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RAMOND-RAMOND FIELDS IN ORIENTIFOLD BACKGROUNDS AND K-THEORY

OSCAR LOAIZA-BRITO Departamento de F´ısica, Centro de Investigaci´on y de Estudios Avanzados del IPN Apdo. Postal 14-740, 07000, M´exico, D.F. E-mail: [email protected]

We review some results concerning the classification of orientifolds and by K-theory, as well as the role played by the Atiyah-Herzibruch Spectral Sequence relating cohomology and K-theory. The non-existencie of certain types of orientifold planes, their fractional charge and the topological obstruction to have non-zero K-theory charges, avoiding global gauge anomalies, are some of the physical consequences of this relation.

1 Introduction

String theories in the presence of orientifold planes have been studied extensively for some time.1,2 There is a variety of reasons for such an effort. In particular, orientifold planes give us a framework where it is possible to study supersymmetric gauge theories with orthogonal and symplectic groups and in their presence, the theory has non-supersymmetric states which are nevertheless stable. On the other hand, the existence of such non-BPS states are realized in a straighforward classification of RR charges provided by K-theory,3 which has been proved to be the correct mathematical tool to classify solitonic objects carrying RR charge (D-branes).4 Recently, G. Moore and E. Witten5 proved that RR fields are also classified by K-theory given the possi- bility to classify orientifolds as well.6 The basic reason to look for a K-theory classification of orientifolds lies in the fact that some of them carry half-integer values of RR charge (accord- ing to a cohomology classification), violating the Dirac quantization prescription. We shall see that K-theory explains this issue. However, not only there are RR fields associated to orientifold planes, but also to lower dimensional branes on top of them.7 A suitable classifi- cation of these branes (by K-theory) must give us new insights of the nature of these objects. In particular, we shall see that the diference between a cohomological and a K-theoretical classification of these branes is physically interpreted as the topological condition to avoid global anomalies8 in the field theory of lower dimensional branes.9

1.1 Orientifolds An orientifold plane is defined as the locus of fixed points under the action of a set of discrete symmetries. Basically they consist in reversing the orientation as well as the transverse coordinates. A p-dimensional orientifold plane is denoted as Op, and we have at least two different types for each p, which are denoted as Op where the  stand for the sign of the RR charge they carry. Actually, they carry a RR charge equal to 25−p in D- charge units (and are BPS states as well). However, an orientifold classification can also be provided by a non-perturbative analysis.10 This classification is given by cohomology. The transverse space to Op (actu- ally the projective space RP 8−p) contains a set of non-trivial homological cycles where the D-branes can be wrapped on. Since an Op+-plane can be constructed by a set up of an Op−

plebanoloaiza: submitted to World Scientific on June 19, 2006 1 For Publisher’s use and a NS5-brane, it is important to study the action of an Op-plane on the B-field (for which the NS-brane is the magnetic source). It turns out that B is odd under the orien- tifold action, which means that H = dB is classified by a torsion cohomology groupa. Hence, 3 8−p [H] ∈ H (RP ; Z) = Z2. The trivial class of the two-torsion discrete stands for the presence of an Op−-plane while the non-trivial one is related to the Op+-plane. e 6−p 8−p In the same way, H (RP ; Z(Z)) = Z2 classifies RR strenght fields G6−p = dC5−p. D(p+2)-branes, magnetic dual of D(p−4)-branes, can be wrapped on normal cycles if the RR e field Cp+3 is normal; when Cp+3 is a twisted form, D(p + 2) can only be wrapped on twisted homological cycles. The trivial class of the above cohomology group is related to the previous two types of orientifold planes Op, while the non-trivial class stands for a different type of  orientifold plane denoted by Op . Afterwards, there are four different types of orientifold planes for p ≤ 6. f

2 Orientifolds and K-theory

The use of K-theory in has been very fruitful in the past five years. We have learnt that RR charges are actually not classified by cohomology but by K-theory 4 (see also 11). The result is, for instance, the very-well known non-BPS spectrum of branes in the presence of orientifold planes. However, what can K-theory tells us about orientifold planes?. This is the question that − Bergman, Gimon ans Suguimoto (BGS) addressed in 6. The problem is that Op -planes have RR charge equal to −2p−5 + 1 , violating the Dirac quantization condition. They showed that 2 f an integral cohomology cannot explain such issues. So, they argued, a K-theory classification must give us an alternative answer. They provide an orientifold classification by classifiying RR fields by K-theory. G. Moore and E. Witten, showed in 5 that RR fields are also classified by K-theory (the groups differ by one order in relation to those classifying RR charges) even for fields not related to sources (source-free). Using this idea, BGS found that the suitable K-theory groups that classifies RR fields related to orientifold planes are given byb:

Op− : KRp−10(S9−p,0) (1) Op+ : KRp−6(S9−p,0) = KHp−10(S9−p,0) . Notice that, by the use of K-theory, we are able to classify Op−-planes and Op+-planes by two different groups. This is interesting since cohomology classification of orientifolds (con- sidering a RR field classification) only give us one group for both orientifold planes. Consider for instance the case of the orientifold five plane. The cohomology groups related to this 3 3 1 3 orientifold plane are H (RP ; Z) = Z and H (RP ; Z) = Z2, while the K-theory groups are KR−5(S4,0) = Z for Op− and KH−5(S4,0) = Z ⊕ Z for Op+. The way to understand this 2e difference between cohomology and K-theory has been well-known by mathematicians. There exists an algebraic algorithm which gradually computes the K-theory group by a sucessive (but finite) series of aproximmations. The first aproximation level is given precisely by coho- mology. This algorthm is called the Atiyah-Herzibruch Spectral Sequence (AHSS). Given the aRoughly speaking, classifies sections of the bundle Ω3 ⊗  where epsilon is the non-oriented line bundle over RP 8−p. bWith Sn,m being the unitary sphere on Rn,m.

plebanoloaiza: submitted to World Scientific on June 19, 2006 2 For Publisher’s use importance of the AHSS, we proceed to explain it in detail.

2.1 The AHSS and orientifolds

The basic idea of the AHSS is to compute K(X) using a sequence of sucessive approximations, starting with integral cohomologyc. Basically each step of approximation is given by the cohomology of a differential dr, denoted as

p p−r Er+1 = ker dr/Im dr (2)

p p p+r where dr : Er → Er . In each step, we refine the approximation by removing cohomology p classes which are not closed under the differential dr. Closed classes survive the refinement while exact classes are mapped to trivial ones in the next step. In the complex case (without 3 3 orientifolds), the first non-trivial higher differential is given by d3 = Sq + HNS, where Sq is the Steenrod square. In the case of string theory, the only possible next higher differential is d5. By the above process we get the associated graded complex GrK(X) which is the ap- proximation to K(X). The graded complex is given by

p GrK(X) = ⊕pEr = ⊕pKp(X)/Kp+1(X) (3) where Kn(X) ⊂ Kn−1 ⊂ · · · ⊂ K0(X) = K(X). Then, the first order of approximation of d p p Z Kp(X)/Kp+1(X) is given by E2 = H (X; ) for even p and zero for odd p. To obtain the actual subrgoups Kp(X) and K(X) we require to solve the following extension problem

0 −−−→ Kp+1(X) −−−→ Kp(X) −−−→ Kp(X)/Kp+1(X) −−−→ 0 (4)

In the presence of orientifold planes, the first approximation is given by

p,q p Z E2 = H (X|τ , ) for q = 0 mod 4 p,q p Z E2 = H (X|τ , ) for q = 2 mod 4 (5) Ep,q = 0 for q odd , 2 e with X|τ the space where the orientifold acts on.

2.2 K-theory classification of Orientifolds

Once we have a process to compare or lift cohomology to K-theory, and knowing the K-theory groups classifyng RR fields related to orientifold planes, it is possible to get a physical picture which interprets the difference between cohomology and K-theory. Let us come back to our example of the orientifold five plane. In this case d3 is trivial for both types of O5-planes, as well as d5. Hence, the approximation ends at cohomology. It is possible to show that the cFor an introductory review of the AHSS see 6 and references therein. Also see 12. dWe are considering the case which is related to type IIA string theory. For type IIB, we get the non-trivial value for odd p.

plebanoloaiza: submitted to World Scientific on June 19, 2006 3 For Publisher’s use extension problem to solve is

− ×2 for O5 + ( Id for O5 ) Z for O5− 0 −−−→ Z −−−−−−−−−−−→ + −−−→ Z2 −−−→ 0  Z⊕Z2 for O5  k k k . (6)

−5 S4,0 − 3 KR ( ) for O5 1 H + H  KH−5 (S4,0) for O5  f

In the case of the O5+-plane, the sequence is trivial while for the case of O5− it is not. This 3 1 means that a half-integer shift is produced in H due to the presence of the flux G1 ∈ H . The physical implication is as follows: cohomology give us a classification of orientifolds that must be refined by K-theory. The refinement is produced by the half-integer shift in the flux G3 or in other words, by a half-integer shift in the RR charge of the orientifold O5−. Afterwards, − the K-theory picture, trough the application of the AHSS, explain why the O5 -plane has precisely, an extra half-integer amount of RR charge than the ordinary O5−-plane. The same − f situation happens for all the lower orientifolds Op . Another interesting result involves the O3-plane. In such a case, the approximation given f + − by the AHSS, does not finish at the first step, since d3 is not trivial for O3 (for Op , d3 is 3 always trivial since HNS = 0 and the twisted version of Sq is trivial as well). Hence, the non-trivial discrete class of H 3(RP 5; Z) (which at the cohomology level suggests the presence + of an O3 -plane) is obstructed to be lifted to K-theory (it is not a closed form under d3). + + The conclusionf is that these both orientifolds, O3 and O3 , are actually the same object. f 3 Branes in Orientifolds and K-theory

Up to now, we have studied the K-theory classification of orientifold planes (actually a RR field classification) and the differencies with a cohomology classification. The next step is to study, in this context, the presence of RR fields asssociated to branes on top of orientifold planes and the difference with their cohomology description. This is the goal of the present section.

3.1 Cohomology and D-branes in Orientifolds

The idea is to obtain the cohomology groups which classify D-branes on orientifold planes. Consider a D(d + n)-brane wrapping an n-cycle of the transverse space RP 8−p to Op. On the spacetime on the orientifold plane this is seen as a Dd-brane (see for instance 13 and 2). Then we must classify the homological cycles where a D-brane can be wrapped on. Such cycles 8−p are classified by the homology groups H∗(RP ; Z(Z)), where the election of a twisted or normal cycle is determined by the dimensionality of the brane we are considering as well as e the dimensionality of the orientifold plane. In order to realize which D-branes are suitable to be wrapped on particular cycles, it is necessary to study the action of the orientifold plane on RR fields. The action is given by

0 untwisted : Cp0 → Cp0 p = p + 1 mod 4

plebanoloaiza: submitted to World Scientific on June 19, 2006 4 For Publisher’s use

Table 1. Dd-branes obtained by wrapping D(d + n)-branes on n-cycles. ∗: Branes that are not related by T-duality to some known D-branes classified by K-theory. ∗∗: Branes not related to known branes but which play an important role to classify different kind of orientifold planes. †: Branes which are also obtained by lower homological cycles.

8−p 8−p Op-planes Hn(RP ; Z) Dd-branes Hn(RP ; Z) Dd-branes R 2 Z Z R 2 Z Z 6 H0( P ; ) = D6 D2 H0( P ; ) =e 2 D4 D0 R 2 Z Z R 2 Z Z H1( P ; ) = 2 D5 D1 H2( P ;e ) = D6c D2c R 3 Z Z R 3 Z Z \ 5 H0( P ; ) = D5c D1c H0( P ; e) = 2 D3 D(−1) R 3 Z Z R 3 Z Z H1( P ; ) = 2 D4 D0 H2( P ; e) = 2 D1**c D5** H (RP 3; Z) = Z D2* 3 c c e c c 4 4 4 H0(RP ; Z) = Z D4 D0 H0(RP ; Z) = Z2 D2 R 4 Z Z \ R 4 Z Z H1( P ; ) = 2 D3 D(−1) H2( P ; e) = 2 D4**cD0** R 4 Z Z R 4 Z Z H3( P ; ) = 2 c D1* H4( P ;e ) = c D2*c R 5 Z Z R 5 Z Z 3 H0( P ; ) = D3 cD(-1) H0( P ; e) = 2 D1 R 5 Z Z R 5 Z Z H1( P ; ) = 2 D2 H2( P ; e) = 2 D3c** R 5 Z Z R 5 Z Z H3( P ; ) = 2 D0*c H4( P ; e) = 2 cD1† H (RP 5; Z) = Z D2* 5 c e c 6 6 2 H0(RP ; Z) = Z D2 H0(RP ; Z) = Z2 D0 R 6 Z Z R 6 Z Z H1( P ; ) = 2 D1 H2( P ; e) = 2 D2**c R 6 Z Z \ R 6 Z Z H3( P ; ) = 2 D(−c1)* H4( P ; e) = 2 cD0† R 6 Z Z R 6 Z Z H5( P ; ) = 2 D1† H6( P ;e ) = cD2 R 7 Z Z R 7 Z Z \ 1 H0( P ; ) = cD1 H0( P ; e) = 2 D(−1) R 7 Z Z R 7 Z Z H1( P ; ) = 2 D0 H2( P ; e) = 2 D1** R 7 Z Z R 7 Z Z \ H5( P ; ) = 2 D0c† H4( P ; e) = 2 D(c−1)† R 7 Z Z c H6( P ; e) = 2 D1†** R 8 Z Z R 8 Z Z 0 H0( P ; ) = D0 H2( P ; e) = 2 cD0** R 8 Z Z \ R 8 Z Z H1( P ; ) = 2 D(−1) H6( P ; e) = 2 D0c†** R 8 Z Z \ H5( P ; ) = 2 D(−1)† e c

0 twisted : Cp0 → −Cp0 p = p + 3 mod 4 . (7)

Hence, twisted RR fields couple to D-branes which must be wrapped on twisted cycles, while D-branes which are sources of normal RR fields can only be wrapped on normal homological cycles. In this way we can know the homology groups classifing the desired cycles where the D-branes wrap. By the use of the Poincar´e duality, it is straighforward to know the cohomology groups associated to the n-cycles. The Poincar´e duality reads

8−p 8−p−n 8−p For p odd: Hn(RP ; Z(Z)) =∼ H (RP ; Z(Z)) (8) R 8−p Z Z ∼ 8−p−n R 8−p Z Z For p even: Hn( P ; (e)) = H ( P ; (e)) . Notice that in the case of d = p we are classifyinge (by cohomology)e the different types of orientifold planes, while for d < p we get the allowed Dd-branes on top of an Op-plane. The results are shown in table 1 where we give the D-branes we are wrapping on n-cycles, as well as the resulting D-branes on top of the orientifold planes.

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3.2 The K-theory classification of branes in orientifolds

Since we already have a cohomology description of branes in orientifolds, now it is necessary to give a K-theory classification of them. In 7 was given the suitable K-theory groups which classifies RR fields associated to d-branes on top of orientifold planes. They are

Op− : KRd−10(S9−p,0) = KRd−p({pt}) ⊕ KRd−10({pt}) , (9) Op+ : KRd−6(S9−p,0) = KRd−p+4({pt}) ⊕ KRd−6({pt}) .

The values of these groups follow from our knowledge of the Real K-theory groups for {pt}, actually,

−m KR (pt) = {Z, Z2, Z2, 0, Z, 0, 0, 0} mod 8 . (10)

3.3 Cohomology vs. K-theory: the physical meaning

It is time to relate the above two classifications of d-branes on orientifolds by the use of the AHSS. For simplicity, we expose the process by one example: d-branes on top of an O5-plane with d < 5. Four-brane According to table 1, the relevant homology and cohomology group for a 4-brane on an O5- 3 2 3 plane is given by H1(RP , Z) =∼ H (RP , Z) = Z2 and we can argue that this 4-brane must −6 4,0 have a discrete Z2 charge at cohomology level. K-theory groups are given by KR (S ) = Z2 + −6 4,0 − for the O5 and by KH (S ) = Z2 for the O5 . Since d3 is trivial in both cases the order of approximation in the AHSS ends at cohomology. The extension problem reads,

0 −−−→ K3 −−−→ K2 −−−→ K2/K3 −−−→ 0 k k k (11)

0 Z2 Z2 ,

which is trivial. The conclusion is that there are not effects on both O5−planes, due to the − torsion flux , i.e, cohomology and K-theory descriptions are equal. For O5 -plane, this is the T-dual version of the D8-brane in Type I theory, while for the O5+-plane, the presence of a topological 4-dimensional object is unexpected. We interpret this brane as the result c of turning on a discrete RR field (without sources) over a 4-dimensional submanifold of the orientifold five-plane. We argue that this is related to a 4-fluxbrane. Three-brane 3 The cohomology group which classifies three branes on top of O5-planes is H (RP3, Z), and −7 4,0 − −7 4,0 + the K-theory groups are: KR (S ) = Z2 for O5 and KH (S ) = 0 for O5 . In the + 0 3 3 3 case of the O5 -plane, the map d3 : H (RP ) → H (RP ) is surjective; this means that the flux G is lifted to a trivial class in K-theory (it is exact). Physically we must understand 3 e that three branes are not measured by K-theory for the positive five-orientifold plane. For

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− O5 , d3 is trivial and the extension problem is given by,

id id 0 −−−→ K4 −−−→ K3 −−−→ K3/K4 −−−→ 0 k k k (12)

−7 4,0 3 3 0 KR (S ) = Z2 H (RP ; Z) = Z2 e The extension is trivial and we conclude that this brane is the T-dual version of the D7-brane in Type I theory. c Two-brane Possible two-branes are obtained by wrapping a D5-brane on the non trivial untwisted and compact 3-cycle of RP 3. The 3-cycle is classified by the untwisted homology group 3 0 3 H3(RP , Z) =∼ H (RP , Z) = Z. However this integral flux has another interesting interpre- tation. As was pointed out in 13,6, this flux is related to massive IIA . In order to look for some correlations, or equivalence criteria, we resolve the extension problem given by the AHSS. In this case for O5− we have,

id 0 −−−→ K1 −−−→ K0 −−−→ K0/K1 −−−→ 0 k k k (13) 0 KR−8(S4,0) = Z H0(RP 3; Z) = Z

This is trivial and admits just one solution (the trivial one). The integer flux described by K- theory (as an image of that captured by cohomology) is just reflecting the presence of massive + 0 3 D2-branes. Moreover, for the O5 -plane there is a surjective map d3 : H = Z → H = Z2 which implies that odd values of G0 are not allowed. This must be related to an in + e the three-dimensional on 2-branes on top of an O5 -plane with odd G0. These two-branes could be related to two-fluxbranes. It would be very interesting to study this system and the possible anomalies related with. 1-brane Essentially we have the same groups that for the five-branes on both kind of orientifolds. However the difference is that K-theory groups are inverted respect to the five-branes. For the O5−-plane we have a D1-brane (the usual one) carrying an integer RR charge and extra one-dimensional brane related to RR fields without source. Again, we argue that this is related to a one-flux-brane. For the O5+-plane we have also the usual D1-brane expected by T-duality, which corresponds to the D5-brane on Type USp(32) string theory, and a fractional 1 integer one-brane, 2 D1-brane. Zero-brane In this case we have the same situation like the case for the 4-branes. The result is that for − + the O5 -plane we have an induced zero-brane with topological charge Z2. For the O5 -plane we have the expected D0-brane. (-1)-brane c Here we have a very interesting result. Let us analyze it carefully. According to table 1, 3 3 relevant cohomology group is H (RP , Z) = Z2. e

plebanoloaiza: submitted to World Scientific on June 19, 2006 7 For Publisher’s use

+ 0 3 3 3 For the O5 -plane, there exists a surjective map d3 : H (RP , Z) = Z → H (RP , Z) = Z and the flux G is lifted to a trivial element in K-theory which means that in K-theory a 2 3 e (-1)-brane must have zero topological charge on top of an O5+-plane. The extension problem −11 4,0 requires a physical interpretation. We found that K3 = KH (S ) = Z2 and K4 = K3/K4 = 0. In order to obtain a trivial sequence K-theory must measure just zero classes. In other words, K-theory is actually classifing discrete-valued (−1)-branes but it requires that these branes must be considered in pairs numbers, i.e., the K-theory discrete charge must be cancelled . Having and extra condition on the fields is expected because we are considering T-duality versions of branes in Type I and Type USp(32) string theories. This resembles the behavior of the D3 -brane in USp(32) theory, where the brane is unstable due to the presence of taquions on 3-9 strings, but cannot decay to the vacumm c because it has a discrete Z2 charge. So it is espected that K-theory measures this charge, but does not allow the presence of a single non-BPS D-brane. As was shown in 9, this is also a property of the D7-brane in Type I theory. The conclusion of the extension problem is that (−1)-branes must be classified in K-theory but in pair numbers in order to the discrete charge c must be zero. This is actually the required topological condition on the D3 -brane on top of an O9+-plane and sitting at a point in T6 in order to cancel global gauge anomalies on c suitable probe branes. Here we have the same condition applied to a T-dual version of such a system (notice that by considering a T-dual version of USp(32) string theory, we are actually compactifying the theory on a torus. In the case of a (−1)-brane, the only posibility is to sti the brane at a point on the torus T4). The study of the (−1)-brane on top of an O4+-plane, give us the same conclusion for the T-dual version of the D4-brane in Type USp(32) string theory 14. The case of the Op− is paradojic. We obtain a non-zero value by cohomology but a zero c one by K-theory. As K-theory is given exactly by the graded complex and then by cohomology, this is in some sense contradictory. We do not know how to explain this feature, although we think that a more depper study on differences at the cohomology level for branes on top of Op− or Op+, could be very helpful in order to explain the above puzzle. Notice however that (−1)-branes given by cohomology actually reproduces the expected (−1)-branes classified by K-theory. Similar results are getting for other orientifold planes.

4 Final comments

In this review we have seen that the use of the AHSS plays a very important role to understand the nature of branes in the presence of orientifold planes. In particular, the presence of a topological restriction to have a zero discrete charge for T-dual versions of the D3 and D4- branes in the type USp(32) string theory consequently avoids the presence of global anomalies c c in lower dimensional systems. Importantly, we have seen that some orientifolds are equivalent to each other in a K-theory classification and that the fractional relative charge among Op− − and Op -planes can be explained by K-theory. Surely, a further study of the role of the AHSS in orientifolds is needed to understand the f presence of branes related to RR fields without sources.

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Acknowledgments

I would like to thank Hugo Garc´ıa-Compe´an and Maciej Przanowski for invite me to partici- pate in this symposium on the ocassion of the 75th birthday of Professor Jerzy Plebanski.

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