We show that the part of the tree-level open string effective action for the
non-abelian vector field which depends on the field strength but not on its
covariant derivatives, is given by the symmetrised trace of the direct
non-abelian generalisation of the Born-Infeld invariant. We discuss
applications to D-brane dynamics.Comment: 13 pages, harvmac. Some confusion about instanton matrix model
  solutions corrected (string solution is interpreted as a D-string bound to a
  large number of D-instantons

A.A. Tseytlin

Abouelsaood

Aharony

Andreev

Argyres

Bachas

Banks

Bergshoeff

Brown

Dorn

Duff

Eguchi

Fradkin

Fridling

Ganor

Gervais

Green

Gross

Hagiwara

Kitazawa

Leigh

Leutwyler

Lovelace

Metsaev

Neveu

Periwal

Russo

Scherk

Schmidhuber

Shifman

Taylor

Townsend

Tseytlin

Witten

Nuclear Physics B

English

arXiv

We show that the part of the tree-level open string effective action for the non-abelian vector field which depends on the field strength but not on its covariant derivatives, is given by the symmetrised trace of the direct non-abelian generalisation of the Born-Infeld invariant. We comment on possible applications to D-brane dynamics

Tseytlin, Arkady A

CERN Document Server

CERN-TH/97-09Imperial/TP/96-97/17hep-th/9701125On non-abelian generalisation of Born-Infeld actionin string theoryA.A. Tseytlin?yTheory Division, CERN, CH-1211 Geneve 23, SwitzerlandandBlackett Laboratory, Imperial College, London, SW7 2BZ, U.K.AbstractWe show that the part of the tree-level open string eective action for thenon-abelian vector eld which depends on the eld strength but not onits covariant derivatives, is given by the symmetrised trace of the directnon-abelian generalisation of the Born-Infeld invariant. We comment onpossible applications to D-brane dynamics.January 1997? e-mail address: tseytlin@ic.ac.uky Also at Lebedev Physics Institute, Moscow.Non-locality of string theory (i.e. the presence of a tower of massive states) impliesthat the low-energy eective action for massless modes is an innite power series of allorders in 0 [1]. In particular, this applies to the tree-level Lagrangian Leff for the gaugevectors in the open bosonic or type I string theory. In the case of an abelian Chan-Patongauge group all terms in the action which depend on the eld strength Fmn but not on itsderivatives sum up into the Born-Infeld (BI) Lagrangian [2,3,4,5,6,7]LBI = c0pdet(mn + T−1Fmn) ; T−1 = 20 : (1)Derivative corrections to this action were discussed in [8,9,10].In the non-abelian case, the tree-level (disc) eective Lagrangian in the open stringtheory can be represented as an expansion in powers of the eld strength and its covariantderivatives,Leff = Tr(a0F2 + a1FD2F + a2F4 + a3F2D2F + :::= L(F ) +O(DF ) ; (2)where L(F ) is the part not containing covariant derivatives of Fmn = @mAn − @nAm −i[Am; An] (F is assumed to be a hermitian matrix with indices in the fundamental rep-resentation of the gauge algebra). Previously, only the terms up to order F 4 in (2) werecompletely determined [3,11] (there was also a discussion of F 5 terms in [10]). The questionwe shall address below is about the structure of L in (2), i.e. of a non-abelian analogue ofthe BI action (NBI action for short).In contrast to the abelian case where the separation between derivative-independentand derivative-dependent terms in Leff (F; @F ) is completely unambiguous, this is not truein the non-abelian case. Since [Dm; Dn]Fkl = [Fmn; Fkl] some of the derivative terms maybe traded for some of non-derivative ones, and vice versa. We shall resolve this ambiguityby assuming that all [F; F ] (‘commutator’) terms should be treated as a part of the DF -dependent terms in Leff and thus should not be included into L(F ) in (2). The eectiveLagrangian will then be dominated by L(F ) under the circumstances when the covariantderivatives of F are much smaller than the powers F .Adopting such a denition of L(F ) or NBI Lagrangian, we shall prove below that, bothin the bosonic and the superstring theory, it is given by the following natural generalisationof the Born-Infeld action (1)L(F ) = LNBI = c0STrpdet(mn + T−1Fmn) : (3)Here mn implicitly includes a factor of the unit matrix in internal space, the determinantis computed with respect to the mn indices only, and STr is the symmetrised trace inthe fundamental representation, STr(A1:::An) 1n!Tr(A1:::An + all permutations). ThisLagrangian is thus equal to the same sum of even powers of Fmn as appearing in the1expansion of BI Lagrangian (1), with each factor of eld strength being replaced by ahermitian matrix F and all possible orderings of the matrices included with equal weight.The same invariant was previously conjectured to be a part of a non-abelian generalisationof BI Lagrangian in [12], where, however, an additional term with STr replaced by theantisymmetrised trace was also suggested to be present.1 The latter is given by the sumof traces of odd powers of F which always contain a factor of [F; F ] (as follows fromFmn = −Fnm) and thus should not be included into NBI Lagrangian according to thedenition given above.Let us rst compare the 0-expansion of (3) (c1 = 202c0)L0NBI = c0STr[pdet(mn + T−1Fmn)− I]= c1STrF 2mn −12 (20)2(F 4 − 14 (F2)2+O(03)= c1TrF 2mn −13(20)2(FmnFrnFmlFrl +12FmnFrnFrlFml (4)− 14FmnFmnFrlFrl −18FmnFrlFmnFrl+O(04);with the known perturbative results. The two leading orders in 0 in (4) indeed give thefull form of the non-abelian open superstring eective action to order O(03) (all 02-termswith covariant derivatives have eld redenition dependent coecients [3]). The F 4 termswere originally found in the STr-form in [11] and in the equivalent Tr-form in [3].As for the bosonic theory, there (4) does not represent the full eective Lagrangian to03-order: the bosonic Leff contains 0F 3 term [1] and the coecients of the F 4 invariantsare somewhat dierent from the ones in (4) [3]. However, it is easy to see that both F 3and the excess of F 4 terms are the ‘commutator’ terms, i.e. they can be represented asTr( 43 i0Fmn[Fml; Fnl] + 202Tr(FmnF rl[Fmn; Frl]) and thus, according to our denition,belong to the covariant derivative part of Leff and not to the NBI part. Similar remarkapplies to the F 5 terms [10]2 and, in general, to all terms of odd power in F .Let us now give the general argument demonstrating that the covariant derivativeindependent part of the open string eective action is indeed given by the NBI action (3).The starting point is the expression for the generating functional for the vector amplitudeson the disc. In the bosonic case [2,3]Z(A) =< TrP exp[iZd’ _xmAm(x)] > (5)1 Some other ad hoc generalisations of BI action to non-abelian case where considered in [13]but because of their dierent trace structure they cannot appear in the tree-level open stringeective action.2 The F 5-terms have the coecients proportional to (3) [10] and should rather not appear inany simple NBI action.2=ZdDx0 < TrP exp[iZd’ _mAm(x0 + )] > ;where x = x0 +(’), 0 < ’  2 and the averaging is done with the free string propagatorrestricted to the boundary of the disc (! +0 is a world-sheet UV regularisation)< ::: >=Z[d] e−12TRG−1::: ; G(’; ’0) =11Xn=1e−nncosn(’− ’0) : (6)As explained in [5,8], the low-energy eective action is given by the renormalised value of(5), computed by expanding in powers of 0, Seff (A) = Z(A(); ).3Using the radial gauge mAm(x0 + ) = 0; Am(x0) = 0 (see, e.g., [15]) we get thefollowing expansion in terms of symmetrised products of covariant derivatives of F at x0,Zd’ _mAm(x0 + ) =Zd’ _m12nFnm +13nlDlFnm +18nlsD(sDl)Fnm + :::: (7)Separating in this way the dependence of Z on covariant derivatives we getZ(A) =ZdDx0L(F ) +O(D(k:::Dl)F ); (8)L(F ) =< TrP exp12 iFnmZd’ _mn> : (9)The path integral in (9) is eectively non-gaussian4 because of the normal ordering of theFnm(x0)( _mn)(’) factors which is non-trivial if the matrices Fmn do not commute. Itmay still be possible to compute it explicitly. In the abelian case the path ordering istrivial and one ndsL(F ) = c0det(mn + T−1Fmn); (10) = −Z 20_G2 = −( 1Xn=1e−2n!0= − 12 +12 ; (11)3 The logarithmic renormalisation of the ‘coupling’ Am corresponds to a subtraction of themassless poles in the amplitudes [14,3,5] (the eld redenition ambiguity in the eective actioncorresponds to the renormalisation scheme ambiguity in this framework [3]). In addition, one is tosubtract (or absorb into the renormalisation of the tachyon coupling) the leading linear divergence.This is equivalent to a subtraction of the SL(2;R) Mo¨bius group volume factor. Power divergencesare absent in the superstring case where the super-Mo¨bius volume is nite [8].4 It may be re-written as a standard 1-dimensional path integral by introducing the auxiliaryelds to represent the path-ordered exponent as, e.g., in [16,8].3so that  = 12 after the subraction of the Mo¨bius volume divergence [5] (which is done eec-tively when using the -function prescription [2]). As a result, one nds the BI expression(1).Since we dened the DF -independent part L(F ) of the eective Lagrangian as notcontaining terms with commutators of F , to determine it we may treat the matrices Fmn in(9) as commuting, or, equivalently, symmetrise over all of their orderings in each monomialFn. Then the path ordering becomes trivial just as in the case of the abelian gauge group,so that instead of (8) we getZ(A) =ZdDx0L(F ) +O(Dk:::DlF ); (12)andL(F ) ! L(F ) =< STrP exp[ 12 iFnmZd’ _mn] > (13)= STr < exp[ 12 iFnmZ 20d’ _mn] >= c0STrdet(mn + T−1Fmn):Since ren =12 , we nish with the NBI Lagrangian (3).This discussion is readily generalised to the superstring case, where the gauge-invariantexpression for the generating functional is given by the following manifestly 1-d supersym-metric expression [8]Z(A) =< TrP^ exp[iZd’^ Dx^mAm(x^)] > : (14)Here x^m = xm(’) +  m(’); d’^ = d’d; D = @@−  @@’and the supersymmetric pathordering P^ is dened by replacing the usual -functions by the supersymmetric ones,^(’^i; ’^j) = (’^ij) = (’^i − ’^j) + ij(’i − ’j); ’^ij  ’i − ’j + ij , so that D^is equal to the supersymmetric -function (’^ij) = (j − i)(’i − ’j). The generatingfunctional (14) automatically includes the contact terms necessary [17] for maintaininggauge invariance. Re-written in terms of the standard path ordering, it takes the form [8]Z(A) =< TrP exp(iZd’_xmAm(x)−12 m nFmn(x)> ; (15)with the [Am; An] term in Fmn appearing due to the presence of the contact ij(’i−’j)terms in the supersymmetric theta-functions in (14). The denition of < ::: > is analogousto (6) with G−1 ! G−1+ K−1 , where K is the restriction of the fermionic Green’sfunction to the boundary of the disc,K(’; ’0) =11Xr=12e−r sin r(’− ’0) :4As a result, the superstring generalisation of (8),(9) has L(F ) given byL(F ) =< TrP exp12 iFnmZd’ ( _mn +  m n)> : (16)Dropping the ‘commutator’ terms to dene L(F ), i.e. symmetrising the trace, we get, asin (13),(10),L(F ) ! L(F ) = STr < exp12 iFnmZ 20d’ ( _mn +  m n)> (17)= c0STrdet(mn + T−1Fmn);where now = −Z 20( _G2 −K2) = (−1Xn=1e−2n +1Xr=12e−2r)!0 =12 : (18)Thus we again obtain the NBI Lagrangian (3), here in completely unambiguous way asthe linear divergence in  present in bosonic case cancels out [7] (which is a manifestationof the niteness of the volume of the super-Mo¨bius group [8]).To summarise, the NBI action (3) is thus a good approximation to the eective actionwhen all products of covariant derivatives of F are small. Since [D;D]F = [F; F ] thatalso means that the ‘commutator’ terms are assumed to be small, i.e. the eld strengthis approximately abelian.5 There may be physically interesting cases in which such anapproximation is a useful one.There is a possible alternative expansion of Leff in which one assumes that all sym-metrised covariant derivatives are small. This does not imply smallness of commutators ofF . In this case, as follows from the discussion above (see (8)), the eective Lagrangian isapproximated by L(F ) in (16) (for which, unfortunately, we do not know a closed expres-sion).It should be noted again that it is L(F ) = LNBI and not L(F ) that reproduces the fullexpression for the superstring eective action at the order 02 (4). This suggests that LNBIis the relevant object to consider as a DF -independent part of Leff . Another indicationof this is provided by the existence of the D = 10 space-time supersymmetric extensionof the Lagrangian TrF 2 + cSTr[F 4 − 14(F2)2] with the symmetrised trace [19]. It shouldbe possible to nd a supersymmetric version of the full non-abelian Born-Infeld action (3)5 There is also another choice for a translationally invariant non-abelian gauge eld: Am =const [18]. It would be interesting to compute the value of the eective action, i.e. the partitionfunction (14),(15) in this case.5generalising the action found in the abelian case in [20]. One indirect approach could be torepeat the above analysis using the light-cone Green-Schwarz formalism with the fermionicpartner of Am included in the world-sheet action (cf. [21,3,22]).Let us now comment on a possible application of NBI action (3) to the description ofD-branes [23,24]. The form of (bosonic, parity-even, part of) the D-brane eective action[25] is essentially determined (via T-duality) by the abelian D = 10 open superstringeective action (see [26,27]). In the ‘small acceleration’ approximation it is thus given bythe BI action for the D = 10 vector potential Am = (As; Aa = TXa) reduced to p + 1dimensions. In the static gauge on a flat backgroundIp = TpZdp+1xp−det(mn + T−1Fmn) (19)= TpZdp+1xp−det(rs + @rXa@sXa + T−1Frs) :In the ‘non-relativistic’ approximation, i.e. to the leading quadratic order, this action isthe same as the dimensional reduction of the D = 10 U(1) Maxwell action for Am [28].As argued in [28], for a system of N parallel D-branes the elds (As; Xa) become U(N)matrices and the Maxwell action is generalised to the D = 10 Yang-Mills action reducedto p + 1 dimensions. This action should, in general, be corrected by higher-order termswhich, as in the abelian case, should be determined by the dimensional reduction of theopen string eective action.6 It is natural to expect that the most important part of thesecorrections is represented by the NBI action (3), i.e. by the following generalisation of (19)Ip = T0pZdp+1x STrp−det(mn + T−1Fmn) (20)= T 0pZdp+1x STrp−det(rs +DrXa(ab + T [Xa; Xb])−1DsXb + T−1Frs)pdet(ab + T [Xa; Xb]):Here STr applies to the products of components of the eld strength Fmn, i.e. Fab =T 2[Xa; Xb] ; Fra = TDrXa = T (@rXa − i[Ar; Xa]) and Frs(A). Expanding in powers of[Xa; Xb] (e.g., assuming that the D-branes are not too close) we nd (cf. [24])Ip = T0pZdp+1x STrp−det(rs +DrXaDsXa + T−1Frs) +14T2([Xa; Xb])2 + :::: (21)6 For a discussion of D-brane equations in the non-abelian case (using the conformal invarianceapproach as in [25]) see [29]. One may also generalise to the non-abelian case the equivalent butmore straightforward partition function approach of [27].6Like the abelian BI action (19) with all higher-order Fn terms included which grasps someimportant features of D-brane dynamics (e.g., a relation between the existence of limitingvelocity and maximal eld strength [26]) the NBI action (20) may also nd some usefulapplications, provided one understands the regions of applicability of dierent expansionsused.I would like to thank C. Callan, I. Klebanov, H. Leutwyler, R. Metsaev and H. Malda-cena for useful discussions at dierent stages of this work. I acknowledge also the supportof PPARC and the European Commission TMR programme grant ERBFMRX-CT96-0045.7References[1] A. Neveu and J. Scherk, Nucl. Phys. B36 (1972) 155; J. Scherk and J.H. Schwarz,Nucl. Phys. B81 (1974) 118.[2] E.S. Fradkin and A.A. Tseytlin, Phys. Lett. B163 (1985) 123.[3] A.A. Tseytlin, Nucl. Phys. B276 (1986) 391; (E) B291 (1987) 876.[4] A.A. 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On non-abelian generalisation of Born-Infeld action in string theory

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On non-abelian generalisation of the Born-Infeld action in string theory

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On non-abelian generalisation of Born-Infeld action in string theory

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