We review orientifold constructions in the presence of magnetic backgrounds
both in the open and closed sectors. Generically, the resulting orientifold
models have a nice geometric description in terms of rotated D-branes and/or
O-planes. In the case of multiple magnetic backgrounds, some amount of
supersymmetry is recovered if the magnetic fields are suitably chosen and part
of the original D-branes and/or O-planes are transmuted into new ones.Comment: Contribution to the Proceedings of the first SP2002 Conference and of
  the 35 Internationsl Symposium Ahrenshoo

Angelantonj, C.

English

arXiv

Angelantonj, Carlo

Institutional Research Information System University of Turin

ROTATING D-BRANES AND O - PLANES

C. Angelantonj

Crossref

Rotating D‐branes and O‐planes

We review orientifold constructions in the presence of magnetic backgrounds both in the open and closed sectors. Generically, the resulting orientifold models have a nice geometric description in terms of rotated D-branes and/or O-planes. In the case of multiple magnetic backgrounds, some amount of supersymmetry is recovered if the magnetic fields are suitably chosen and part of the original D-branes and/or O-planes are transmuted into new ones.We review orientifold constructions in the presence of magnetic backgrounds both in the open and closed sectors. Generically, the resulting orientifold models have a nice geometric description in terms of rotated D-branes and/or O-planes. In the case of multiple magnetic backgrounds, some amount of supersymmetry is recovered if the magnetic fields are suitably chosen and part of the original D-branes and/or O-planes are transmuted into new ones

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CERN-TH/2002-269hep-th/0212066Rotating D-branes and O-planesCarlo AngelantonjTh. Division, CERN — CH-1211 Geneva 23AbstractWe review orientifold constructions in the presence of magnetic backgroundsboth in the open and closed sectors. Generically, the resulting orientifold modelshave a nice geometric description in terms of rotated D-branes and/or O-planes.In the case of multiple magnetic backgrounds, some amount of supersymmetry isrecovered if the magnetic fields are suitably chosen and part of the original D-branesand/or O-planes are transmuted into new ones.1. IntroductionHomogeneous magnetic elds provide an interesting example of a non-trivial deforma-tion compatible with two-dimensional conformal invariance. The study of their eect inString Theory is relatively simple for string coordinates are still described in terms of freeelds and the magnetic deformations only aect the world-sheet dynamics by modifyingboundary and periodicity conditions for open and closed strings.Despite their simple ro^le in world-sheet dynamics, magnetic elds lead to new in-teresting phenomena in D-branes and orientifold constructions. Considered at the verybeginning as an interesting tool to break supersymmetry[1, 2], deeper investigations haveshown how they can aect the geometry of branes and O-planes. As usual, vacuum ex-pectations values of gauge elds, after appropriate T-dualities, admit a very elegant andsimple geometrical interpretation: while Wilson lines correspond to brane displacements,magnetic backgrounds translate into brane/O-plane rotations[3, 4, 5, 6, 7]. As a result,the corresponding brane/O-plane congurations are no longer BPS and typically breaksupersymmetry. In lower dimensions, however, some amount of supersymmetry can berecovered if background elds, i.e. rotation angles, are chosen appropriately.2. Open-string magnetic fieldLet us start with open-string magnetic elds. As already stated, the study of theireect is relatively simple, for they interact only with the string ends [8].For simplicity, let us consider the bosonic string in the presence of a uniform magneticeld H in a plane with coordinates (X1; X2). The variational principle for the world-sheetactionS =120∫d∫ 0d @Xa@Xa (1)+ iqLH∫d abXa@Xb∣∣∣∣=0+ iqRH∫d abXa@Xb∣∣∣∣=;yields the wave equation (@2@ 2− @2@2)Xa = 0 ; (2)together with the boundary conditions@Xa − 20qLH ab@Xb = 0 ;@Xa + 20qRH ab@Xb = 0 : (3)Eq. (3) admits an alternative geometric formulation in terms of rotated branes[3].Indeed, after performing a T-duality along the X2 direction, so that  and  derivatives1are interchanged,@X2 ! @Y 2 ; @X2 ! @Y 2 ; (4)the boundary conditions become standard Neumann and Dirichlet ones, say, at  = 0@(cos  X1 − sin  Y 2)= 0 ;@(sin  X1 + cos  Y 2)= 0 ; (5)for a rotated set of coordinates, where the angle  = tan−1(20qLH) is determined bythe magnetic eld H .The interesting feature of intersecting brane constructions that has triggered a massiveinterest is that, in general, chiral fermions emerge from strings with charged ends whilesupersymmetry is generically broken.From eq. (5), however, it is not hard to see that something special can happen if oneintroduces a second magnetic eld on a dierent two-plane, or, alternatively, if one furtherrotates the branes along another pair of directions. In this case, the choice 1 = 2 =12yields a conguration of orthogonal D7-branes that preserve half of the supersymmetries.As a result, one can use magnetised, or rotated, branes to construct, for example, newT 4=Z2 type I vacua as in [9].As a remark, rotated branes are recently playing a pivotal role in brane world con-structions, where the Standard Model degrees fo freedom live at their intersections [10]3. Compactification on Melvin spacesWe can now turn to the case of closed-string magnetic backgrounds, correspondingto non-vanishing vacuum expectation values for the graviphotons[11]. This backgroundbelongs to the class of Melvin metricsds2 = G(γ; R)dXdX = d2 + 2(d +γRdy)2+ dy2 + dxidxi ; (6)where γ is related to the background magnetic eld, (; ) are polar coordinates on a two-plane, y is the compact coordinate on a circle of radius R and xi are spectator coordinates.According to eq. (6), the string dynamics in the Melvin space{time is governed by theworld{sheet ActionS = − 140∫dd G(γ; R) @X@X : (7)This model has a very simple equivalent description after introducing the complex coor-dinateZ =  ei’ =  ei(+γRy) ; (8)2that actually corresponds to a free boson(@2@ 2− @2@2)Z = 0 (9)with the twisted boundary conditionZ(;  + 2) = e2inγZ(; ) ; (10)where n is the winding number in the compact y direction.As a result, we can use all the techniques of orbifold compactications to solve exactlythe model in eq. (7).The background we are considering has the nice property of preserving the invarianceof the IIB superstring under the world{sheet parity Ω. In the spirit of [12], we can thenproceed to construct the associated orientifolds, that, as we shall see, have interestingfeatures [7].Aside from the standard Ω projection, that due to lack of space we shall not discusshere (see, however, [7]), one has actually the option of combining world{sheet parity withother geometrical symmetries as, for example, parities along the angular ’ and compacty coordinates, ~Ω = Ω’y. The resulting action on the world{sheet coordinates (y; Z) is~Ωy(; )~Ω−1 = −y(;−) ; ~ΩZ(; )~Ω−1 = Zy(;−) ; (11)and, together with the identication (10), implies thaty = −y + 2sR ; ’ = −’ + 2sγ + 2m ; (s; m 2 Z) (12)whose xed points(y; ’)1 = (sR; sγ) ; (y; ’)2 = (sR; sγ + ) ; (13)identify the positions of orientifold planes. Actually, we are now dealing with two innitesets of rotated O-planes, the angle being proportional to the twist, or magnetic back-ground, γ. Moreover, O-planes in dierent sets undergo a relative  rotation. As a result,they carry opposite R-R charge, a result that we shall soon recover from the correspondingKlein-bottle partition function.In this orientifold construction, although the string partition function accounts for thewhole spectrum of physical states, built from the vacuum with string oscillators and/orKaluza-Klein modes, a proper derivation of the divergent contributions to the tree-channelamplitudes is quite involved. In fact, as we have seen, a key feature of Melvin compacti-cations is that the twist on the complex coordinate Z is coupled to the number of times3the closed string winds the compact coordinate y. This means that each winding state hasa dierent correpsonding \twisted sector", and, as such, in the partition function the char-acteristics of the theta-functions are shifted accordingly. For example, the direct-channelKlein-bottle amplitude readsK  17=22 6∑nK (n) q12′ (nR)2; (14)withK (0) = 1211=22 2∑;12#4[]4;K (n) =∑;12 e−iγn(2−1) #3[]3#[+γn]#[1=2+γn1=2] : (15)(Here  = (−1)2+2+4 denotes the standard GSO projection.) Therefore, one cannot perform explicitly the S-modular transformation, and the resulting amplitude doesnot manifest the usual physical interpretation.However, divergent contributions originate only from massless states. Hence, to extracttheir tadpole it suces to restrict ourselves to the massless Hamiltonian and to its Kaluza-Klein modicationH =02−1@@(@@)− 12@2@2− @2@y2+1R2(k + iγ@@)2 : (16)As a result, the leading contributions to the transverse-channel Klein-bottle amplitudecan be extracted fromhcje−‘H jci = ∑ΨhcjΨie−‘EΨhΨjci ; (17)where HjΨi = EΨjΨi. Taking into account only the zero-mode contributions to theboundary state jci, one gets[7]hcje−‘HjciNS−NS ∑m;k[1 + (−1)m][1 + (−1)k]e−′‘2R2(k−γm)2 (18)for the NS-NS sector, andhcje−‘HjciR−R ∑m;k[1− (−1)m][1 + (−1)k]e−′‘2R2(k−γm)2 (19)for the R-R sector, where k is the momentum quantum number and m is the quantisedangular momentum on the two-plane. From eq. (18) we can then extract the non-vanishing dilaton tadpole (for m = 0 and k = 0), as well as informations on the geometryof the O-planes. The factor 1 + (−1)k suggests that in the y direction there are O-planes4sitting at 0 and R, while the projector 1 + (−1)m implies that, in (; ’) space, thereare pairs of orientifold planes rotated by a  angle. Furthermore, from eq. (19) one canread that O-planes diering by a  rotation have opposite R-R charge, in agreement with(13). This phenomenon is reminiscent of the intersecting brane models discussed in theprevious section. Although the overall R-R tadpole vanishes, cancellation of the NS-NSone calls for the introduction of D-branes, as in standard orientifold constructions [7].Also in this case, one can generalise the construction considering multiple two-planessubject to independent twists. We have now the option to couple each twist to the sameS1 or to dierent ones. While the latter choice corresponds to a trivial generalisation ofthe single twist previously reviewed, the former turns out to be quite interesting and leadsto amusing phenomena.To be more specic let us consider the case of two two-planes with twists γ1 and γ2.A simple analysis of Killing spinors on this Melvin background[13] shows that wheneverthe γi are even integers all supersymmetry charges are preserved. However, one has theadditional possibility of preserving only half of the original supersymmetries ifγ1  γ2 2 2Z : (20)This is very reminiscent of the condition one gets for orbifold compactications. Moddingout by Ωy’1’2 then introduces O-planes at the xed points of the involution. Partic-ularly interesting is the case γi =12. The resulting model is the traditional (C2  S1)=Z2orbifold, where the Z2 acts as a reflection on the C2 coordinates accompanied by a mo-mentum or a winding shift along the compact circle, and mutually orthogonal O6 planesare generated[7], as pertains to a (C2  S1)=Z2 orientifold.Acknowledgments. It is a pleasure to thank the organisers of the rst SP2002 confer-ence, and of the 35th International Symposium Ahrenshoop on the Theory of ElementaryParticles for their kind invitation.References[1] E. Witten, \Some Properties Of O(32) Superstrings," Phys. Lett. B149 (1984) 351.[2] C. Bachas, \A Way to break supersymmetry," arXiv:hep-th/9503030.[3] M. Berkooz, M. R. Douglas and R. G. Leigh, \Branes intersecting at angles," Nucl.Phys. B480 (1996) 265 [arXiv:hep-th/9606139].5[4] R. Blumenhagen, L. Go¨rlich, B. Ko¨rs and D. 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Rotating D-branes and O-planes

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