S-duality and noncommutative gauge theory

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Citation Gopakumar, Rajesh, Juan Maldacena, Shiraz Minwalla, and Andrew Strominger. 2000. “S-Duality and Noncommutative Gauge Theory.” Journal of High Energy Physics 2000 (6): 036–036. https:// doi.org/10.1088/1126-6708/2000/06/036.

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This content was downloaded from IP address 65.112.8.139 on 27/09/2019 at 16:36 JHEP06(2000)036 =4 N June 23, 2000 , Accepted: June 6, 2000, [email protected] , [email protected] , Received: D-branes, String Duality, Non-Commutative Geometry. It is conjectured that strongly coupled, spatially noncommutative [email protected] HYPER VERSION Jefferson Physical Laboratory, Harvard University Cambridge, MA 02138 E-mail: [email protected] Abstract: Yang-Mills theory has a duala description near critical as electric astrings. field, weakly and coupled Evidence that open for this this stringpoles dual conjecture theory in theory is in the given is non-planar by fully one-loop theviewed decoupled open absence as string from of living diagram. closed physical in The closed a open string geometry string in theory which can space be and timeKeywords: coordinates do not commute. Rajesh Gopakumar, Juan Maldacena, Shiraz Minwalla and Andrew Strominger S-Duality and noncommutative gauge theory JHEP06(2000)036 1 -duality 4 S Investigations to date have largely concentrated on theories with purely spatial 3.1 Spacetime3.2 noncommutativity The free3.3 spectrum Worldsheet correlators5.1 Holes5.2 Handles7.1 Open7.2 string dipoles and Thermodynamics UV/IR7.3 Generalizations to 7 other dimensions 8 7 14 15 12 12 15 2.2 The scaling limits 5 2.1 Born-Infeld noncommutativity (see however [2]). While suchspace, theories they are interestingly are nonlocal local in inand time, a admitting quantum state. familiar notionsmore Noncommutativity like of far-reaching that a consequences, of time-like and the coordinate itexist. hamiltonian is should natural have to even ask whether or not such theories Noncommutative field theories have a richof and these fascinating structure. theories The intorelevant embedding string to theory understanding [1] the inevitable suggestsand breakdown time that of at this our short structure familiar distances may notions in be of quantum directly gravity. space 1. Introduction 4. The one loop diagram5. Higher loop diagrams6. Supergravity duals7. Discussion 8 11 13 14 3. The classical NCOS theory 7 Contents 1. Introduction2. Inducing S-duality 3 1 JHEP06(2000)036 ). N -duality on S -dual of NCYM, S = 4 Yang-Mills)? This N = 4 YM, and the induced duality N -duality induces an S 2 on the open string worldsheet does not commute 0 X or in a space-time with noncommuting space and time 3 , 2 This open string theory can be viewed either as living in a -duality. S 1 field in IIB string theory [3]. IIB B Since the closed string sector of the IIB theory is decoupled in the scaling limit, This paper is organized as follows. In section 2 we derive the Related work will appear in [17]. In this paper we give one answer to this question by asking another: What is the The existence of a scaling theory at near critical electric fields,The and critical its value relevance of to the temporal electric field arises when the force pulling apart the charges at either The low energy sector of the open string theory is ordinary 2 3 1 with the spatial zero modes.same as The the scale effective associated open withinextricably string this tied scale. noncommutativity up is Thus with the the the effects of usual the stringy noncommutativity nonlocalities. are the dual open stringof theory an open does string not theorystring have without poles a a appear closed closed stringaddition string in sector of sector. open is asymptotic string striking. closed The stringwe loop Ordinarily appearance states. analyze closed diagrams, (following In and [11]–[16]) order the unitaritybosonic to nonplanar better case. then one understand We requires loop this find the open point cancellation that string the of diagram temporally all for noncommutative the cancellation phases the in lead closed fact to occurs string a for precise family poles, branes of of in non-commutative any open accord dimension, string with indicating theories. the our existence expectations. of a This near critical electric field, reduces to the standard coordinates. A precise statementis of that the the temporal spacetime zero noncommutativity mode in this theory noncommutativity was emphasizedcommunications). to The us scaling by to the N. critical Seiberg, electric field L.end was of Susskind also the considered string and just inBeyond balances N. [4, this the 5]. Toumbas value string the tension, (private so spectrum that contains the a string tachyon is and effectively the tensionless vacuum [6]–[10]. is unstable. which we refer to astheory. NCOS In (noncommutative section open 3 strings),critical we by electric show embedding that in field. it string decoupling is of In a closed decoupled section strings open by 4theory computing string with we the theory two give non-planar incoming with annulus and evidence anary for analysis two near- at of bosonic outgoing the string the general tachyons. higher Sectionare one loop found. diagram; 5 loop no contains In obvious level a closed section string prelimi- our 6 for singularities open we the string make theory. some We commentswe conclude regarding with concentrate the some on supergravity discussion the duals in U(1) section of theories 7. but For simplicity our results generalize easily to U( question can be addressedbranes in with a the description of NCYM as a scaling limit of three- strong coupling dual of NCYM (spatially-noncommutative the NCYM theory, mapping theopen strongly coupled string NCYM theory theory. to a weakly coupled JHEP06(2000)036 . 2 . ) 2 k 0 (2.1) . ,and 0 πB α on the . /θ o /B = 2 B o 0 G ij =(2 to infinity,  G 0 str 1 = ) k str π = 0 g 23 ij F 0 -duality; in fact -dual version has θ =(2 S ,g S 2 ij k B − and 0 = /θ 0 ij . In order to obtain noncommu- =1 1 9. (We will reserve unprimed no- B 3 directions, in a background geom- , ,B , B/k 0 o 2 2 ,..., , G 1 MN πk , δ =4 3 = 4 SYM may be deduced as a consequence and = =2 0 2 θ 0 o N 0 MN fixed (we will refer to this as the NCYM limit). G M,N , noncommutativity parameter ij ,B δ 2 2 field, together with the modified zero slope limit [3]. ) ,g 3and ,k 1 , 1 k ,where πB ij µν 2 k δ o 0 duality of IIB theory in flat space in the presence of D3- (2 we will consider the NCYM limit described above in an 1 B , i.e. it is the inverse of the original open string coupling, =2 0 o k 0 o S 2 πG = 0 G /G α i, j 0 ij =2 = G ). =1 , 0 ij 0 ij 2 o YM 0, keeping B µν g G η 1with → 0 , α = in the bulk. But, in such a background, the open string coupling that ,g =0 0 µν 01 duality of IIB theory in the presence of D3-branes in the zero slope limit. In µν -dual of F η G S seeming to indicate thatpicture any will be description strongly of rather brane than dynamics weakly coupled, obtained independent in of this S µ, ν = = These difficulties may both be circumvented. In order to avoid having to deal Consider a D3-brane, extended in the 0 In order to obtain a weakly-coupled dual description of the noncommutative ) The S-dual of the NCYM limit takes the closed string coupling ) It seems to involve branes in the presence of an an RR 2 form potential (the = 4 SYM, using the b a 0 µν 01 -dual picture. Before describing this in detail we note that the g tation for the S-dual variables toin be introduced [3] in that the the next sub-section.)gating decoupled on It theory a was on shown four the dimensionalof space brane with is [3]) (open noncommutative string) U(1) metric SYM (we use propa- the conventions etry gauge coupling branes and a background in the limit tative field theory propagating on a space with unit metric we choose S In terms of the field theory couplings with RR fields, we gaugethe away the S-duality. constant This bulk NS-NS gauge potential transformation before induces performing a magnetic field governs the strengthto of the interactions closed betweenbackground string brane is coupling. modes is It not turns directly out related that the open string coupling in this Here of the this section we will determineN the Olive-Montonen dual of spatially noncommutative The Olive-Montonen dual of ordinary 2. Inducing S-duality gauge theory at large two potentially unpleasant features: the D3-branes, whichan is electric converted field into thatfield an approaches may its electric in critical field turn valueB be by in gauged the the into scaling a limit. constant background This NS-NS electric two form potential JHEP06(2000)036 ij F (2.2) (2.3) (2.4) ,the 0 o G .Weare 3 ,L 2 . , it is sufficient, ,L 2 . 1 01 is the size of the 01 . F L i 2 F 01 ) L 00 F duality. ( µν g 00 i F S 11 0 g is 01 . Thus at large g is nonzero and constant, 11 F 2 1 ) πα g 0 π/L 00 A 2 2 01 g ) 0 F πα →∞ − 11 , whose value is given by the g = 1 for simplicity. πα 01 µν str g g F N 1+(2 over any nontrivial cycle of the torus det( 1+(2 p − g q Adx g − q R 4 x − √ i 4 3 e √ d L x will be reviewed in this subsection (see [18]). 2 . This quantized momentum is the electric flux 4 i Z L d L 1 BI str L S Z g 2 str 0 str 1 α 1 g πg 3 2 ) 0 2 , the momentum conjugate to background of the spatially noncommutative theory maps, π α 1 appears in the lagrangian only through 3 01 = 0 ) (2 23 1 F 1 π ˙ F A duality on = (2 -duality NL S BI S = S = S 1 P , but we will mostly stick to the case is negative) N duality, to a background with constant is a periodic physical ‘coordinate’, with period 2 00 is zero. Since g i S ij A F We now consider this limit in more detail. We could consider any finite number In order to work out the expression for the quantized electric flux, consider the The action of Thus, for constant Thus the constant spatial direction). Consequently, the momentum conjugate to the zero mode of is quantized in integral units of i -duality, map to a quantized electric flux. Recall why electric flux on a torus -duality transforms a constant magnetic field on the three-brane to a constant th effective description is anoncommutativity weakly in the coupled time noncommutative direction! open string theory, with of branes, 2.1 Born-Infeld for the purposes of computing canonical momenta in such backgrounds, to set i interested in background field configurations in which is a gauge invariant observable, implyingfield that the zero momentum piece of the gauge that is interchanged with the quantized magnetic flux under theory (2.2) on a rectangular torus, with spatial coordinate radii under A Consider a gauge theory ontwo a cycle torus. of The the fluxS of torus the is magnetic integrally fieldis on quantized, any quantized. and nontrivial so The must,infinite constant space under is piece electromagnetic physically (zero unmeasurable,however, as on momentum it a mode) can torus, be of as gauged the a away. Wilson This gauge line is field not true, in flat to zero in the(recall lagrangian. For a diagonal metric the Born Infeld action simplifies to and therefore remains finite despite the fact that but S electric field. Constant fieldsaction on a single D3-brane are governed by the Born-Infeld JHEP06(2000)036 θ str are /g (2.5) (2.8) (2.6) (2.7) (2.9) . µν AB 9 0 g , dual de- 8 = , S 7 . (the symbol , 0 µν 6 3 g , , , AB 5 2 , , str e G 1 , /g =4 , , =0 =1 . 0 AB 1 θ ) 0 str  . B g 0 = 2 B 0 ) the effective open string scale ) 0 23 πα g str F πα 0 while the open string metric ( 0 eff 1 g of the NCOS theory. ,M,N 2 0 / α +2 = ,A,B 3 1 → , +2 0, are summarized in table 1. ) g πα AB 11 2 2 ( 0 τ 01 2 g g e 0 , 2 ( G det → / α F / 0  1 0 00  0 1 0 eff , θ α g 00 α πα ) crit α ( 0 01 AB g 2 − F det 33 11 Θ =0 5 = 01 g πα √ g i 2 str F 2 22 g ) = g 0 AB 00 + g =(2 = 2 G ) πα crit 11 2 o 01 1+ τ g F AB G p ln( and the (spatial) non-commutativity matrix Θ ,A,B is similarly read off from the coefficient of the gauge 1+(2 = 0 o 3 +Θ AB o , G p e 01 G G 0 AB F g α =2 e G str − 0 − g √ πα are the background metric and field strength in the 2 )= τ µν ( F ,i,j B . Prior to any scaling limit, an open string metric 1 X , µν and B (0) =0 A µν , the noncommutativity parameter and open string coupling in the (S-dual) g X o will be reserved for a rescaled open string metric defined below) and a non- , open string coupling µ, ν G Here As discussed above, in the NCYM limit, is unchanged). The associated open string quantities may then be read from their 0 AB 0 AB α In table 1 weand have expressed all open and closed string quantities as functions of where The open string coupling one finds 2.2 The scaling limits Consider IIB theory with apotential, D3-brane in the presence of a background NS-NS 2-form scription. In terms of the critical value of the electric field held fixed. We would nowtype-IIB like theory. to We study will this callclosed scaling this string limit the backgrounds in NCOS transform the limit. in S-dual the Under description usual an of fashion, S-Duality, the type-IIB G theory action. These are related to closed string quantities by the formulae [3] G and the rescaled open string metric ( solutions to the equations commutativity parameter Θ can beworldsheet deduced boundaries from disk correlators on the open string definitions in (2.9). The results, in the limit NCOS theory. We have also defined the quantities, JHEP06(2000)036 (2.11) (2.10) in the 0 o G  2  1 0 2 o x πα θG 2 ∆ . As a consequence, it  0 eff 0 eff 1 2 µν 1 α πα − θ 4 MN 1 − δ AB 0 0 θ 4 o η =  4 o = πα πα 1 . G θG 2 2 = x crit 2 µν µν ) 0  ∆ 4 o F = = 2 AB o 0 0  µν = πα G θG ij G η πα of the NCOS theory attains its critical =0 0 01 0 δ MN 0 2 (2 4 o θ 0 θ ≡ µν 2 g F 0 o ij π 01 0 o πα θ θ − F 2 θG πα 1 01 G 2 -dual NCOS limit F = G F 2 6 = AB S = = π = =0 = = = e = G = 2 crit 01 Table 1 0 MN o µν ij µν ij 0 eff 0 eff F µν ij str α is the inverse of the gauge coupling − α g g B B g G Θ Θ G α The 0 o . 1 α G 0 eff  α MN π 01 δ ij 2   0 1 = θ = 0 ij − 0 δ 0 ij =0 p θ πα 2 MN  0, the electric field 0 ) = 0 2 0 AB 2 g 0 θ 2 0 µν η → 0 o 0 ij 0 o θ 0 µν πα F − 0 = F G η α (2 G = α =0 ======AB MN µν ij 0 0 0 0 0 o 0 0 µν ij 0 0 0 0 str µν ij B B g G G Θ Θ G α The NCYM limit g g so these open strings have an effective tension set by value the 1 direction is given by (recall that the ends of an open string are charged) NCYM limit. will turn out thatpart in of the the NCOS decoupled limitmass theory excited scale on open is the also string brane set oscillator in by states the are NCOS limit, and that their To summarize, strongly coupled spatially noncommutative Yang-Mills theory has Note that 2. The energy per unit coordinate length of an NCOS open string stretched in 1. In the limit 3. The open string coupling an effective description as ain weakly the coupled presence open of stringtheory a theory are near living listed in on critical D3-branes, table electric 1. field. We will The explore parameters this of theory in this the open rest string of this paper. JHEP06(2000)036 , θ . (3.2) (3.1) 0 eff α → 0 α ,orequivalently 0 /α 1 product is Witten’s star | 2 0 w k ∗ 00 g | , 0 , α N decouple from closed string modes. µν 0 there the = , decouple from the closed strings. As α 1 directions (and using the appropriate 2 p iθ A B , √ m k / w decouple from closed strings. However, the 1 7 ∗ ]= 0 ν AB is the same as that of non-commutativity A α |  G 2 0 w ,X A √ 0 eff 0 k ∗ µ k / k α 00 1 E 0 X A 0 eff g [ α | α  + = 4 YM multiplet. 0 k N AQA R = S 1, open string modes with . − 0 eff Since the effective string scale Disk diagrams in the NCOS theory are very simple. As argued in [19, 3], open Now consider the same limit in the NCOS picture. The argument above ensures We first consider the scaling limit in the NCYM picture. Near the NCYM limit = α 00 open string metric and coupling).the NCOS Thus the limit classical may actionclassical be for action obtained open into string star by modes products.field turning in theory In all action other products words in if the we think usual about open the string open string product [20] then the onlyproduct change which is just that adds we in replace the Witten’s Moyal product phases, by and a modified of course we replace Einstein frame energies obeying g open string oscillator statesopen in string this metric picture in obey the the RHS mass of shell table condition 1 set by the as may easily be seen from (2.8). string correlation functions onthe the equivalent disk correlation in functions theaddition in NCOS of the theory noncommutative phases theory may in be without the obtained the 0 from electric field, by the the noncommutative phases are nonof negligible string only oscillators. for energies of the order of3.2 those The free spectrum In this subsection we will arguethe that 3-brane, the as NCOS open limit string definesthe oscillators an spectrum do open in not string theory decouple the on inby free this NCOS limit. theory We and will see examine that the effective scale is indeed set 3.1 Spacetime noncommutativity In the NCOS limit,field. open This strings results on inzero the temporal modes noncommutativity, brane obey in propagate the in sense a that background the open electric string 3. The classical NCOS theory The decoupled theoryinequality, includes namely all just the brane excitations with energiesthat that open string obey modes with this one has weakly coupled closedstring strings excitations coupled with gravitationally string to frame open energies strings. obeying Open JHEP06(2000)036 0 A M G X X (3.7) (3.6) (3.5) (3.4) (3.3) restricted A . 3 X , . 2 , 1 MN , δ 0, and are given by N =0 → 0 andwehaveextranon- ∂X α M 0 eff . α ∂X → MN Z δ 0 ,A,B 4 0 2 N α ) 0 , M G τ 0 ( 1 α πα 0 eff X  ∂Y 4 α π 0 eff θ M 0 2 AB α  α 2 0 θ = i G ∂Y 8 2 = . Correlation functions of such vertex oper- 0 eff N N α M + = Z 0 eff 2 are finite in the limit α X ∂X = ) M 0 eff . A τ 2 M 0 1 Y k α πα X ln( 4 ∂X = . AB MN M G S G Y 0 eff α Z replacing 0 − 0 eff 1 πα 3. This implies α 4 , )= 2 τ 0. Thus the decoupled theory on the brane includes all open string , ( = 1 B → , S 0 X α =0 . (0) A M X X A, B Thus all correlation functions of NCOS vertex operators on the disk will be the The boundary correlators of The vertex operators representing physically normalized states are functions of In terms of the rescaled fields On the other hand, correlation functions involving the transverse directions ators may be computed given the two point correlators of the free fields oscillator states! The massbrane, spectrum except is with exactly the usual free spectrum on the three in the limit to the boundary offields the world sheet, as well as the two point functions of the free with same as in usual open string theory except that commutative phases appearing as in [3]. The open string coupling constant is and it is finite. 4. The one loop diagram One loop open stringrequires graphs usually that contain closed closed strings string poles, be and included unitarity as then asymptotic states. In this section we are derived from the sigma model and normal worldsheet derivatives of with 3.3 Worldsheet correlators Nontrivial vertex operators are functions of tangential worldsheet derivatives of the normal derivatives of JHEP06(2000)036 ) (4.1) (4.3) (4.2) (4.4) is/π ( , η ] . )) 12 sB ν , k (  4 (singularities) k − · AB 4 3 12 , k G ν ) in (4.1) as 0 eff (2 s rA 2 α ν s k k 2

× − ) 1 = ) k r s − ν  k )) · = 34 transverse momenta. 0 eff N , is/π r ν , is/π ( t 2 α N. rs  34 d ν ν (0 − − ( + ( 34 0 11 2 3 ν 11 = θ × k ) θ (2 j ,k 4

4 B N. k k k k + , ij sB − × = 2 2 s 3 δ k + AB i k k k 2 g · [ = 3 k r i A 2  Ψ k AB k k in the bosonic string with incoming mo- π . Then we get for a D-3 brane in bosonic B e 2 θ s 9 4 0 0 eff Θ 4 k 2 + 12 A α  α 0 2 2 ,k rA X − Ψ + θ

AB 3 k k e 2 A 2 0 , ) g ν k πα ) . We thus find non analyticities ik 2 ∼ Ψ A k G + 24 2 = k e 1 · k o 1 − Ns 1 0 s µν  Ψ k k − 2 k η  G α 4 ( , is/π e µ 0 eff 4 + , is/π π is α k × rs δ dν 2 = annulus 1 2 2 2 ν (0 3 4 o r

i 0 k  ( ) − ) T 0 11 ) 4 η ) dν 10 θ V = θG k πα 2 θ 0 eff ( 11 2

0 α T dν k 2 4 in (4.1). This factor comes from the number of transverse di- ds , , 1 πs V , is/π ,k (4 ) , is/π 3 2 =1 =3 2 4 o 12 Y 3 dν s r − k ν (0 k s 1 ∞ 4 ( ( 0 0 0 11 + T GG s/ 11 Z Z θ → 1 V − θ √ ) k e 2 × × i 2 / and outgoing momenta t k d = = = ( ∼ .) Closed string singularities arise in the integral over 2 − t T 1 k s d 12 V ,k Ψ ) 1 Ψ 1 = k k ( 11 In the NCOS scaling limit, this condition may be written as Nonplanar diagrams for spatial Θ were computed in [11]–[16] — we will fol- The expression for the annulus amplitude in (4.1) is written in the closed string Singularities on the real axis occur at a squared energy T − These singularites are 10 dimensional poles integrated over V 4 s h with may be expanded in a series in in the amplitude when menta consider the nonplanar annulus diagramphysical in closed the string NCOS poles. limit, andbe This show produced demonstrates that that in it an collisions has on of no shell open closed strings. string cannot low [14]. Thetinuation. nonplanar For diagram simplicity considertachyon for vertex the our operators case case of can two initial be and obtained two by final analytic open con- string channel. (The expression forof the superstring would be similar except that the factor mensions string theory [14, eq. 2.17]. JHEP06(2000)036 . is 0 eff 0 α (4.5) s/α =0.For k branes with p ··· 2( + q replaced by ≥ · 0 t d α tβp + 2 . tq has closed string poles, 0 eff N k 0 eff πα 2 M − k πα =0,with e 2 q B × MN ip g 2 θ< as in our NCOS theories, may be + dte 0 eff 0 eff 1 ∞ πα 0 0 limit is manifestly smooth when Z N/α q 10 is 2 4 → dependence from standard string theory with d 0 θ +2 0 is made increasingly small. In the strict limit α is fixed in the NCOS limit, it is easy to see that 2 3 α Z 0 k α ) are very similar to those induced by one loop graphs in ∼ AB 2 i is unstable. + = 0. It is possible that stronger IR singularities appear at k ) G 2 2 . The momentum integral for this diagram takes the 2 i 0 eff k θ ln( k q, p 2 M ( πα − 2 k t 2 t d =0 d ) θ 2 d i I k q ( θ> × Z ip ∼ 2 qe Recalling that 4 d 5 , and so from all open string oscillator states. Apart from the non- Z s . This different dependence is responsible for the absence of physical closed B 2 the amplitude diverges at 0, open string one loop amplitudes factorize on singularities of the form The absence of closed string poles in a non-commutative open string theory, Consider the simple non-planar diagram represented in figure 1, in an open It is instructive to contrast the behaviour of (4.1) in both the NCOS and the 7 in the supersymmetric case), this amplitude, though non analytic, is finite at These singularities 5 ≤ → 0 t α However, the exponential termclosed in string (4.1) channel has coming a from different momentum flowing along the while the theory with understood more directly, asthat we a explain non below. commutative This open line string of theory reasoning with also suggests zero string poles. whose non-commutativity parameter form string field theory. Letnoncommutativity the parameter open string theory in question be noncommutative, with the amplitude (4.1) isin finite the (except superstring). of Itthere course is are for also no the straightforward physical tachyonvertex using poles pole the operators. for which results any is of numbers absent HigherGreen [14] of functions mass initial to on and show vertex the that finalspoil operators annulus. open the These involve string finiteness are additional tachyon of finite thewe powers in amplitudes. expect of the that Although NCOS the the we limit behavior have and not of so worked the out will superstring the not NCYM is details, limits. similar. In the latter case the held fixed. This forces onestring into a states corner of are the decoupled modulifrom [11]–[16]. space in finite In which the the massive NCOS open commutative limit phases (4.1) receives the contributions oneform as loop the open corresponding string diagram in diagram a (4.1) theory with has almost the same that becomes arbitrarily large as As these singularities arephysical states. never in the physical region, they do not correspond to higher loops, specially for high dimensional branes. d spatially noncommutative field theories, as found in [21, 22]. Notice that if p< JHEP06(2000)036 is = and θ θ q in (4.5) -channel p s is slightly · θ q . When ) 2 1 p − 2 0 t p ( q 0 eff does not reveal any 2 α Nonplanar open string θ 2 0 π 8 α s − along the loop. e p + channel, the integrand has the = 0 value only by the additional p Figure 1: diagram. In openory string we field would the- build itbic from vertex the and cu- westates would carrying consider momentum q θ dependence of the exponent. Here π/t is 1 × = p and ,p β (4.6) 0 s ( t p q (terms that would produce the 11 ) ). Note that terms of the form ··· t + =0and 0 eff 2 1 is integrated . This may be seen by shifting the integral over p 2 θ + p q πα 2 0 2 p θ (4 − surface in the string loop expansion is weighted by ( / . − is the total proper ν g ) 2 0 eff t p α t = s 0 eff and so high genus surfaces are suppressed at weak open µν ρ − p πα 0 e Θ i (8 νρ G p/ + Θ ◦ p µ µν q − e Θ µ = p 0 µ ) is the integrand at diverges in the NCOS limit, a perturbative expansion in genus seems q − 2. We have exponentiated the prop- / p, q = str ( µν g were zero) multiplied by the additional factor p Θ =0 ◦ θ θ µ this extra factor exactly cancels the I p q µ .As p 2 Thus the integrand of (4.5) is modified from its 0 eff − g to one over = . As in [21], the effect of noncommutativity on 2 πα str exponential factor (4.6). On shiftingusual to terms the of the form this integral is anthe extra form term in the exponent of β over we get the diagram as a function of 2 q difficulties. Naively, a genus poles if are unaffected by the shift due to the antisymmetry of Θ. we have used the factexponent that we in are (4.6) in is lorentzianless signature the that so opposite its that to critical thebigger value, the final than sign then one of its (4.6) in the critical does euclideanof value not signature. the then cancel instability all the If of closed closed the string string system. poles poles. turn If tachyonic, a reflection where string coupling. 5. Higher loop diagrams In this sectionWe we will will not examine attemptbut to higher we prove loop that will string the observe limit diagrams that is in a nonsingular the for simple NCOS arbitrary counting diagrams, limit. of powers of q where impossible. However, weweighted shall argue by below powers that of both holes and handles are really some other Schwingerintegrated parameter, over; which we isin have also supressed (4.5) this for integral simiplicity). When time along the loopthe and form we have of explicitly the given leading dependence on agators in the diagramtime using representation, a Schwinger where proper g JHEP06(2000)036 is 2 eff (5.1) λ .Italso str . g 9) ,whichcanbe B ,..., a V ) with an additional ) =0 B 2 φ a 2 a V xx I,J m ( , + 2 o G IJ , = φg 2 a J ) m with spacetime action B 0 φ∂ φ + = I , we obtain a worldsheet ) with two extra closed string insertions. ∂ J 1 πα A ( (a) (b) g k ( g I A 12 2 k / √ +2 , with open string boundary conditions cor- 1 x IJ g A ( g 10 2 det k d / 6 1 d Z det 4 Z 0 α str 1 2 eff g ). Hence the total weighting of a hole is 2 str λ g g a A = ( 2 ,V / a = 1 V − A 0. S S → )det 0 X α B 0 B = = denotes the amplitude on πα B a Adding a handle to a worldsheet S ,V a +2 V A g ( S 2 / 1 and is finite as leads to an additionalthe integral loop. over the Asdet zero shown mode in momentum [6], circulating these around integrals have a measure factor proportional to Here handle can be factorized inamplitude the reads closed schematically string channel as along the handle. The resultant responding to a 3-brane. The amplitude on a worldsheet ( 5.2 Handles Consider an open string world sheet determined as follows: A closed string mode Figure 2: 5.1 Holes The addition of a hole in the world sheet is accompanied by one power of The integral is over thetions momenta (momentum of is not the conserved intermediate in states these in directions). the The transverse effective direc- coupling represented as coming fromthe the worldsheet. propagation We of sum closed over string all states closed string between states. two points of JHEP06(2000)036 ) N . N A (5.2) (6.1) S .The N ··· )+ 2 3 k + 2 2 k ( 0 eff α 1 + is of the same order as N a 2 ,V / 2 + eff 1 a 0 / egN V 5 0 α π A . . 4 0 α S MN 3 1 δ 4 0 , 3 = 0. Then we do the following scaling G , 2 eff 0 0  N G 2 e x k 4 0 e α x u 23 0 = 0 eff 0 0 4 0 M 2 α G 0 0 α 0 B 0 k α α 2 0 b 0 b G e e 0 2 13 α 0 4 0 eff α b e e g α √ √ √ / α α α 1 ≈ k g str = = =fixed=4 = = =  6 g A B 0 0 4 d 3 1 r g , , S S θ α 2 0 R = x x Z 0 eff α cosh λ = 2 a m + J 1 k I k IJ g k , as they would have been for an ordinary weakly coupled open string 6 4 0 d G Z 6 Putting it all together, we find that See also [23] for a discussion of diagrams with many holes. = 4 SYM we expect that it should have a supergravity dual for large 6 Finally, in the normalization we have adopted, is of order Thus we conclude that extrapressed handles, nor in enhanced thefactor NCOS in limit, of the are NCOS neither limit. infinitely sup- They are instead really weighted by a of parameters 6. Supergravity duals The considerations of the previous sections generalize to open string theories on coincident 3-branes. In that case since we are dealing with a deformation of U( in the NCOS limit. The integral has effective coupling N theory. relevant supergravity solutions were written inversion of [24, 25]. the solution We [25, start eq. from (2.3)], the with lorentzian JHEP06(2000)036 (7.1) (6.2)  2 5 Ω d 2 in this case is u . u 0 + in section 2.2. We α 2 0 √ du /α solution as we expect, ∼ 5 )+ S r = 4 SYM case we rescale 2 3 e x × d N 5 the metric becomes different + + (oscillators) u 2 2 σ AdS e x ν d p ( 1 µν − Θ f 1 π = 4 SYM at low energies. In particular we )+ + 14 2 1 τ e x N µ d p + 0 eff .Asweincrease 2 0 2 0 iα . The fact that we have e x . G u d 0 we recover the usual = − α +2 here ( u B µ 0 e g 4 4 0 ∼ 1 x u R , . − field as in the previous sections, in [25] it was normalized differently πα r u  1 f 4 4 , 4 4 and we also see that the dilaton becomes large. This suggests 2 − B 4 )= u / R 4 u =2 4 R f 1 1 eg u R 1 eg f f 2 0 0 0 0 2 σ, τ MR ( AdS α eg α α α µ B we should do an S-duality to analyze the solution. After we do the , ======1+ 0 X 7 u u φ f 01 23 πα 2 2 str e A B 0 ds 0123 F πα 2 We obtain The particular scalings that we have to do to reproduce this solution are, up to Here we normalize the 7 see that we should identify since the open string theory reduces to related to the fact that the closed string metric has a factor of 1 the radial coordinate as see from (5.2) that for small 7.1 Open string dipoles andFree UV/IR open string statesopen in strings propagating the in NCOSthe the presence limit same of metric, behave background despite fields, quitein (as having differently discussed [14] the for for from same example the spectrum. in ordinary magnetic [7, In 8, case) 9] the and mode especially expansion reads 7. Discussion byafactorof2 than the metric of that for large S-duality we obtain a solutioncorresponds which to is the a samesee D3 as [25, the brane eq. supergravity with (2.7)]. solutionwe which spatial are This studying non-commutativity suggests would that innon-commutativity. have at the a very directions dual high description 23, energies in the terms open of string the theory theory with spatial constants, the same as thosethe in scaling section 2.2. of the The radial only scaling coordinate. that Notice is that not so in obvious the is JHEP06(2000)036 =4 (7.2) SYM, is the N ,is 1 replaced N X 0 AB k G . This is minimized 1; in that limit [25, 24], . 4  / 1 2 o ) 2 πN/L G . As we argue below, the . 0 (see (2.11)). The energy of λθ + 1, 2 ( o 0 / L X 0 eff  1 G 1), may be expected to reflect eff λ T  πα  0 4 , = k / 0 k N E 2 o 0 eff =1 G duality) argument for the decoupling of ,is πα eff = S T N 15 λ =2 the NCOS theory reduces to ordinary , even at large . At intermediate energies, the thermodynam- 4 1 4 0 T eff T X α ∆ p / , as in (7.2). dualizing spatially noncommutative SYM. We presented E 0 eff S . It is plausible that this result to continues to hold at strong πα θ √ =2 / 1 N  0 eff 0 α k with a crossover scale renormalized by a function of the coupling. If p π 8 =2 However, as argued in this paper, the weakly coupled NCOS theory has a dual It would be interesting to investigate this issue further. It is easy to extend our construction of the NCOS to other dimensions, even The invariant energy and proper length of an oscillator state may be estimated L This statement is true at least in the ‘supergravity’ limit 8 and so scales with temperature like plus oscillator contributions which time average to zero. (Note that ∆ its Hagedorn density of states. description as a strongly coupledtheory, NCYM at theory. weak coupling, In planar a diagrams spatially [26]for dominate noncommutative energies over field nonplanar diagrams [21] for proper length of the string is given by a formula analogeous to (7.2) with 7.3 Generalizations to other dimensions In this paper westring have theory, ‘derived’ NCOS, the by existenceevidence of that, a independent decoupled ofand four weakly this dimensional coupled derivation, open over the a range resultant of theory parameters. isthough well we defined, do not have an independent ( coupling, SYM, and its free energy scales like by the centre of mass energy of the state.) as follows. The tensionalmost of canceled an by open a string negativeIn aligned contribution the with from its NCOS a dipole near limit, interaction critical the with electric effective the field tension field. is distance between theworldsheet endpoints time of rather the than string along worldsheet a along line of a constant line of constant ics of a weakly coupled NCOS theory ( 7.2 Thermodynamics At low energies, compared to 1 For strings in thethe NCOS field limit between this the implies two ends that of the the distance string, along as the measured direction in the of metric true, this assertion impliesnoncommutative that, SYM at is proportional high to temperatures, the the free free energy energy of of ordinary spatially large- such a string, with an oscillation number supergravity suggests that planar diagrams dominate for JHEP06(2000)036 , , ]. B 294 Phys. Lett. Phys. Lett. , , J. High Energy loop equation , hep-th/9711162 . In other words, Nucl. Phys. strings N , 1) n, ( ,...,p (1998) 003 [ String loop corrections to beta ]. from 0 02 Dynamics of A, B . The annulus amplitude is finite in brane is, once again, defined by table p p Noncommutative geometry and matrix the- and Space/time non-commutativity and causality 16 ]. hep-th/0005015 are finite. In the NCOS limit, open strings ]. ]. 2 ,...,p / 3) − (1987) 525. J. High Energy Phys. p eff ( , 0 α Pair creation of open strings in an electric field 2 0 (2000) 044 [ String theory and noncommutative geometry G B 288 06 run from 2 hep-th/9908142 ∼ hep-th/9209032 hep-th/9711112 i, j . 2 YM A matrix string interpretation of the large- g Open string instability in background electric fields Nucl. Phys. (1999) 032 [ , (1992) 77 [ (1998) 64 [ 09 ory: compactification on tori J. High Energy Phys. B 296 hep-th/9705029 functions (1987) 427. B 423 Phys. This work was supported in part by DOE grant DE-FG02-91ER40654. 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