JHEP01(2007)107 and a hep012007107.pdf , January 8, 2007 January 9, 2007 January 31, 2007 uge theories on November 20, 2006 Revised: Accepted: act Calabi-Yau threefold Published: Herman Verlinde Received: 9, based on a D3-brane at a bc n particular, we work out the [email protected] , inside a compact CY threefold, a, nstein-Institut, http://jhep.sissa.it/archive/papers/jhep012007107 /j (1) gauge factor. We apply this insight to the U David R. Morrison, ∗ a [email protected] , Published by Institute of Physics Publishing for SISSA [email protected] , Dmitry Malyshev, a ad Intersecting branes models, Superstring Vacua, D-branes. We report on progress towards the construction of SM-like ga [email protected] singularity, and present a geometric construction of a comp On leave from ITEP Russia, Moscow, B Cheremushkinskaya, 25 8 ∗ Department of Physics, PrincetonPrinceton, University, NJ 08544, U.S.A. Center for Geometry andDuke Theoretical Physics, University, Durham, NCDepartments 27708, of U.S.A. Physics andSanta Mathematics, Barbara, Univ. CA of 93106, Californi Max-Planck-Institut U.S.A. Gravitationsphysik, f¨ur Albert-Ei Potsdam, Germany E-mail: [email protected] SISSA 2007 b c d a c Martijn Wijnholt ° Matthew Buican, Abstract: the world-volume of D-branes attopological a conditions Calabi-Yau on singularity. the I embeddingthat of select the hypercharge singularity as the only light with all the required topological properties. Keywords: proposed open string realizationdP of the SM of hep-th/050808 D-branes at singularities, compactification,hypercharge and JHEP01(2007)107 7 8 9 3 6 6 7 1 2 16 20 22 24 28 29 11 13 14 21 28 m-up approach to string es on D-branes near CY sin- hich the string and 4-d Planck een the Planck and TeV scale, g realizations of an increasingly ry can be isolated from the closed singularity 16 – 1 – 8 dP (1) hypermultiplet U (1) breaking via D-instantons masses via RR-couplings U (1) 4.1 A standard4.2 model D3-brane Identification of hypercharge 5.1 Local Picard5.2 group of a Construction CY of singularity the CY threefold 6.1 6.2 Eliminating extra Higgses 3.2 Some notation 3.3 Brane action 3.4 Some local3.5 and global considerations Bulk action 3.6 Coupling brane and bulk 3.1 The 2.1 D-branes at2.2 a CY singularity Symmetry breaking2.3 towards the SSM Summary U 5. Constructing the Calabi-Yau threefold 6. Conclusion and outlook 4. SM-like gauge theory from a rich class of gauge theoriesit [1 – is 3]. natural Given to thephenomenology, hierarchy that make betw aims use to of findgularities. this Standard In technology Model-like this and theori setting, the pursuestring D-brane physics a world-volume in theo botto the bulkscale via are a formal taken decoupling to limit, infinity, or in w very large. 1. Introduction D-branes near Calabi-Yau singularities provide open strin Contents 1. Introduction 2. General strategy 3. JHEP01(2007)107 ult, we (1) vector the Cartan U ij A s follows: singularities Pezzo 8 singularity m. We begin with 2 with enology. Our set-up ncrete construction of A ij arates the task of find- t lead to realistic gauge A ed to build semi-realistic sless gauge symmetry. To h 2 pairs per generation): − and matter content of the on 5, we present a concrete how the local singularity is lt challenge of finding fully l the required properties. ectrum of light . In subsection 2.1, we give ng geometrical properties: r a suitably chosen del Pezzo = cles of the singularity. Several se local constructions lead to the compact embedding of the j ns. ng of a CY singularity inside a metry breaking patterns can be uplings of the world-volume the- α · surface is spanned by the canonical i 8 α swampland program” of [4], that aims to ns near CY singularities to full-fledged string consistent UV completions with gravity. dP (1)-symmetry factors survive as massless U 1 – 2 – are degenerate and form a curve of (1) bosons acquire a mass of order of the string scale. The with intersection form 2 U i α α are non-trivial within the full Calabi-Yau three-fold 4 and α 1 α (1) gauge symmetries beyond hypercharge. As our first new res U . With this notation, our geometric proposal is summarized a 8 E and eight 2-cycles K singularity, such that only hypercharge survives as the mas More details of this proposal are given in section 4. In secti The world-volume gauge theory on a single D3-brane near a del In this paper we work out some concrete aspects of this progra In the second half of the paper, we apply this insight to the co In scanning the space of CY singularities for candidates tha The clear advantage of this bottom-up strategy is that it sep The general challenge of extending local brane constructio 8 1 (ii) all 2-cycles except (i) the two 2-cycles has, for a suitable choice of moduli, K¨ahler the gauge group SSM shown in table 1, except for an extended Higgs sector (wit geometric recipe for obtaining a compact CY manifold with2. al General strategy We begin with ashares many summary ingredients our with general the D-brane approach approach to of string [6 – phenom 11] embedded in a compact Calabi-Yau threefold with the followi matrix of class state this condition, recall that the 2-homology of a a brief review ofgauge the theories general from set branesmodels with of at extra ingredients singularities. that Typically can the be us bosons and the number ofembedded freely inside tunable the couplings, full depend compact on Calabi-Yau geometry. left-over global symmetries are broken by D-brane instanto an SM-like theory given in8 [5], based singularity. on We a specify single adP D3-brane simple nea topological condition on gauge symmetries. The other identify the general topologicalcompact conditions CY on threefold, the that embeddi determines which compactifications represents a geometric component of the “ determine the full class of quantum field theories that admit theories, one is aided by theory fact are that controlled all by gauge the invariantenforced co local by geometry; appropriately in dialing particular, the sym other volumes properties of compact of cy the gauge theory, however, such as the sp ing local realizations ofconsistent, SM-like realistic models string compactifications. from the more difficu JHEP01(2007)107 , 0 zo we -th Y and n (2.1) b ¯ a nonical , which h X = 6 blown up at is defined by b ¯ 2 2 a σ d i / P R 1 H − a positive first Chern with where u is not unique but actually i n σ H inciple, good candidates sheet CFT, they are in X et a cone over dP . (More general del Pezzo + 0 tical description; the basic ied in [5, 14, 15]. 5, Y e combinations of D-branes, s singularity. For most of our c i ularities. The reader familiar dψ cone over a del Pezzo surface ≥ ν 1 3 counts the 3 generations llections of sheaves, supported n into so-called fractional branes. rspective, fractional branes may cal specifications. and sometimes called “the i iant open string boundary condi- ess, however, we specialize to the = 2 X n η , for c i ds e 2 dP r can be represented as + 1 2 1 0 1/2 2 P / i η ` 1 2 × r − 1 P + – 3 – 2 c i ¯ 31111 1 d dr = singularity and denote by 3 1/3 2 Y near the singular point.) To specify the geometry of / c i n ¯ 3 2 u 0 ds Y − . Introduce the one-form b ¯ i a is the angular coordinate for a circle bundle over the del Pez 3 21121122 Q 1/6 iR π , one obtains a smooth non-compact Calabi-Yau threefold. If 2 n be a metric K¨ahler-Einstein over the base By placing an appropriate complex line bundle (the “anti-ca = 6 dP L b ¯ C b ¯ Y 2 a The matter content of our D-brane model. dz = ω a (1) <ψ< X U dz SU(2) SU(3) b ¯ a h and 0 Table 1: = 8 complex moduli represented by the location of the points. ω − 2 X 8 generic points; such a surface is denoted by n There are many types of CY-singularities, and some are, in pr A del Pezzo surface is a manifold of complex dimension 2, with ds This terminology is unfortunately at odds with the fact that = 2 2 ≤ . D-brane theories on del Pezzo singularities have been stud first Chern class let surface. The Calabi-Yau metric can then be written as follow singularities are asymptotic to has 2 del Pezzo surface”. bundle”) over then shrink the zero section of the line bundle to a point, we g we will call the conical del Pezzo n dσ a quick recap of somewith relevant this properties technology of may D-branes wish at to sing skip to subsection 2.2. 2.1 D-branes at a CY singularity D-branes near Calabi-Yau singularitiesFractional typically branes split can up bethat thought of wrap as cycles particularone-to-one of bound correspondence the stat with allowed local conformallytions. invar geometry. Alternatively, by In extrapolating to termsbe a of large represented volume the geometrically pe world- on as corresponding particular submanifolds well-chosen withindiscussion, co the however, local we Calabi-Yau willproperties not that we need will this use have abstract relatively mathema simple topologi for finding realistic D-brane gaugesubclass theories. of For concreten singularities whichX are asymptotic to a complex class. Each del Pezzo surface other than JHEP01(2007)107 s r F (2.4) and counts and the del s i F F of an exceptional j t of F ij counts the number of χ and a s i p . ≡ F denotes the canonical class j o-called exceptional collec- ) F mpact CY manifold, with i em to find consistent bases of the gauge multiplet. The for- K F anes . ! n particular ensures the absence and rity. We will consider both cases. !5 a i α ditions. For del Pezzo singularities, F ) (2.2) ) deg( j (where s F ) (2.3) , and is equal to the D7-brane wrapping , q s s K !5 · a F !3 s rk( F ) ( i 1 F − c , p ( s are extracted from the first Chern class of ). Geometrically, a ) 1 r j α governs the number of massless states of open c ) is the 2nd Chern character of a s X F Z s ( – 4 – p − F ij 2 ( = χ 2 ) = ( H may )= s a s i , and F p ) deg( wraps the 2-cycle F i X Y = ch F s , each characterized by a charge vector of the form ch( s F s q F -coordinate has infinite range. Alternatively, we can think r forms a four- : the del Pezzo ) = rk( j X X F , i F is given by deg( #( i F . The number X ) is the rank of the fractional brane s F denotes an integral basis of . The allowed fractional branes, a X = rk( α s ) and equals the intersection number between the D5 componen r For a given geometric singularity, it is a non-trivial probl The del Pezzo surface X strings that stretch between the two fractional branes Pezzo surface. The intersection number Here the degree of on collection reads [5, 14, 15]. fractional branes that satisfy alla geometric special stability con class oftions consistent [28, bases 14, are 15].of known, These adjoint in matter satisfy the in special form themula properties, of for world-volume that the gauge s intersection i theory, product besides between two fractional br times the D5-brane component of via where the D3-brane charge. Finally, the integers number around Here cycle within the full three-manifold however, typically do not correspondgle to branes sin- wrapped on some givenratherto cycle, particular but bound states For the non-compact cone, the of the del Pezzobeing singularity the local as radial a coordinate localized distance from region the within singula a co itself supports several non-trivial two-cycles. Now, if we considerdel IIB Pezzo string singularity, theory wefind on should a a expect basis of to the fractional complete branes homology that of spans 4-cycle itself may beber wrapped of by D7-branes, any any num- 2-cyclebe within wrapped by one orthe more point-like D5 D3-branes branes, occupy and thewithin 0-cycle JHEP01(2007)107 3 4 1 ). , (2.5) 0 , . Upon ij χ )=(0 i F ch( i n i iven node, the total ) gauge group factor | i P kes the form of a quiver n labi-Yau, such that the llection is not unique. | ( ank of the gauge group at tring compactification: all sufficiently separated from U cting a globally consistent ifold. In general, these tad- or which the charge vectors the fractional brane charge l mutations that act within ermined by the value of the ion, they will not contribute . , one associates to the quiver i can be reached via mutations, ons, known as mutations [28]. ble global constructions [6 – 11]. F n), all global tadpole constraints 3-charge conservation. It is then awing the quiver diagram are: orm of Seiberg dualities [14, 15]. on of fractional branes constitute nes that intersect the CY singularity. We are such that i . n y conditions on branes, and determines a region of the collection, i = 0 F j n ij 0 by an oriented line with multiplicity – 5 – χ > = 1 gauge theory, free of any non-abelian gauge j ij X N chiral multiplets in the bi-fundamental representation can be freely adjusted, provided the resulting world χ i ij n χ to each fractional brane i n , as well as i F ). The multiplicities j ¯ n , i n anomalies. volume theory is a consistent ( for every node diagram a quiver gauge theory. The gauge theory has a assigning a multiplicity Absence of non-abelian gauge anomalies is ensured if at any g In general, one could allow for local brane configurations, f The world-volume theory on D-branes near a CY singularity ta For a given type of singularity, the choice of exceptional co Each collection corresponds to a particular set of stabilit These rules can be generalized by including orientifold pla 3 4 (ii) connect every pair of nodes with (i) draw a single node for every basis element number of incoming andthe outgoing other lines end (each of weighted by the the line) r are equal: This condition is automatically satisfieda if single the D3-brane, configurati in which case the multiplicities In this special case (which isare the already one satisfied we by will the consider later localvery o construction, easy except to for D embedone the needs brane to construction do insidetotal is a D3-charge adjust of complete the the s string 3-form compactification fluxes vanishes. or topology of the Ca add up to somestring compactification non-trivial then fractional requires braneneeds to a charge. be bit cancelled Constru by morepole other conditions care, regions impose within since important theHowever, compact restrictions as man on long the possi asthe the other singularity fractional that branes supports and the fluxes local are SM-brane configurat in moduli K¨ahler space where it is valid. will comment on this possibility in the concluding section. Different choices are relatedHowever, only via a simple subset basislead of transformati to all exceptional consistent collections, world-volumethe that gauge subset of theories. physicallyWhich relevant The collections of all specia the take Seiberg thegeometric dual moduli f that descriptions determine is the appropriate gauge is theory couplings det gauge theory. For exceptional collections, the rules for dr JHEP01(2007)107 (1) factors of s it possible to U s on the topology some of the fractional ong all ingularities (and other base of the singularity, atter content to that of stic theory from a single we look for a well-chosen etry. mpactification, however, lly, one obtains one such al contraints of the string ocally, all gauge invariant at follows, whether or not zed set of steps: ve. Their value will need to massless gauge symmetry. ble SM-like sector. This ge- ing deformations of the local en the CY singularity is not s several components [3], and d in two steps. First we look is mechanism in some detail in etric insight into the symmetry quiver gauge theory is just rich y, such that the corresponding roaches to string phenomeology rying all local couplings: only within the local CY singularity e also section 4). The final model tions for the bottom-up approach (1) bosons that remain massless are in one- non-trivial U – 6 – (1) factors besides the hypercharge symmetry. Such U (1) gauge bosons may acquire a non-zero mass via U within the full CY threefold. This insight in principle make (1)’s actually survive as massless gauge symmetries depend U trivial (1)’s are characteristic of D-brane constructions: typica U The interrelation between the 2-cohomology of the del Pezzo In a string compactification, The above procedure was used in [5] to construct a semi-reali the D-brane gauge theory, only the hypercharge survives asand a the full CYcouplings of threefold the has D-brane other theorygeometry. can relevant This local be consequences. tunability varied is via oneto L of correspond the string central motiva phenomenology.typically The introduces embedding a intothose topological couplings a obstruction that full descend against string frombe va moduli fixed co of via the a full dynamical CY moduli survi stabilisation mechanism. 2.3 Summary Let us summarize our general strategy in terms of a systemati to-one correspondence with 2-cycles, that are but are ensure — via the topology of the CY compactification — that, am coupling to closed string RR-form fields. We will describe th the next section, where we will show that the extra of how the singularity is embedded inside of a compact CY geom D3-brane on a partially resolved delof Pezzo [5], 8 singularity however, (se still has several extra factor for every fractionalthese extra brane. As will be explained in wh any low energy degreesometric of freedom separation that between interact thecompactification with local is the central SM visi to physics ourhave and set-up. been the Similar investigated glob in local [7, app 13, 12]. 2.2 Symmetry breaking towards the SSM To find string realizations offor CY SM-like singularities theories and we brane nowenough configurations, procee such to that contain the the SMsymmetry gauge group breaking and process matter content. thatthe Then reduces Standard the Model, gauge orisolated, group at the and moduli least space m realistically ofthe close vacua symmetry for to the breaking it. D-brane we theorybranes need Wh ha is move found off on ofbranes a are the component replaced by in primary appropriate bound which singularbreaking states). point allows This geom us along to a identifyD-brane an curve theory appropriate looks of CY like s singularit the SSM. JHEP01(2007)107 . ζ s of (1). = 2 (1)- (3.2) (3.1) (3.3) U U N variant ζD. F − roperties, that contains discuss the St¨uckelberg ∧ ) (non-isolated) CY singu- y contains several naturally follows from its F dρ 2 upergravity on a Calabi-Yau branes at CY singularities. iζ . π − , produce an SM-like theory. θ f these steps and thus set up a . Together with the mass term, + 32 A ng to the RR-form fields or get ( ρ V + ercharge as the only massless (1) vector bosons remain massless, = ∧ ∗ F ) . U and a Faillet-Iliopoulos parameter ζ dρ ∧ ∗ ρ − and enumerate the resulting quiver gauge F i 2 A F ( g 1 , and find a suitable basis of fractional branes 1 2 4 0 Y – 7 – − = 2 , S = (1) gauge sector on a single fractional brane. Let )) 2 U to each α π g 4 i V i W n 2 α + − W ¯ S π θ 2 π τ − 8 S = θ ) covariant complex coupling, that governs the kinetic term 2 τ Z d , (Im( = 2 supersymmetry. Closed string fields thus organize in Z 4 1 θ N Im to a chiral superfield, we can write a supersymmetric gauge in 4 d S Z hypermultiplet in (3.1) combines a field St¨uckelberg (1) S U masses via RR-couplings on it. Assign multiplicities . denotes the auxiliary field of the vector multiplet is the usual SL(2 0 i Look for a compact CYY threefold, with the right topological p theories. Use the geometric dictionary tolarity. identify the corresponding F (1) gauge boson via (omitting fermionic terms) τ D (1) The complex parametrization (3.1) of the D-brane couplings U U (ii) Look for quiver theories that, after symmetry breaking (iii) Identify the topological condition that isolates hyp (i) Choose a non-compact CY singularity, all others either acquirea a mass mass St¨uckelberg via through the the coupli mechanism Higgs in mechanism some [1, detail. 26, 27, 20]. We3.1 will The now us introduce the two complex variables factors, one for each fractional brane. Some of these To set notation, we first consider the computer-aided search of SM-like gauge theories based on D- 3. In principle, it should be possible to automatize all three o The quiver theory of a D-brane near a CY singularity typicall The field After promoting Here mass term for the gauge field via [1] the Here we observe a Faillet-Iliopoulos term proportional to embedding in type IIB stringthreefold theory. Without preserves D-branes, IIB s JHEP01(2007)107 , . S K . A , we (3.7) (3.6) (3.5) 0 = X , that . The i Y and 0 X α α τ base ’s are all non- β can be chosen as a (1) symmetry is non- τ the world volume per- U gree zero two-cycles half the supersymmetry, the components of the so called closed string hypermultiplets. eze into fixed, non-dynamical d the CY singularity. In this setting, om the 10-d fields, via parates from the open string dy- ogether as the scalar components ,...,n . The remaining 0 (1) gauge factors. In our geometric Y = 0 ) (3.4) ! U 2 ij ij , 0 A A, B τB φ A . The canonical class has self-intersection completely decouples, simply because the − − n − + S is generated by the canonical class non-compact ie E n 2 , describes the class of the del Pezzo surface spans the space of compact, even-dimensional = n C 0 + 0 − 0 j ( – 8 – α 0 + 3 independent fractional branes, and a typical =1 superfields with scalar components Y d dP 9 α b a ∗ C · δ n à N i -indices. + 3 separate = = A = α τ = n b must enter as a dynamical field. In a decoupling limit, dS β ab S · η a satisfying α . Using the intersection pairing within the threefold b i β α , which coincides with the even-dimensional homology of 0 Y thus contains supports a total of 0 -th del Pezzo surface 0 Y to raise and lower n appear to stand on somewhat different footing: Y ab be a non-compact CY singularity given by a complex cone over a S η 0 = 0, have the intersection form Y , dual to the canonical class i 0 α and β equals minus the Cartan matrix of · τ 0 ij orthogonal 2-cycles α singularity A . In the following we will use the intersection matrix n n In spite their common descent from the hypermultiplet, from n itself, and forms the only compact 4-cycle within dP becomes a fixed constant as expected, while (1) gauge boson stays massless. − and its inverse 9 where satsify compact and extend in the radial direction of the cone. The de X The cycle introduce the dual 4-cycles plus 2-homology of the homology cycles within anomalous and one starts from aτ D-brane on a U 3.2 Some notation As before, let complete basis of IIB fractional branes on A The hypermultiplet of a single D3-brane derives directly fr dictionary, we need to account for a corresponding number of spective non-dynamical coupling, whereas one would expect that all closednamics string on dynamics the strictly brane, se andcouplings. thus that This all decoupling closed can string indeed fields be fre arranged, provide D-brane theory on multiplets [17, 18]. The fourof real a variables in single (3.1) all hypermultipletdouble fit that tensor t multiplet appears [19]. afterhypermultiplet Since dualizing splits adding into two two a complex D-brane breaks JHEP01(2007)107 . , ) ) φ to T N ( ( s ˆ ˆ A A (3.8) F , with r s via the ) th F F s e even from F p through the Tr( ngs of the D- s → F .) From each of . ; it in particular ) p s F B − F , with e ∗ s 8 i p strength − C s of the CY singularity dC F 10 ctional branes e me of X ∗ p . In case the D7 brane ective, in which the local 5 s sonic fields: the dilaton C = r e contributions, as they do via its world volume action, . The goal of this section is ∗ s p i rant, we omit factors of order s ate. en couplings and closed string F anton number charge. branes all carry a non-zero D7 may take half-integer values. p dC n over a corresponding compact ssions below are extracted from P s ol. We will not attempt to solve q Z ern-Simons terms that would need to be ) (3.10) s n character by [21] Tr( -form potentials ) + F p ) (3.9) s s B , ey do not affect the main line of argument. ∧ F ∗ s F corrections are substantial, and this gauge s i 0 − F α − ) Tr( s a B Tr( F α – 9 – 1 2 + Z = X G = and the RR s Z ( a s F ∗ s J i p = s q det( p φ ∗ s are represented by fluxes of the field strength i − . s e F X Z is larger than one, we need to replace the abelian field streng are integers, whereas the D3 charge = s a r S s = 1 units. For a more precise treatment, we refer to [20]. , the 2-form K¨ahler p . These scalar fields parametrize the gauge invariant coupli s 0 B ` Y , we can think of them as D7-branes, wrapping the base s denotes the pull-back of the various fields to the world-volu r ∗ s i The D5 charges of The D-brane world-volume theory lives on a collection of fra The D-brane world volume action, since it depends on the field In general, there are curvature corrections to the DBI and Ch 5 times. We can thus identify the closed string couplings of s Analogously, the D3-brane charge is identified with the inst various 2-cycles within a non-abelian field-strength and take a trace where appropri wrapping number given by the sum of a Born-Infeld and Chern-Simons term via r properties as summarized in sectioncharge 2. Since the fractional encodes the information of the D7 brane wrapping number combination theory/geometric dictionary is onlythis partially under hard contr problem andcurvature will is assumed instead to adopt beto a small give large a compared volume basic to accounting persp thefields, of string rather the scale than geometric a dictionarythe full betwe quantitative leading treatment. order All DBInot expre action. affect the Moreover, main we conclusions.1 drop To and keep all the work curvatur formulas in transpa these fields, we can extractcycles a 4-d within scalar fields via integratio brane theory. Near a CY singularity, however, Here The D5 charges taken into account. They have the effect of replacing the Cher 3.3 Brane action The 10-d IIB low energy field theory contains the following bo We will ignore these geometric contributions here, since th the NS 2-form 0 to 8. (Note that the latter are an overcomplete set, since JHEP01(2007)107 by s F (3.12) (3.11) (3.15) (3.17) (3.14) X urface , and vice versa. ional brane, that s ) of the fractional s y. F ge is given by the F ( number that tells us Z into angular variables. s can be used to restrict 2 +Λ, with Λ any one-form. rge t under (3.12). B s s condition provides a useful A the D-brane charges via .10) the field strength of , harge vector for the fractional a variant notation. A convenient a → a arges the boundary state of the n ζ ) (3.16) b a into fluxes of s s a a n F b s A s e B s p r , r i 1 2 Λ, ) . The gauge transformations thus have 1 2 ) (3.13) s d Tr( s − − F ) Y + q ( − a s a s , 2 iJ ζ r a Q a B. p a + b a B b s · a a a a s B ζ p α a ( n s r n n remain unchanged. But the only restriction is , Π s ∗ s Z → p s i r s + − + r − 2 − = – 10 – 1 2 F + e a s a s a a )= B s s b q p b s X − . Evaluating the integral gives q p of F Z ) = ( ( s a s Y = = → → → a s s q p Z s F q s a s a p ( )= q b p )= s Q s F F ( ( Z Z -periods for us are those along the 2-cycles of the del Pezzo s and B s r = denotes the class K¨ahler on . The central charge is an exact quantum property of the fract an a priori arbitrary set of integers. These transformation s s some vector that depends on the geometry of the CY singularit to the interval between 0 and 1. r J a F Λ belongs to an integral cohomology class on Π n a d In the large volume regime, one can show that the central char Physical observables should be invariant under (3.12). Thi The charge vector is naturally combined into the central cha b : s The integral gauge transformations act on these periods and The relevant This integral gauge invariance naturally turns the periods where F that an integral version, that shifts the integral periods of If Λ is single valued, then the fluxes where can be defined atwhich linear the combination level of of right-brane and the preserves. left-moving worldsheet superch It CFT depends linearly as on the the complex charge vector: with following expression: [22 – 24] brane the check on calculations, whenever doneway in to a preserve non-manifestly the in invariance, isbranes, to introduce obtained a by new type replacing of in c the definitions (3.9) and (3 with is invariant under gauge transformations The new charges can take any real value, and are both invarian JHEP01(2007)107 ) s F ( Z (3.25) (3.24) (3.23) (3.22) (3.21) (3.18) (3.19) dent gauge alized) B-field, expanding the CS-term o fractional branes are e central charge n for the couplings of the 6 . ges are equal. Deviations of all respect the integral gauge aking, and such a deviation is 4 C 7) for the central charges C X rg field, which arises by dualizing , Z X 0 Z 2 C = s C ) (3.20) = q , ), we can extract the effective gauge s s s | X q F ) X + F c ( s ( Z F a J . + s Z ( θ a F a Z a | α s c ∧ φ Z p a s s − = – 11 – Im log p e C + a 1 . π ζ X = , θ + 4 θ 2 = s 2 C s π X r C g s a 4 . In the large volume limit, this relation directly follows c ζ a α s = s α s Z r F θ Z = = = a s a θ that couples linearly to the gauge field strength via C c s C -angle reads +3 different fractional branes, with a priori as many indepen . From the central charge θ s n F +4 independent continuous parameters: the dilaton, the (du n there are n With this preparation, let us write the geometric expressio The couplings of the gauge fields to the RR-fields follows from dP contain only 2 Note that all abovesymmetry formulas (3.12). for the closed string couplings 3.4 Some local and globalOn considerations From the CS-term we read off that from the BI-form of the D7 world volume action. The phase of th couplings and FI parameters. However, the expressions (3.1 coupling via In addition, each fractional brane maythe support a RR St¨uckelbe 2-form potential with fractional brane with which equals the brane tension of of the action. The gives rise to themutually FI supersymmetric parameter if the ofthe phases relative the phase of 4-d generically their gauge givestherefore central rise theory naturally to char interpreted [23]. D-term as SUSY an bre Tw FI-term. JHEP01(2007)107 (1)’s (3.26) U . In this Y 6= 0 would (1) factors. A U by extending later discussion 0 ear combination Y with d fields. (1) factors remain a U , do not intersect any . The corresponding c (1) factors. The two i 0 (1) gauge anomalies. α . Correspondingly, we U Y U ct cone (1) factors become infrared -angles and St¨uckelberg X . We conclude that there nges for the compactified θ U n e ce ycles on-compact limit, all closed alizable kinetic terms; these tion of associated D-term equations dP heory interpretation of these iated with anomalous responding branes wrap dual mixed non-abelian dynamics of a D-brane ion of the dual scalar field, the ompact CY threefold omalous FI variables and gauge Their presence implies that two . qn (1), the normalizable 2-form can be e IR, the (1) gauge couplings needs to be defined 6 X . U θ 0 beyond the energy scale set by the size of Y may extend to normalizable harmonic = 0 zero 0 , all non-anomalous Y 0 Y +1 2-cycles in n – 12 – -independent forms over (1) couplings develop a Landau pole. Via the holographic r U , ρ to X ρ X (1) gauge rotation of one generates an additive shift in = U , which would have a vanishing kinetic term. We are thus a 0 on associated with the two anomalous ρ supports two compact cycles for which the dual cycle is also θ i . The normalizable 4-form is its Hodge dual. 0 ˜ X ω ) for every of the η Y θ 8 a ∧ ζ quiver gauge theory always contains two anomalous r , dr a cut-off at some finite value Λ. This subtlety will not affect our n and 2 b r 0 (see formula 2.4). This correlates with the fact that the lin − dP θ ω 7 Y ˆ 4 1 r , with 0 Y = X , and all corresponding closed string modes are dynamical 4- ω Y -angle of the other. This naturally identifies the respectiv As we will show in the remainder of this section, the story cha We obtain non-normalizable harmonic forms on the non-compa The non-compact cone θ (1) charges associated to degree zero 2-cycles do no have any There is a subtlety with this argument, however. Whereas the All other D5-brane components, that wrap the degree zero 2-c Using the form of the metric of the CY singularity as given in e 8 7 6 U (1)’s are dual to each other: a forms on massless. This is instring accord dynamics with decouples. the expectation that in the n setting, for D-branes at a del Pezzo singularity inside of a c case, a subclass of all harmonic forms on the cone led to conclude that for the non-compact cone carry infinite kinetic energy.would-be variabel This St¨uckelberg obstructs the introduct 4-d RR-modes are non-dynamical fields: any space-time varia on a finite cone of the compactified setting, provided all Landau poles occur compactification. free. Towards the UV, on the other hand, the dictionary, this suggests that the D-brane theory with non- the other harmonic 2-forms U the closed string modes survive asare dynamical the 4-d two fields axions with norm The geometric origin ofintersecting these cycles. identifications is that the cor fields via on a del Pezzo singularity flows to a conformal fixed point in th compact, namely, the canonical class and the del Pezzo surfa other branes within expect to find a normalizable 2-form and 4-form on are not freely tunable,are but automatically dynamically satisfied. adjusted This socouplings. adjustment that relates the the an As emphasized for instance in [25], the FI-parameters assoc of must be two relationsrelations restricting is the that couplings. the The gauge t found to be and a pair of periods ( JHEP01(2007)107 X are . On σ (3.30) (3.29) (3.28) (3.27) ) p,q ( on Calabi- acts via H ) denotes the i,j O ( ± − H ates the constant inverts the sign of tion into harmonic p and Ω ) , where L ) is the involution acting i,j F . , ( + . ) ) p,q σ 1) ( that were initially present tion is determined by the − H b a ibed by IIB string theory Z Z δ − H include the type of D-branes ce b b Y , Y , t of [20]. = ⊕ ( ( cohomology spaces nes do not intersect the base ness, however, we will consider ) b , 2) 1) , , ¯ ∂ ˜ ω , decompose as: α p,q (2 (1 − − ( + σ ω ∧ . The orientifold map p H H ) H a O Ω x σ ˜ , ∈ ω ∈ ( L . p α F α b a a Y Y / a Ω are scalar fields. Similarly, we can expand ρ ˜ ˜ ω Z ω ω ω 1) L a ) = ) ˜ F + . − θ x x ( α 1) a Y ( a α a ω ω − ω = ( b ζ – 13 – ) ) ˜ and ) ˜ , , , since the operator ( x = = x , x O 2 ) ) α ( ( = ( 2 ( β J α ρ a α Z Z B a δ is world-sheet parity, and B c c θ O = p b b Y , Y , = = = on the various forms present. The fixed loci of β ( ( 4 2 and 6 ω ∗ 2) 1) C C , , ∗ C 6 σ ∧ (2 (1 σ + + C α will denote the Poincar´edual 2-forms to the symmetric and , H H ω 2 X ∈ ∈ C Y ω Z α α , respectively. ω ω X and ˜ and NS B-field as X ω J . Note that the orientifold projection in particular elimin are two-form fields and b Y a c , which we may identify as elements of the is left fermion number, Ω Y and L F α c . It acts via its pullback The RR sector fields give rise to 4-d fields via their decomposi The orientifold projection eliminates one half of the fields Y anti-symmetric lift of In what follows, We can choose the cohomology bases such that the form K¨ahler Here The relevant RR fields, invariant under eigenvalue under the action of all these fields. forms on the orientifold, we need to decompose this space as Yau manifold zero-mode components of orientifold planes. We willof assume the that del the Pezzo orientifold singularity. pla Our treatment here followson tha the full Calabi-Yaudimensions of space. the corresponding even Which and odd fields cohomology spa survive the projec on where 3.5 Bulk action The most general class ofat string singularities compactifications, discussed that here, may the is sub-class F-theory. ofcompactified F-theory For on concrete compactification an orientifold that CY can threefold be descr JHEP01(2007)107 ) nd Y ( 2 (3.35) (3.33) (3.34) (3.32) ), thus H Z Y, ( 1) , p from (1 ± . . . may have fewer H . The cohomology → Y Y 0-d fields expanded ) p Π between the 2- n-zero field strength . X ) and ( b rees of freedom. A first 3 lowing kinetic terms for n stabilizing the various Z . ˜ ω ] (3.31) H β gh scale. The St¨uckelberg c X, , the harmonic forms on the β e will assume that a similar ∧ ∗ ( d c a → a d all the above fields, that fixes 1) ˜ ω , ) αβ ˜ ω a (1 Y ∧∗ G α may have non-trivial cohomology Y ( α H Z Z 3 c = Y of the singularity, in general do not d H = = α X αβ a a → ab dρ G Π ) ∗ + . The overlap matrices b X Y/X c ( d 3 – 14 – . Conversely, , , G H ∧∗ a Y , β a c c → ω α d d ω ) ab still play an important role in deriving the low energy are related to the above 2-form fields via the duality a ∧ ∗ ab X G α ( 0 α G . For instance, the 2-cohomology of α 2 Z θ − [ ω ρ X = = H b Y Z = a α Z , in which case there must be one or more 2-cycles that are = Π → dθ naturally leads to an exact sequence X = and S ) X ρ , when restricted to base ∗ b X αβ Y Y θ ( G denote the natural metrics on the space of harmonic 2-forms a 2 but trivial within and H and ab X α ρ G Y → ) is referred to as the ‘relative’ k-cohomology class. The ma is the 4-cycle wrapped by the del Pezzo in the CY 3-fold, . . . and Y Y Y/X ( -form fields ⊂ αβ k p G X H As a geometric clarification, we note that the above linear ma The 4-d fields in (3.27) and (3.29) are all period integals of 1 The IIB supergravity action in string frame contains the fol where cohomologies of typically have both a non-zero kernel and cokernel. space effective field theory, however. 3.6 Coupling brane and bulk Let us now discussobservation, the that coupling will between the becompact brane CY important and manifold in bulk what deg follows, is that in harmonic forms.with Each some of quantized flux. thegeometric 10-d moduli These fields of fluxes the play maytype compactification. an also of important support In mechanism role will the a i their generate following no expectation w a values stabilizing and potentialand renders for axion them fields massive at some hi relations: span the full cohomology of the RR generators than that of 4-forms on non-trivial within when viewed as linear maps between cohomology spaces where The scalar RR fields elements that restrict to trivial elements on JHEP01(2007)107 (3.39) (3.38) s . As a A Y ype thus that is common nce in our case adds up to that s 1) , F (1 s mbinations of gauge k gically available local H mediate repercussions s hes. These correspond alently, integrating out P . β in the next section. l period integrals of certain ρ d (3.29), as for the vector bosons . = 0 α anes α ogy spaces reduces the number ct embedding typically reduces n is independent from the issue ζ c elative 2-cohomology. Similarly, decouples from the normalizable α a ubspace of α α ∧ ∗∇ a a s obstruction. Π α A a ρ ) (3.36) s s = Π = Π k p coker[Π] (3.37) ∇ ) is dual to the space of all 3-cycles in s s a a s k ⊕ c αβ Y/X Q P ) ( G s s 2 Y/X Y P k = ( mechanism for fixing the couplings, whereas ( 3 topological + H 3 that is trivial within the total space s A P H X H ∼ = – 15 – X ρ = , ζ , ∼ = . a a Q ) b θ X a a a ∧ ∗∇ , a ker[Π] ⊂ dynamical X Y/X ρ Γ =0 ( = Π = Π s 3 ∂ ∇ r a a H s 0, we deduce that b θ k XX (1) generators (1) vector boson ∼ = G 0, we have from (3.35): s U ) U P ∼ = X ( ) 3 X H ( 1 . The number of independent closed string couplings of each t H X and 0 and ) in the exact sequence is given by our projection matrix Π. Si , for which the linear RR-coupling (3.23) identically vanis Y s ∼ = X ( A ) 2 Let us compute the non-zero masses. Upon dualizing, or equiv As a special consequence, it may be possible to form linear co By using the period matrices (3.34), we can expand the topolo This incomplete overlap between the two cohomologies has im Y ( H plus all 3-chains Γ for which 1 result, the corresponding the 2-form potentials, we obtain the mass St¨uckelberg term RR-modes, and remains massless. This lesson will be applied to linear combinations of such that of a D5-brane wrapping a 2-cycle within coincides with the rank of the corresponding overlap matrix fields The charge vector of the linear combination of fractional br to H to both or, in words, the kernel ofusing our the projection matrix fact is that just the r Y In other words, the relative 3-cohomology By construction, the left hand-side are all elements of the s for the D-brane gauge theory, sincethe it implies space that the of compa gaugeharmonic forms, invariant and couplings. any reduction of Theof the associated allowed couplings deformations cohomol are of the al of gauge theory. moduli This stabilization, truncatio which is a the mismatch of cohomologies amounts to a couplings in terms of the global periods, defined in (3.27) an JHEP01(2007)107 i E (1) (1)’s U (3.41) (3.40) U dual to X ω It lifts all . This matches the Y 10 mains unbroken. to the massless . . X is (via the intersection pairing idering compactifications with 3-brane on a del Pezzo 8 = 0 ω es that correspond to frac- construction of a Standard und in [5]. b i 0 acquire a mass from vacuum X ifications. s E (1)’s must be non-anomalous. p ∧ ∗ · rning on FI-parameters. It is a U s X . plus eight exceptional curves p . This is the central result of this s ω 2 b β Y Y A P Z Π α is the norm of the 2-form a , a α in ,H s ume that the compactification manifold is of = Π = 0 Π ij A a δ s XX H α s blown up at 8 generic points. It supports r p αβ ζ − XX G 2 a α G s s P = P singularity Π P j a + 8 s − − 0 – 16 – E p s · X α s r dP i s k r dρ dρ s = = P , G α XX X 2 ρ ρ G (1) factors for which the above mass term (3.39) vanishes, C , E (1)’s must always be massive. ∇ ∇ 0 U = U α associated with the canonical class on 0 Z = 1 2 ss k = m = . In particular, it is therefore always non-trivial within H 0 9 X · : X α ρ X H ) dual to Y is the axion field of the D7-brane and (1) masses can be made much lower than the string scale by cons (1) bosons thus remain massless, as long as supersymmetry re U U X ρ As a final comment, we note that the charge vectors associated Besides mass via terms, St¨uckelberg vector bosons can also Recall that the 2-cycle The 9 10 with intersection numbers nine independent 2-cycles: the hyperplane class We now apply the lessonsModel-like of theory the of previous section [5],singularity. to using Let the us the string summarize world the volume set theory up4.1 of — more A a details standard D are model fo D3-brane A del Pezzo 8 surface can be represented as vector bosons from thetional low branes energy that spectrum, wrap except 2-cycles for that the are on trivial within the FI parameter cancels must label branes that wrap degree zero 2-cycles. Massless 4. SM-like gauge theory from a expectation values of chargedworth scalar noting that fields, for triggered the same by tu section. These and is of the order of the string scale for string size compact the del Pezzo 4-cycle The vector boson mass matrix reads with on the threefold Here string size. large extra dimensions. For our discussion, however, we ass field theoretic fact, that anomalous JHEP01(2007)107 , 2, 12 13 on . K the =1 − − (4.1) W N = 1 c . , which each are 3 i le in the quiver di- E n [5] (presumably F serving symmetry − 11 = 1 supersymmetry, 13] as a possible starting e generations of bi- 2 . branch. From a non- 9 E N e dictating expectation t has an excess of Higgs − (1) belian gauge symmetries, e the superpotential 1 U del Pezzo 8 singularity have . As first discussed in [3] E ve the model one step closer × ional branes auge group (1)-factors. We would like to − li space), this leads to an onfiguration assigns multiplicity U h are such that the charge vectors (3) . ination with suitable non-commutative 7 i U . = n i g duality to the world volume theory of a 8 × (1) E α U =1 (6) 8 i × U , while preserving P G (2) = + 0 U G H 7 – 17 – × 3 (1) gauge symmetries except hypercharge from the − . The 8 simple roots, all with self-intersection (3) of singularities will emanate, each having a generic ), the elements with zero intersection with U ,..., 8 = ) that indicate their (D7, D5, D3) wrapping numbers. U i Z i E F = 1 = K X, i ( G 2 H , consistent with local tadpole conditions. The constructi i , orbifolds), when a Calabi-Yau singularity is not isolated, F +1 2 i Z E contains Yukawa couplings for every closed oriented triang × − orbifold singularity — the model considered earlier in [12, i W 2 , one of the ADE singularities. Z 27 E i ∆ R = / 1). For the favorable basis of fractional branes described i 3 i , C α 0 , To effectuate the symmetry breaking to As shown in [5], this D3-brane quiver theory allows a SUSY pre Exceptional collections (=bases of fractional branes) on a This particular quiver theory is related via a single Seiber The superpotential See also [30 – 33]. to each fractional brane 11 12 13 i moduli space of D-branesisolated on singularity, several that curves Calabi-Yau Γ has more than one singularity type (in the context of The D-term and F-termvalues equations that can result then in both the be desired solved, symmetry whil breaking pattern low energy theory. The quiver diagramfundamental is fields. The drawn D-brane inand model the thus fig same has 1. quark thefields and same non-a lepton — Each content two as line pairs theapply Standard the represents per Model. new thre generation insights I obtained —to in reality, and by the eliminating several previous all section extra the to extra mo breaking process to a semi-realistic gauge theory with the g quiver gauge theory with the gauge group it is necessary turn on a suitable set of FI parameters and tun corresponding to a specific stability region in modu K¨ahler add up to (0 of [5] starts from a single D3-brane; the multiplicities been constructed in [28]. For an given collection, a D-brane c characterized by charge vectors ch( A del Pezzo 8 singularity thus accommodates 11 types of fract can be chosen as is isomorphic to the root lattice of The degree zero sub-lattice of D3-brane near a The canonical class is identified as point for a string realization of a SM-like gauge theory. agram, can bedeformations tuned [29] via of the the del complex Pezzo structure surface. moduli, in comb JHEP01(2007)107 (4.2) 8 . That i ,., = 5 singularity in i 2 tained in [5] is A 0) , i 1 ) , 0) E i , are frozen to zero, and from the original brane 4 4 E =1 P i i i space, the branes move e second entry gives the 2 E very high energy like the . But there are additional + α R lly following [5], is this: by − d become part of the bulk − fractional brane classes are singular points — such that K states which are independent H 2 ectors, one easily obtains the 2 − . As shown in [5], the above he branes have the minimum 2 and ith the required e FI parameters are identified α heory is free of adjoint matter i ,H , , A 1 F -fractional branes along Γ α i e complex structure) we can find a singularity on the (generalized) del and R 2 1 ) = (1 : on such a branch, the FI parameters ) = (1 ) = (1 i i α A 4 9 F F Our symmetry-breaking involves moving F ch( ch( 14 ch( – 18 – fractional branes have moved out along the curve i singularities. In particular, these branes can be taken ) R 1 2 2 , A 4 ) E 2 1 − − i , i E 2) E 8 − =5 P i , 4 =1 i P have been blown down to an i + E 2 − K α 8 =5 P 2 i H 3 , − , , and . The remaining bound state basis of the fractional branes ob 1 2 α α )= (3 ) = (3 ) = (3 2 3 1 F F F and Making this choice removes the branes supported on The strategy for producing the gauge theory of fig 1, essentia In this non-isolated case, on one of the branches of the modul ch( 1 ch( ch( We will give an explicit description of a del Pezzo 8 surface w and their positions give new parameters. α 14 i specified by the following set of charge vectors spectrum, and replaces other branes inof the spectrum by bound to be very fartheory: from any the effect primary whichrest singular they of point have the on of bulk the theory. interest, physics an will occur at free to move along the curve Γ of Pezzo surface where it meets theonto singular the locus. Γ branch in the moduli space, where the the classes Γ appropriately tuning the superpotential (i.e.,Calabi-Yau varying with th a non-isolated singularity — a curve Γ of is, on this new branch, some of the the next section. new parameters arise which correspond to positions of which would normally be used to blow up the ADE singularity freely on the Calabi-Yauwith or “blowup modes” its which (partial) specify how resolution,branches much of blowing and the up th moduli is done space associated with each Γ collection of fractionalnumber branes of is self-intersections rigid, andbesides in the the corresponding the gauge gauge sense multiplet. t that From t the collection of charge v Here the first and2-cycle third around entry indicate wrapped the by D7 the and D5-brane D3 component charge; of th JHEP01(2007)107 . K and (4.3) 1 α singularity 2 A modes for those represents three gener- ge group corresponds .2) have vanishing D5 o the charge of a single a single D3-brane placed he first two roots 1 C C C C C C C C C C C C C C A urface with an ds 8 and the canoncial class orm a complete basis. ding fractional brane, and has , .., y the geometric condition that isolates E = 3 i (1) gauge symmetry. ν U , the remaining 2-cohomology of the del 2 u 6 α 1000000 1000000 1000000 1000000 1000000 1000000 with H − − − − − − i 2 2 3 0 111111 2 2 2 2 3 0 111111 α d U(1) and – 19 – U(2) − − − − − − − − H 1 1 1 1 1 1 1 3 00 3222222 0 α − − − − − − U 0 B B B B B B B B B B B B B B @ D QL )= j F , i F -fractional branes. 2 #( A U(3) U(1) The MSSM-like quiver gauge theory obtained in [5]. Each line hypercharge as the only surviving massless , since we have converted the FI parameters which were blowup Y 8 E (1) After eliminating the two 2-cycles Note that, as expected, all fractional branes in the basis (4 U of 2 is 9 dimensional, and that the fractional branes (4.2) thus f Note that the total cohomology of the generalized del Pezzo s Pezzo singularity is spanned by the roots cycles into positions for α Figure 1: which gives the quiverto diagram the drawn (absolute fig value 1.been of chosen the) The such multiplicity rank that ofD3-brane. of the weighted each In sum correspon gau other of words, chargeat the the vectors gauge del adds theory Pezzo up 8 of t singularity. fig 1 arises from wrapping numbers around the two 2-cycles corresponding to t ations of bi-fundamentals. Inthe the text below we will identif matrix of intersection products via the fomula (2.4). One fin JHEP01(2007)107 6 d 15 ed (1) (1) (1) U (4.6) (4.7) (4.4) (4.5) , and U U 1 geometry (1) and U 8 u dP (1) U , resp. The the 3 (1) ¨cebr mechanism, t¨uckelberg U (1) factors in the quiver he right by or the 2-cycle associated . Of the remaining eight U on . ´ tot X j b ´ i u mentarily. and nly the hypercharge survives y. We denote the nine Q i Q the form Q 2 F , where each r eturn to discuss possible string 8 n of fractional branes. Instead, we ro degree is free of anomalies. 3 d − , , selects a corresponding linear 3 =1 7 P i i b (1) , 16 Y ect compact cycles within the del urface P (1) represents a trivial homology class U Q − Q =6 i U mology class of the linear combination of i d Y = the direct Seiberg dual of the the calculation + L, Q Q i ij b s − -terms, the supersymmetric mass terms F and 3 =1 9 µ P i Q , B 3 u 5 , ³ s = P 2 1 1 X (1) =4 i Q − U − = (3) nodes by 2 1 3 – 20 – 1 , Q U . -orbifold. j u F Q tot − Y 27 1 6 Q 3 Q − (1)’s. First we discuss how to make them all massive. (1) 3 Q − − . The total charge U F U 3 1 1 } i u , . In other words, any linear combination of charges such ³ Q Q i d (2) and 1 2 X 1 2 U (1) charges that defines (1) factors in the quiver of fig 1, and identify the linear = ,Q (1) anomalies. As discussed in section 3, these are associat = U = U i u (1) charges are U Y ij u,d Y U F ,Q Q Q 3 ,Q 2 ,Q 1 Q { (1) generators add up to (1)-factors of the U U Let us take a short look at the physical relevance of the extra The linear sum (4.4) of Hypercharge is identified with the non-anomalous combinati The field theory verification of this claim is easily made, via With this equation we do not suggest any bound state formatio 15 16 sum of fractional branes, which we may choose as follows of fig 2. If unbroken, they forbid in particular all within the full CY threefold. We will compute this 2-cycle mo for the extra Higgs scalars.mechanisms In for the breaking concluding the section extra 6, we r such that only the particular 2-cycle associated with generators, two have mixed with the canonical class within to linear combinations of thePezzo fractional singularity, branes i.e. that that wrap inters either the full del Pezzo s that the corresponding fractional brane has zero rank and ze node at the bottom divides into two nodes generators by acts on the matter fields of the corresponding generation onl the overall decouples: none of the bi-fundamental fields is charged unde We would like to make certainas that, a among all low these energy charges,we gauge o now symmetry. know From that our this study can of be the achieved stringy if S we find a CY embedding of combination that defines hypercharge. We denote the node on t outlined in [12] for the quiver theory of the ∆ simply use itbranes, as whose an intermediate step in determining the coho 4.2 Identification of hypercharge Let us turn to discuss the The other non-anomalous together with four independent abelian flavor symmetries of JHEP01(2007)107 , 9) , 6 olds , (4.8) 1 , denote 1 , J (1 P is the one singularity. W 8 Y the weighted are non-trivial where Ω Q 4 dP α J is fixed via the GVW

) Ω singularity embedded Y J singularity which forms 8 2 τm singularity with all the dP A enerator en added to the insights 4 + 8 o 8 singularity, such that e J t contain a invariant both Weyl reflections, low for the formation of an n dP − J 5 moduli of the compact geom- SM quiver theory of fig 1, the ctors =( attractive geometrical conclu- ill present a general geometric e representing a trivial cycle and y coordinate regions where the e 18 hypersurfaces W 4 = lex structure of . Examples of this type are the α 4 8 invariant under the diffeomorphisms that , develop an α 2 dP α (1)’s, but also keeps a maximal number the form (with U and 1 17 α ´ 1 2 Requiring non-triviality of all other 2-cycles ex- singularity. , – 21 – factors except hypercharge acquire a St¨uckelberg 4 2 α A − (1) then has an extremum for Ω , U α α α α α 0 W . 8 ³ 4 . α 2 α α )= 2 3 4 5 6 7 Y α , one natural route is to look among realizations of CY threef F and 3) has the required 1 , 2 ch( 8 α , 1 , dP 1 (1 singularities on the CY, and all remaining 2-cycles except P α 2 A W surface can be constructed as a hypersurface of degree six in 8 Our proposed D3-brane realization of the MSSM involves a dP not only helps with eliminating the extra represents a trivial homology class. 4 4 we can arrange that all extra α α We consider this an encouragingly simple result. Namely, wh This can be done without any fine-tuning, as follows. The comp singularity within the del Pezzo 8 geometry 17 2 CY threefolds obtained by resolving singularities of degre projective space Since a as hypersurfaces in weightedCY projective equation space, degenerates and into identif that of a cone over desired topological properties. 5. Constructing the Calabi-Yau threefold It is not difficult to find examples of compact CY threefolds tha A simple calculation gives that, at the level of the charge ve within the full CY. part of a curve of all other cycles beingprescription nontrivial). for In constructing the a next compact section, CY we embedding w of the of gauge invariant couplings inetry. play In as particular, dynamically to tunable accommodatecomplex structure the moduli construction of of the the A compact CY threefold must al cept obtained in the previoussion: section, we arrive at the following mass, provided we can findonly compact CY manifolds with a del Pezz which is the locus where represented by the simple root inside a CY manifold, such that two of its 2-cycles, the periods of theact 3-form Ω. like Weyl Now reflections choose in the integer fluxes to be We read off that the 2-cycle associated with the hypercharge g Figure 2: superpotential, which for given integer 3-form fluxes takes JHEP01(2007)107 1 σ. P p . So iated since Ω of the Y slightly L except F 8 . The 1) 1 dP P − ), are divi- X d space and consider the = ( , one of which σ ∞ 2) curve. The Weyl O ◦ − X ) has an automorphism , but as a generator of 4 8 b Y α space of complex di- ( ompact CY manifold) dP w and -linear combinations of sidered in [35], where it ed cover of a = as the ( Z 0 only two classes, K¨ahler ample of a CY orientifold btain a suitable compact ρ 4 . ngularity, and the effect it X pace of the fibration is the α Y . On a nonsingular variety, , denoted Div( } hat all 2-cycles of has only two generators, most is eliminated as a generator of he projection X ) on the homology lattice. One way 4 4 Y α ed assumptions about the specific = 0 α e so that the ( ), are the w f -fibration over any del Pezzo surface. { ) 2 X ( T X 1 − singularity with d the del Pezzo surface. The covering space as a 1 Z Div( still lifts to the cover space A / Y . X X ) Y – 22 – X , but acts non-trivially on the cohomology and the string on ( (1) , which exchanges these two sheets, and the IIB Y 1 4 U σ − α Cartier divisors on . On the other hand, this example illustrates the d ) with Y Z , denoted X ). Therefore, the FI-parameter and field St¨uckelberg assoc develop an b Y X 18 K ( X 1 , ⊕ 1 1; the + 7-planes are at the same locus as before, but the monodromy is H X O − model, and thus combines well with our general bottom-up O d ( 8 P The surface must in fact be trivial within dP ρ. singularity, based only on the geometry of the neighborhood . This set-up looks somewhat more promising for our purpose, 8 p are manifestly preserved as 2-cycles within the orientifol 8 Y , i.e. which acts as the Weyl reflection Ω dP 4 ) instead of L . We may then define a new holomorphic involution ), so the quotient group X dP α F b Y Y ( X 1) Weil divisors on 1 , − 1 − → − H 4 =( α 0 , the is a (local or global) algebraic variety or complex analytic O d has a holomorphic involution is represented in hyperelliptic form, that is, as a two sheet ) = Div( -fibration over the del Pezzo has two special sections, X 1 2 ) while all other 2-cycles are kept, would yield a concrete ex X Y are projected out, leading to a massless P ( T If Rather than following this route (of trying to find a specific c A potentially more useful class of examples was recently con Y 1 4 lift to non-trivial cycles within A concrete proposal is as follows. Tune the complex structur ( of − α 2 18 d 4 ˆ odd homology to different. The harmonic 2-form associated to Z can be contracted toorientifold a geometry del Pezzo singularity [35]. The total s The philosophy. 5.1 Local Picard group ofTo a begin, CY we singularity discusshas the on local things Picard such group as of deformations. a Calabi-Yau si mension H with the desired global topology. we will insteadembedding give of a the general local prescription for how to o singularity. This local perspective doesUV not completion rely on of detail the all 2-cycles within Y orientifold orientifold on this CY surface is obtained by implementing t fibration takes the form reflection then acts trivially on the Calabi-Yau subvarieties of dimension theory spectrum on to get such an automorphism is to let a suitable modification of the construction, so that only which maps The sors which are locally defined by a single equation α 2-cycles within the basic phenomenon of interest: since the 2-cohomology of was shown how to construct a CY orientifold considered in [34].and therefore This can class not of satisfy CY our manifolds, topological requirement however, t has JHEP01(2007)107 , 1 ∼ = X X ) is ) CP ∈ ), or S S × P . K 1 X, P X contracts ): it may such that − ) with the CP e in Pic( in e X X 0 X, P P ⊂ ] are the same X, P 1 D on 1 of 0 CP CP U D × in a Calabi-Yau space in a rational double point, }× nt depends sensitively 1 of the familiar conifold S S cal version of the group ow do we determine this h would yield CP dimesion 3 were studied in n which case Pic( ing divisor athematics literature. o consider only the “small” unction defined on all of point d singularity in the form orhoods = , { ) S urface lassification theory of algebraic U ), are the divisors which can be ; however, we could also choose . 1 ) 1); on our case, the normal bundle is X ( , and the image of − X ] and [ 0 , Div( CP ( } 1 X 0 / ) is finitely generated. In our context, ] are distinct; when they are the same, ) − 1 ( is. U S ( Div O point 1 , CP X / X, P = 0 (5.1) − ) , the classes will be distinct; on the other X is a neighborhood of a singular point d × { X S }× zt was Z ∈ 1 X ). The rank of this image is always at least one: − e – 23 – (with the generator corresponding to X P ← In this case, we can identify Pic( S , denoted Div CP point Z xy in . There are two possibilities for Pic( X { 19 . S 1). The “small” blowups exist exactly when the two ) = lim Pic( X ∼ = ) = Div( on X − ) ∈ X , → 1 ) → X, P ] and [ P is closely related to one of the key examples from Mori’s − } e . ( X Pic( X, P e 1 Z S X : for a point Pic( : is the quotient K CP ∼ = × point − π 1 ) X CP ×{ is simply a cone over 1 O X, P X CP to have a curve of rational double point singularities, meet is called “generalized”, following the terminology of the m e X S local Picard group 20 principal Cartier divisors ), in which case Pic( Picard group of 2). is the divisor of a meromorphic function on e − X , S ( is sufficiently small, this is trivial, and one introduces a lo 2 2 The calculation of the local Picard group near a singular poi To take a simple, yet important example, suppose that Local Picard groups of Calabi-Yau singularities in complex The We allow In Mori’s case, the normal bundle of + − to a point via a map H ( . Note that if X 0 19 20 2 S e we are in Mori’s situation where small blowups do not exist. H from the equation? If we can write the equation of the conifol with normal bundle homology classes [ to simply blow up the singular point in the standard way, whic Z original pathbreaking paper [37] which startedthreefolds. the modern c on the equation ofsingularity. the It point. is common Mori’sblowups in example of the was such study in a of fact singularity Calabi-Yau a (which spaces form replace t it by a hand, the case Pic( it may happen that those two homology classes are distinct, i in happen that the two homology classes [ image of the natural map Pic( which is obtained by contractingX a (generalized) del Pezzo s it follows from the adjunction formula that there is always a D to a Calabi-Yau singular point the anticanonical divisor and the If called the where the limit is taken over smaller and smaller open neighb detail by Kawamata [36], who showed that Pic( we are mainly interested in the case where O written as the difference of zeros and poles of a meromorphic f is one measurement of how singular the variety which is why JHEP01(2007)107 , i 2 4 ’s × P 45 1 D D 1 D D CP to the = 0 or and could be z 45 := 1 0 P at the four 1 through D S = S 1 transversally s x 45 S 3 ` ’s ruling which of flops. First, 2 C to the del Pezzo , (2) blowing up 23 8 lift to exceptional D 8 E C n the equation, the 45 , ngs are homologically ion can be embedded P tart with 7 divisors C meeting , ..., , the two homology classes 4 6 3 e glued together to form a as before. en the two homology classes E 22 C D ted ones, including examples 3 , 8. We also attach a rational cts the factorizability of (5.1) P , (3) blowing down two out of , 45 ch a 3-chain Γ. , 1 7 alytic change of coordinates, but that . The transformed surfaces , and consider local divisors 2 C , ed in figure 3. P 5 8 6 P d in [39]. S P , , singularities, as was crucial for the analysis 1 1 to curves P bi-Yau. A singularities. We label the fiber of 8 , and = 45 8 P 2), is similar. 2 7 i P − A this is Mori’s case. The case of a del Pezzo P , , ’s which pass through 7 2 , 1 , and we label the fiber of P 21 6 singularity. Note that the line − 2 , ..., , P ( 2 4 CP – 24 – 6 C 1 , is sufficiently large, the difference between the two P be a generalized del Pezzo surface obtained from A P , and that the same del Pezzo surface CP 1 0 , which has the effect of blowing up 45 infinitely near to by 45 8 × S P 1 3 E 1 C CP ] are the same, there is a 3-chain Γ whose boundary is P : in that case, one would blow up a point CP 1 D . This is all illustrated in figure 3. × 1 2 1 O ∩ C CP CP 1 and , and CP 7 C 2 × C . We also consider a local divisor }× P 3 1 yielding a del Pezzo surface , in the flopped curves, as indicated in figure 4. 6 C 8 5 CP C P S point , by { 45 2 C and D , and then blowing up 7 ∩ 5 which transversally meets the first of a pair of ruled surface P 2 E ] and [ , 1 } C 6 P now meet transversally at another point, for P and two points i 8 and , meeting transversally at observing that the two original , at 1 1 C 4 D point 45 P S 1 45 lifts an an exceptional curve , = 0, respectively. However, if there are higher order terms i E P C 7 E 5 × { t by (1) blowing up 5 distinct points embedded in a Calabi-Yau neighborhood such that the two ruli P D Each of the curves and surfaces we have used in thisWe construct now pass from this structure to the one we want by a sequence We give an embedding into a Calabi-Yau in the following way. S If the neighborhood of the 1 More details about the topology of this situation can be foun In that case, the small contractions would yield a curve of The factored form can always be restored by a local complex an 1 2 , = 6 23 22 21 CP D points we flop the curves in a Calabi-Yau neighborhood,Calabi-Yau and neighborhood those of neighborhoods the can entire b structure illustrat pass through away from its intersection with meeting ruling which passes through surface curve CP equivalent in the Calabi-Yau. We attach rational curves which together can be contracted to a curve of curves curve obtained starting from and a point CP the last three exceptional divisors to an 5.2 Construction of the CYFrom threefold this simple example, we canof easily the obtain type more complica we are interested in. Let of [38]. the difference between theare two. distinct, so Such an awill analytic 3-chain have change cannot the of exist topological coordinates if effect which of affe creating or destroying su cases can be detected by the[ topology of the neighborhood. Wh change of coordinates may fail to extend over the entire Cala then the two small blowups are obtained by blowing up the Weil x contraction, with normal bundle nicely factored form (5.1) may be destroyed: JHEP01(2007)107 , 45 g , 45 f has been 1 P 45 in the flopped g 6 S 45 g desired topology. The . The curves 45 8 45 45 C C D C 45 meets P 45 1 D and 6 D 45 (“infinitely near” to the first f 6 C 7 6 P 6 45 2 f D 6 , C D P C on which the point 6 6 , C S 7 45 C C 7 7 P 7 C 7 D in a point D . 6 1 S S 8 C . – 25 – 8 3 3 5 8 C D S D P 8 8 D D meets 3 3 2 C C C , yielding a del Pezzo 1 2 C 2 1 D 1)-curves on D 1 g 2 − g 2 C C 1 1 C C 1 P 1 P 1 are all ( 1 D 8 D 1 C f are fibers in the two rulings on 1 f i g The CY threefold after flopping the curves Starting point of our construction of a CY threefold with the , and 7 and C i , f 6 C Next, we flop the curve , 45 curve, and the transformed curve blown up, as indicated in figure 5. The transformed surface C Figure 4: Figure 3: curves JHEP01(2007)107 , 8 3 is 3 to S C 2 C 8 C = . We S + 2 0 2 C S 1)-curves C − = = 45 2 2 g 45 α e g ), and when , 2 3 , as indicated in P 2)-curve, and C and 7 − S 1 + 45 45 C 2 D C 45 45 C D C = are additional ( el Pezzo surface + 1 singularities, yielding the 1 1 α C 2 6 C to yield 45 A 6 D f 6 2 45 C = 6 D f P , and 1 C 1 has become a ( e g 1 , 1 C 8 so that f 7 (“infinitely near” to , 7 C 7 3 7 , D P 7 C D to a curve of in the most recently flopped curve. , as indicated in figure 7. (The transformed = 6 2 7 8 j S S 8 D C . The curve is blown up at – 26 – 8 2 8 for . The curves 3 1 6 C C D D j in a point and 8 S C 7 C meets 1 D S to yield 3 2 3 D = C P 3 D j 3 P e 3 1 C D 2 g meets C 1 3 2 g 2 D C , and C 2 2 P 45 D g 1 C is now flopped, = in the most recently flopped curve.) To match the curves on 1 2 5 C 8 e C is blown up at 1 S 1 , . D 7 D 7 45 S 1 S f f 1 = meets f The CY threefold after flopping 4 The CY threefold after flopping 3 e ). When D , 1 3 P C . To achieve our final desired singular point, we contract the d The transformed curve 1)-curve on 6 S = − 3 configuration illustrated in figure 8. can now contract the transforms of e the standard basis for cohomology of a del Pezzo, we set is now flopped, point figure 6. The transformed surface surface a ( on Figure 6: Figure 5: JHEP01(2007)107 1 → S ) X remains 1 C 45 45 g g 2)-curve, generate a subgroup 45 45 − 45 45 D C 8 D C k 6: the anticanonical D struction, the surface , singularity, embedded into a 7 6 2 D 45 6 enerator, the map Pic( 6 D f 45 A , nt. D f 6 C 6 e it allows the fractional branes C D , has become a ( ogous. 2 would have had the same property, 45 C 1 7 D 7 S 7 C singularities which emanates from our , C 7 D 3 2 D D A 8 C 8 – 27 – . The curves 8 C 3 D 8 . C 8 D S to move off of the singular point we are interested in, 2 3 C α 3 1 D 2 g C and 1 1 1)-curve on g 2 α D − 1 C 3 C 3 is a ( D and the transformed divisors 3 1 1 S C D f 1 K f ≡− The final configuration: a del Pezzo 8 surface with an The CY threefold after flopping 2 0 D A ) would be surjective and the original surface Second, the local Picard group of this singular point has ran First, it is not isolated: there is a curve of 2)-curve, and S − so that its local Picard group would have had rank 2. But by con Pic( divisor into the bulk of the Calabi-Yau manifold. of the local Picard group of rank six; if there were a seventh g singular point. This is onewhere of were the features supported we on needed, becaus Calabi-Yau such that two of its exceptional curves are homol to a point. Let us analyze the properties of this singular poi a ( Figure 8: Figure 7: JHEP01(2007)107 , 4 n, R their are in , s iθ + 2 s /g 2 π 8 − gs, depend on the e , which subsequently 4 ing preserves the exis- α ich the linear combina- truction. The outlined he presence of a 3-chain s that wrap compact cy- is no longer a conserved n constructing a compact Standard Model, however [5], aimed at constructing eserve the same supersym- ory, such as the spectrum 4 neling process, however, is gauge theory is given in fig visors. We can identify this ility can be made exponen- s of D-brane tension) of the α ns. n 3, we have worked out the a Calabi-Yau singularity. We y chosen Calabi-Yau threefold e understood at large volume. pping ngularity and does not rely on tion. ranes: they are localized in 2). This is not entirely obvious, f our bottom-up perspective, since (1) symmetries. Here we make some basic U – 28 – . Apart from the exponential factor s F (1) factors besides hypercharge are massive. (1) gauge symmetries get lifted. As a direct application of , which therefore does not exist as a homology class in the U U 4 (1)’s are approximate global symmetries, which, if unbroke -terms. Fortunately, the geometry supports a plethora of D- E µ U − of the CY singularity. The ‘basic’ D-instantons of this type 5 E X = 4 α breaking via D-instantons (1) vector bosons and the number of freely tunable of couplin U (1) U The simplest type of D-instantons are the euclidean D-brane Thus, via the above geometric procedure, we have succeeded i An important physical assumption is that the compact embedd contribution is independent of moduli K¨ahler and can thus b but otherwise have the samemetries Calabi-Yau as boundary the state fractional and branes pr self-annihilates by unwrapping alongsuppressed the because 3-chain it Γ. istially non-supersymmetric small The and by tun the ensuring3-chain that probab Γ the is 3-volume large (measured enough. in unit 6. Conclusion and outlook In this paper, werealistic further gauge developed theories the on programhave the advocated seen world-volume in of that D-branes severalof at aspects light of the world-volume gauge the compact Calabi-Yau embedding ofstringy the mechanism singularity. by In which sectio would in particular forbid this result, we have shown how to construct a supersymmetric with some extra Higgs fields, onwith a single a D3-brane del on a Pezzo suitabl 2, 8 where singularity. in The addition final all result extra for the6.1 quiver At low energies, the extra comments on the generic form of the D-instanton contributio cles within the base instantons, that generically will break the 1-1 correspondence with the space-time filling fractional b quantum number: onetion could (4.7) imagine of a fractional branes tunneling combines process, into a in single wh D5 wra full Calabi-Yau. CY threefold withstrategy the furthermore properties preserves the we main characteristicsit need o only for refers to our theunnecessary D-brane local assumptions cons Calabi-Yau about neighborhood the of full the string si compactifica tence of all constituentsince, fractional branes in listed particular, in the eqn D5 (4. charge around the trivial cycle had a local PicardΓ, group with of boundary rank 1. equaldifference to We the with thus difference have demonstrated of t two exceptional di JHEP01(2007)107 3 26 µ , the (6.2) (6.1) f stanton otential N with ter gauge − The result 4 c SM, like ours, C N α 24 Σ singularity still gets lifted from R i 8 α ρ − dP ) ton effects, it seems in general contains an α s that probe the full CY. In θ ue of the fluctuation de- α come fully realistic. Most Vol(Σ ding node. In our gauge the- (1) symmetries. rm that, from the low energy 3 ed on the µ U transforms non-trivially under ively depends on e in [41, 40, 42]. a angle form α α ranes wrapping the dual 4-cycles = ly with the volume of the 4-cycles iρ iρ s e S ual cycles Σ iθ )+ contributions (6.2) are not governed by the α + y get generated via D-instantons. contribution to the superpotential, that helps nifold. 2 s n case one would consider more than one single roup, and thus may contain terms that at first -terms are small compared to the string µ /g 2 Vol(Σ π (1) symmetries is analogous. The relevant 3 terms, in particular, can be viewed as baryon-type 8 (1) gauge rotations. Instanton contributions µ U µ − − U is the field, St¨uckelberg we observe that the e e – 29 – α -terms can be generated via this mechanism. ρ µ . The (Φ) N = A = Ω(Φ) 4 = C Since the phase factor α (1) symmetry. )). It is important to note, however, that the form of the D-in δW Σ δW U R 25 N -terms can not arise in oriented quiver realizations of the S µ will get fixed by minimizing the potential, and , in which case the one-instanton contribution to the superp α c (1) rotation, the pre-factor must be oppositely charged. Af ρ U > N f . The classical D-instanton action reads N Y (Φ) denotes the perturbative pre-factor, the string analog A While we have not yet done the full analysis of these D-instan The story for the non-anomalous global within In [47] it was argued that For more recent discussions, seeNote [44 that, – 46]. unlike all classical couplings, the D-instanton , it would naturally explain why (some of) the 26 24 25 α α has several issues that need to be addressed, before it can be reasonable to assume that the desired the low energy spectrum. Whatperspective, remains breaks is the a superpotential global te Since the D-instanton contribution decreasesΣ exponential scale. 6.2 Eliminating extra Higgses From a phenomenological perspective, the specific model bas sight would not be allowed for general rank contributions sensitively depends on the rank of the gauge g the corresponding operators for SU(2), and thus one can easily imagine that the fixing, the value of Here terminant, of the D-instanton. D3-instanton contribution to the superpotential takes the the D3-brane tension. Since to the effective action thus generally violate the anomalous D-instanton contributions are generated by euclidean D3-b Σ axion field, that is shifted by the anomalous is of the schematic form where Ω(Φ) is a chiral multi-fermion operator [43]. The thet number of colors minus the numberory of we flavors for have the correspon brane (so that SU(2) becomes SU(2 fact, eqn (6.2) is astabilize direct all generalization geometric of moduli the of famous the KKLT compact Calabi-Yau ma because they seem forbidden by chirality at the SU(2) node, i agrees with the expected field theory answer [43], and sensit local geometry of the singularity, but depend on the size of d In this limit, the analysis has essentially already been don JHEP01(2007)107 example. 8 aints, such as the dP ur 02-17383 of the Russian when part of this work irement (on all D-brane orld-volume theories in an stein, Michael Douglas, gs fields. An example of onal Science Foundation find an orientifolded CY search Fellowship (M.B.). ndations expressed in this U(1) via various mechanisms: via asses of all extra Higgses and e see no a priori obstruction to the fact that supersymmetry is ear orientifold planes, D-branes eproduce this quiver. The extra de should have an equal number atural to look for generalizations vy, and Michael Schulz for helpful cs, etc. The structure of the SUSY ν ). With this generalization, one can E N ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¡ ¡ ¡ ¡ ¡ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¡ ¡ ¡ ¡ ¡ u ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¡ ¡ ¡ ¡ ¡ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¡ ¡ ¡ ¡ ¡ H ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¡ ¡ ¡ ¡ ¡ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¡ ¡ ¡ ¡ ¡ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¡ ¡ ¡ ¡ ¡ d ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¡ ¡ ¡ ¡ ¡ ) or Sp( ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¡ ¡ ¡ ¡ ¡ H SO(2) Sp(2) – 30 – ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¡ ¡ ¡ ¡ ¡

N D QLU U(3) -terms are strongly restricted by phenomenological constr µ An MSSM-like quiver gauge theory, satisfying all rules for w The presence of the extra Higgs fields is dictated via the requ (1) factors in fig 7 can then be dealt with in a similar way as in o Acknowledgments It is aShamit pleasure Kachru, John Elias to McGree Koll´ar, Kiritsis, J´anos thankdiscussion Vijay and Balasubramanian, comments. David Thisunder grants work Beren PHY-0243680 was and supported DMS-0606578,Foundation by by of grant Basic the RFBR Research Nati (D.M.) 06- and byD.M. an NSF would Graduate like Re towas thank done. the Any IHES opinions, for findings, hospitality and and conclusions support or recomme Figure 9: immediately noticeable is the multitude of Higgs fields, and unoriented string models. unbroken. Supersymmetry breaking effects mayfluxes, get nearby generated anti-branes, non-perturbative string physi breaking and suppression of flavor changing neutralthe currents. existence However, of w mechanisms thateffectively would eliminate sufficiently them lift from the m the low energy spectrum. constructions on orientable CY singularities)of that in- each and no out-going lines.among To gauge eliminate theories this on feature, orientifolds itcan of is CY support n singularities. real N gauge groups like SO(2 such a quiver issingularity drawn and in fractional fig brane configuration 7.U that would It r should be straightforward to draw a more minimal quiver extension of the SM, with fewer Hig JHEP01(2007)107 ]; . 7 , . 02 3 Phys. Lett. 6-branes JHEP , Chiral , ]; (2000) 002 Phys. Rev. Lett. (2004) 069 (2002) 029 , hep-th/0011073 hep-th/0508089 08 , 04 09 hep-th/9603167 , D = 4 chiral string ]. . JHEP ]. JHEP JHEP Adv. Theor. Math. 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