,extending along x1,y3,...,y25directions.We will now show that,it is actually a D24-brane living in the T-dual geometry where T-duality is performed along the x1direction. The closed string geometry T-dual to Eq.(16)is given by

ds2=1R21dx1dx2+ 1+B2

soliton of codimension-one,Eq.(18).Begin with noting that the pull-back of the T-dual geom-etry on the D24-brane worldvolume is given by

ds2=1

g,

where T24is the tension of a T-dualized D24-brane,given by

T24=T24g st/ g st=T24R1/?st.

Using the fact that dx1√

the transformation relates non-BPS noncommutative D-branes in Type IIA/IIB to those in Type IIB/IIA via the T-duality and hence o?ers a new kind of D-brane descent relations.

Recall that in the commutative limit,Sen has given descent relations among D-branes via worldvolume kinks or vortices formed out of tachyon condensation.The noncommutative D-brane descent relations we will present are di?erent from those of Sen’s.Moreover,we will?nd that the new series of descent relations provides a?rmer support to our interpretation regarding the U(∞)gauge transformation as a T-duality between Type IIA and IIB strings through an analysis of the Chern-Simons coupling and Ramond-Ramond charges,an aspect which was not available for bosonic D-brane setup.

Let us?rst recollect Sen’s descent relations in the commutative limit.Begin with non-BPS D2p-branes in Type IIA string or D(2p?1)-branes in Type IIB string.These non-BPS branes are unstable,as represented by a tachyon?eld on the worldvolume.The tachyon is a real-valued?eld and it can condense in the form of a topological kink on the worldvolume of the non-BPS D-brane.As proven by Sen,the kink turns out precisely the same as BPS D-brane.Thus,in Sen’s descent relation,in Type IIA string,BPS D(2p?1)-brane can be formed out of kink formation on the worldvolume of non-BPS D(2p)-brane and,in Type IIB string, BPS D(2p?2)-brane can be formed out of kink formation on the worldvolume of non-BPS D(2p?1)-brane.

In in?nitely strong noncommutative limit,solitons formed out of tachyon condensation can come with two varieties:(1)codimension-two solitons studied in[6,7]or(2)codimension-one solitons proposed in the present paper.

For the case(1),even though the solitons are objects of codimension-two,they ought to be identi?ed with circular ring of codimension-one,BPS object.This is because,as shown in[7], the Chern-Simons term on non-BPS D p-brane involves a coupling

1

2T0 M p+1T0d P0(r)∧C RR p=1

For the case (2),we will now argue that the soliton ought to be interpreted as the non -BPS D(p ?1)-brane of the T-dual,Type IIA/IIB string.To see this,let us evaluate the Chern-Simons coupling

(21)for the codimension-one soliton pro?le T sol (r )=T 0P p 0

(x ).We get 1

R 1dδ(x 2)∧C RR p =1

2T 0 M p dδT ∧C RR p ?1.

This is indeed the correct form of the Chern-Simons coupling for a non-BPS D(p ?1)-brane

localized at a point along the T-dualized x

1-direction.Again,we have shown that the T-duality between Type IIA and IIB superstring theories,which is known as a discrete symmetry,is part of the U(∞)gauge symmetry arising in the large noncommutativity limit.

We are grateful to M.R.Douglas and S.R.Wadia for useful discussions,and J.A.Harvey and E.Witten for important suggestions.We acknowledge warm hospitality of Theory group at CERN during this work.

References

[1]A.Connes,M.R.Douglas and A.Schwartz,J.High-Energy Phys.9802(1998)003,

hep-th/9711162.

[2]M.R.Douglas and C.M.Hull,J.High-Energy Phys.9802(1998)008,hep-th/9711165.

[3]N.Seiberg and E.Witten,J.High-Energy Phys.9909(1999)032,hep-th/9908142.

[4]R.Gopakumar,S.Minwalla and A.Strominger,J.High-Energy Phys.0005(2000)020,

hep-th/0003160.

[5]A.Dhar,G.Mandal and S.R.Wadia,Mod.Phys.Lett.A73129(1992),hep-th/9207011;

A.Dhar,G.Mandal and S.R.Wadia,Phys.Lett.B32915(1994),hep-th/9403050.

[6]J.A.Harvey,P.Kraus,https://www.sodocs.net/doc/1812507515.html,rsen and E.J.Martinec,J.High-Energy Phys.0007(2000)

042,hep-th/0005031.

[7]K.Dasgupta,S.Mukhi and G.Rajesh,J.High-Energy Phys.0006(2000)022,

hep-th/0005006.

[8]E.Witten,“Noncommutative Tachyons and String Field Theory”,hep-th/0006071.

[9]A.Sen,J.High-Energy Phys.9912(1999)027,hep-th/9911116.

[10]J.A.Harvey and P.Kraus,J.High-Energy Phys.0004(2000)012,hep-th/0002117.

[11]D.J.Gross and N.Nekrasov,“Monopoles and Strings in Non-Commutative Gauge The-

ory”,hep-th/0005204.

[12]L.Alvarez-Gaume and S.R.Wadia,“Gauge Theory on a Quantum Phase Space”,

hep-th/0006219.