JHEP05(2001)047 May 24, 2001 May 24, 2001 Revised: Accepted: April 2, 2001, Received: D-branes, Differential and Algebraic Geometry. We analyze the role of RR fluxes in orientifold backgrounds from the [email protected] [email protected] [email protected] HYPER VERSION CIT/USC Center for Theoretical Physics,Los Angeles Univ. CA, of USA SouthernE-mail: California School of Natural Sciences, InstituteEinstein for Drive, Advanced Princeton Study NJE-mail: 08540, USA California Institute of Technology, PasadenaCIT/USC CA Center 91125, for Theoretical USA Physics Univ. and of Southern California,E-mail: Los Angeles CA, USA Abstract: point of view ofing K-theory, these and fluxes demonstrate incertain some K-theory physical fractional rather implications shifts than ofnaturally in cohomology. describ- explained RR In in charge particular, K-theory. quantizationwhich we are We due considered show also to distinct that show in discretethe that the K-theory RR some cohomology description, fluxes orientifold classification, while backgrounds, become are others become equivalent in unphysical. Keywords: Shigeki Sugimoto Eric Gimon Oren Bergman Orientifolds, RR torsion, and K-theory JHEP05(2001)047 )27 0 , 6 S ( KH 1 )and 0 , 6 S ( KR using a spectral sequence 25 K A.3 Extending theA.4 AHSS to Illustrative K-theories example: with freely acting involutions 26 2.1 Basics2.2 Cohomology torsion2.3 variants Brane realization of torsion3.1 Real K-theory3.2 RR3.3 fields in orientifold backgrounds Comparing with cohomology4.1 Charge quantization4.2 Equivalent orientifolds4.3 Unphysical orientifolds5.1 K-theory5.2 Anomalies5.3 4 Components of type-I vacuua 6 9 A.1 Defining theA.2 associated 10 graded Computing complex 3 Gr 8 12 15 17 21 24 17 19 1. Introduction Orientifold planes are intriguing objectsbatively, in to many be ways. void They of appear, dynamics,the at and other least in pertur- hand this respect orientifoldwith they planes the are are similar same to also dimensionalities, orbifolds. carry similar RR On to charge, D-branes, and in have a that non-vanishing they tension. appear 3. Orientifolds in K-theory4. Applications5. The orientifold 6-plane6. Conclusions and open questionsA. AHSS in complex and 8 real K-theory 12 17 23 24 Contents 1. Introduction2. Review of orientifolds 3 1 JHEP05(2001)047 ,as- p 2charge / 1 ± ,therelevant 2 +1 / p 4 are fractionally C ), respectively. The +1 p ≤ 5 5 ) shifts the charge by 5 5 − p 8 − − − − p p p p p − P 8 2 2 R P − +2 − +2 charge ( variant, and yet another to is not an allowed value for p R ( − 5 − p 6 − − -plane in the reduced space of p 6 f H O4 p 2 H ± . The generator of this group has = 6, both of these cohomologies are Z ) ) , and have traditionally been classi- p +1) p n n )and ) plane. So in the first case it seems to p − n n 6 − )= + gauge group 8 )-brane, and can be expressed in terms of p G p p 2 − P p 8 -plane is charged under R − SO(2 USp(2 SO(2 USp(2 D P p ( 3 R ( 5), depending on the values of the discrete fluxes H p -plane charge )-form p − p 8 ≤ p . One could imagine that the latter is due to the term − H variant + + − + − p p p p p f (for O4 f f O O O O O Orientifold plane variants and their charges. p -branes). This appears to be at odds with Dirac’s quanti- ) p p 2) / − 6 2) 1 0) , , 2 there are actually additional variants, which are associated with / 2 2 0) 1 plane, but not for the O Table 1: / , / , 2chargeof -flux [2]. H, G − / p p< (1 ( (0 (1 (0 p is the “sphere” that surrounds the O ,andtheRR(6 − 2 p H plane is consistent with an anomalous phase in the D2-brane path integral − G in the D2-brane action, but there is no obvious candidate for the former -brane charge, so the O 8 ± p 1 P C R ∧ 4. In addition, the presence of RR torsion in There are actually a variety of orientifold planes for each dimensionality In some cases the violation can be attributed to an anomaly in the path integral One immediately observes that orientifold planes with , so the fluxes are torsion classes. This means that there are four variants of the B ≤ 2 of the O4 charged (relative to D zation condition, suitablyor generalized equivalently to with higher thehomology. rank idea that anti-symmetric In RR tensor charges particular, fields, and since fields the take values O in integral co- a discrete spacetime fields [3], but this is not generally the case. For example, the Z the orientifold projection. Except for the case fied by the (integral) cohomologies 1/2 for the O explain the +1 cohomology group is given by space contribution. For other orientifoldanomaly planes contribution of one [3]. does not even have an analog of the R However the latter twosome differ cases from the their tension of D-brane an(and values orientifold tension) plane in is is two negative, fractional. important and in ways. some cases In its charge sociated with the possibilitytensor of fields turning [1, on 2]. discretethree-form fluxes These of fluxes certain come anti-symmetric in two varieties corresponding to the NSNS orientifold plane for each (table 1). For lead to a further violation of Dirac’s condition. for the dual magnetic object, i.e. a D(6 p coming from the fermionto determinant the [3]. phase in However order one to needs explain another the contribution 0 charge of the a unit D JHEP05(2001)047 -flux. For p − 8 .Thusweare G 2 G -brane is characterized not only by its p alters the quantization of p 3 − 6 G and the half-integral part of 0 G , but also by a gauge bundle, which in turn determines +1 p C ) is integral rather than discrete, but there is an analogous 0 G The rest of the paper is organized as follows. In section 2 we review what is In this paper we investigate some of the consequences of describing RR fluxes All this indicates that integral cohomology is not the appropriate mathematical = 6 the former ( able to compute thethe discrete shift RR in flux. thethe quantization absolute We condition charges are ofcohomological however. not the one able In RR to we comparing charge establishwhile also the due others the discover to K-theory are quantization that obstructed. condition classificationsome some for of with discrete This the the lower fluxes has orientifold interesting are planes. physical actually implications trivial, concerning known about orientifold planes, andcohomology. their In classification section in 3 terms wedescribing describe of RR fields discrete the in necessary fluxes orientifold backgrounds, K-theory in also and machinery compute explain required the how relevant for to groups. comparesion We the of K-theory the results Atiyah-Hirzebruch spectral within sequence cohomology the (AHSS), using appendix. which a is In real furtherK-theory section ver- elaborated and 4 we cohomology discuss in threeRR orientifolds. implications charge quantization, The of the first the second difference concernstifold is between variants the the which identification question are in of distinct K-theory intain proper of cohomology, orientifold certain and variants as orien- the unphysical. third Section isthe 5 the orientifold is 6-plane, dismissal devoted which of to in cer- a many separate ways discussion was of the original motivation for this work. p correlation between the parity of 2. Review of orientifolds 2.1 Basics Orientifolds are defined bycludes gauging a a reversal of discrete the symmetry string world-sheet in orientation string (denoted Ω). theory which The loci in- of points its charges under lowerfields rank RR take values fields. in It K-theorycreated has as by D-brane recently well sources, [7, been since 8]. suggestedas the the that This field source at the is which infinity intuitively RR creates should obvious it.RR belong for fluxes, But to RR the are the idea fields same classified is class in that all K-theory. RR fields, includingin source-free K-theory rather thanexplanation integral of the cohomology. observed fractional In charges;in particular, K-theory such correlates this a different RR way, provides that fields a the discrete partial flux of framework for RRproperly fields. RR charges, We are take[4, 5, values familiar 6]. in with This K-theorycharge the follows under rather from idea the than RR the that field in fact D-branes, that integral a or cohomology D more JHEP05(2001)047 2 + ). P p n R (2.1) (2.3) (2.4) (2.2) =+1. )cases, R 3mod4 and O n F , + − L 6) is to note p , F . =2 ) ≤ 1) p J p − projection ( ≡ (hence the sign in 2 2 for ( Z +  L Ω) p 2 F n ) and U Sp(2 I × n 1) 0 n . I 3mod4 1mod4 − i =( , , for O , 2 5 ) n 1 2 − =2 =0 L p × F n p p 0 I -planes (as compared with those 1) i p ), and in the second case U Sp(2 − L − +3mod4 n +1mod4 F  for the SO(2 p p Ω( -plane. A non-trivial holonomy con- 1) 2 p + ,and+2 I = = − p or − ( 1 0 0 p -brane invariant, and in fact preserves the p p 4 n  p 2 · =0 or × n Ω and O 2 π for O 0 0 B p I 2 p p − 5 − 2 9 C B C = p − P I p + R obius strip amplitude with the cylinder amplitude, / 2 Z p amplitude, and therefore exchanges the O − − 2 9 = −→ −→ − −→ − P R b 0 0 field [1], R p p B 1 on fermions, but ( × M, M C C B T − ,p -branes (and their images), there are two possible actions on ) in type-II string theory is given by the obius strip amplitudes, and therefore also the vacuum 1 which surrounds the O λ p p 1 = R p D − 2 − is odd under the projection, which maps opposite points in the 8 n 1inthe M Ω) P − B 2 R : I → = p λ -plane (O ⊂ O p πib 2 -branes) is given by 2 Since is even in IIA and odd in IIB. The presence of ( P e p 1 R p If we include It isn’t obvious at this stage which variant should be identified with trivial holonomy, and which -planes. These are denoted O 1 p the open string Chan-Paton factors [9], where In the first case the resulting gauge group is SO(2 The corresponding M¨ where In particular, the projection leavessame the 16 D supersymmetries. can be understood from theFor requirement example, that the generator ( square to 1 on fermions. The action of the orientifold on the antisymmetric tensor fields is given by the notation). 2.2 Cohomology torsion variants Another way to see that therethat are (at the least) orientifold two variants background (atto least admits the for a holonomy of discrete the degree of freedom corresponding tributes planes. amplitudes, have the oppositeO sign, so we see that there are actually two types of which are fixed underorientifold this symmetry are called orientifold planes. In particular, an respectively. Comparing theone M¨ finds that the RRof charge the and D tension of the O with non-trivial holonomy. This will be clarified in K-theory. JHEP05(2001)047 . b and ] U(1) (2.7) (2.6) (2.5) , but plane [1]. If − + .These − − 2 Z f O3 , corresponds + is related to an torsion, both of . 3 f + O3 2 ] U(1)), where G , Z p − = 8 plane as an O3 and , rather than P ) − Z R , , ) theory corresponding to H 4 5 ( ] 2 U(1)) are in turn classified , , n 2 2 1 f O3 , 2 3 Z )= p , , H − ) theory is self-dual. Z 8 , )theoryofO3 n 5 the relevant holonomy is is interchanged with p )= P =0 =1 n -planes, and is therefore similarly − e Z + R ≤ 8 − p p , ( p p P 2 p f − O R 8 H ( 0 P . Unlike the previous case, the charges of R H ) for ) for ± ( =0 or 5 p e 3 Z Z ∈ p , , f O ] p p H − π 0 5 . S-duality in type-IIB string theory exchanges − − 2 2 8 8 ∈ -field, G C -plane with p ] Z P P p p − 6 R R − . This exchanges the orthogonal gauge group with 5 ( ( dB G p p P − )= R − − e Z 6 6 Z f , are invariant, and O3 O3 is denoted ]=[ 5 5 (see table 1). H H c = + ). The theory is perturbatively the same as for O3 P H ( [ c n ≤ bundle. The choice of torsion coincides with the holonomy R f ( O3 p ∈ 3 Z , the holonomy is an element of ] p p H − − 8 and 6 P torsion on − G [ R = 6 is special, since [ H (see also [17]). p − = 4 supersymmetric gauge theory. The SO(2 ,soO3 3 f O6 is the twisted N G + 1) theory [10]. We can therefore interpret the e Z n is invariant under this duality, but the U Sp(2 . However, we shall see in section 5 that there is another variant, which has the The case There are analogous strong coupling arguments for type-IIA O2 planes [11] and RR fields can also have a discrete holonomy, which leads to additional variants + − and O3 with an additional fractional (1/2) D3-brane. The fourth variant, which take values in The choice of a non-trivial is the twisted U(1) bundle. The elements of and is identified with a torsion the BPS dyon spectrum is different [2]. This U Sp(2 O4 planes [12]–[14] usingpattern persists M-theory. for all However, it is clear from T-duality that this O6 we include D3-branes, string S-duality becomesvolume Olive-Montonen duality in the world- these variants cannot bebackground. computed in Instead, perturbation one theory, mustexample, since appeal the they to O3-plane involve arguments has a four involving RR variants strong corresponding coupling. to For topologically by the characteristic class classes correspond to themassive allowed type-IIA (integral) supergravity values [15,distinct of 16]. O6-planes, the but It cosmological rather does constantsit. different in not One IIA might appear, backgrounds therefore at in conclude that this which the an stage, O6-plane O6-plane to only can has label two variants, O6 where of the orientifold plane. For the O a symplectic one, U Sp(2 to turning on H SO(2 denoted double cover of properties (namely charge and tension) of the JHEP05(2001)047 . p 1, Z , and (2.8) (2.9) ,and =0 inverts (2.10) (2.11) } H )= p 2 6 Z Z , ,x 6 +2)-brane 1 p − P p In both cases R ( 2 0 H ). We denote the ,wherethe ,...,x Z 2 1 , )inthecaseof Z , and therefore forms 0 x 8 / c } { P m 1 -plane. The NS5-brane field, or a D( + R − l p . ( p 2 R H 2 1 2 1 H ≡ field (figure 1). = p + 2)-brane interchanges the O l,m p ,...,x − 1 R − p π 6 5 x 2 )and 2 1 G { C e Z p 1 2 , = − 7 5 p = P P − π R 6 R =0 or π 2 Z 6 ( H 2 G . 1 p 1 1 for the case of the 1 − , π H − 3 1 p, l 2 } S =∆ − P C 5 6 Z S R p π Z B − 2 ∼ 1 c O0. When both RR torsions are turned on we get planes, and the D( 2 P 0 P R l, for the case of the background, which takes values in ,...,x + R S Z 1 } 0 5) situated along the directions p Z 9 -plane. The brane is “fractional” in the sense that it is x = G p { ∆ ≤ 0 ,x c 8 Now remove two small disks at the intersection of the sphere c p O1 and c f O0. The charges of these variants were not known. Using K- 3 and O ,x 7 − p ,x 1 − -plane ( p c f O1 and p to denote the sphere in the reduced space l,m +2)-brane. S = 0 and 1 there is another possible holonomy given by p ,...,x 1 p planes. x p { are trivial (as differential forms) at infinity, but the above result shows that f O For p A similar construction can be made forWe the additional use RR torsion ( 2 3 − 6 of the coordinates. In particular more variants, theory, we will be able tovariants determine are their removed charges. when We will oneone also interprets could see RR that also fields some consider of in these a K-theory. For the O2-plane, This is related to the torsion groups for the NS5-brane, and corresponding variants along the brane intersects the orientifold plane along interchanges the O there is relative torsion between the two pieces of the O a “domain wall” onits the own O image underorientifold the plane. orientifold It projection, is therefore and a can source therefore of 1/2 not unit move of off flux the in the reduced space, i.e. for the D( either an NS5-brane along and with the orientifold plane, and apply Stoke’s law to get for the difference in the holonomy between the two sides. The field strengths The situation is similarsection to 5. the O6-plane case above,2.3 and Brane will realization also of be torsion addressedPure torsion in fields can be realizedConsider physically an by simple O brane constructions [18, 19, 2]. using an intersection with a D6-brane and D7-brane, respectively. G l JHEP05(2001)047 . . ± is 2 at p +1 P 2 p f O R w +2 C p (2.12) ∧ R F is integral p R 0 p 5-p ~ ~ O O G RP is created by a 6-p,1 0 S (p+2) G D torsion. p − 6 b (p+2) G ) D b , + 2)-brane satisfies [20, 17] p Z 5-p p O torsion, ( RP , and “opens up” into a flat field also has 1/2 unit of flux on mod p H O ) is special, in that B b ) a →∞ 7 6 1 2 x could only change by an integral amount due = at + + p π p 2 2 F 2 p ,sincethe O P ~ 2 O RP +1 R P p R 3,1 × C S Z p ∧ is not a homology cycle in the bulk, so the wrapped part plane would shift by 1/2, due to the coupling R is non-orientable, so the question remains open. 2 B − (p+2) 2 P p D R P Brane realization of ( a R Deforming the intersecting brane to a wrapped brane. R NS 5 2 Figure 1: p Figure 2: RP - O = 6 the D-brane picture (figure 1 p p + 2)-brane wraps O p We can try to use this picture to understand the relative charges of the For two coincident branes the flux is 1, but they areFor now free to move off the = 0 (figure 2). The 6 rather than torsion. the additional coupling D8-brane, which fills the entiremove transverse off space. the orientifold The plane, D8-branes and can this therefore is never consistent with the fact that On the other hand the charge of O This would therefore explainresult the similar observed to relative (2.12) charges was of obtained these in two [21] variants. for A orientable manifolds for which x of the brane shrinks toassume zero that size, the giving world-volume the gauge original field intersection on configuration. the If D( we non-trivial. However planes. Consider a “deformation” ofthe the D( intersecting D-brane configuration, in which then the charge of the O orientifold plane (in oppositetrivial, directions). and the The orientifold torsion plane part remains unchanged. of the field is therefore JHEP05(2001)047 ) ) X X ( ( (3.1) (3.4) (3.3) 1 ), and − KR X K .These ( ± KH are related by KH One way to think KH 4 )and projection on closed X 2 and ( ) in IIA, and by Z ± X ( KR , KR ) (3.2) . K ) , ) depends only on the difference X ) p,q ( , and similarly for X X ( R p,q ( +1 p,q ), respectively. ,q ,q +4 × R n X +1 +8 KR ( − X p p 1. In K-theoretic terms, one is given ( K − ). In addition KH KR KR 8 KR X . Physically this corresponds to the inclusion ( q F )= )and )= )= − )= p X X X X ( ( X = 3 mod 4 in (2.1)), which exchanges the branes ( ( 1 ( n and p coordinates in − , and an antilinear involution on the bundles that − KR p,q p,q p,q E p X K KR )= KR KR KR X ( (e.g. for p,q L . The first implies that ), respectively [7]. This complements the fact that RR charges, F D9 system in IIB, and non-BPS D9-branes in IIA. ), or more precisely by the kernel of the natural maps from these groups to KR X 1) ( X KH . This defines the real and quaternionic K-theory groups 1 ( − on the space τ − τ cpct K K ), depending on whether the involution on the bundles squares to 1 or X ( is non-compact, then D-branes are actually classified by K-theory with compact support, )and )and X X X ( ,sowedenote ( KH q If 1 4 K − 1 [22]. Higher groups can then be defined by (assuming compact support) cpct − where the involution acts on of the operator ( about this, which will beinto two useful regions in using generalizing aand to domain orientifolds, non-BPS wall is D8-branes made to in of dividethe IIB. D8-branes the two and Differences space regions anti-D8-branes in are in theby then IIA, the values in D8-brane of one-to-one system, the correspondence and are RR with therefore fluxes the classified between RR by charges carried K 3.1 Real K-theory The orientifold backgrounds of interest are defined by a and antibranes. The corresponding groups are denoted an involution commutes with strings that includes world-sheetPaton parity factors Ω, and that an squares action to on either the open 1 string or Chan- p In ordinary (non-orientifold) type-IIAin and -IIB backgrounds RR fluxes take values 3. Orientifolds in K-theory satisfy the periodicity conditions Another variant of realalso K-theory exchanges the can two be bundles defined by considering an involution that and similarly for in IIB. The groupswe are start exchanged with relative a to D9- the D-brane classification since there and or D-branes, are classified by the ones without compact support. − JHEP05(2001)047 , 8 p S − 8 ). In (3.5) ,and ), and P X − ( X R 2 p ( Z ± K .InIIAwe KR p − ) for O 8 0 S p, − 9 ). Orientation reversal S ( X ) rather than ( , respectively. This is , 2 10 ± X Z ) − ( ), depending on whether p K 0 ± 1. Following the general X , ( 9 -plane we divide the trans- K KH − 2 → p S KR − ( . The former gives ) -plane, and the latter to the 0 1 ± , , − 2 1 and R R KR ± KH × and )= X ( 2 KR 2 , 2 , )or 0 0 Z 0 , R Ω in IIA maps a D8-brane wrapping R 9 K 9 S 9 . Note that, as expected, the groups which ( × I + ± / p 9 X ( R KR KR ) for O 0 p, )= − 9 X (3.2), and fails for ( S 1 ( ± , 6 1 5 − R . p KR ± ). The latter can be understood as the failure of ordinary Bott periodicity [23]–[25]. For example, D-branes in an O8 background (type IA) 1 , planes the groups are also KR 1 9 by wrapping D8-branes on the transverse sphere KH R − + p ). p × )= − X 9 0 ( X p, ( ± R KH 2 − Z 9 and O8 S exchanges the two spinor chiralities, and therefore gives K KH ( and 2 − 10 9 R )= − − -plane. p The appropriate groups for the other orientifold planesUsing (3.2)–(3.5) can we be can re-express similarly these deter- groups as For example, the O0 projection The usual 8-fold periodicity of orientifolds (and real K-theory) implies that for p X This relation is the analog in real K-theory of Hopkins’ relation in complex equivariant K-theory, ( + 5 ± O0 the action onarguments the of Chan-Paton [9], bundle we squares associate to the 1 former or to the O0 somewhat counterintuitive, in thatand a not D8-brane an gets anti-D8-brane. mappedsphere, to However which an recall in image that thismeans D8-brane the that case D8-branes the is wrap orientation of just theThe the transverse two projection “wrapped” points D8-brane maps on isother the reversed either side, on D8-brane side the so to other it of side. a looks the like D8-brane O8-plane. a of “wrapped” This themined. anti-D8-brane. same The results orientation are on summarized the in table 2. again consider both D8-branesD8-branes. and anti-D8-branes, The and orientifold in projection IIB (2.1) we acts use freely the on non-BPS the sphere, giving and the corresponding K-groups areacts obtained on by the keeping track D8-branes of and how their the bundles. projection KH classify RR fields are shiftedKR by one relative to the groups which classify RR charges, and similarly for verse space with a given orientationIt to therefore a exchanges D8-brane wrappedversa. D8-branes wrapping The with it group wrapped with is anti-D8-branes, therefore the either and opposite vice- orientation. 3.2 RR fields in orientifoldTo backgrounds classify RR fields in the background of an orientifold higher versions are defined preciselyreal as K-theory in (3.1). as follows, These groups are related to ordinary the O8 K when one tries to construct the Dirac index map: on the the real case it’s exactlyBott the periodicity opposite, works since for complex conjugation also exchanges the two chiralities. the latter JHEP05(2001)047 6 (3.7) (3.8) (3.9) (3.6) relative − 2 2 ). In the real Z f O5 Z 6thereisan Q . To compute Z Z , ⊕ , 2 ⊕ − X, Z Z ⊕ Z Z ⊕ ( 3) Z Z Z =2 Z Z ≥ KR . even )= )= flux of p )= )= 0 0 )= )= )= 0 0 , , H , , 3 m 0 0 = 0 8 )= 2 , , 6 4 )= , 0 9 1 5 0 , G S S ' , S S ± . 7 ( ( 3 S S S ( ( 2 1 1 ( ( ( 1 1 S Q . S .For ( − − ± ± ± Z − − ( ± ± mod 8 p KR − + × )( − ) ⊕ 8 } 0 pt. K-group KH KH KH KR KR KR KR KH KH X G Z ( ( , n 0 K , − for O5 for O5 + + + + + + + + + + 0 )= 0 1 2 3 4 5 6 7 8 , e Z flux of the O5-plane, and the second 2 Op O O O O O O O O O Z KR , , 3 3 Z 0 ⊕ P G , ⊕ ) 2 R , which also agrees with cohomology. The ( 10 Z Z Z 0 1 pt. , ( 2 G H Z 2 +1 )= )= , 2 ⊕ n 0 0 -plane itself, i.e. Z , , Z Z − p ) 4 4 { Z Z S S ⊕ m f Z O5 variants. Let us once again stress that these ⊕ ( ( , Z Z 1 1 ⊕ Z Z 3 ⊕ Z Z Z − − )= P KR Z Z R Orientifold K-theory groups for RR fields. )= )= )= )= pt. ( 0 0 )= )= )= KR 0 0 ( , , 3 KH , , 0 0 0 )= 6 4 )= , )= 2 8 , , m 0 0 5 1 9 0 H , S S , S S − 3 ( ( , whereas RR fields take values in 7 S S S ( ( m, 1 1 ( 1 ( ( 1 1 S S S ( − − − ± ± ( − − ( ± ± ± KR n Table 2: − KR KR KH KH KH KH KR KR K-group KR KR KR − − − − − − − − − − ) part of K-theory always agrees with cohomology. This corresponds 2 3 4 5 6 7 8 0 1 Z O O O O O O O Op O O . In K-theory we find torsion, which gives the − 1 For example, the relevant cohomologies for orientifold 5-planes are This is a consequence of the Chern isomorphism, e.g. G 6 to the RR flux of the orientifold 3.3 Comparing with cohomology The above results agreeThe with free cohomology ( in some cases, and differ in other cases. The first term corresponds to the integral to O5 How should we interpret the absencewe of need the torsion to subgroup understand in the better first how case? to Clearly relate (or “lift”) cohomology to K-theory. cohomologies do not incorporate the fractional shift in the additional free part corresponding to are classified by the groups we used the following decomposition isomorphism due to Atiyah [22] is difference between K-theory and cohomology can only be in the torsion subgroup. and our knowlege of the real K-groups of a point, case the cohomologies willcorresponding be RR twisted field or (2.2). normal, according to the action of the orientifold on the JHEP05(2001)047 3 1)- g Sq − is not (3.10) (3.11) (3.12) p 3 . d 5 -exact are d r ,where d H + . 3 0 ), as . g Sq X → ( = ) itself, requires solving ) K odd even 3 X X d , ( ( is the Steenrod square [27]. p p ) )= ). This is defined in terms of 3 K +1 The basic idea of the AHSS p X X X ( Sq ( ( 0 7 /K +1 K ) p ) for K ) which are trivial on the ( maps between ordinary and twisted X Z ( /K X 3 p ) ( d X, ,where K ( X K p ( H p → ⊆···⊆ 0 for H 11 + K ) ) p 3  X X ⊕ -plane. We shall assume that the first term is ( ( Sq p + 1 )= ), and in particular p − = K )= n X X 3 ( ( K X d p → ( +1 ). Each successive step is given by the cohomology of ) K p ⊆ K Z ) . As we have seen before, this provides the difference X 2 ( /K Gr X ) Z X, is just the usual co-boundary operator acting on co-chains ( +1 ( p n ∗ 1 X ( d K K p H ) using a sequence of successive approximations, starting with )= K ), e → Z X , -plane and the O ( X 0 p ( that acts on classes in the previous step. Thus classes which are not − − K p r 8 K ) (or any of its variants), denoted Gr d P ) is the subgroup of classes in X R ( X ( . The successive ratios appearing in (3.10) are precisely the objects one ( 3 p K X H K ∈ Actually, what one really computes in the AHSS is an “associated graded com- There is in fact a systematic algorithm which does exactly this, known as the The AHSS can be extended to real K-theory as well. In the appendix we de- Introductory reviews of the AHSS for complex K-theory can be found in the appendix, as well (which is untwisted) is trivial in all cases. ] 7 -closed are removed in the new approximation, and classes which are 5 r H trivial in both cases, andd justify this assumption in hindsight. We shall also see that trivial. For example, The only other differential which may be non-trivial in ten dimensions is cohomologies, and must therefore itself be twisted. The precise form of is to compute known, but it can presumably be expressed in the form between the O is a twisted version[ of the Steenrod square. Here the second term is torsion, since (which are the zero’th step), so thedifferentials, first if step they is ordinary are de-Rahm non-trivial, cohomology.cohomology Higher will classes, refine and the trivializing approximationin others. the by complex removing The case some first is given non-trivial by higher differential d as in the appendices of [27] and [28]. plex” of Atiyah-Hirzebruch spectral sequence (AHSS) [26]. scribe how toappearing do in this table 2. incase, In the this with case case the the modification of AHSStion that follows a can the a cohomologies be freely similar appearing twisted pattern actingfields in to or (2.2). involution, the the normal, complex first Correspondingly, in such approxima- the accordance as differential with the the ones orientifold action on the RR a differential where integral cohomology To get the actual subgroups tree of computes by the AHSS. In particular, in the first order approximation extension problems of the form a filtration of JHEP05(2001)047 ) 5 P R (3.14) (3.13) (3.15) ( 2 H ) will differ X ( . K 0 ) → 5 ) P 5 R P ( R 4 ( 2 ). If all the extensions are 3 H Z X , ⊕ ( /K = 4 ) ). In other words, K-theory is ) 2 5 Z 5 +1 p X P P H ( R R )= ( /K K ( 5 2 ) 2 , for which . P K 5 X H 2 ( R P p )). However it will be extremely useful ( Z ⊕ → R e K 12 ) X K )=Gr ⊕ ) ) 5 ( 5 5 = 2 ⊕ P X K P P ( Z ) )= R X R R ( 5 K ( ( 0 X ⊕ P 2 1 ( H Z R K K +1 ( p 1 = → K K )= 5 ) 2 5 P Z P )= R ( kk k = R X ( 4 ( K 3 p H K Gr K ) (as opposed to Gr → X 0 ( ) become correlated in K-theory; twice the generator of the former cor- K 5 P R ( 4 H In practical terms, one often needs additional information to determine whether Consider for example the space On the other hand it is known (see for example the appendix of [28]) that responds to the generator of the latter. an extension iscomputing trivial or not, which precludes the AHSS as a useful method for and 4. Applications Having elaborated the K-theory machineryorientifolds, needed we to would classify now RRfact like fields that to surrounding that demonstrate RR three fields important are implications valued in of4.1 K-theory the Charge rather quantization than cohomology. As was previously stated,proper there RR are charge two quantization. separate issues The related first to has the to question do of with the absolute fractional If the above exactto sequence be splits, i.e. trivial, if and it is exact both ways, the extension is said The relevant extension is given by so this extension is non-trivial. The RR 2-torsion and 4-torsion appearing in trivial we deduce recursively that in our case, sinceable we to determine already all know theK-theory the extensions data precise (3.12) with unambiguously, groups cohomology. and (table thereby compare 2). the We are therefore isomorphic to the ringAHSS. On of the cohomologies other that hand, survive if some the of higher the differentials extensions are of non-trivial the from its graded complex.different from In that particular, of theof the different additive cohomology degree classes. structure become in Physically correlated. this K-theory means will that be RR fields JHEP05(2001)047 . + p (4.2) (4.4) (4.3) (4.5) and O − p , but both are 0 G and 0 2 ) (4.1) G → X . ( 1 . Z ) ). The first issue encompasses K Z Z Z , respectively, we can determine + = /K , p + 0 3 0 ∈ P K )= )= H . R e Z Z ( Z , , 3 → 2 2 Z and O6 = P P H Z ,x , 0 for O ⊕ 2 0 − R R ) − ( ( 3 ⊕ 13 Z K x ) is obviously trivial in this case, the AHSS p d 0 2 K 5 −→ d H H = = ) ch( e 0 1 Z b = =0 = → A 4, for example), and the second with the relative , ) for O6 K K 3 1 2 2 0 ZZ , (and p could be in the map ≤ P 3 K 1 3 3 R S p /K /K = = kkk d ( ( d 0 1 K 2 0 ) K K x H H π ( KR 2 G → 0 )and + 1) in these cases [17]. 0 , 3 -planes with n S p ( KH = 0 K The first case where this procedure gives a non-trivial result is the O5-plane. The O6-plane background supports two RR fluxes, Note first that for the O8 and O7-planes K-theory is in complete agreement Using does not have ahow unique the solution, two so RR one fluxes are cannot correlated. determine We in willThe this return cohomologies way and to whether K-theory the and for O6-planeonly the in possible O5-planes section were non-triviality given 5. of in (3.8) and (3.9). The However the extension given by the filter groups uniquely, the second, and should in principle be resolved by an analog of the relation [7] integral. Since the differential In particular the O8-planebrane). (O7-plane) This cannot is support consistentgauge group with a SO(2 other fractional arguments D8-brane which (D7- rule out the possibility of a terminates at the first approximation, i.e. charges (of the O charge induced by RR torsion (1/2 for O for real K-theory.centrate Unfortunately on the the generalization issueprevious is section, of not is the known, that relative sovia in non-trivial K-theory charge. we extensions. RR will fields The con- of main different idea, degree become aswith correlated alluded cohomology. to There in isnon-trivial the only extension. a single There RR are flux, only and two therefore variants no of possibility these for a planes, O JHEP05(2001)047 is 3 d (4.9) (4.7) (4.6) (4.10) . The AHSS + . In the latter plane, and has − ) (appropriately , i.e. − p p − − 8 . 2 , P and O5 − + planes. The relevant H maps to the generator = 0 (and we assume 0 R − ( 3 − . We therefore see that p 0 H 1 , which are appropriately → p H − , 2 7 . p for the O5 2 H ) Z for O5 for O5 → for both ) ) (4.8) − p H 3 Z 7 p p − 2 − − 8 G 2 2 =0 =1 = 8 8 Z )= =2 P /K Z p p p p P P p p /K R = − − − R R ( 1 8 ⊕ 6 6 1 ( ( p for for P p p Z K − Z H for K H − − R 8 6 11 ( Z p → )= → H H H is trivial since )= − 0 0 2 2 6 , p , 1 is trivial since 3 4 )= 4 Z Z Z − 14 d H . The generator of S 5 6 K S Z could arise from the (twist preserving) maps ( −→ −→ −→ ) is trivial for both O5 ( d − , 1 K 5 5 1 3 ) ) ) flux, i.e. the charge of the orientifold plane, is − → d d − )= )= )= p p p P and p e − − − plane. Z Z Z → R Z 8 8 8 − , , , ( Z 8 6 7 8 + KR KH 3 P P P 2 ZZ , but to twice the generator for O5 = kk G (and P P P + R R R p K  H 3 ( ( ( = 3 R R R )= − p p p ( ( ( d p 7 kk k p H = = 0 1 2 − − − − − 6 3 1 planes, and K 1 3 8 8 H H H maps to non-trivial torsion in → − P H H H K K torsion. for O5 H 1 p R 0 p 1 ( = = K − p 2 there is an additional RR cohomology → K 6 0 2 − )=0,so 8 0 ≤ G K K e Z H p , , but non-trivial for O5 3 + = 0 for O P R ( H 0 H = 0). Possible non-trivialities of Actually, for The story repeats along similar lines for the lower O torsion produces a half-integral shift in the flux of 3 1 g so the half-integral part of the is trivial for O5 again terminates at cohomology, and we find Sq twisted or untwisted. The differential case the generator of no effect on the charge of the O5 This time we find that the extension given by G however in all cases the maps are found to be trivial. The relevant extension is cohomologies are trivial since However determined by the As with the other cohomologies, these lift unobstructed to the graded complex; of (the free part of) JHEP05(2001)047 − Z are The ⊕ + (4.14) (4.13) (4.11) . 8 Z − f O3 plane . 0 + case above; C and − and O0 + ) transformation − ,whichis with Z planes, as we will 0 0 , K + ω plane, the extensions p . = + charge (and tension) as 0 p − -torsion. This will modify ) (4.12) 2 → + 1. On the other hand, the e H Z K p 0 , same 5 − ω 3 P p )theoriesonO1 − R → 2 k /K ( n ,and , 0 p 3 , H 0 − Z ω 2 . In particular, for the O3 H  ω case is similar to the O6 3 , such that K = d .AswiththeO5 ∪ 0 H − Z p ∪ torsion does not shift their charge. ω −→ 11 01 → H − 15 p 3 )  p − Z )= 2 K − = , 0 2 = , torsion therefore lifts to a trivial class, and the 5 G + 1. We therefore identify 6 3 K 3 P S 0 T ( G therefore have the G R C 1 1. The O2 ( → . − ± planes with non-trivial 0 − , 2 p → H − Z − c O0 0 3 p : KR =0 , cannot support the end of a D-string. C 3 to − K . We will return to this problem in section 5. For O1 p d 0 Z and G → for − 0 2 Z c O1 3 through the differential variants are related by a shift ⊕ ≤ + Z p fluxes are equivalent in K-theory, the two variants O3 3 the extension is trivial, so G − and a D7-brane for O1 − =2,and planes correspond to O The story becomes even more interesting for the lower O p This is also consistent with the fact that the branes which create this torsion, i.e. a D6-brane + 8 p for NSNS and RR 3-torsions of the two variants are related by an SL(2 which also shifts the axion see next. 4.2 Equivalent orientifolds O equivalent as well.implies Physically the this existence can of be a constant understood 0-form as follows. Equation (4.12) is a surjective map from there is no uniquedetermine solution the to effect the extension, of so one must use a different method to and the two O3 the AHSS for are trivial, so thethe charge doesn’t two shift. However, the new feature here is that, since One can show that in all three cases variants we called the original torsion-free planes.how) It the would gauge be theories on interestingorientifold D1-branes to variants and differ investigate D0-branes from whether in the (and the ordinary presence SO(2 of these exotic and O0 normal or twisted) is trivial. There is also an additional extension to solve, in type-IIB string theory given by [1, 2] graded complex is simply Gr for O0 JHEP05(2001)047 − + + + is f and O2 + case, plane (4.17) (4.15) (4.16) 9) are + f O2 planes, + p + 3 with flat + O f O2 1 ,..., ω The =3 . and one 9 i 9 ( + x i x )= 9 x planes correspond to an ( 2 + can be written as a derivative , respectively (figure 3), which ), where n i + dξ . G x ( ,ξ , ∧ plane, respectively (figure 3). The ξ ) torsion class which is not trivial p 2 planes, but in the second case only + π 0 − − B planes. This is unlike the 3 G 2 + ω < f <π and O3 O4 + = 9 f 9 O2 ∪ )= + 3 i torsion can be written as . On the other hand, the M-theory shift x = 2 there is again an interesting physical H 16 p f O3 π will not correspond to just the cup product instanton bundle on U ( and O(2 ) known as the parity anomaly. It was shown in [30] that a complex 3 / π − was considered in [17]. In this case the gauge group on the D2-branes is .ForO6 iD − . The integrals of the corresponding CS forms defined in (5.9) transform − is the mass of the fermion. A similar computation in seven-dimensional Z ), and the global symmetry corresponding to the D6-brane gauge group is is the appropriate winding number of the gauge transformation. The relative f )). Since this bundle is pseudoreal it survives the projection to U Sp, but O6 4 m N ), and a pair in the antisymmetric representation ( w S ), as well as a pair of states in the symmetric representation of the gauge group ) for O6 1 ( )= There is an additional contribution to the CS action from the fermion deter- Consider first , n e G K , ( 7 odd for of the sum of (5.11) and (5.14) therefore requires of can be any integer. This is exactly what we obtained in K-thoery. the associated bundle on π brane world-volume theory consists( of a state in the bi-fundamental representation where now the traces are in the fundamental representation of ( must be doubled to get the generator for O. For O(2 USp(2 minant det( fermion in a representaion CS term as follows as follows where gauge theory gives case of O6 and the factor ofbranes 1 move away from6, the as orientifold these plane. are The the cases only of cases interest (in are ten dimensions) with integral winding, global symmetry is U Sp(2 in this case there is no parity anomaly. Furthermore, we see from (5.12) that generator of where factors of 2 can be understood by considering U Sp and JHEP05(2001)047 − − ). − had and f O6 f − O2 (5.17) π 2 N / 0 G (and planes [31]. + 1 for + − Z 2 ∈ flux, for which we π 0 2 . and O7 . The world-volume / planes. Supersymme- G ) 0 − + − m G is an arbitrary integer. ) (5.16) O6 π 2 m + / 0 N G has additional disconnected com- for O( for U Sp(2 3 even), and T ,and m 7 , as we saw in the previous section). ( + 4 7 ω − ω 7 G planes and , and T-duality, yields a background with 7 Σ 3 charge quantum is not correlated with the 21 − Σ Z T Z 6 O6 G 8) for O6 − +8) Z − N 2 m m ), but allows any integer value for O2 ∈ = 256 configurations. However not all of them are + π planes, in much the same way as one describes the eight- 8 =( =(2 2 f O2 + / 0 adj adj 7 7 G ω ω 7 7 (and , since otherwise anti-D6-branes would be required for tadpole Σ Σ + , but we made no statement about O2 Z Z + and O6 + N − ≥ − for O2 N (only with the torsion class of planes and sixteen D6-branes. On the other hand it is known that the Z 0 2 − G , there are naively 2 ). ∈ + Z π , The same approach can be applied to the O2-plane with Consider a configuration with Some of the additional components can indeed be described in this way, by The same result can also be obtained by considering the world-volume theory on 2 odd). There is no such correlation for O6 / 0 m to be even for O2 only had partial K-theory results. In particular, we determined that theory on D2-branes in(four) the of background complex ofno fermions parity an in anomaly. O2-plane the Invariance containsG under adjoint an large representation gauge even transformations in number therefore all requires cases, so there is value of eight O6 In the fully compact casewhich the seven-dimensional are orientifold located contains eight at O6-planes, the vertices of a cube. Since each plane can be either O6 dimensional theory without vector structure in terms of O7 try requires Together with (5.13) wetransformation see requires that, in the first case, invariance under a large gauge consistent, and manySL(3 of them are related by the symmetry group of the 3-torus This implies, in particular, that the ponents, with gauge groupsto of describe lower these rank differentorientifold [31]–[35]. components 6-planes. in One the of T-dual our picture, original using goalsincluding D6-branes was both and O6 or O6 ( cancellation. Furthermore, consistency of the orientifold projection requires 5.3 Components of type-I vacuua In many ways the orientifold 6-planepactification was of the type-I motivation for string this theory investigation. on Com- moduli space of type-I (or heterotic) vacua on the D6-branes. Insentation this of case the there gauge is group.are a related The single to CS complex those 7-forms defined fermion defined in in in the the the fundamental adjoint adjoint representaion representation repre- by JHEP05(2001)047 ). ). b Z ,and n, n (5.18) (5.20) (5.19) i odd on ··· 1 )equiva- i π  Z ) are thus 2 , 2) case the / + , 0 G ,N − ) N g − )( 1 =0.For − ). In the (6 , is invariant under SL( g 0 n a i 3 3) v g G , T ) 2 g j − ··· , T 3 )( 1 g 6= i g . v T 2 i 1 ) =1 n ( g i − j i T 1 g ··· . ( 4) configurations the total charge of 1 g 2 i 6= 3 ,  g , and change the number of D6-branes − i gg 2 )-invariant property. j − g Z g − = i i 1 , f correspond to translations along the torus, )( O6 T 22 1 g gg gg gg ) i , =0 ( vectors 1 − g − 3 g by . If we denote the action of the projection on g } n g 2 1 j ) transformations, and the rank is 8 (figure 4 − 2 g = =+ =+ =+ − = 1, it follows that the Levi-Civita tensor Z g 3 ,g gg 1 , i T T T T ) ) ) g gg − g M i j 3 gg ( ( { g ,where g i − − − 2 gg } ( ), and the other to configurations where they are not g 3 gg c 1 = = = ( ,g 2 gg T ( ) ,g 3 1 g 2 10 4) (figure 4). In the first case the configuration is unique and g ) satisfies det , 1 ). g,g Z d { gg 2) configurations, which yield new components with a rank 12 and n, ( , to be even. In particular, they exist for SL( π ∈ 2 / 2), and (4 4) case there is no gauge group, and there are actually two SL(3 0 M , , G 0) and (6 , and the seven others are O6 , + = 3 this means that co-planarity is an SL(3 f Given the previous results, we now understand that the above configurations to be even. For example, let us assume that the plane at the origin of the cube O6-planes is +2, and can only be cancelled by anti-D6-branes.) Of course since Since n 6= 0 in these cases, there is a non-vanishing vacuum energy, and supersymmetry 0), (6 , 10 + 0 co-planar (figure 4 lence classes. One corresponds totype configurations are in which co-planar four (figure O6-planes 4 of the same configuration is unique up toIn SL(3 the (4 G corresponds to the original rank 16 component (figure 4 a rank 5 gauge group, respectively. (In the (4 the the other hand, we must replace O6 (8 is O6 then we have N appropriately to cancel thethe tadpole. (8 This can only be done supersymmetrically in From the first three relations we deduce that the CP factors by require in contradiction with the last relation. The allowed values of ( is spontaneously broken. On the other hand this implies For therefore also the pseudo-scalar product of JHEP05(2001)047 − )eightO6 a ) two inequivalent d seen in the AHSS), 5 )+( d c and 3 ,d 1 d brings us from one to the other. We planes (rank 8), ( planes (rank 0). 3 + d + 23 )-inequivalent configurations of O6-planes: ( Z planes and two O6 differential has been explicitly worked out [26, 27]. For , planes and four O6 − 3 d − )sixO6 b Consistent SL(3 ab c d One way to understand the extension problem is to realize that while the graded The equation of motion for fields described by generalized differentials involve the The AHSS was particularly powerful for our set of examples because we already which in general willsource not of be the particular zero RR-fields. but The can novelty also here give is a that K-class a K-class corresponding corresponding to the configurations of four O6 would like to conclude bymatter making of some extensions. comments regarding the other obstacle: the complex generated by the AHSS yieldsus the with elements of their a K-group,extracted additive it by structure. does thinking not of provide We thevalues of propose K-group generalized elements that differentials on in this theform the asymptotic additive bulk. consists sphere structure For of as our can limiting a purposes,action a be vector which general of can differential differential increaselines, forms the we of refer rank different the by reader ranks more to with than the a ongoing one differential projectgeneralized (for derivative in some (naively [36, a ideas 8]). combination along of the these KR- and KH-theory, we didcan not work see these that out, adding although (along the the torsion lines H-fields of to [21]) we knew the relevant K-groups. However, useThe of first the AHSS obstacle in involves generalable faces computing two to higher obstacles. circumvent differentials this in obstaclecomplex the by K-theory, sequence. using the our We exact were knowledge of the K-groups. For 6. Conclusions and open questions In this paper we haveon demonstrated how the K-theory nature generates very ofbackgrounds. useful We discrete information saw RR that certain G-fluxesfied G-fluxes due in are to correlated, discrete string that transformations, theory,very some and useful finally specifically become tool that identi- in for some this are orientifold generalization process disallowed outright. of was it. the A Atiyah-Hirzebruch The Spectral AHSS(even Sequence ties torsion and together our ones) our in understanding terms of of separate RR-fields single K-classes. Figure 4: planes (rank 16), ( JHEP05(2001)047 1, 2)- − − (A.3) (A.2) (A.1) p -plane p ,hasa − 2 ≤ 3 S . O m } 1 − p with , X } m ) . When we wrap a 4 G Z x 3 } x ⊕ , reflecting the fact that ) ( 3 , Z Z ) x trivial on 4 2 -brane charge and D( x x p x 2 1 | )= )= x 2 x ) 2 ( ( S the group K(X). Define S , , ( X ( ) is the subset of this triangulation ) 2 ( 0 3 4 p x K x K 2 2 X filter x x ( 1 . xK , )= { x ˆ ) 2 X ( 4 . For example the two-sphere, . , S = 24 x ) ( p 1 4 ) 1 x x X K ( 3 ,wehave . , x 2 ) 1 } 3 S Dim( 4 x x 2)-brane charge. ≤ ( 1 p , = ,x x ) − 3 ≤ ( 4 1 , p X x | ,x ) 3 ) 2 2 x x 2 X ,x 1 ( x 1 x p ( . Consistency (commutativity) of the additive structure with could be a source for any field strength ( x 4). Although some earlier work on this exists [37], there does equal to { 1 { { / p − 2 1 ,K J p , with a triangulation , these two charges correspond to D = = ) X 2 − G X 1 0 S X = the classification of consistent and inequivalent O6-plane configura- ( X X ˆ K X = 0) in subsection 5.3 also appeared in [38]. . 0 = ) G X ( 0 -form current p K Another outstanding issue concerns the absolute charges of the orientifold. In -brane on this p not yet exist a satifactory answer withinNote the added: K-theory framework. tions (for triangulation starts with charge Goingbacktoourexample and so with simplices of maximum dimension instead of just for these groups capture only D( We are now going to use the triangulation to the differential action willgeneralized differential put forms, useful which will restrictionsof in the on turn K-group the help at determine additive infinity. the structure group of structure the this work we have onlymake used use K-theory of to the compute fact jumpsto that in argue K-theory the that contributes the charges, an jumps and integral arebe one index anomaly can determined, to free the [7]. the path-integral bare This charge leaves us of with the one constant orientifold still plane to (for example, an A. AHSS in complex andIn real this K-theory appendixsequence we will (AHSS), summarize which thetriangulation. can notion be We of will used show thefirst to how step Atiyah-Hirzebruch this compute in spectral generalizes using K-theory the to groups AHSS real is variants whichA.1 to of have define Defining K-theory. a an the The associated associated graded graded complex. Consider complex a space to a D brane charge. Thus in this example, JHEP05(2001)047 , (A.5) (A.6) (A.4) (A.8) (A.7) (A.9) .The (A.10) X ’s, but we p i filtration σ + 1)-skeleton, p associated graded and thus ˆ , X )) ), there is, as it turns , ’s define a p p pt . X ( ( − ) K p p i p, . 2 to any other K-theory, such − K ˙ σ p E ’s. Summing up the integers , , G p K i N +1 = ,the . σ . p N i ( X, K p 2 =0 K = σ N ( )) p )) M p should be K E i pt +1 pt p . ( = p ( ⊕ , and it consists of K-classes on the p p G p C 1 )wegetthat − − yields, with appropriate orientation, a +1 E p pt which usefully connects with much of the )= ⊇···⊇ ( p i +1 )) with p p 1 i ˙ σ /K 25 K −→ − X, K i X, K pt σ p ( -skeleton to a class on the ( K ( ( p ⊕ p K p q K )) . This allows us to define an , ⊇ C H p i N pt 0 =0 σ = ( )= i p = M = p p p K X, K − p ⊕ 1 p 2 ( S . ( = p ( = E E e and extend it to the K H K K K p 1 X, K we can label as ), etc. = = ( E Gr p p 1 p )= X using a spectral sequence p i ( p,q 2 C E G ˙ σ ± : filtered complex E K , 1 p i ,K d σ ) ( )a ) doesn’t capture all the structure of , we are going to use a sequence of successive approximations. The X K X X ( ( K ( trivial on their boundaries. In other words, K i K p σ ,KR ) + 1)-skeleton. This is precisely the cohomology differential which maps X p ( 1 − Now the first approximation gives us a candidate K-classes on the For a finite dimensional space X of dimension To continue with this construction, we define elements K -simplices i.e. take every element in need to make surefirst that step they is can to be extended extend to a K-classes class on on the from each component of the boundary ˙ winding number which ifto non-zero the acts ( as an obstruction to extending the co-chain which implies that the next approximation to Since we our usingappear. complex K-theory for now, only the even-dimensional co-chains p A.2 Computing Gr To compute Gr Although Gr Of course, since which makes first approximation to complex out, a simple algorithm for computingphysicist’s Gr intuition about fields asextend seen the in notion cohomology. of Also theas associated note graded that complex we Gr can easily JHEP05(2001)047 ) X ( s steps. (A.14) (A.16) (A.15) (A.11) (A.12) (A.13) C r )isnot. 1 − to another x up . Start with 2 − 0 χ r x p, ) to those for E S 1 + X x ( s − . ) K and repeat. We have . X r ) ( ) d . can naturally be used to s maps co-chains in X ) with a different addition ) + ( ) using a slight modification p p j ∓ d ( X p,q ’s for r − ( X − . ∗ E ( p, . For finite dimensional spaces, n p,q p, n s 3 is allowed to range freely. With 1) H E − E E − q S. r Sq ,...,x ( . − .Wethenhave i ∓ / ∈ )= KO )= ∅ x n r,q ( X i ∓ X + ( = ( p r x p p s i E +1 G level in the scheme, hence we can view this )and −→ − p is mapped by the involution σ 26 2 ⇒ ) X p i ( E j ± ∩ /K )= s −→ σ S ) − p i X correspond to. In complex K-theory, only odd X p,q ( r ∈ σ K ( ), we need to make some restrictions on the trian- 2 s E +1 0 i ± ,...,x − p all map from the p : E p, x i ± r S K r . The simplices that make up the triangulation consist x d ( /K d } s . ) = is the Steenrod square and p ± − :( ) X 3 )suchas 1 ( χ d p d X ) is a 2-simplex in this triangulation, but ( X to be the obstruction to lifting a K-class in ). KR ( 3 + , as the cohomology of the operator s K 1 r X x − p ,...,x ( d E 1 ± 2 − G +1 r x +1 x s { E 1 + C x of these points (with appropriate orientations) such that such that p S i ) just as the regular cohomology differential ranges from 0 to the dimension of X, but σ X p ( +1 As an example, we can give a simple such triangulation for the There are two features to this construction that are of particular interest. The s − these elements, the wholefollows. approximation scheme We can define then be refined recursively as of all sets We then define the involution to to take Formally, this procedure terminates at some finite order construction of K-classes as a sort of refinement of the set of points already seen what where which clearly satisfies eq. (A.14). differentials appear, and compute the associated gradedinvolution complex on for the space other K-theories (as long as there is no simplex For example, ( We then define gulation, namely that every simplex of the coefficients in eq. (A.10). For example structure. The second is that the same coefficients K A.3 Extending the AHSS toIn K-theories order with to freely extend actingspatial the involution, involutions AHSS e.g. construction above to K-theories with a freely acting first is that cohomology appears at the to co-chains in Note that the differentials JHEP05(2001)047 . , i.e. gives )isa χ X (A.22) (A.21) (A.23) (A.17) 0 2 , 6 E S ( 0 , 0 2 . E ) )meansthatit p i 3 ˙ ¯ σ E ) is real (or pseudo ⊕ i ˙ σ p i , ˙ . σ 2 i , 2 term given in eq. (A.6). terms are: Z σ p i ( Z ¯ 2 1 σ K E E 2 )= ⊕ can act non-trivially: Z Z 0 )= (and thus , p i 0 ) 6 , 3 σ 2 0 6 S , d ( ( 6 S E )= )=0 )= 2 ( S e 4 Z Z Z ). The ( − = 2 mod 4 (A.19) odd (A.20) = 0 mod 4 (A.18) , , , , 0 − KR 3 , , 2 5 5 5 i q 6 q q -groups, 4 2 P P P E S ⊕ KH ( E R R R ( ( ( → KH 0 2 4 ) for ) for . → )) = 27 KH e Z Z H H H 2 and p i Z , , on X which switch signs with the map ˙ Z ¯ = 0 for σ χ χ ) | | 0 -and , ⊕ )= )=0 )= )=0 )= )=0 6 )= = 2 mod 4 (the conjugate spinors have opposite X X 0 0 0 0 0 0 p,q 1 , , , , , , )= p 0 S i )and ( ( , i 6 6 6 6 6 6 ( 0 0 p p 6 σ , E , KR S S S S S S 6 (˙ -groups we use 6 S C C i ( ( ( ( ( ( ( S S 0 1 2 3 4 5 , ( ( ⊕ KR 0 − − − − − 2 2 0 , , , , , , = = , KR 1 2 3 4 5 − 2 2 2 2 2 ) 0 E , 2 p i 1 E 2 E E E E KR ¯ σ E p,q p,q 1 1 E : is identical with the usual differential operator, so E E ⊕ : 3 local coefficients p i 1 3 d σ d d ( i ⊕ ( KR = 0 mod 4 but not for = refers to the i p 1 takes twisted chains to un-twisted chains and vice-versa and so is not the e Z E 3 d Let us look at the case (A.22) more closely. The generator of Given a proper triangulation then, we can continue to follow the procedure above, The differential where twisted and un-twisted cohomologies. Onthat the other hand, we can see from eq. (A.18) bundle of rank one over the 0-skeleton. Being in extends to a rank one bundle over the 2-skeleton. Let’s try to extend it to a rank For example, for computing Steenrod square like inand the KH-groups. complex case. It provides forA.4 a Illustrative difference example: between KR- chirality) . This implies that for real) for To illustrate the AHSS inas the context an of example KR-groups the and groups KH-groups, we can examine There are two possible places where the differential the co-chain takes opposite value for opposite simplices in the covering space except now there should be a slight modification of the At this point it useful to bear in mind that generator of JHEP05(2001)047 ”. 3 P (1998) R (A.24) is only J. High , 3 07 ’s (which d 2 ) but that (1998) 019 0 P , 6 R S J. High Energy J. Geom. Phys. ( 12 , , ) which looks like KR e Z , 5 P Adv. Theor. Math. Phys. 0in R , ( 3 → C z J. High Energy Phys. , 2 Z takes . ]. 3 ⊕ Z d Z J. High Energy Phys. ”, but in KH-theory the rank one bundle , 28 )= )=2 ]. 3 0 0 , , The final result is then: P 6 6 . S S R ) ( ( 0 ]. , 6 ]. S ( KR KH hep-th/0003025 K-theory and Ramond-Ramond charge KH Further analysis of the same kind reveals that On orientifolds, discrete torsion, branes and M theory . ) ]. hep-th/9710230 ) = 0). In KR-theory, the constant rank one bundle over 0 , e Z 6 , S 5 ]. (2000) 013 [ ( hep-th/9609122 P . 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