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Home , Faheem Hussain, Graviphoton, Orbifold

INTERNATIONAL THEORETICAL

INTERACTION OF MOVING D- ON ORBIFOLDS

INTERNATIONAL ATOMIC ENERGY Faheem Hussain AGENCY Roberto Iengo Carmen Nunez

UNITED NATIONS and EDUCATIONAL, Claudio A. Scrucca SCIENTIFIC AND CULTURAL ORGANIZATION CD MIRAMARE-TRIESTE 2| 2i CO i IC/97/56 ABSTRACT SISSAREF-/97/EP/80 We use the boundary state formalism to study the interaction of two moving identical United Nations Educational Scientific and Cultural Organization D-branes in the Type II compactified on orbifolds. By computing the and velocity dependence of the amplitude in the limit of large separation we can identify the International Atomic Energy Agency nature of the different forces between the branes. In particular, in the Z orbifokl case we THE INTERNATIONAL CENTRE FOR 3 find a boundary state which is coupled only to the N = 2 multiplet containing just a graviton and a vector like in the extremal Reissner-Nordstrom configuration. We also discuss other cases including T4/Z2. INTERACTION OF MOVING D-BRANES ON ORBIFOLDS

Faheem Hussain The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy,

Roberto Iengo International School for Advanced Studies, Trieste, Italy and INFN, Sezione di Trieste, Trieste, Italy, Carmen Nunez1 Instituto de Astronomi'a y Fisica del Espacio (CONICET), Buenos Aires, Argentina

and

Claudio A. Scrucca International School for Advanced Studies, Trieste, Italy and INFN, Sezione di Trieste, Trieste, Italy.

MIRAMARE - TRIESTE February 1998

1 Regular Associate of the ICTP. The non-relativistic dynamics of Dirichlet branes [1-3], plays an essential role in the coordinate satisfies Neumann boundary conditions, dTX°(T = 0, a) = drX°(r = I, a) = 0. understanding of theory at scales shorter than the Planck length [4,5]. It also provides whereas the three uncompactified coordinates satisfy Dirichlet boundary conditions. some evidence [2] for the existence of an underlying eleven dimensional theory [6]. D- X1(T = 0, a) = 0,X'{T = I, a) = Y'. The boundary conditions are implemented by suitable configurations have also been used to calculate the Bekenstein-Hawking formula for black boundary states. hole entropy from a microscopic point of view [7]. To evaluate the interaction between moving branes we calculate the amplitude

In this work we use the boundary state technique [8] to compute the interaction between A = J°°dl , (l) two moving identical D-branes in a TypellA or IIB superstring theory compactified down to where we have taken one of the branes to be at rest whereas the other is moving with velocity four dimensions on orbifolds. We then evaluate these amplitudes in the limit of large sepa- V. The boundary states in position space are given in terms of the momentum states as ration of the branes, that is in the field theory limit. Further by looking at the dependence d3k of these amplitudes on the rapidity we can unambiguously identify the various contributions coming from the exchange of the graviton, vector and scalar fields. The cancellation of the \B,V,X* = ' (2) force at zero velocity for the toroidal compactification comes from the exchange of the N = 8 In principle these should contain sums over the discrete compactified momenta pn (they are graviton multiplet containing the graviton, the vectors and the scalars. In the case of the Z% zero for Neumann boundary conditions in the corresponding directions whereas for Dirichlet orbifold there is a particular BPS boundary state, corresponding to a D3-brane in TypellB, compactified directions one has a wave function like for the uncompactified case). Since later for which the no force condition comes from the cancellation due to the exchange of the we will be interested in the field theory limit, I —> 00, and since each internal momentum N = 2 graviton multiplet containing the graviton and the graviphoton. Thus, the classical p pn will be weighted by a factor e~' " from the compact Hamiltonian, only the pn = 0 term solution corresponding to this configuration would be the Reissner-Nordstrom . of the sum will be relevant (similarly we ignore the winding states). Thus, the momentum When the D-branes are on the fixed point of the orbifold, this being possible for DO-branes content of the boundary state at rest will be effectively (k° = 0, k,pn = 0). in TypellA, there are contributions from both the untwisted and twisted sectors, the latter We consider the case of two branes, one of them moving along one of the uncompactified corresponding to the exchange of vector multiplets containing vectors and scalars. For the space directions, say the X1 direction. We make pairs of fields, XA = X° + X1 and XB = case of the orbifold T4/Z2 x T2 we find for parallel branes that the BPS cancellation of the X°-X1, and pair the X2 and X3 into the complex fields X2±iX3. The compact directions force is always like in the toroidal case, apart from when the branes are on fixed points are treated in terms of three pairs of complex cordinates [9]. The net effect is as if the b — c where also additional vector multiplets contribute. ghosts cancel the contribution of the pair of coordinates, X2 and X3, orthogonal to the We begin by considering a system of two D-branes in a superstring theory compactified boost. Similarly the /? - 7 contribution cancels the contribution from the fermionic down to four dimensions in the interesting case of the Z3 orbifold, which breaks the super- pair ip3 and ip* for each spin structure. symmetry down to N=2 (the branes will further break it to N=l) [9,10]. In the closed string In order to get the contribution to the boundary state of the D-brane moving picture the interaction between two branes is viewed as the exchange of a closed string be- lv J with constant velocity V, let us consider the boost [11] \BV >— e~ ' >\B >, where V = tween two boundary states, geometrically describing a cylinder. In the present work we use tanh v, (v = |i^|), is the velocity and J*1" is the Lorentz generator. r for the coordinate along the length of the cylinder, 0 < r < I, and a as the periodic coor- The full amplitude is a product of the amplitudes for the bosonic and fermionic coordi- dinate running from 0 to 1. We will always consider -like D-branes, that is the time nates. We first consider the bosonic coordinates. With the standard commutation relations 2 2n 2 for the bosonic oscillators [aJJ,a"m] = [&£,2!lm] = ??'"'5nm, the oscillators, (an,pn), for the where f(q ) = n£°=i(l - Q ) with q = e" "'. A B l X and X fields, respectively, are now defined as an = a^ + a n,Pn = a° — a^, etc., with the Let us now introduce the standard Z3 orbifold [9], that is compactifying the ;J =

commutation relations [an,/?_m] = [5n,/Lm] = — 2<5nm, and the other commutators being 4,5,6,7,8,9 coordinates on a 6- and identifying points which are equivalent under 2 !Zo r zero. pa = e " rotations on pairs of them, with J?4]5 = z$j = ±1/3 and 28ig = —24,5 - *G,7 f° 4 5 4 6 7 6 7 8 9 8 9 The Neumann boundary conditions for the time and Dirichlet for the space coordinates the pairs X ' = X + iX\ X ' = X + iX , X - = X + iX respectively. The request, 4 translate into of the same Neumann or Dirichlet b.c.s for both members of a pair is, with ft'}/' = a + mjj, P^' = a4 - ial etc. {an+p.n)\B>=0, (3)

\B >= 0 {P™ ± 0™n)\B >= 0 (9) Here \B > is the unboosted bosonic spacetime part of the boundary state. Under a Lorentz boost in the 1 direction the oscillators transform as and the corresponding boundary state is

a ' n>l

and similarly for the &n,Pn- The bosonic spacetime part of the boundary state of the moving This is the same as for the torus compactification, except that in the orbifold case there brane is then could be a twist in the u-direction giving noninteger moding, which we discuss in a while. . However the presence of the orbifold opens new possibilities for BPS states. In fact, let us \BV >= exp ]T{-( Ln/Ln) + oIB.aIB}|0 (5) n>0 consider D3-branes in TypellB theory with Neumann b.c.s for X* and Dirichlet b.c.s for where aj denote the oscillators of the directions orthogonal to the motion of the brane. Xi+1 (i = 4,6,8) at both ends T = 0, /. That is The momentum content of the boosted state will be q° = sinh{v)ql, q = {cosh(v)ql ,q±) (Pn + tl)\B>=0 (P™ + Pt )\B >= 0 (11) Therefore from momentum conservation in eq. (1) we will get q1 = k1 = 0, q± = k± and n

thus we get the amplitude at fixed impact parameter Y± as and the corresponding boundary state is

3 ikj Yi A = j d kj_e - - H dlM(l, kL). (6)

In the following we write M = ZBZF where ZB,F are the bosonic, fermionic contributions. The physical boundary state, which is required to be Z3 invariant, is We start by computing the matrix element representing the bosonic spacetime coordinates 2 contribution to the amplitude \Bphys >= i(|B, 1 > +\B, 9 > +\B, g >) (13)

H a a a ZB =< Bv\e-' \B > (7) where we have introduced the "twisted boundary state", {g P° + g *P Jn)\B,g >= 0:

a 2 tt a 2 where H is the usual closed string Hamiltonian. \B,g >=Y[exp—'£,{(g )' P-nP -n + {g ') PTnPTn)\0 > . (14) The oscillator part of the bosonic spacetime contribution is computed to be [11] Since is generically an element of Z we will write, in the following, gPP for g2PP. One ™ 1 2f(q2)q^Hsinhv 3 (8) 2nlz gets for a pair of coordinates {g*ag'a = e ") l lH 1 2f(q?)q /*sin(nza) + irtfLn)\B,r, >= 0, (V; - min)\B,r, >= 0. (20) =Y[ (15) n=l Here TJ = ±1 has been introduced to deal with the GSO projection. For the longitudinal Taking now into account all the contributions from the compactified directions as well as A B B A coordinates these can be rewritten as {ip + ir)ip n)\B >F= 0 , (ip + i7]i> n)\B >F= 0 the spacetime sector and the normal ordering term from the Hamiltonian (q~2?3), it is seen Now to construct the moving boundary state we note that under the boost in the A'' that the oscillator part of the bosonic amplitude is direction the fields 4>A and 4>B transform like the bosonic coordinates ipA —» e~"ipA ,^i" —, 1 3 i4 q / sinh(v) sinlnz ) evtl>B. Thus the fermionic spacetime part of the boundary state of the moving brane is found to be On the orbifold cr-twisted sectors are also possible. We will be concerned with this sector only in the case when the branes are on an orbifold fixed point, since only then the twisted \Bv,r, >= (21) closed string is shrinkable to zero. Thus we consider this sector only for Dirichlet b.c.s on The zero modes of the four uncompactified coordinates ip^, with fi = 0,1,2,3 for the R-R every compact coordinate (thus for DO-branes in TypellA) and the corresponding boundary state can be identified with the 7-matrices 7" = i\/2ip% and 7^ = iv^V'oi with {Y,l") = state is the one of eq. (10). In this sector the pairs of fields in the compactified directions -lif", which act on a subspace which is a direct product of two spinor spaces. We define may be diagonalized [9] such that 0 1 0 1 2 3 2 :i a = (7 + 7 )/2, a' = (7 - 7 )/2 and 6 = (-i7 + 7 )/2, b' = {-i-y - 7 )/2 and similarly

a b 2 u a b a b for a,b and a vacuum by a|0 >= 6|0 >= a\0 >= b*\0 >— 0. Under the boost we have X - (a + 1) = e * °X ' ( e a* leading to the boosted boundary state \Bv,r/ >= with (a,b) = (4,5), (6,7), (8,9). This leads to fractional moding of the oscillators. The e"7 7 /2\B, v = 0 >, giving for the RR zero mode part of the moving boundary state oscillator part of the bosonic amplitude for a pair of cordinates Xa'b becomes

BV,V >= ^-re (22) lH 2 < Ba\e- \Ba >= (1 - (18) The GSO projected state in both the NS-NS and RR sectors is Combining with the spacetime part and converting to Jacobi theta functions we get the full 1 |S >NS,R= 7;{\B, T) >NS,R ~\B, —Tj >NS,R} • (23) bosonic amplitude to be lH 2 4 We now compute the matrix element < Bu|e \B >p (with the appropriate Hamiltonians ZB{a - twisted) = 2 [f(q )] (19) for the NS-NS and RR sectors [10]) for the . Thus the spacetime part of the fermionic Now we consider the fermionic modes' contribution. Again here we combine the fermionic amplitude turns out to be coordinates ip° and ip1 into a pair of coordinates ipA = tp° + ip1 and ipB = ip° - ip1. The B , \e-'"\B,r,' >= IJ(1 ± eV")(l ± (24) = = v V oscillators satisfy the anti-commutation relations {ipwi'n} {^m>V"i?} ~2i5mn with n>0 appropriate half-integer or integer moding for the NS-NS and RR cases. The other directions where n is an integer or half integer for NS-NS or RR, 7777' = ± for the two possible cases of are combined into complex pairs (2,3), (4,5), (6,7), (8,9) as before. With the appropriate the GSO projection and Zo(±) = 1 for NS-NS, Zo(+) = 2cosh(v) and Zo(-) = 0 for RR. normalisation the spacetime modes' b.c.s for the D-brane at rest are given by [8] (Neumann The tp2'3 contribution is cancelled by the /? — 7 ghosts, but in the RR 177/ = — 1 (odd spin for time and Dirichlet for space) structure) case there remains the zero mode of this pair giving zero.

6 For the compactified coordinates one again combines the fermions into pairs Xn'5 = still to sum the product of the bosonic (16) and fermionic (28) amplitudes over the three l ipn + iip\, etc. Let us describe the fermionic part of the previously introduced twisted possibilities for g and g' (actually only three possibilities for g~ g> are distinct). boundary state (14) (mixed Neumann/Dirichlet b.c.s for each pair). The condition for the On using the Riemann identity it is easy to see that the amplitude (28) behaves for small twisted boundary state is now velocities as V2, if g ^ g' and as V4 if g = g'. In the limit I —> oo (where the bosonic: part

(16) is za independent) the leading behaviour of this amplitude is proportional to (/Xn + i9a'vXa-n)\B, 9a,i? >= 0, (25)

{Acosh{v) .za) — (cosh2v + (29) for each pair of coordinates and the state satisfying this condition is (for generic a and g) We observe here that the first term with cosh(v) is the contribution of the RR vector (26) \B,g,n >= exp \ -+ X-nX-n + 9*X*-nX-n whereas the rest are the contributions from NS-NS exchange. We note that the amplitude 2 Using these states we find for a pair of fermionic coordinates in the compactified space, (28) vanishes at v = 0 for all the twists (l,g, g ) individually. cr-untwisted closed string sector, that Just as for the bosonic amplitude, when the position of the brane is on the fixed point of the orbifold here we also have to include the closed string N$= f[ [l ± e-«<"-§)][l ± e-<«(»-^] . (30) n=l as the spacetime contribution and the normal ordering terms for both the NS-NS sector For the RR sector (in this twisted sector there are no zero modes) we have here and the RR sector we find, for the twisted boundary state (cr-untwisted sector), that the tH fermionic amplitude is, in terms of Jacobi theta functions, B,ri\e- \B,7]' >R= f[ [l ± e- (31) 71 = 1 The net result in the twisted sector is that the full fermionic amplitude, including the <71/3/(<22)4 spacetime sector and the appropriate normal ordering contributions, can now be written in ,iv. (28) terms of Jacobi theta functions as

In making the algebraic sum of the RR and NS-NS sectors, we take the same signs which l Bv\e- "\B, >P= hold for the partition function on the torus. The first term is the contribution from the 3 3 {-\ il) - •di(-\2il)04(-2il/3\2il) ) . (32) RR sector (even spin structure) whereas the second and the third are from the two NS-NS n

GSO projections. Observe that the compactified parts of eqs. (16) and (28) agree with the Recall that in the twisted sector for the Z3 orbifold there has to be a relative positive •d-function expressions on the torus found in ref. [9]. This is consistent with viewing the sign between the two NS sectors because of invariance under the modular transformation cylinder as half of a torus, and also the < i/nl> > correlators computed with the b.c. (25) T —* r + 3. At low velocities this amplitude goes like V2. would agree with those on the torus with Minahan's twist [9] in the r-direction. Since the Let us now compare the large distance interactions of the two moving branes found

Z3 invariant, physical boundary state is given by the linear combination of eq. (13), one has from the string formalism with the field theory results. At large distances we look for the

9 Feynman graphs representing the exchange of massless . We can have the exchange 1.) Toroidal compactification. The boundary states can be of the form of eqs. (10) or either of scalar, or vector or graviton. The scalar and the graviton give attraction while the (12) and in any case one gets eqs. (28) and (29) with za = 0. Thus in the field theory limit vector gives repulsion, since we consider two branes of the same nature. The net result for the amplitude is proportional to zero velocity is zero, since the branes are BPS states, and this is what is obtained from the Acosh(v) - cosh(2v) - 3. (30) Riemann identity in the string formalism [12]. But when the velocity is different from zero, Take in particular the case of two DO-branes. If we consider eq. (36) as the result for the the various contributions are unbalanced. By comparing the velocity dependence with what ten dimensional uncompactified case we write it as Acosh(v) — (cosh(2v) + |) — |. The first we get from Feynman graphs we can tell which kind of particles are actually coupled to the term is the contribution from the exchange of the RR vector whereas the second term is from branes, in various compactification cases. the exchange of the graviton in 10 dimensions (as is seen from the field theory calculation We treat the branes as spinless particles of mass and charge equal to 1. The exchange above) and the last is the scalar exchange contribution. We have normalised to the graviton of a scalar gives then exchange. Our DO-brane in 10 dimensions is like the classical 0-brane solution [13] of the S=± (33) bosonic part of the Type IIA 10 dimensional action. If we now view eq.(36) from the point of view of toroidal compactification, with no SUSY where k is the momentum transfer between the two branes. In the so-called eikonal approx- breaking, in four dimensions the graviton exchange gives the contribution cosh{2v). Our imation in which the branes go straight (which is the standard setting, which we follow, for toroidal compactification corresponds to Steile's vertical reduction of the 10 dimensional describing the branes' interaction at nonsmall distances), k has only space components, and 0-brane solution to four dimensions [13]. In this case the relations between the masses, it is orthogonal to V. the electric charge and the scalar coupling of the 0-brane are Q2 = 4M2 and a2 = 3M2, The vector is coupled to the current, which in the eikonal approximation is proportional precisely the relations that we have obtained. Further, in both these cases, uncompactified to the momentum, J^iV) = (cosh(v),sinh(v)). Note that J^k^ = 0. Taking one of the and toroidally compactified, the force between the branes goes to zero like V4 as expected branes at rest, the vector exchange is as there is no breaking. Following the work of Pollard [14] we also see that

(34) the DO-branes in this case are extremal Dobiasch-Maison blackholes [15]. This extremal blackhole has zero horizon radius and zero horizon area. The graviton is coupled to the brane's energy-momentum tensor T'"' = J^J". Therefore 2.) Orbifold compactification,

10 11 Acknowledgements. The authors would like to thank E. Gava and K.S. Narain for very graviton with no scalar exchange. In terms of the N = 2 SUSY theory these systems couple valuable discussions. C.A.S. also thanks F. Morales and M. Serone for useful exchange of only to the graviton and its N = 2 partner, the graviphoton. In this case the amplitude ideas. C.N. would like to thank the Abdus Salam ICTP for hospitality during the completion behaves like V2 for small velocities. Prom the pattern of cancellation [14] we see that these of this work, which was done within the framework of the Associateship of the Abdus branes correspond to classical extremal Reissner-Nordstrom blackholes. Salam ICTP, Trieste, Italy. R.I. and C.A.S. acknowledge partial support, from EEC contract 3.) Orbifold compactification, cr-twisted sector. In this case the I —> oo limit of eq.(32) ERBFMRXCT96-0045. gives

cosh(v) — 1 (38) REFERENCES

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