5 April 2001

Physics Letters B 504 (2001) 55–60www.elsevier.nl/locate/npe

Tachyon condensation in large magnetic fields with backgroundindependent string field theory

Lorenzo CornalbaÉcole Normale Supérieure, Paris, France

Received 4 October 2000; accepted 23 October 2000Editor: L. Alvarez Gaumé

Abstract

We discuss the problem of tachyon condensation in the framework of background independent open string field theory. Weshow, in particular, that the computation of the string field theory action simplifies considerably if one looks at closed stringbackgrounds with a largeBab field, and can be carried out exactly for a generic tachyon profile. We confirm previous resultson the form of the exact tachyon potential, and we find, within this framework, solitonic solutions which correspond to lowerdimensional unstable branes. 2001 Published by Elsevier Science B.V.

1. Introduction

The problem of tachyon condensation on unstablebranes has received lately a lot of attention [3–5,7–9,11–15,21–25], and considerable progress has beenmade following the initial conjectures of Sen (see [14,15] and references therein). The problem is most easilystudied in the context of bosonic open string physics,where the relative simplicity of the theory has led bothto numerical as well as exacts checks of the conjecture.

Questions on tachyon condensation are most nat-urally addressed within string field theory [10,17],since we are interested in processes which connecttwo distinct vacua of the theory, and which are bytheir very nature off-shell non-perturbative phenom-ena. More precisely, Sen’s conjecture states that weshould find a variety of stationary points of the stringfield theory action, corresponding to branes of variousdimensions. Moreover, the value of the action at these

E-mail address: cornalba@lpt.ens.fr (L. Cornalba).

stationary points should equal the energy of the corre-sponding brane configurations. Finally the minimumof the string field theory action should be given by aunique transitionally invariant vacuum which containsno open strings at all.

From a practical point of view, two are the questionsthat need to be answered. First of all, one needsto compute the tachyon potential, and show that thedifference in energy between the maximum and theminimum corresponds to the energy of a single 25-brane. Second, on needs to find, within the string fieldtheory, classical solitons which correspond to branesof lower dimensions.

These questions have been mostly addressed withinthe framework of Witten’s cubic Chern–Simons stringfield theory [10], and have been studied in the two dif-ferent closed string backgrounds, either with vanish-ing NSNS two-form fieldBab, or with an infinitelylargeBab. In the first case, most results are numerical[2–5], and have led, using the level truncation scheme,to computations of the tachyon potential which con-firm Sen’s conjecture to approximately 1%. Tensions

0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0370-2693(01)00266-0

56 L. Cornalba / Physics Letters B 504 (2001) 55–60

of branes are more difficult to obtain in this case, sincethe corresponding solitons are expected to have a char-acteristic size of orderls , and therefore all derivativecorrections to the string action will then contributewith the same weight. Indeed, solitons can be foundeven restricting the action to two derivatives [7–9],but the energies obtained are usually only qualitativelycorrect. Still in theBab = 0 regime, an analytical con-struction of the stable vacuum has been given in [6].In the limit of largeBab nothing new can be said aboutthe tachyon potential, but solitons corresponding to thevarious branes can be explicitly characterized and en-ergies can be computed exactly [23,25]. This is thecase since the presence of a magnetic field introducesa new length scale which becomes much larger thenthe effective string scale, and which controls the sizeof the solitonic solutions. Thereforeα′ derivative cor-rections to the action become irrelevant in the compu-tation of the energies. The unique information whichis needed in order to compute tensions exactly is thevalue of the tachyon potential at the extremum points,which is assumed from the conjecture of Sen.

A different approach to the problems discussedabove has been proposed in [21,22], and is basedon an alternative background independent formulationof open string field theory, which was originallyproposed in [16] and then subsequently developedin [17–20]. The basic data is a two-dimensionalfield theory defined on a disk string worldsheetΣ(the disk being equipped, for definiteness, with thestandard metric). The action is conformal in theinterior of Σ , and is the one defining the closedstring background. On the boundary, on the otherhand, one can generically insert any operator (ofghost number zero) built from the fields of the theoryand their derivatives. Every single boundary operatorwill come with its own coupling constantg and weshould consider the space of all coupling constantsas the configuration space of string field theory. Theaction functionalS(g) is then very simply computedstarting from the partition functionZ(g) of the abovetheory, as well as from theβ-function RG flow in theconfiguration space of coupling constantsg.

Already in the early work of [16] it was conjecturedthat a stable vacuum could correspond to a breakingof Lorentz invariance fromSO(1,25) to SO(2) ×SO(1,23), or, in modern language, to a decay froma 25 to a 23-brane. More recently, in [21], the

authors compute, for generic tachyon profile andfor vanishingBab, the effective action up to twoderivatives. In particular, they are able to computeexactly the tachyon potential, precisely confirmingSen’s conjecture. Moreover, in [22] the authors haveshown that, if one works with the same closed stringbackground withBab = 0, and if one considers as aboundary operator a tachyon profile which is quadraticin the spacetime coordinates, then one can computethe action to all orders in derivatives (based on [18]).Moreover one can describe explicitly the solitonscorresponding to the lower dimensional branes, andcompute exactly their energy.

In this Letter we show that, if one considers theprocedure of [21,22] but concentrates, as a closedstring background, on the limit of largeBab, thenone can compute the action for generic tachyonprofiles extremely easily. We recover the expectedform of the potential, and we compute energies ofsolitonic solutions. All results agree with the knownexpectations, and we make some contact with theusual formulation [24,25] of the problem in cubic openstring field theory in the largeBab limit.

2. Large magnetic field limit

We study a bosonic open string moving in Euclideanspacetime, and we take the metric to be, for simplicity,equal toδab. We will concentrate, in particular, onthe motion in two of the Euclidean directions, withcoordinatesx1 andx2, and we shall collectively callthe other 24 perpendicular coordinatesy⊥. We willfinally have, as a background field, a constant NSNS2-form potential in the 1–2 direction given by

B12 = B.Units will be such that 2πα′ = 1.

We consider, following [17,19,20], string actionsdefined on the unit diskΣ ⊂ C (endowed with thestandard flat metric) of the general form

I = I0 + 1

2π

∫∂Σ

dτ V .

L. Cornalba / Physics Letters B 504 (2001) 55–60 57

In the above equation, the actionI0 is given by

I0 = 1

2δab

∫Σ

dXa ∧ �dXa + iB∫Σ

dX1 ∧ dX2

+ terms inY⊥ and ghost fields,

and is the usual conformal action defined on theinterior of the diskΣ which describes the motionof the string in the chosen closed string background.On the other hand, the operatorV inserted on theboundary is a general linear combination

V = Vigi ,where the operatorsVi span the set of all possibleboundary operators built only from the matter part ofthe theory (the most general construction, which doesnot assume a decoupling of matter and ghosts, is givenin [17], but we will not need it in the sequel).

In the rest of this Letter, we will actually restrict ourattention, as indicated at the beginning of the section,to boundary operatorsVi built only from the matterfieldsXa and independent of the transverseY⊥. Thismeans that we are considering physical configurationswhich are invariant under translations in the transversedirectionsy⊥.

The main result of [20] is that, if one considers thepartition function

(1)Z(g)=∫DXe−I (X),

then the classical background independent effectiveaction which describes open string boundary interac-tions is given by

(2)S =(

1+ βi ∂∂gi

)Z,

where the functionsβi = dgi/d lnM are the usualβ-functions (M is a mass scale) associated to thecouplings gi . To first order in the couplings, theβ-functions are given by

βi = (∆i − 1)gi + o(g2),

where∆i is the dimension of the operatorVi .We will use the above general prescription to

compute the tachyon field effective action in the limitof large B → ∞. Before doing so, let us thoughdiscuss some issues regarding the normalization (seealso [11]) of the path integral (1), by considering the

actionS(0) at zero value of the coupling constantsgi .On one hand, it is well known that the value ofthe action for a brane (maximal 25-brane, since weare considering open strings with Neumann boundaryconditions) with a constant magnetic fieldB12 = B isgiven by the Born–Infeld value

S(0)= T25V⊥∫d2x

(1+B2)1/2

,

whereV⊥ is the volume in the perpendicular directionsy⊥ (we can think of the perpendicular directions asbeing compactified on a torus) andT25 is the tensionof the 25-brane. On the other hand, at the pointg =0 in parameter space clearly all beta-functionsβi

vanish, and thereforeS(0) = Z(0). This implies thatthe normalization for the path-integral which definesZ(g) for general values of the couplings is given by

〈· · ·〉I0 =∫DXe−I0(X) · · ·

→ T25V⊥∫d2x

(1+B2)1/2

×∫Dξ e−I0(ξ) · · · ,

where the integral over the fieldsXa = xa + ξa isgiven by an integral over the zero modesxa and anintegral over the non-zero modesξa , normalized sothat

∫Dξe−I0(ξ) 1 = 1.

We now consider a generic tachyon profile given byan arbitrary functiont (x). The associated boundaryoperator is nothing but

V = t (X).We will show that, in the limit of large magnetic fieldB → ∞, we are able to compute both the partitionfunctionZ(t), and the corresponding actionS(t). Tothis end, we first recall [1] that, for generic functionsf1(x), . . . , fn(x), and for generic pointsτ1, . . . , τncyclically ordered on the boundary of the worldsheetΣ , one has that, in the limitB→ ∞,⟨f1

(X(τ1)

) · · ·fn(X(τn)

)⟩I0

� T25V⊥∫d2x B(f1 � · · · � fn).

In the above equation, the�-product is given as usual[1] by f � g = fg + i

2θab∂af ∂bg + · · · , with θ12 =

θ = B−1. Therefore, in the largeB limit, one can

58 L. Cornalba / Physics Letters B 504 (2001) 55–60

compute the partition function

Z(t)=⟨

exp

(− 1

2π

∫t(X(τ)

)dτ

)⟩I0

� V⊥T25

∫d2x B

(1− t + 1

2t � t + · · ·

)

= V⊥T25

∫d2x Be−t .

We now rewrite the above result in terms of operatorsacting on a one-particle quantum Hilbert space. Thisis easily done using the standard function-operatorcorrespondence (for example, see [24]), which as-signs to each functionf (x) an operatorQf such thatQf ·Qg =Qf�g and such that Tr(Qf )= 1

2π

∫d2x B f .

We can then rewrite the partition functionZ as a func-tion of the operator

T =Qtas

Z(T )= 2πV⊥T25Tr(e−T

).

We now compute the actionS as a function ofthe tachyon field. This is extremely simple in thelimit of large B, since all the anomalous dimensionsof boundary operators go to zero [1]. To quicklyrealize this fact, one must simply recall that theanomalous dimension of composite operators comesfrom the logarithmic divergence of the two-pointfunction〈XaXb〉 on the boundary of the string world-sheetΣ , which is proportional [1] to the so called openstring metricGab � −(B−2)ab. In the largeB limit,Gab vanishes, and so do all anomalous dimensions.This implies that theβ-function associated to thetachyon field is simply given by1

βt(x) = −t (x).Therefore, Eq. (2) implies that the action is given by

S(t)=(

1−∫d2x t (x)

δ

δt (x)

)Z(t)

= V⊥T25

∫d2x B(1 + t)e−t

1 Also, no other field will be generated under RG flow, ifit is not present from the start in the boundary action. It istherefore consistent to restrict the string field space to only tachyonconfigurations.

or, in operator language, by

(3)S(T )= 2πV⊥T25Tr[(1+ T )e−T ]

.

This result could have been expected. One can in factguess the above equation by considering the actionproposed in [21,22], and by replacing, as usual [1],ordinary products with�-products. In the largeB limit,the derivative corrections become negligible, and onerecovers (3). The point of this letter is to show thatthe formalism of background independent open stringfield theory actually confirms this expectation.

We now wish to describe solitons of the abovetheory which represent a single 23-brane. To thisend, we follow the reasoning of [22] and consideronly tachyon configurations which are quadratic in thecoordinatesxa (and also, given the symmetry of theproblem, rotationally invariant)

t (x)= c+ uBr2,with r2 = (x1)2 + (x2)2. Recalling the function-operator correspondence

1√2θ

(x1 ± ix2) → a, a†

(where [a, a†] = 1), we quickly see that the corre-sponding tachyon operatorT is given by

T = c+ u(2N + 1),

whereN = a†a is the number operator. Therefore, thepartition function is given by

Z= 2πV⊥T25e−c−u Tr

(e−2uN )

= 2πV⊥T25e−c−u

1− e−2u.

Correspondingly, the actionS is

S =(

1− c ∂∂c

− u ∂∂u

)Z

= 2πV⊥T25e−c (u+ 1+ c)e−u + (u− 1− c)e−3u

(1− e−2u)2.

We now wish to find the stationary point ofS whichcorresponds to the 23-brane solution. This will surelycorrespond to a saddle point, since all branes areunstable to decay into the vacuum solution. In fact, wecan first find amaximum of S with respect toc at the

L. Cornalba / Physics Letters B 504 (2001) 55–60 59

following value

c�(u)= −u(

1+ e−2u

1− e−2u

).

We can then follow the value of the action as a functionof c�, u

S(c�(u),u

) = 2πV⊥T251

1− e−2uexp

(2u

e2u − 1

),

which clearly decreases to the value

S(c�(u),u

) → 2πV⊥T25 = T23V⊥

at the saddle point obtained atu→ ∞, c→ −∞, withc � −u. This shows, like previously in [22,25], thatthe energy of the soliton is exactly that of the 23–brane, as expected. Moreover the saddle point is atinfinite values of the couplingsu andc. This is again tobe expected (as also noted in [22]) since infrared fixedpoints of RG flows always occur at infinite value ofthe couplings if one chooses coordinates in couplingspace such that theβ-functions are linear.

Some comments are in order:

(1) First of all, the 23-brane soliton corresponds tothe highly singular tachyon profilet � u(−1 + Br2)for u→ ∞. This might, at first sight, contradict theintuition [24,25] that the solitons describing lower-dimensional branes in the large magnetic field limithave a characteristic size of order

√θ — i.e.,Br2 ∼ 1.

In the limit u → ∞, since the size of the solitonBr2 ∼ 1/u is going to zero, one might worry that,even in the highly non-commutative limit, derivativecorrections to the tachyon action are actually impor-tant in determining the energy of the solitonic con-figuration. This is not the case, as can be seen if oneconsiders [22] the (highly non-local) field redefinitionΦ = e−T/2. In terms of this new variable the minimumof the tachyon potential is atΦ = 0 and the maxi-mum atΦ = 1. Following the reasoning of [24,25], wethen expect the soliton to be simply given byΦ = π0,whereπn = |n〉〈n| is the projection onto the harmonicoscillator nth state. This is indeed the case, since atachyon configurationt � u(−1 + Br2) correspondsto an operatorT � 2uN , or to

Φ � e−uN → π0

for u→ ∞.

(2) More generally, following the considerations of[24,25], and using the intuition from the above remark,we see clearly that a solitons withn coincident 23-branes can be described, within this framework, witha tachyon profile which is a polynomial of degree 2nin the coordinate variables. Let us illustrate this fact inthe case ofn= 2. It is simpler to proceed backwards,starting from the result of [24,25]

Φ = π0 + π1.

We can write the above equation as the limit, foru→ ∞, of

Φ = e−2uN(N−1).

Therefore we conclude that the correct tachyon con-figuration is given by the operatorT = 4uN(N − 1)or, usingr2 � r2 = r4 − θ2, by the function

t = u(B2r4 − 4Br2 + 1)(u→ ∞)

which is quartic in the coordinate variables, as claimed.

3. Conclusions

We have computed, using background independentopen string field theory, both the tachyon potential,as well as the explicit form and the energies ofsoliton solutions corresponding to lower-dimensionalbranes. This was achieved by considering a closedstring background with largeBab field. In this case,the partition function which defines the theory iseasily computed. Moreover, all anomalous dimensionsvanish in the limit of largeBab, and theβ-functionflow in the space of couplings, which is needed tocompute the string action, is extremely simple, and canbe analyzed explicitly for generic tachyon profiles.

One now has a handle, within background indepen-dent open string field theory, on the theory in two dif-ferent closed string backgrounds. On the other hand, itis conjectured [12,13], that the end-points of tachyoncondensations actually correspond to the same pointin configuration space of string field theory. In one de-scription (Bab = 0) all derivativeα′ corrections count,but in the other (Bab → ∞) none of them do. Howthis independence on the backgroundBab is achievedat the level of string field actions is still unclear, and isan open problem for future research.

60 L. Cornalba / Physics Letters B 504 (2001) 55–60

Acknowledgement

This research is supported by a European Post-doctoral Institute Fellowship.

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