COURSES AND PLENARY TALKS REVISTA MEXICANA DE FÍSICA 49 SUPLEMENTO 1, 19-29 JUNIO 2003

Aspects of Born-Infeld theory and string/M-theory

G.w. Gibbonsf).A.M:U~, Cambridge University

WillJelforce Road, Cambridge CH3 OWA, U.K.

Recibido e125 de abril de 2001; (lceptado el3 de junio de 2001

Keywords: Bom-Infcld theory

Descriptores: Tcona de Bom-Infcld

PACS: 11.25.Yb

1. Introduction A" at lhe ends of lhe slrings. In lhe Polyakov approach onehas an aelion of lhe form

whcre

(6)

(4)

(3)

(5)

ami

3. Dirac-Born-Infeld actions

whcrc the embcdding of lhe string world sheet al -t M isgivenbyya = ya("A),A = 1,2anda = 1,2, ... ,n =dimM, all(l Gab and Bab are lhe spacetime melric andNeveu-Schwarz lwo-form respeclively.

One obtains an effeclive aetion for a D-hrane if 'lne "inte-grales OUl"all possible string mO!ions subjecllo lhe Dirichlclboundary conditiorl.

The resulting aClion depends on Ihe posilion of lheD-brane and lhe pullhack lO lhe D-brane of lhe metricand Nevcu-Sehwarz two-fOnTI.It also contains the veclorfield Aw

One oftcn defines

are lhe pull-hacks of lhe melric 1Jab and Neveu-Schwarl. lwo-form Bab lo lhe world volume "»+1 of lhe p-brane.

The world-volume field F"" is given by

This is governs lhe embedding y: E»+1 -+ M given in 10-cal coordinalesby ya = ya(x"),wherea = 1,2, ... ,n =dim Jl,1ami J.L = O, 1,2, ... ,p. It is

2. Open strings and D-branes

Bmnes may be incorporaled in string theory if olle contem-pIates opens strings whosc cnds are constrained (by Diriehlctboundary condilions) lo lie on a (p+ l)-dimensional subman-ifold ap+l. Now open strings can couple minimally lo veclor

M/slting lheory is lhe currenlly mosl popular approach lo aunified quanlUm lheory of gravily and lhe olher inleractions.We still lack a complele fonnu!ation of lhe lheory, bullhereis a general consensus that whatever finally emerges it willinvolve in some way or to sorne degrec of approximation,p- brall~s,i.e. p + l-dimensional Lorenlzian subrnanifoldsa of a Lon::llizian spacetime manifold M. In M-theory onesupposes lhal M is eleven dimensiona!. In slring lheory ilis usually laken to he ten dimensional. Branes may crudelybe sub-divided inlo lwo lypes Heavy amI Ligh!. In lhe formercase one is usually thinking of many coincidcnt branes whosegravitational ficld and hence lhe ambient spacetime metric isnon-trivial. Semi-c1assically these may be studied using su-pergravity techniques. The olher extreme is to sludy a sin-gle isolated brane moving in nat Minkowski spacelirne as aSolulion of lhe Dirac-Born-Infeld equatious of mOlion. Thiswill be lhe approach lakeu in lhese lcclures. It is well suiled10 ncwcomcrs lo lhe subject hecause, as 1 will try lo show,considerable insighls into slring theory can be gained by ask-ing sorne of lhe simplesl physical questious. There is Iittleneed for lhe full heavy lechnical machinery of supergrav-ily or superstring lheories. Thus lhe malerial is well suiledfor presenlation al a Schoo!' 1 have deliberalely lried 10 keeplhings simple. This runs Ihe risk lhal experls may fcel lhat Ihave nO!done full jusI ice lo lhe subjecl or indeed lheir con-tributions lO it. If so, 1 apologize bul 1 repeal my aim waslO provide lhe begiuner wilh a rapid survey of lhe subjec!. 1will mainly assume lhal lhe branc is Oa!. It is fairly slraighl-forward lo eXlend the prcsent cirelc of ideas to the case ofa curved background. In lhe ca,e of Born-Infcld lhcory lhercader is referred lOReL 15

20 G.\V. OIBBO;o...'S

This is invariant under a Ncvcu-Schwarz gauge lfansfonna-tion B --+ B - de whcrc e is a onc-form, ir we transfonnF --+ F + de. One may check lha! lhis is eonsislen! wilh lhehchaviour of the opcn string mctric.

4.1. Constituti\lc rclatioIls

To c10se the systcm Dne needs constitutive relations 1IH(E. B) and D = D(E, B) whieh. if one has a LagrangianL = L(E, B). lake [he form

D is the canonical mornenturn density. NOle also that (he con-served elcclric charge is given by Ihe flux of D ami nO( as isoflen assumed. lhe flux of E.

[n wba[ follows we shall den ole by K.v lhe Amp~re 2-fonn with componeIlls (DI H) ami refer 10 FJ.!1I a~ the Fara-day 2-form. Thus Ihe equations of motion without sourcesare

D= aL8A'

3.1. Monge gange

To procccd wc fix sorne of the gallgc-invariancc associatcdwi[h world shee[ diffeomorphisms of [he eoordinales X. hyusing what is usually, ami mislcadingly callcd static gaugc(sincc il applies in non-stH.tic situations) and which is moreaccumlc!y and with more justicc callcd Monge gaugc. In cf-[cel wc project onto a p + 1 planc by scuing ya = x~ l yi amiuse the n - p - 1 height fUllctions yi. i = 1, 2, ... , n - p - 1as sealar ficlds on lhe world volume. [n lhe lheory of min-imal surfaces this is callcd a lloTl-pararncLric rcprcscntation.For Monge's work see ReL 1. Of eourse lhere may nol bea global Mongc gaugc, and wc shall encannter this siluatioTllater.

Thc determinant thcn bccomcs (wc use unit'i in which2"0' = 1).

Because

R = _ aLaB'

dF=O,

aLD= DE'

d*K = O.

( 12)

( 13)

( 14)

The symmetry of the eTlergy momenlum lensor TOi = T¡oand hence the uniqueness of lhe Poynling vector rcquires thatlhe laller be given by

It is evidently consistent to set the scalars to zero yi = O amiwe lhen oblain [he Lagrangian of Bom and [nfeld whieh is aspecial form of Non~Linear Elcctrmlynamics.

....2. I..orcntz-ilwariancc

ExH=Dxll. (15)

(8)

This \ViIIfollow if L is eons[rueled from [he lwo Lorenlz in.varianl~

The cOllstitutive relations will permit the obvious rotationneeded [o ro[ale lhe lw() seL, of Eqs. (!O) and (11) itllo [hem-selves

(18)

( 19)

(16)

(17)

x = ~(B2 _ E2),

y=E.ll.

E + iH --+ e'9(E + iR),

D + in --+ e'9(D + in),

wilh 8 constant if

4.3. I>uality invariam.'c- J d"x J- det (rya' + Fa')'

\Ve make[he ansalZ Aa = [A. (x') .y' (x')J andoblainlbeMongc-gauge-fixcd Dirac-Bom-Infeld action

The previous section result has a sort of converse. \Ve couldstart with apure Born-Infeld actiotl in n flat dimensions amidirnensionally reduce lo p + 1 dirnetlsions. \Ve begin with

Thus all solutions of lhe Dime Bom-Infcld action are solu-(ions of(hc Born-Infeld aClion. Intcrcstingly in the case p = 1we get a string aClion from (he pure Bom~lnfeld aClioTl.

3.2. Dimensional reduction

4. Non-linear electrodynamics

There are advantagcs in viewing (he theory in lhis context. Anexccllcnt account of the theory is givcn in Ref. 3. Thc gCTleralIhcory in four-spacc(irne dimensions (p = 3) has equations

NOle that what we are encountering hcre is a non-linearform o( lIJe famiJi:lr /ine:lr Hodge (}[lalily. This gives a con-straint on possihle thenries. For example if the L1grangiandcpends arbitrarily on the invariants x and y il gives rise lOa Lorentz-invariant theory. Imposing dualily invariance re-duces lhis freedom to tllat of a funcrion of a single variable.For more dctails on duality invariance sec Refs. 11, 12 and 3whieh was no[ known lO lhe aUlhors of Refs. 11 amI 12 whenthey were writlen.

aBenriE = -at'

aDeurlR = -at'

divB = O.

divD = O.

( 10)

(11 )

E.B=D.R. (20)

Re\'. Mex. Fú. 49 SI ('2003)19-~9

ASPECI'S UF BORK.INFELDTHEORY ANO STRING/M-TIIEORY 21

4.4. Hamillonian densily

One has

1{= Too = E .D - L, (21)

lhan 1 will quiekly breakdown and lhe eleetrie field will bereduccd 10 a value Icss than anc.

Note that if ane restares dirncnsions and unito;; the criticalfield strength Ee is given by

[n lhe zero slope Iimit Q' -f O there is no upper bound amiin the strong coupling limit a' --+ (X) the critical field gocsdowll lo zero. Lalcr we will invcsligale lhe behaviour of lhelheory in lhis limil.

one may lhink of 1{ = 1{(B, D) as the Legendre transformoflbe Lagrangian and is thus expressed in lerms oflhe eanon-ieal variables B and D whose Poisson Braekels are

4.5. Born.lnfeld

1Ee= --211"0"

(27)

We have 6. BIons

5. The maximal electric field strength

A eonstant ha, becn addcd to make lhe zero field have zeroenergy. This is nol strietly neeessary in lbe theory of banessinee lbe notion of world volume energy is nol well definedbecause Lhere are no privileged coordinatcs 00 the branc.However it is convenient whcn making comparisons wilhstandard nat spaee field theory. To do so we musl howeveruse Monge gauge.

Lorentz ami Dualily invarianee are elear. Before the ad-venl of stringlM-theory lhe laller was ralher mysterious.Nowadays il may be lhoughl of as a manifestalion of S-duality. In lhis way we see how Bom-Infeld theory consid-croo sui generis has important Icssons foc M/string thcory.Conversely M/slring theory throws lighl on Bom-Infeld lhe-ory. We shall see more examples of lhis mutually symbioliebehavior later.

1{ = VI + B2 + D2 + (B X D)2 - 1. (24)

(31)

(28)

(29)

(30)

(32)

DvI +D2

E=

Too = 1{= E .D - L.

The maximal elcelrie field was originally invoked lo ensurelhe exislence of a e1assieal solution representing a ehargedobjeel wilh finile lotal energy

/.""3 d3x Too < oo.

Becausc

D= Dvl-D2'

lhe eleelrie induetion D diverges at the origin and so does theenergy densily

lhe clcctric ficld achicves ils maximal value at lhe centrc.NOle thal

This can be aehieved by setling

TlJus rJJis soJurion is not a smoorlJ soliton soJurion willJou(sourccs. In facl thefe is a dislributional source

(23)

(25)L=I-VI-E2.

L = 1 - VI - E2 + B2 - (E. B)2

and

IfB = O, lbe Bom-Infeld Lagrangianis

If we use a gauge in whieh Ao = O, we have div D = 4,,-qJ(r). (33)

The analogy wilb speeial relativity is e1ear. There will be anupper bound 10 lhe eleetrie field strenglh. The speeial rela-tivistie analogy may also be understood from the point of T-duality.

In slring lheory the existenee of a maximal eleelriefield strength may be underslood dynamieally as follows. Astretehed open string of lenglb L has, in our unils, e¡astie en-crgy L. Ir it has chargcs +1 al Dne cnd and -1 al the othcr itwill, in an e1ectrie field have energy -EL. This if E> lonemay gain energy from the background elcetrie field by ere al-ing open strings. This an clcctric field with strength grcatcr

L=I-VI-Á2. (26) Finile energy bul singular Solulions like this of nOIl-linear lheories with distributional SOUfcesare a sufficientlydistinct phcnomenon from the familiar finite energy non-singular lump solutions without sources a~ lO dcserve a dif-ferenl name. The suggeslÍon has been made [7] lhal lhey beealled Blons. From lhe string poinl of view lbe souree has anatural inlerpretation as being assoeiated wilh a string endingon a three-brane. In faet one retums in lhis way lO a pielllrevery elose to lale nineleenlh eenlury speeulations in whiehan clcctron is rcgardcd as an "clhcr-squirl" on a 3.surfaccembedded in four dimensional spaee [2]. The applieation toslrings is contained in Rcfs. 7 and 14. Thc present accounl islarge[y based on.ReL 7.

Rev. Mex. Frs. 49 SI (2003) 19-29

22 G.w. GIBBOSS

6.1. Masimal spaccJikc hypcrsurfaccs

Anolher interpretalion of the statie solutions may be obtainedas follows. Qne introduccs the eleelrostatie potential <1>= Aoand finds the Lagrangian dcnsity 10 be givcn by

is just that which would be obtaincu ir one sought a maximalspaeclike hypersurface of Minkowski spacelime where <1>isnow thoughl of as a timc funclion 6.4. Thc III'S solulion: S.duaJily

which are manifcslly invarianlundcr an obvious 0(1,1) ac.tion analogolls lo the wel1.known Harrison transformalioTl ofslalic Einstein Maxwcll thcory. Using this aelion one muyeonslmel everywhere smoOlh ehargcd eatenoids, (he c)cctriefield lines passing through lhe neek or throat in a way similarlo lhat diseusscd by \Vhccler in lhe ea'ie of Einstein.Maxwelltheory. This family 1 eaU under-extreme. They are obviouslyanalogous lO under extreme Reissner-Nordstrom SOIUlioTls.One may a)so excite lhe sealar field of lhe BIon solution.The original Oat lhree-bmne aequires a eusp as if it werebeing puUed. AU of these solutions are singular. 1 eaU themover.exlreme. They are obviously analogous to over extremeReissner-Nonlstrom solutions.

(35)

(34)1 - JI - ('V<I»'.

The Euler-Lagrange equation

6.2. Calcnoids and D-D soJulions

Thc maximal hypcrsurfacc bccomcs null al the criticalfield strength.

Rathcr Lhancxciting lhe clcctric field we can excite a singlesealar y. Wc gel as Lagrangian dcnsity

Xo(x) = <I>(x). (36) In the Jimit of infinite 0(1, 1) pararneler one obtains an ex.treme solulion analogous lo extreme Reissner-Nordstrom.This solution is in fael supersymmetric. It may be interpreledas a fundamental (F-) or (1,0) string ending on a three-brane,Using lhe cleetric.magnetic dualily one may easily ootain amagnetic monopole solution which represenlli a D-string or(0,1) slring ending on a three-brane. lu faet using SL(2, Z)and lhe Dime quantization condition we can gel dyon or(p, q) strings ending on a three-brane.

is thal govcming lhe hcight function of a minimal surface infour spaee-like dimensions. Qne readily check s that Mongegauge is not global. In the spherieally symmetrie case. Thereis a branch 2.surfacc al a finite radius. One ncclis twoMonge patehes. The reso1ting two sheeted worm-hole or bet-ter Einstein-Rosen bridge type sorfaee looks like two parallelthrcc planes a wilh finitc scparation joined by a neck. Thc so.lution is nol stable and therefore one thinks of it as a Brane-Anti-Branc pairo

7. Opcn string causality

In string theory, open string states propagating in a back.grouno :Fp.", ficlo do so aceording to a different metric fromlhe Einstein metric gp.1-' feh by closed strong slales.

One has

(44)

(43)

(42)(_1_)"" = C"" +8""9 +:F '

where C"" = C("") ami 8"" = 8[""1. If

Cp.>.C>.p. = ó~,

then

(38)

(37)1 - JI + ('Vy)'.

d' [ 'Vy ]IV JI + ('Vy)2 ,

The Euler-Lagrange equation

6.3. Chargl-d calcunids: 0(1, 1) symmclry rclaliugcalcuuids aud lIIuus

Note that even if Bp.1-' = O so thal :Fp.1-' = Fpl-" lhe metrieG pI-' is not invariant under electric-magnelic duality.

Including both eleetrie and sealar fields gives a Lagraugian 7.1. lIoillal mclries

1 - JI + ('Vy)' - ('V<I»'.

Jt and the Eulcr-Lagrange equations

d' [ 'Vy ]IV JI + ('Vy)' _ ('V<I»' ,

and

(39)

(40)

One muy investigale the propagation of small dislurbancesof veetors. A" sealars y and spinors 1/J around a Bom.lnfeldbackground using lhe method of characterislies. This wasdone in greal dctail hy Boillat for a general non. linear clec-lrodynamic theory. He found that in general, because of bi.refringence, lhere are u pair of characteristic surfaces Sconstant salisfying

div [~==='V=<I>====] = OJI + ('Vy)2 - ('V<I»' ,

(41) (45)

Rev. Mex. Fú'. 49S1 (2001) 19-29

ASPECrs OF BORN-INFELD THEORY AND STRING IM~THEORY 23

7.2. Hooke's' Law

This mcans that if J.L is positive then CEinstein lies insideor touches CSoillat. Remembering that duality reveres inclu-sions one finds then that the Einstein cone CEinstein lies out-side or touehes the Boillat cone CSoma'. Note !hal what weare ealling a cone here is the solid eone. The Iight eone is lheboundary of !his solid eone.

where T~:xwell is the Maxwell stress tensorconstructed formPIlII' Of course the stress tensor T#II of the non-linear elec-trodynamic theory is different from T~:xwell' The quantityJi = Ji(x, y) satisfies a quadratic equalion whose eoeffieienL'depend upon firsl and second derivatives of !he LagrangianL(x, y) with respect to x and y. Boillat finds il conveniem tofix !he arbitrary con formal resealing freedom in the charac-teristic co-metric by setting

C.v _ l (. T?v) (46)- . / 2 2 2 JiY + Maxw,ll ,vJi -x -y

with inverse or metric

C-1 - l (.V _T.v ) (47).v - . / 2 2 2 JiY Maxw,ll .vJi -x -y

In general lhe boundaries of !he lwo Boillat conesCSoillat: C¡;;vJ.lvll 2: O, va > O and the Einstein coneCBoillat: 91lllVllVll 2:, va > Owill touch along the two prin-cipIe null direetions of F.v . Dne sometimes find lhat one atleast of !he Boillat cones Jies ouL,ide !he Einstein cone. Inother words smaII fluctuations can travel faster than gravita-tional waves whose specd is govemcd by Y.v'

To check causality we examine lhe Boillat co-conesCSoilJat: CPIlPIlPII 2: O,Pa > O and the Einstein co-coneCEinstein: CJ.lIlPIlPII 2: O,Pa > O in the cotangent spaceT*O'p+l. Suppose that l. is the co-normal the Einslein eo-cone CEinstein

(54)

(55)

det C.v = det y.v .

lC-1 = G.v -V-=I=+=2x=-=y2= .v.

This follows form another useful idenlity is

The con formal factor is relaled to!he Lagrangiao:

..j-det(r¡.v + F.v) = VI + 2x - y2. (56)

From Hooke's Law it is ea,;y to see, since T#II for Born-Infeld theory salisfies the Weak Energy Condition, thal lheBoillat cone lies inside or touches the Einstein cone. In otherwords small nuctuations travcl with a speed no greater lhangravitational waves. Beeause!he Bom-Infeld energy momen-tum tensor is invariant under electric-magnelic duality rota-lions, the Boillat metrie, unlike lhe open string metrie G.v isalso invarianl. One ha,;

7.3. Hooke's Law, lhe Monge-Ampereequalion and pulseintcraclions

The reason for the name Hooke's law is lhal Hooke a,-serted, in the days when the archive waq in Latin that eiiiou-"enssstt. In !his way he hoped to make bo!h a priority claimand preserve his discovery for his own later use. Bearing inmind thal " v are not distinguished in Latin, lhe earth shal-leeing discovery !hat he wished to hide was !hat "t tensio sicvis. In olher words stress is peoportionallo strain. A standardmeasure of strain in non-linear elaqticity theory is the differ-ence of lwo metrics. More precisely, the configuration spaceof an elastic medium is a map fmm an elastic manifold loan embedding space. There is usually a rest or un-deformedconfiguration and one takes as a measure of stress the differ-ence belween the pullbacks fmm !he embcdding space lo theelastic manifold in the straincd and unstrained configuration.

What we have is an expression involving lhe difference oftwo co-melrics but the idea is similar. Onc is the co-metric in-duced on the brane fmm !he Einslein co-metric and !he o!heeis a measure of the vector fleld excitations.

(48)

(49)

(50)

y.VI.lv = O.

The weak energy condition implies

Thus

Bom-Infeld is exeeptional in that there is jusI one solutionfoe Ji:

Thus there is no bi-refringenee. Morcover one finds that theBoillat co-metrie satisfies the remarkable idenlity

I caH this identity Hookc's Law for real50nswhich will beexplained below. Another strikiog idenlity is

Ji=l+x.

C~~= g#1I + TJ.l1I.

(51)

(52)

The steiking delerminantal identity has an inteeesting appli-cation to lhe propagation of pulses in Bom-Infeld theory.

In flat two dimensional spacetimc, the conservation lawfoe the sleess tensor implies !hat it is given by a single feeefunction, the Airy stress function,p, such !hat

Ttt = ,p", Tzz = ,p", T,z = ,pzt. (57)

Wrinen in terms ofthe Airy stress function, the detenninantalidentity becomes the Monge-Ampeee cquation

det (o~+ T::) = 1. (53) (58)

Rev. Mex. Fú. 49 SI (2003) 19-29

24 G.W. GlBBONS

This can be solved exaeUy (see Ref. 8 and referenees therein)by a Legendre transform under whieh il heeomes D' Alcm-bcrt's cquation with rcspCCl lo a ncw set ofvariablcs T and Z.

One has

CI-'II. One may also consider fermion fields 1/;. OmiUing four-fermion tenns, lhey eouple in a typieal Volkov-Akuiov f"h-ion

- J dxP+1Jdet(g"v+i;¡rr~\lv1/J+F"v+B~v), (66)

where

(68)

(67)IIJIII + IIIIIJ = 29IJII'

Let's define Boillat gamma malriees by(61 )

(60)

(59)T" = A + B + 2AB1-AB '

T" = A + B - 2AB1-AB '

T"= B-A1-AB'

where A = A(T + Z) and B = B(Z - T) are arbitraryfunctions oC thcir argumcnls. The rclation bctwccn the ncweoordinales (T, Z) am! usual eoordinales (t, z) is most eon-vcniently cxprcssed using null coordinatcs. Let v + t + z, ti =t - z, E = Z - T, r¡ = Z - T. The asymmetrieal definilionof TI is so a~ lo agrcc with prcvious work c¡lOO in Rcf. 8. Qnehas

and

One has

1"1V + 1v1" = 2G~V. (70)

Becnuse the lending derivative tenn in the action is

dv = dE - B dr¡, du = -dr¡ + A dE. (62) (71 )

Thus

(I-AB)dr¡=Adv-du, (l-AB)dE=dv-Bdu. (63)

one ehecks that

dT'- dZ' = dt'(1 - A - B + AB)

- dz'(l + A + B + AB) - 2dtdz(A - B)

= e-1dx"dx"u (M)"V ,wherc

(65)

Thus we see lhat the Legendre transformation to lhe new eo-ordinates (T, Z) used 10 solve lhe Monge-Ampere equalionin cffcel passcs lo Om incrtial coordinatcs with rcspcct LotheBoillal metrie. It should he noted lhal one does nnl expeel lheBoillat mClric LObe llat in general.

Thc general solution consists of two pulses, onc right-rnoving and onc lefl moving which pass through-anotherwilhnul dislnrtion. In {erms of lhe usual eoordinates (t, z)they lwo pulses experienee a deJay Thal is measured withrcspcct lo the c10scd slring mctric. Howcvcr with rcspcct LOlhe Boillal eoordinates. that is measured wilh respeel lo lheBoillat metrie, lhere is no delay.

7.4. S('¡llars and fermions: 0pl'n string cquivl.llcnccprincipie

The coupling of scalars has already been given aboye. Itis easy to check thm the Boillat C(Hnetric determines theirlluctuations around a background, Lhey are in fact govcmedhy lhe D' Alemhert cquation eonstmeled fmm the eo-metrie

It is c1ear that the characteristics of the fenniotls are alsogivcn by lhe Open Slring metrie or equivalcntly lhe Boillatmetric.

Thus we have a sort of world sheet equivalence principieor universalily holding: all open string fields have the samecharacteristics and hence the same maximum specd.

8. Tolman redshifting of lhe I1agedornlemperalure

As an application of the equivalen ce principie it is intcrcstingto considcr open strings at finite tcmpcraturc in a backgroundelcetromagnetie field. This was done for lhe neutral basonieslring in Ref. 5. If lhe free energy densily in lhe absenee ofa haekground is F = F({J) where {J is the inverse tempera-ture, lhen the free energy in a background is obtained by thereplacement

where GjJ.1I is lhe open strong metric. The f¡rst factor maybe Lhought of as a redshift and volume contraction factor.The resealing of the argument is essentially the Tohnan cf.fcet whcrchy in order to retain local equilibrium in an ex-ternal sLatic or stationary metric G ¡J.II' the local temperaturemusl vary as 1/ VGOQ. Note that Goo = 1 - E' and so lheredshifting is indeed red shifting ami it depends only 011 theelcetrie ficld, lhe effeel diverging al lhe erilieal c!eetrie fieldstrength,

AlLernatively, olle may reganl the effeel as being due tothe faet thaLfi!lite temperature physics corrcsponds to work-ing in imaginary time with a period givcn by the inversctcmperalUre, Ir the global time variable is idcntifled with pe-riod {J, the local period will he {JVGOQ, Thus lhe loeally mea-sured tcmperature will be higher. If more than one metric is

Rev. Mex. Fís. 49 SI (].003) 19-29

ASPECTS OF BORN-INFELOTHEORY ANO STRING/\1-TIIEORY 25

¡nvolved, lhen lhe lemperalure of states in local equilibriummay differ, sinee eaeh will be redshifled by lhe appropri-ate Tolman factor. In the present case one has closed stringslates at lemperalure 1/(3 and open slrong states al temper-alure 1/({3,¡r:;;). The redshifling of open slring slates isuniversal, was confirmed in Ref. 6.

In lhe absenee of a background field lhe open slring has,in perturbation theory, the free energy has a singularity atthe Hagedorn tempcraturc TJlagedorn = l/,6I1agedorn' This isrcprcsents a maximum possible temperature bccause ahoye ilthere are so many massivc string states that thermal equilib-rium heeomes impossible. In a background eleetrie field Ihemaximum temperature is redueed to

TllagedornVl- E2. (73)

9. Strong coupling behavior of Born-Infeld

There are (al least) two interesting strong eoupling Iimits ofBOfIl-Infeld theory.

• A \Veyl-invariant duality invarianl theory whieh ap-pears to he related to a Ouid of massless magnelicSchild lype slrings a",1 may describe string lheory nearcritical c¡ectrie field strengths.

• A massive theory which is related lo a Ouid of massivestrings and may be related to current ideas about D-Dannihilation and taehyon condensares.

In bOlh cases the key lo underslanding lhese limiling lhe-ories is passing lo the Hamiltonian formulation. It also helpsto hear in mind some faets ahout:

This cffeet has been interprcted as being dlle lo a redue-tion in the cffectivc slring tension in an elcctric field. This isecrtainly true bUl olle cannot derive the exael formulae fmmthat assumption alone whereas everything follows rather nnt-urally by an appliealion of the equivalenee principie, as longas one uses the open string mctric.

9.1. Simple 2-forms, distrihutions and string f1l1ids

A 2-form O is simple if

O = a /\ {3, (76)

equivalenlly

In particular since the matrix of components has 01-'''' has ranktwo:

8.1. Shoeks and execptionalily

Loosely speaking, shoeks can oeeur of lhe speed of wavesdepends on lhe phase or amplitude in sueh a way lhat dif-ferent waves surfaces S = constant can catch up and fonncausties. More prcciscly onc assurncs rhe ansatz

O/\ 0= o.

detO"" = O.

In four spacctime dimensions n is simple if

O"" * O"" = O.

(77)

(78)

(79)

and /(5) an arbitrary funetion. The surfaees 5 = eonstantare hyperplanes and are to be lhoughl of as surfaces of eon-slanl phase. If lhe phase spced v depends non-lrivially on Ihephase 5 !here will he shoeks along the envelope of lhe hyper-planes. Theories wilhoul shocks for whieh v is indepemlenlof 5 are called exeepliona!.

Boillal has shown that lhe only form of non-linear clee-lrodynamies with a sensible weak field limil is that of Bom-Infeld.

Theories wilh shoeks are essenlially incomplete. In asense, like General Rclalivily they prediellheir own demise.By eontrast Bom-Infeld, like elassieal Non-Abelian Yang-Milis lheory scems to be a perfeel example of a elassieal the-ory. As far as one can lell il appears lo possess lhe propertywhich is known to he truc for Yang-MilIs theory,that regularBorn-Infcld initial daL.'1with finitc encrgy may cvolvcd for alltime to givc everywhcrc non-singular solutioTls of the fieldequations. For a more dctailed diseussion and refereTlces tothe originalliterature sec Ref. 10.

with

5= n. x - v(n,5)t,

(74)

(75)

Of eourse a and {3 are nut unique bul a field of simple 2-forms defines the uniqlle two-dimensional sub-spacc whichIhey span in the eOlangenl spaee T,M al every poinl of spaee-time. Hence, given a metrie, a simple two form is cquivalcnlto a simple bi-vector 01-'''' which defines a distribution D of2-planes in the tangent space TM. Raising indices with themetric, the simplicity condition beeomes in terms of the bi-vector

(80)

One may think of lhe dislribulion D a, a suh-bundle of Ihetangent bundle with two-dimensional fibres. The 2-planeswiJl be limelike, nuJl or spaeelike depending upon whether01-'",flJ.w is negative, zero or positivc respectivcly. (Note thatthis statcment is signature indepbedent). In the timelike caseone may ehose a 10 be limelike and {3lo be spacelike. [n thenuJl case one may choose a 10 be null and {3to be spaeelike.

In general the distribulion D will not be integrable.That is neighbouring 2-planes will not mesh IOgelher 10form the tangent spaces of a co-dimension lwo family of 2-dimensional surfaces. If it is, then if two veelor fields X andy helong lo D Ihen lheir Lie brackel [X, YJ musl belongto D. Sllch an integrable distributioI1 may he identificd as agas or SOllP, pcrhaps more accllralely a spaghetti of strings.

Rev. Mex. Fís. 49 SI (2003) 19-29

26 G.w. GlBBONS

Thc condition [oc intcgrability may be cxprcssed in variousways. For lIS lhe simplest condition is in lerros ofthe bi-vccLorand is If OIledefines

P=L. (90)

(81) T2 = (0</»2, (91)

it is natural 10 regard the pressurc as a [unetion of the tem.pcraturc T bUl the cnergy dcnsity as a function of the cntropydensily s. In fael they are relmed hy a Lcgendre transformoOne finds lhat

Note thal if J is a smooth funetion, lhen n and Jn definelhe same disLribution and if Lhe firsl is integrable thcn so islhe sccond. Morcovcr, Lhe partial dcrivalivc in (81) may hercplaccd by a torsion free covariant dcrivativc.ln foue spacc-time dirncnsions wc may rc-cxprcss Lheintcgrability comti-tion as p+P = sT (92)

(82) aodWc may rc-wrilc this as

n 1\ ón = O, (83)

OPs= 8T'

T= Op8s

(93)

may he re-inlerpreted as a (vorlieity free) string fluid. Differ-enLLagrangians corrcspond to diffcrcm cquations of s1atc.

aUlomatiea!ly defines an inlegrable distribution. In olherwords non-linear clecLrodynamic. theoey supplcmcntcd withlhe conslraint

where ón = *d * n,Now ifwe lake for n lhe Ampere tensor K"" of any non-

linear clcctrodynamic theney. Wc sec tha1any simple solutionof lhe cqualioTls of motiorl

11is an illuminating exercise 10convince oncself that find-ing lhe speed of small flueluations by lhe ealculating lhesound spccd

(94)

(95)

e, = jop8p

L = 1 - vII - (8</»2

is equivalent lo calculating lhe charactcristics, that is theBoillat rncLric.

Thc masl intcrcsting case fmm thc prcscnt poiot of viewarises when one lakes lhe sealar Bom-Infeld Lagrangian

(85)

(84)\1K"" = O

F-I\K = O,

9.2. O.hranc f1uids

This seetion is based on part on ReL 8. The silualion de-seribed above should be eompared wilh lhe familiar caseof a non-linear sealar ficid lheory wilh a Lagrangian L(O</»eonlaining no explieil dependenee on lhe sealar field </>. Thecquations of moti no muy be cast in the form

(86)

and

P = 1- VI-T2

1P = v'f+S2 - 1,

(96)

(97)

whcrc UPo is a normalizcd timelikc vector givcn by

(87)

which has a maximum tcmpcrature reminisceot of the Hage.dom temperalure. However the delailed cquation of slale isdifferen!. One has lhe equation of Slale

and

U" _ 8Ls - 8(0"</»'

(88)

and hence

P= -p-l+p

(98)

may be intcrprctcd as a conscrvcd cnlr0py currcnt. The quao-lily s eorresponds lo lhe entropy densily and p lo lhe localcncrgy dcnsity. Thc cncrgy momcntum tensor takes the per-feel nuid form

1cs= --o

l+p(99)

One has

T"" = (p + P)U"U" _ Pg"". (89) Note that one need not regard the conserved current as anentropy current if one does no1wish 10. One could rcgard ilas a conscrvcd particlc numbcr.

Re\'. Mex. Fís. 49 SI (2003) 19-29

ASPEcrs OF BORN-ll\'FELD TlIEORY I\ND STRIj\'G / M-TIlEORY 279.3. The Weyl.invariant Bialynieki-Birula Iimit

Thc Hamiltonian density. with units rcstorcd is

One can take the limil T ~ O to get

H = ID x BI.This givcs the constitutive relatiolls

(100)

(101)

is folialed by two-dimensionallighllike surfaees whieh maybe interpreted as the world shcets of magnetic null or Schildstrings. In other words, in this criticallimit which may be in4

terpreted as dcseribing Botn-Infcld theory near erilieal fieldslrenglh, lhe system dissolves into a gas or nuid o[ Sehildslrings.

Sincc elcctric-magnetic dualily is maintained in the Iimil,one can of cnurse pass lo a dual descriplion in terms ofK Jl.v' This amounts to the observation thal I-I:.! = D2 andH. D = O, i.e. KJl.vKJl.Jl. = O and [<muv * KJl.V = O.

E = -11 X n, H=nD, (102)9.4. Coyariant formlllation of UU] lIsing auxiliar,)' liclds

wherc we have dcfincd a unít vector in the dircction of thePoynting vector D x B

The Weyl-invariant limit was ealled by Bialynieki-Birula [3,4], Ultra-Bom-Infeld. Lel us follow him amI eon-sider

Remarkably lhese eonstitutive rdOlions (whieh arise as theIimiting form of lhe eonslitutive relations of lhe full theory)imply lhe eonstraints

n=DxBIDxBI

(103) £- I'¡;: F""+vF F"" (111)- -4 Jl.V '4 ¡.J.V* 1

where I-t ami v are dimensionlcss auxiliary fields, variationwith respectto which gives the constraints

(1 12)

T"'"" .", =HI"'I"' ... 1"'. (107)

Defining a null vector lIJ. = (1,11), the cncrgy momcntumtensor bccomcs

The eonstraints (104) tell us lhal the Faraday tensor F""is simple

VarialioTlwith respect lo Ap. givcs lhe ficld cquation.Note lhat in axion-dilaton Maxwell theory, lhe auxiliary

fields eould be funetions of lhe dimensionless dilaton 4' amIaxion X, l' = 1'(4', X), v(4',X) ehosen in sueh a way thal thesyslem was 5£(2, IR) invarianl. The dilaton and axion pro-vide a map [rom spaeelime inlo 5£(2, IR)/50(2) and onewould, in general, have a non-linear sigma model type kineticterm [or them (see e.g. Ref. 12). For dimensional reasons itmust be multiplied by T. In the limil we are eonsidering lhekinctic term vanishcs ami lhe axion and dilaton becomc aux-iliary fields.

9.5. Tach,}'on c()ndcnsation

We are now kecping o/ fixed and using units in which21TQ'=1. Now it is bclievcd lhat V has a critical point away[rom zero a whieh V vanishes. It is also believed lhat dynam-ieally lhe system will relax to the state wilh V = O. a so-eaHed taehyon eondensOle. One may thus ask, whOl happenslO lhe Botn-In[eld vector in lhis limil. Again the Lagrangiandcnsity causes confusion: il vanishes identically in the limil.However lhe Hamiltonian density is

This subseetion is based on Ref. 13 where referenees 10 theslring lileralure may be found. The basie idea goes baek toAshoke Sen. In lhe presenee of a taehyon field the Bom-¡nfeld Lagrangian density is believed to be modified by thetaehyon pOlential V 10 lake lhe form

£ = V - V JI - E' + B' - (E. B)2 (113)

(106)

(105)

(104)

(108)

(109)

E.B=O.

detF"" = O,

Tt =0,

T"" =HI"I".

lt follows lhOl lhe lraee vanishes

and null,

and henee lhe limiting theory is Weyl-invarianl. lt may beeheeked [hat is Lorcntz-invariant and invariant under clcctric-magnetie duality rotations. One may also check from lheequation of motion that there are infinitely many conscrvcdsymmetric tcnsors

Thus F"" defines a two plane whieh is null. lhal ¡s, the two-plane is tangent to the light eone along lhe lighllike vector 1"and

( 110) H = JV'(1 + B') + D' + (D x BJ2 - V. (114)

Thc cquations of mOlioo lcllus thallhc two-planc distributionin the tangent space defincd by the Faraday two-form FJl.v isintegrable, that is surface forming, and hence that spacetime

ami the limiting forOl is

H = JD' + (D x BJ2. (115)

Rel, Mex, F{s. 49 SI (2(X}3)19-29

28 G.W. GIBBONS

K"v * K"V = O. (121)

Thc rcsulting cncrgy morncnturn tensor is given by

T"v = K"" Kv " (122)V-!K17-rK(n

The trace is given by

Ti:= -V-2K",K"'. (123)

and lherefore lhis is cenainly nO!a conformaIly invariant Lhe-ory.

LocaIly one may P'L" lo a resl frame in whieh B = O.Then

(126)dH =0.

HU. Ilianchi idenlily

10. Thc MS-branc

Consider the simplest situation: just the 3-fonn in a fixcdbackground Einstein-metric glJv. Thus

Now in six-dimensional Minkowski spacetirne and aCling onthrec-fonns the standard linear Hodge dualily is an involu-tion oCorder two: ** = 1 ami in linear thcory self-duality isa consistent field equation, In other words c10sure of H andlhe self-dualily eon"ition give lhe complete sel of equationsof motion. For the M5-brane a very remarkable non-linearseJl'<iua/ily eondition is possihle which fulfills lhe SaIne pur-pose. This was firsl discovered by Perry and Schwarz and itscovariant form wrilten down by Howe Sezgin and \Vest. Onedoes not sccm to be ahle to construct a covariant Lagrangianjust lIsing the 2-form A. Non-covariant varialÍonal principiesexist ami a covariant action principies ha,; been written downusing an additional sea lar field which aets as a time funetian.For lhe time being we need in lhese leelures only lhe cqua-tions of mOlion.

This remarkable eondilion is pcrhaps mosl expcditiouslywriuen as

HU, Non-linear self-dualily

Locally lherefore <lnehas H = dA, for some 2-form A.

[o lhis concluding scclion I will indicale how many of lheideas described above exlend 10 lbe lheory of lhe M5-brane.To paraphrase Hooke 111 /)3-brane sic M5-brane. Indccd fmmlhe M-lheory poinl of view one should perhaps have reversedlhe logie, sinee one may regard lbe equations of Bom-Infeldlheory a\i lhe dimensional reduclion of lhe M5-brane cqua-lions. The lheory and it's cquation have a repulalion for com-plexily and so I will try lo presenl lhem in as direcl a wayas possible. The inleresled reader may find refercnces lo lheoriginal papers and lhe slalemenls made below the paper onwhieh seclion is based 191.

One is oCcourse considering a 6-dimensionai non-linearlheory involving sea lar ami spinor fields and in addition andclosed 3-rorm H"rl>' In wha! follows. 1 shall follow lhe orig-inal papers except that J1, = 0,1, ... ,5. In particular in lhissection [ shall Collow thcir Icad in this section be using themainiy positive signature convention.

(117)

(116)

(124)

(118)

(119)

ti. = IDI.

D.H=O.

D' - H' > O.

Thc rcsulting constitutivc rclations are

H = BD' - D(B .D)VD' + (B x D)2

E = D + DB' - n(n .D)VD2 + (B x D)2

They lell us lhal E x JI = D x B, and lherefore lhe lheoryis LorcJ1ll.-invariant. Qne may check that electric-magneticduality invariance is IOS1in this limit. The constitutivc rcla-tions also imply lha!

but

It foIlows lha! lhe Ampere tensor K"v wilh eomponenlsD, H, is simple bul timelike. Thus lhe lwo-form K"v il de-fines a 2-planc distrihution in the tangent space. As discusscdaboye the cquation of moti no ror K implics thal the distribu-tion is integrable.

The limiting lheory maybe expresscd in lerms of lheAmpCrc tensor K JIol/' One way 10 proceed is lo consider a dualLagrangian. \Ve defineG = *K. The field equation d*K = Obecomes lhe Bianchi-Idenlily dG = O. \Ve now sel G = dCand considcr the Lagrangian

L= J~G"vG"V = J-~K"vK"v. (120)

The action is now varied with respecllo C bUI subjecllO(he conslrainc that

Ol" course in Lhe limit of smal! H we have H ~ *H. Asstated aboye. ir one reduces to five spacetimc dimensions lheequaLioTlsreduce to the standard Bom-Infeld cquations.

This is preeisely what one eX]JC<:Lsof eleetrie flux lubes wilhan energy pmportionallo lhe lcnglh and lO lbe 10lal flux ear-ried by lbe lube.

ln this rcst frarne Orle finds [hat

T"v = (~

O O ~)-7 O (125)O O

O O O

This is just what orle CXPCCLSfor a string fluid.

1

J1+ ~H'x [(I+~H2)J~-4(H')~]H,rh' ( 127)

Rev. Mex. Fís. 49 SI (2(X»)) 19-29

ASPECTS OF BORN-INFELD TIIEORY AND STRING/M-THEORY 2910.3, Boillal cone and Hooke's law

One may introduce lhe analogue of!he Boillat co-metrie:

Thc characteristics of the sealar, spinor and 3.form cqua-tiaos of rnotion are determined by the Boillat mctric CJlll.

Morcover ane may introduce an cnergy morncnturn tensorT"V whieh satisfies Hooke's law: (135)

(134)

(, Bij) ,det O" +4- -TlJ T 1

1i = T2

E_ID1iij - 16 DBij'

The quanlily 1i is lhe encrgy densily Too.One may now restore dirncnsions by sctting

The full non-linear self dualily constraint has a, solulion

To elose lhe syslem one need a eonstitulive relalion. To thiscnd one defi "es

where T ha, dimension mass cubcd. The hmil T~is noweas-ily taken. More interesting than lhe general formulae for Eijare !he resulIs for !he energy momentum tensor. [llakes lhcnull maltcr form

1i = Vdet(Óij +4Bij) -1. (133)

(130)

(129)

(128)

(131 )T¡J.V _; 1/ = O.

3 ( 2 ,)Q=-- l+-H _H' 3

Q [yoo (1+ ~H') _ 4(H')OO](2 - Q) 3 '

where

and is conscrved

10.4, Weyl,invarianl slrong coupling Jimil

Note the sigo change in Hookc's law bccausc ofthc signatureehange. (However TOO 2': ° in bOlh convenlions).

One may prove that T¡J.v salisfics lhe Dominant EnergyCondilion and hence, a, wilh Bom-Infeld theory, lha¡ lheEinstein con e never lics ¡nside the Boillat conc. In generalthe two eones touch along a cirele of dircctions.

In general lhe trace of!he cnergy momentum tensor Tt: docsn01vanish. Thc lheory is nOlWeyl-invariantcxccpt al vanish-ing field slrenglh. However il becomes Weyl invariant in lhelimil of slrong couphng. As wilh Bom- Infeld, the mosl direcIroule lo !his resull is lhe non-covarianl [in oor case 50(5) e50(5,1) symmelric] fonn of lhe equalions. One defines apair of two-forms Eij = HOij and Bij = -ieijpqrHpqr.Thc Bianchi identity may be writtcn in an obvious notationas

( 136)

The three form H is in general nol sclf-dual and is thesum of two tOlally simple three-forms. One factor is lhe nullDne fonn L~dxP., and one has HJ•wulJ1 = O, *Hp.vulJ1 = O.

Thc quantum mechanieal natore of lhis myslerious eon-formally invarianl theory is an inleresting challenge for lhefuturc.

whcre, 1" is again a null veclor in lhe direction of lhe Poynt-ing nux. Thus, jusI a, is lhe case wilh Bom-[nfeld in foursspacctimc dimensions, we attain Wcyl-invariancc in this I¡miland lhe theory has infinilely many conservalion laws.

Poinl wisc, Dne may skew diagonalize Bij. In general ithas rank four and lwo dislincl skew cigenvalues Bl and B,respcctively. Of eoorse lhc basis in which Bij is skew diago-nalizcd will in general vary with position. Pointwisc one findslhat if 1" = (1,0,0, O,0,1)

H = (di - dx') A (B2dx2 A dx' + B¡dx' A dx'). (137)

(132)divB = O.DB7ft + curlE = O,

1. G. Mongc, Application de l'analyse ala geométrie (1807).

2. W. Rouse Bal1,Messenger o/ Marhematics 21 (1891) 20.

3. I. Bialynicki-Birula, "Non-linear Electrodynamics: Variationson a Thcrnc of Bom and Infeld", in Quanlum Theory of Fieldsand Parlicles edíled by B. Jancerwicz and J Lukierski, (WorldSceinetific, Singaporc, 1983).

4. 1. Bialynicki~Birula, Acla Physica Polonica B 23 (1992) 553.

5. EJ. Fcrrcr, E.S. Fradkin, and V. de la Incarra, Phys. Lel!. B 24S(1990) 281.

6. A.A. Tseytlin, Nud. Phys. H4('0 (1998) 69.

7. G.W. Gibbons Nud. Phys. 514 (1998) 603. 9709027.

8. G.W. Gibbons, Pulse Propagation in Bom-Infeld Theoryhep-th/0104015.

9. G.W. Gibbons and P.e. West, 1. Malh. I'hys.hep-th/0011149.

10. G.W. Gibbons and CA.R. lIerdeiro, Phys. Rev. f) 63 (2001)064006-1 - 064006-17 hep-th/0008052.

11. G.W. Gibbons and DA Rashced Nud. Phys. 11454 (1995) 185-206hep-th/.

12. G.w. Gibbons and D.A. Rashhcd I'hys. Leus. B 365 (t996) 46.hep-th/.

13. G.W. Gibbons, K. lIori, and P. Yi, String Fluid from UnstablcD~br.mc,hep-th/0007019.

14. c.G. Callan and J. Maldacena, Nucl. Phys. USl3 (1998) 199hep-th/970814 7.

15. G.W. Gibbons and K. Hashimoto, Nonlincar Elcctrodynamicsin Cutvcd Backgrounds hep-th/ 0007019.

Rell. Mex. Fis. 49 SI (2003) 19--29