JHEP05(2000)006 May 3, 2000 Accepted: March 27, 2000, -field. The result relies in Douglas proposal — B Received: D-branes, Superstring Vacua, Conformal Field Models in String We find the orbifold analog of the topological relation recently found by [email protected] HYPER VERSION Department of Physics, CaliforniaPasadena, Institute CA of 91125, Technology USA,Caltech-USC and Center for Theoretical Physics,Los University Angeles, of CA Southern 90089 E-mail: California Theory. which we derive from worldsheetrepresentations consistency on conditions open — stringdiscrete of Chan-Paton torsion. embedding factors projective The when orbifold consideringalgebraic action orbifolds on K-theory with open group strings — gives usinging a twisted Ramond-Ramond natural cross charges definition products of in —correspondence the orbifolds responsible between with for fractional measur- discrete branes torsion.interesting and Ramond-Ramond fashion We fields from show follows the that inclosed way the an strings. that discrete torsion is implemented on openKeywords: and Abstract: Freed and Witten which restricts thewith allowed a D-brane topologically configurations of non-trivial type-II flat vacua Jaume Gomis D-branes on orbifolds withand discrete topological torsion obstruction JHEP05(2000)006 (1.1) . We will ignore this Y is trivial and the orbifold Y , their formula constraints X ⊂ Y . A careful analysis [7] of string Y , [7] in the context of orbifolds. Let us 1 1 ]=0 is zero — imposes, for a class of backgrounds, H [ ∗ i dB = 2 and a submanifold H X Perhaps surprisingly, orbifold CFT [1, 3] seems to be able to realize topological The original motivation of this work was to find the analog of the topological Several aspects of thisThis topological formula relation can had have an already additional been term considered that depends by on Witten the in topology [5, of 6]. 1 2 briefly explain their resultsGiven in a space-time a manifold language that will be convenient for what follows. relations satisfied by geometricditions. compactifications A from nice worldsheet example consistencyimposed of by con- this modular phenomenon invariance is asthe the the space-time interpretation analog manifold [4] of to of the have a topological a restriction constraint vanishing requiring second Stieffel-Whitneyformula class. recently found by Freed and Witten 1. Introduction, results and conclusions Orbifolds [1] in stringdescribe theory perturbative provide vacua. a In tractablegeometry the and arena topology more where provide conventional powerful CFT geometric techniquesapproximation in can compactifications, to describing string be the long theory. used wavelength In to provide a sense, a geometric sort compactifications and of orbifolds topology dual is description. less manifest. In Ontion the the [2] latter, other lacks hand, an CFT conventional exactanalysis techniques Calabi-Yau CFT compactifica- of are formulation the available but corresponding but a supergravity rich approximation. mathematical apparatus aids the Contents 1. Introduction, results and conclusions2. Open and closed strings3. in orbifolds with Examples discrete and torsion D-brane charges 5 1 9 correction since it vanishes when the second Stieffel-Whitney class of the configuration of D-branesworldsheet global allowed to anomalies in wrap B-field the — so presence that the of curvature the D-branes following and topological a relation flat Neveu-Schwarz model satisfies a relation which can be identified with the vanishing of this class. JHEP05(2000)006 . ) 0 n Z ]is ]is Z such Y, H H × ( [ 3 ∗ n m i Z H ]—ofthe ∈ U(1)). This ]] H , 5 ] is not trivial, H [ (Γ ∗ 2 H i H 0, the charge of the ) used whenever [ ' X ]] .Since[ H X [ ∗ i [ ⊂ · Y m times bigger than in a background . Therefore, in a background with m X is ⊂ For example, whenever [ Y 2 Y [4], we show that the charge of the minimal 3 4 -field is trivial. This suggest that discrete torsion B so that Γ acts cristallographically were found. 0 n ) determines the topological class of the B-field and D-branes wrapping Z ] to the D-brane worldvolume and H 0, the anomaly relation (1.1) can only be satisfied whenever there n m X, Γ with discrete torsion ( / 3 ' 6 ]] H T -field in a geometric compactification and the conventional orbifold cor- H B ∈ [ ∗ ] i [ · H ) [6, 13] instead of the more conventional group K( X m ( A crucial ingredient in deriving this result is a careful treatment of open strings In this note we find the analogous phenomenon for orbifold models. Given a com- ] The simplest supersymmetric orbifold model with discrete torsion appears whenIn the many examples orbifold the relation is not direct and the correspondence between torsion in homology The work of Sen describing D-branes via tachyon condensation of unstable systems — see for 4 5 3 H [ alone suggest the correlation between discreteWe torsion show and that projective representations. worldsheetthe consistency orbifold conditions group uniquelyspecified. on determine Chan-Paton the This factors action result once of can a be closed derived string by orbifold demanding model that open is and closed strings in string theory is intimately related to torsion in homology [15]–[19]. D6-brane configuration wrapping theminimal orbifold charge is when an one integer considers multipleThis conventional orbifolds bigger result (without parallels than discrete the the torsion). consequencesing that on stem discrete from torsion (1.1).non-trivial in Roughly the speaking, orbifold turn- corresponds to turning on a flat topologically responds to the case where the is a torsion class in cohomology, so that there is a smallest non-zero integer that the restriction of [ where [ is three complex dimensional. For concreteness, we will study in sectionand 3 discrete the torsion case is Γ only = visible after an irrelevant perturbation, as in for example [17]. example [9, 10, 11]See [12] — for was a crucial type-IIA discussion. in making the identification between D-branes and K-theory. type-IIB D-brane charges areK classified by the twisted topological K-theory group in orbifolds with discreteshould torsion. be implemented Douglas onof [20] open the has strings orbifold proposed byand group that embedding on discrete projective a torsion Chan-Paton representations projective factors. depends representation on Whether its Γ cohomology admits via discrete torsion are a multiple of a topologically non-trivial flat B-field such that minimal D-brane configuration wrapping string background. Intion some [8, cases, 6] this of Ramond-Ramond fact charges. is realized by the K-theory classifica- with trivial B-field. TheD-branes anomaly in relation string (1.1) theoryconfigurations expresses are in may not topological depend pure terms on geometric that discrete constructs, choices but — that like the the allowed choice of [ In [14], the values of trivial. pact orbifold JHEP05(2000)006 9 ). X ( (1.2) ] H [ Γ is larger / 6 U(1)) corre- , T (Γ 2 H ∈ c Γ is representated by 1. ) is the algebra of continous 7 ∈ X ( i one-dimensional projective g C 10 of Γ acting on the Chan-Paton is free, the algebraic K-theory of 8 X Γ, where ∝ , ) | U(1)) and defines and associative group ring R , = 1 and a brane stuck at the singularity by an Γ X d | ). See [22] for a more complete discussion. ( (Γ 3 Q 2 C X = ( ] H c Γ [ Q ∈ ) coeffitients. The Grothendieck group of the twisted -representation c X ( R C ) is isomorphic [23] to the twisted topological K-theory X ( ] c [ Γ is the order of the group. The minimal charge is therefore obtained by Γ) which classifies D-branes in a background with topologically non- | ,istwistedbyacocycle Γ X/ | X ( approach to equivariant K-theory via cross products [22], one incorporates ] is the dimension of the 6 H [ R d The effects of discrete torsion can be incorporated to define a K-theory group We show that the minimal D-brane charge for six-branes wrapping In the more conventional setup of branes transverse to a non-compact orbifold, aOne bulk always brane has is the trivialAny representation group where each Γ element can have projective representations. The important issue is whether a given See [21] for priorThe use multiplication of law the of algebraic the approach cross to product K-theory in string theory. 8 9 6 7 10 twisted cross products K for orbifolds with discreteputing torsion than the for D-brane conventional orbifoldsamplitude charge. with by an explicitly insertion com- The ofoperator. the D6-brane corresponding The charge untwisted same Ramond-Ramond result can vertex canism be be [25]. obtained as extracted As in shown [24] fromsix-brane using in the a charge [26, boundary is 24], disk state given the formal- by properly normalized untwisted Ramond-Ramond sponding to the choice ofthat discrete whenever torsion the and action projective of representation. a It finite turns group out Γ on where factors and As shown in section 2,ing particular projective with representations orbifolds must with bejective used discrete representations when torsion. is deal- that A there simple are and no important non-trivial property of pro- interact properly in theinteractions. orbifold, A so careful that account of the whatand is orbifold we discrete group shall torsion is Γ present essential its is in this description conserved derivation in by section their 2. which measures Ramond-Ramond chargesalgebraic in orbifolds with discrete torsion. In the discrete torsion by twisting the cross product by a cocycle described by the regularirreducible representation representation so and that carries fractional untwisted charge [26]. projective representation can bein redefined section to 2. become a vectorial one. This will become more clear generated by the elements of Γ with taking the smallest irreducible representationorbifolds of one Γ. must For use opengroup standard Γ, strings the in (vectorial) smallest conventional representations irreducible of vectorial representation Γ. is always one-dimensional. For any discrete trivial flat B-field. The— use of provides projective a representations definition — of and K-theory therefore which of is cocycles the orbifold generalization of K group K functions on crossed product yields the K-theory group K JHEP05(2000)006 (Γ), (1.3) R ' ) 3 C ( Γ Γ(we take Γ abelian for simplic- in the presence of discrete Γ. The charge vector of any ∈ 11 / j . 3 g . ∀ C ) i ,g j convent g ( Q c proj R 4 )= d j ,g = i . g ( c tors . dis Q -regular if c Γis elements of Γ [30]. Moreover, the closed string spectrum in the ∈ 12 i g -regular c 1 is dimension of the smallest irreducible projective representation of Γ. (Γ) is the representation ring of Γ. The CFT argument goes through in the R U(1)), the charge of the minimal D6-brane configuration when discrete torsion > , The correlation between discrete torsion in the closed string sector and the use A group element See section 2 for more details. (Γ 2 12 11 proj R presence of discrete torsion.tion That in terms is, of any the zero-braneThe states state question associated has is with a the ifirreducible unique irreducible there projective decomposi- projective are representations. representations, as socharge many that lattice massless one Ramond-Ramond to can one-formsince each associate fields the irreducible a as projection generator representation. in of the the This twisted a sector priori is seems different non-trivial is turned on iswhen given discrete in torsion terms is of turned the off charge by of the minimal D6-brane configuration zero-brane state lies in airreducible charge representation lattice of generated by Γtice. a is basis This associated of result charge with can[8, vectors. be a 6] Each shown basis to both vector D-branespectrum from of charges CFT yields using the [26, equivariant a charge 24] K-theorysector, lat- and massless [6, the but Ramond-Ramond 28, K-theory there one-form 29]. approach so are potential The that as for closed indeed many each string oneducible twisted twisted can representation. sectors associate as A achoice irreducible of particular generator representations representation zero-brane of of of state Γ thecan Γ, on is be charge its uniquely lattice uniquely Chan-Paton decomposed factors, with specified intothe but a each by states any particular irre- representation the associated sum of of with Γ zero-brane its the irreducible states. irreducible ones. representations This Therefore, can is be realized used by as equivariant a K-theory basis since of K d torsion and generically thebetween projection massless removes Ramond-Ramond these fields andfollows massless irreducible fields. in projective an The representations interestingturns matching way out from that the thenumber number algebraic of of properties irreducible of projective discrete representations of torsion. Γ equals It the where representations. Therefore,H given a model with orbifold group Γ and non-trivial Thus, the charge ofone a consider D6-brane orbifolds wrapping the withorbifolds. entire discrete orbifold torsion is than always when larger when one considersof conventional projective representations ontional open branes strings [27, provides 26] in aat these natural a models. description point For of simplicity, in let’s frac- a consider D0-branes conventional sitting non-compact orbifold JHEP05(2000)006 ) j g i g ( γ ) j ,g i g ( c )= j g to the correspond- ( -regular elements. See γ c ) i 10 g ( (Γ) denotes the module ] γ c [ R =1for  Γ — with abelian Γ — is found ). i X/ ,g . j c g ( /c ) j 5 (Γ), where now ,g ] i c g [ ( c R ' )= -regular elements is identical ) j 3 c ,g C i ( g ] ( c [ Γ  K We will start by briefly explaining the essentials of discrete torsion 13 is a cocycle determining the projective representations c The organization of the rest of the paper is the following. In section 2 we explain The spectrum of closed strings on an orbifold Recently, [44, 45, 19] have considered the gauge theory on branes on an orbifold with discrete 13 of projective representations of Γ with cocycle the inclusion of discrete torsionthe in topology closed of string orbifolds thederive and orbifold from relate group worldsheet its consistency Γ. propertiessentations conditions to when We the analyzing analyze open necessity strings D-branes to inwe in use orbifolds find with projective these the discrete orbifold repre- orbifolds torsion. analog and amples In of and section the describe 3 the result charges by of Freed fractional and branes Witten in orbifolds [7], with present discrete several torsion. ex- by quantizing strings that are closed up to the action of Γ and projecting onto Γ ing twisted sector spectrumwhich in do the have conventional masslessrepresentation orbifold Ramond-Ramond is (without fields. associated discrete with torsion) Thus, anon-trivial generator each discrete in irreducible torsion. the projective This chargeerties intuitive lattice of result, even cocycles, when which follows there ties from is instrings. algebraic a From prop- nice this way the CFTof effects result, twisted of one cross discrete is products torsion naturally on led open to and closed conjecture that the K-theory ity), where The dynamics of ahow D-brane the at geometry an of orbifolda space-time singularity partial is provides list encoded a of in simple referencessee the example [42, see D-brane 43] of [31]–[41] worldvolume ). and theory The foropen (for low interesting and energy applications closed gauge to strings theory AdS/CFT on onters the the in orbifold brane the is [31]. gauge found theory ClosedOpen by such string string quantizing as modes modes both in provide appear gauge Fayet-Iliopoulos as termsthe fields parame- and brane. and scalars in In which the this describe superpotential. sectionand the we fluctuations closed will of strings show require that embeddingorbifold consistency an of group appropriate interactions on projective between the representation open ofdiscrete open torsion. the string Chan-Paton factors when studying orbifolds with 2. Open and closed strings in orbifolds with discrete torsion twisted sectors associated with torsion. and describing the closeddetermining string the appropriate spectrum projection of on these open models. strings. This will be crucial in sections 2 and 3 for more details. and the discrete torsion phase JHEP05(2000)006 | Γ | of Γ (2.5) (2.1) (2.4) is the 16 ) j ,g i g ( Z . Γ directions of the and ∈ τ k 15 phases satisfying the ,g j Γwithaninsertionof and 2 | ,g σ ∈ i Γ | i g in the single string Hilbert Γ (2.3) g ∀ i ) (2.2) i ∈ into account when computing , k s j | )  j g ,g i ,g , . ∀ ). i ) g ) ) g Z one must quantize and project j i ( ( j ,  ) transformation Z ) ,g ,g ,g 14 ) j i i j Z i j g g g , i SL(2 ( ,g ( ( i s ,g c c c | i ∈ g ) ) g ( k  6 j (   ,g ,g )= j Γ i j cd ab g ∈ g i )= j . (  ,g k g }  ,g i X i ( i g g g g j c ( j =  g i ,g = i i must furnish a one dimensional representation 1. = g )= s )where Z (  k j d j  ≡ ·| g g i ,g c j i  j g ) must be a one dimensional representation of the stabilizer subgroup ,g g ,g j Γ b j ( g ,g c ∈ i a ) i g j g k ( ( g g   { j = ,g ) is invariant under an SL(2 i )= j i j g g Γ in the trace. ( relations ,g N ,g c Γ. This provides a natural way to take i i 3 ∈ g g | ( ( ∈ j  Γ  U(1) is a two-cocycle, which is a collection of i | g g ∈ c Γ, where terms corresponding to all the possible twists along the Discrete torsion is intimately connected with the topology of Γ via [4] As first noted by Vafa [4], modular invariance on higher genus Rieman surfaces In [4], Vafa showed that ∈ In this paper we shall consider Γ abelian only. ItThat is is straightforward to generalize to non-abelian For non-abelian Γ, 2 twisted sector is obtained by keeping those states i | 14 15 16 g i Γ action of States satisfying (2.3) transform in aguage, one the dimensional spectrum representation of of conventional Γ. orbifolds transformrepresentation In in of this the trivial lan- Γ one-dimensional where such that Here the closed string spectrum.g In orbifolds with discrete torsion, the spectrum in the following space that satisfy partition function of a string closed up to the action of together with factorization ofthe allowed loop phases. amplitudes Orbifolds imposesreferred models as very admitting orbifolds these severe with non-trivial restrictionswhether discrete phases on torsion. are a usually particular As wetopology orbifold shall of model briefly the explain admits discrete in group such a Γ. a moment, generalization depends on the closed strings. This is| reflected in the partition function of the orbifold by it having invariant states. When Γ is an abelian group, for each worldsheet. One loop modularto invariance be allows multiplied each by term a in phase the partition function groups. N JHEP05(2000)006 . : of n d +1 n n d (2.8) (2.9) (2.7) (2.6) Im d / is a one +1 n  d U(1)) = Im , -cochains forms an (Γ n n B U(1)) = Ker , (Γ n , H , so it is a 2-cocycle. Moreover, b 0 3 b d , ) i ) g j . ( g ) . -cocycles and i γ j )=1. n g ) i i 0 j ( ,g b g i ,g 0 γ of i j ) g g g j ( ( ( c +1  s, a γ = 0 and write down a corresponding complex. ,g j n ) · i -cochain. The set of all j c d ij i i g n c ∝ +1 ( c ,g g 7 i | n c ) 0 g d 1 j ( g  − aa ◦ U(1)). Summarizing, given a discrete group Γ ( ) )= i n )= , j γ is a Chan-Paton state. Consistent action on the j d g ) ( g i ,g (Γ ( i g γ 2 ab ( g γ U(1)) = Ker U(1) is an ( ) γ , H 0 = i c g )=1and i (Γ of Γ. Moreover, the discrete torsion phase (2.4) is the U(1)). i ( n , −→ γ ,g Z 17 i { (Γ Γ g s, ab 2 ( group |  × H 18 U(1)) such that ... , n }| so the equivalence classes of cocycles with (2.6) as an equivalence relation -th cohomology group is defined as usual as Γ (Γ n ij U(1)) under multiplication. One can construct a coboundary operator × +1 , /c Γ z n j U(1)). Cocycles in the same conjugacy class are therefore cohomologous. (Γ c C , : i n c c (Γ C 2 −→ H )= j is an oscillator state and ,g s i U(1)) g = 2, a 2-cochain satisfying (2.5) maps to the identity under ( , Placing D-branes in these vacua requires analyzing both open and closed strings c One can also show that The map n (Γ ◦ 17 18 n -coboundaries. The 2 there are as manynumber possibly of different elements in orbifold models that one can construct as the This seems lo leavetations. some arbitrariness since As Γ weare may will also have show several taken classes shortly intorepresentations of account. the where represen- group arbitrariness In multiplication is particular isgeneral realized removed Γ such only once may representation up is have closed to given several a strings by classes phase. of The projective most where open string state andembedded completeness on Chan-Paton of factors Chan-Paton by wavefunctions matrices demand the Γ represent to Γ up be to a phase in the orbifold.graphs The in closed a string language spectrummost that general was will action summarized on be interior an the convenient on open when last the string considering few string isThe open para- obtained open (the strings. by string oscillators) letting spectrum The andcombined Γ action is its act of found end-points both Γ by [46, (the on keeping 31] the Chan-Paton all in- factors). those states invariant under the One can show from the definition of discrete torsion in (2.4) that indeed dimensional representation same for cocycles inorbifold the models that same one conjugacyof can class. cocycles. construct is Therefore, given Topologically,its the by equivalence second number the classes cohomology number of of of different conjugacy cocycles classes of Γ are determined by The set of cocyclesrelation can compatible be with split (2.5) into conjugacy classes via the following equivalence abelian group is given by d One can also define the group n For C JHEP05(2000)006  (2.12) (2.10) (2.11) U(1)). .This , c )onthe i g (Γ ( 2 γ H Let us consider ). Therefore, the i . 19 g i to satisfy the cocy- ( on the closed string. γ c i defined in (2.6). The (1) c  . µ 0 ) c i V g )= i ( , (0) g γ i -th twisted Ramond-Ramond ( i ) β i 0 j ] g V γ g ( (1) µ γ (0) i ) α V j V h ,g ) i (0) i j g β ( ] V , which includes the action of ,g c i i g 8 g (0) ( i  α )= i V )) h j ,g ) satisfies (2.5), so does g j ( λ g c as ) ( i j c λγ g ) g ( 1 j − γ g ) ( j γ tr( g ) ( i γ g is the vertex operator for the photon. Consistency requires ) ( i of the open string gauge field. This amplitude is completely µ γ g ( V λ γ tr( are the right and left moving parts of the i β U(1). Associativity of matrix multiplication forces f V , ∈ i α c V We will now show that once we make a particular choice of discrete torsion A similar restriction was imposed by Polchinski [32] in orientifold models. 19 -th twisted sector and photon arising from the open string ending on a D-brane i follows from a worldsheet CFT conditionThe demanding Γ action to of be a Γinteractions. symmetry on of the We open OPE. already and knowclosed closed strings. that strings this One is must isthat also consistent the demand is only case consistency Γ if of for has open-closed Γ interactions to string is involving be interactions, only conserved conserved by by a open-closed string amplitude. determined by Lorentz invariance Moreover, the invariant openpends string on the spectrum cohomology class (2.7) ofone of the chooses. cocycle the and orbifold not This onthat model is the projective particular only complete representations representative should de- analogy be used withtorsion. when the describing orbifolds closed with discrete string discussion indicating in (2.3) for the closedis strings, uniquely that the determined action to on be the a open string projective Chan-Paton representation factors (2.9) with cocycle for concreteness the transition betweeng a Ramond-Ramond closed string state in the different classes of projective representations of Γ are also measured by corresponding representation is trivially found to be invariance of this amplitude underis the not action specified of until Γ. we As choose mentioned a earlier, the particular model discrete torsion Therefore, taking intousual account adjoint action how (2.7) Γ ontransforms acts open under strings the on Chan-Paton action closed factors, of the string amplitude states (2.10) (2.3) and the Chan-Paton matrix vertex operator and cle condition (2.5). Moreover, if where where transverse to the orbifold.arises in the To disk. lowest Thecreates order closed a string in cut vertex operator the fromjump is string across its built the out location coupling cut of by inside this a the orbifold twist the amplitude action field disk which to the boundary of the disk. Fields Invariance under Γ requires settingthe equal discrete (2.10) and torsion (2.11), phase which in gives terms after of writing cocycles the following constraint JHEP05(2000)006 c (3.1) (3.2) -regular -regular Γ — for c c ∈ is given by i c g )isthegreatest 0 n, n . ) j . The allowed classes of 0 ,g n i g =gcd( Z . (  d different orbifold models one Γ × Γ d A group element n ∈ ∈ j Z j ,g X U(1)) is non-trivial. The simplest i g 21 g , ∀ | arise from a two-loop effect on closed ,where (Γ 1 Γ d |  2 Z H = ) i ' ) ) 9 i j ,g j ,g ,g g i j ( g g c U(1)) 20 ( ( , c c Γ (Γ )= ∈ 2 j j ,g H X ,g i i g g | . Thus, a priori, there are ( 0 1 Γ c | n -regular elements of Γ [30]. = c c 22 and N n -regular if c of irreducible projective representations with cocycle class c N A basic definition in the theory of projective representations is that of a It is interesting to note that constraints on This is very different to the caseWe of use vector the representations, fact for that any which non-trivial there one-dimensional representation are yields as zero many when one irre- sums There seems to be some arbitrariness in the projective representation one chooses. The world- 21 22 20 element. The numberequals of the irreducible number projective of representations of Γ with cocycle This definition is independentnumber of the representative of the cocycle class. Thus, the can define. common divisor of We have used (2.4) to writeIt the above is formula clear in terms then of that the discrete the torsion closed phases. string spectrum for the sectors twisted by abelian Γ — is abelian group admittinggiving non-trivial rise projective to discrete representations torsion in — orbifolds or — is equivalently, Γ = representations are labeled by the following formula 3. Examples and D-brane charges In this section we willrelevant properties work of out projective a representations general ofshow class discrete the of groups results that examples anticipated are and in neededgroup develop section to Γ some 1 admits of and projective the 2. representations if As mentioned in section 2 a discrete This constraint is satisfiedrizing, by we have choosing shown a fromdiscrete projective simple torsion worldsheet representation (2.4) principles (2.9). that thatmined the given Summa- by action an the of orbifold corresponding the with cocycle. orbifold group on open strings is deter- ducible representations as there are conjugacy classes in theover all discrete group group. elements. sheet consistency condition onlya determines the particular cohomology representative. classmodel. This of the freedom, cocycle however, but does does not not affect pick the spectrum of the orbifold strings but that consistent open-closedaction string on interactions the at tree open level strings. determine the JHEP05(2000)006 ) a X ( ] (3.5) (3.3) (3.6) (3.4) c [ Γ for each blocks of b /p 0 are integers. the generator n and b be the smallest 2 different phases a g p d and and , a and 1 n/p n − elements. Usual vector Z p , for which the sum over p . . The discrete torsion phases ,where 0 0 b 2 ,...,d n 2 1) are one-dimensional and all g p a Z 1 nn , g ≡ respectively, the total sum yields =0 × 0 = c , b n 2 =1 a Z p 2 R 2  d . Since there are and 2  c p 0 0 =1 ,m N 2 a a 10 ). As expected there are X d ) p 0 b nn 0 πimp be the generator of are multiples of a = 2  0 1 − | and 0 0 n, n b  g Γ ab | b 1 ( nn irreducible projective representations with cocycle m exp 2 =1.Ifweperformthesumoversay and α = and 0 p /p c =gcd( 0 a a N d )= 0 nn . For our purposes, we only need to find the number of b 0 0 .Ifwelet n = 0 n Z c -elements just gives )and Z p ab, a N × (  × n n πi/d Z Z of . The dimensionality of each irreducible representation can be obtained 0 n 23 labels the different irreducible representations. Thus, each representation Z =exp(2 a × a general group element can be written as α R n over a block of 0 Z b n The study of D-branes in these backgrounds require analyzing the representa- Let’s consider in some detail the example -dimensional. Usual vector representations ( This example has also been considered recently by [19]. Z p elements for the sum over 23 for irreducible projective one are bigger. block we get 0 except when Therefore, there are of p where irreducible representations in a givencan cohomology class use and (3.2) their and dimensionality.non-zero (3.3) We integer to such find that how many of them there are. Let where one can associate to the closed string partition functiontion (2.1). theory is and then the sumrepresentations (3.2) correspond can to be split into sums of blocks of from the fact that theones regular representation can be decomposed in terms irreducible appearing in the closed stringresentations partition of function correspond to one dimensional rep- c elements is the same as forrepresentations conventional orbifolds, as so massless that we Ramond-Ramond have fields asthe many of section irreducible a 1, this given prediction rank. should As follow from explained the in algebraic K-theory group K Then, the allowed discrete torsion phases are of twisted cross products. JHEP05(2000)006 B (3.7) Nucl. Phys. , Ramond-Ramond i g (1986) 285. ). The computation can Vacuum configurations for i g The conformal field theory of B 274 appears in the vertex operator. 6 Strings on orbifolds, 1 ...µ 0 µ , C | R Γ d | Nucl. Phys. 11 , = Q (1985) 46. (1987) 13. B 258 B 282 Strings on orbifolds, 2 Nucl. Phys. , Nucl. Phys. , 2) picture to soak the background superghost charge on the disk. In this / is the dimension of the representation considered. This formula applies both 1 (1985) 678; R − d , 2 261 orbifolds superstrings The conclusion stating that the minimal charge of a D-brane configuration wrap- / 3 [1] L. Dixon, J.A. Harvey, C. Vafa and E. Witten, [3] L. Dixon, D. Friedan, E. Martinec and S. Shenker, [2] P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, − be easily computed by conformallythe mapping appropriate onto boundary the conditions. upper half The plane final and result imposing is [26] picture and the Ramond-Ramond potential field one multiplies by the trace of the representation for for conventional orbifolds ascase well one as must use for projective orbifolds representationsSince and with the in discrete the smallest second torsion. irreducible vectorsmallest vector representations. In representations irreducible the of projective Γ first isminimal representation one-dimensional D-brane is but charge larger, the allowed thisfor for conventional shows orbifolds orbifolds. that with indeed discrete torsion the are bigger than where References Acknowledgments I would like to expressGukov my and gratitude E. to Witten D.E. forDOE Diaconescu, under enlightening M. grant discussions. Douglas, no. J.G. E. DE-FG03-92-ER is 40701. Gimon, supported S. in part by the The amplitude is multiplied byfactors the for trace the of identity element the (to representation compute acting the on charge under Chan-Paton the ping the entire compact orbifoldever is discrete an torsion integer bigger iswant than non-trivial to the compute can minimal the be charge charge undersponding when- verified the to by untwisted a sector a Ramond-Ramond wrapped simple fieldsix-brane D6-brane. corre- disk Ramond-Ramond This amplitude. can vertex be operator We putation computed on and by the inserting refer the disk. to( untwisted [26] We for will more sketch details. the com- The vertex operator has to be in the JHEP05(2000)006 . , ]. Phys. (1995) (1998) , B 273 B 441 J. High theories . , 07 15 (1998) 007 2) , (1998) 019 06 J. High Energy J. High Energy J. High Energy , 12 =(2 , , hep-th/9907189 , Phys. Lett. N Nucl. Phys. Discrete torsion and , , hep-th/9406032 . ; ]. Adv. Theor. Math. Phys. hep-th/9909089 J. Geom. Phys. , , , ]. J. 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