PoS(cargese)001 http://pos.sissa.it/ ∗ [email protected] Speaker. A pedagogical introduction to string theory andquantum string strings phenomenology are is introduced given. using The a classical and pedagogicalwith supersymmetric and field pragmatic theory. approach The for emphasis is a on readerlogical deriving familiar and properties, understanding the chiefly phenomeno- thetheories. spectrum and An overview interactions of of somegiven. the recent phenomenological 5 applications perturbative and supersymmetric developments is also ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

Cargèse Summer School: Cosmology and ParticleJune Physics 30-August Beyond 11 the 2007 Standard Models Cargese, France Steven Abel Cargese lectures: a string phenomenology primer Institute for Particle Physics Phenomenology, Durham E-mail: PoS(cargese)001 (I 8 E × agogical 8 ways that E Steven Abel , yielding as ich has been m turns into a ls one of which rotic nomenology. The subject, but not in (UV) divergences. e moment you just ls and there is little gravity. Thus even s much higher than the expectation was bt that string theory et more MSSM-like 2, 3]. The concepts I s have evolved based ts in phenomenology. st important property , and especially since at cannot be addressed There is no doubt that I’ll get to later – ensure ncepts, in enough depth pen string theory) which actually means, or at least what it  b A 2 phenomenology f f  a Vacuum polarization in field theory. A Figure 1: The UV divergence corresponds to the “loop becoming small”. At energie Because of this I think it important to lay out what I believe to be the two important In this brief set of lectures I give an overview of String theory and its phe To start, I would like to discuss what string string phenomenology influences our thinkingof in string less theory specific is ways. thatthough The it the fir remains final theory our may only orhas candidate may shown for us not how a resemble theories theory string of theory, quantum ofin there gravity field quantum deal is theory. with The no various most dou problems obvious th exampleConsider is the of vacuum course polarization the diagram taming and of its Ultra-Violet includes string equivalent the (in annulus, an as o shown in figures (1) and (2) respectively. will summarize the properties of the 5need supersymmetric to theories know shortly but that for this th it is does one GUT of models them) as seemedthe well second as to string MSSM-like give revolution, models. tantilising a support large Inon number to recent so of years this called new however idea D-branes model in building type technique indication II models. in These string can theory also as yieldmodels MSSM-like then to mode seems which a is less the productive enterprise correct than route. it once The did. search for y means to me, because Iin believe the the early emphasis days has of changed stringthat theory in string (during recent theory the years. would first yield stringwould a revolution if closely unique you or resemble will) at the most Minimaleloquently small Supersymmetric outlined number Standard of to possible Model you mode (MSSM) by wh Hitoshi Murayama in these lectures. Indeed Hete lectures will be aimed at therather newcomer than familiar comprehensive. with field My theory goalthat and is the my to interested aim introduce reader is the will toso be be basic much ped able physical depth to co that progress I to simplydevelop end the will up many be repeating sufficient textbooks these to on already then the excellent go texts on [1, to sketch the more recent developmen the string scale the loop istiny much cylinder. smaller However, than the the conformal typical properties string of length string and theory the – diagra which 1. Introduction: what is string phenomenology anyway? String Phenomenology PoS(cargese)001 he familiar Steven Abel , is that the cm 33 orresponds to long − th to annulus-radius theory. Many ideas le extra dimensional lds are confined to a ways positive.) The ing theory or at least parently large Planck is in its interplay with 10 of the field theory, such posal by Arkani-Hamed × agation of a closed string ies. f the theory. The general ivergences are much easier 61 . 1 m (in this case closed strings) = itself. (This is possible because, 3 Pl c / M Gh is the scale at which we used to think q ) = Pl GeV L 19 3 → 10      × 2 − 22 . s 1 1 is vastly different from − = N kg Pl 3 G 1 Vacuum polarization in an open string theory M m − Js 11 34 ≡ ms − − 8 10 10 10 × × × Figure 2: 673 055 . 997 . . 6 1 2 = = = G h c What can go wrong with this estimate? The crucial point, emphasized in Ref.[4] The second important area where string theory has influenced our ideas Consider how we used to estimate the fundamental scale of quantum-gravity. T quantum-gravity effects would first make themselves felt. alone among the forces, the effect on gravity of adding extra masses is al that the overall size of theis cylinder physically is meaningful. meaningless: The diagram only can the thenformed ratio be by of seen the as string-wid combination a tree of level two prop range open ones. Infra-Red UV (IR) in propagation the in “opento string the understand channel” “closed c (and string cancel) channel”. sincelesson they IR here depend d is that on in the aas global theory UV properties of divergences, o quantum to gravity, one be expectswhich cured pathologies pop-up by in the new theory physical at degrees Planckian of (or more freedo precisely1.1 stringy) energ On large extra dimensions The resulting Planck length ( estimate is a dimensional one, based on measured constants of nature energy scale at which we measure String Phenomenology field theory. A case in point iset again extra al dimensional in field theory. ref.[4] The that pro scale, large was extra supported dimensions by can thesubspace fact be of the an a such explanation higher a for dimensionalthat construction, the model, are in ap have now which a common, matter natural such fie been realization as inspired in large by string extra it. dimensions,field Conversely have theory ideas arisen have couched often from guided purely str subsequent in string terms theory developments. of for examp PoS(cargese)001 (1.1) some s, but te that we arameters Steven Abel is therefore as planation as to . To summarize, fundamental scale follows. The size same extra dimen- ompactified vacua, ture of string theory nstant can be much s), with the Standard They could feel a volume. Indeed the have extra dimensions tly live is shown as the the extra volume would be much lower than the rmines all of the physics the relationships become as a plane and extra small dimensions 4 M D G 4 vacuum degeneracy problem 1 − − D Gravity 4 V SM 6 ∼ K 4 Gravity G









































































































































































10 then we would require the extra dimensions to be of order (in fundamental lengths) gives the required enhancement factor = 32 D 10 ∼ 4 − D V . 1 − is the volume of whatever extra dimensions our theory happens to have. (No 4 TeV Brane world picture with 4 large flat dimensions represented − 1TeV then 5 D to the Planck mass. If V ∼ 10 extra dimensions are a reasonable thing to consider only because of a fea 16 s This type of set up is a natural possibility in string theory with its 6 extra dimension The generic picture for significantly changing the scale of quantum gravity On the other hand gauge forces cannot consistently be allowed to feel the × M that we used to regard as a problem, namely the the problem is that stringor theory even gives the no hints number aswhy of to there compactified the are dimensions. shape 4 or large So size flat for of dimensions. the example c we More have specifically no this ex can be stated as large shown in Figure 3. The largeflat flat plane. 4 Blowing dimensional up space any in portion(supersymmetry, which of particle we it content apparen reveals and an so internalPlanck on). space scale that The if dete fundamental gravity scale feelsModel can a (SM) fields large being internal confined volume to (denoted some by restricted subvolume. green blob of 10 determining the different scales of nature. will for simplicity only consider flat extraof dimensions.) If for example we have a where Figure 3: String Phenomenology implicit assumption is thatentering in into between the physics. the In two particularthat for scales are this there discussion, much are in larger theories noweaker which than abnormally than the large expected fundamental because p scale, thenaive the relation gravitational is measured force Newton’s is co diluted by the extr large volumes however, in which case there isa some little rescaling required more and complicated but similar.) sional volumes. This is because gaugejust couplings lead are to dimensionless either so nonperturbatively that large or immeasurably small couplings. ( few PoS(cargese)001 . 4 M different rst need a Steven Abel ferent corners heterotic), so- ry point in radical idea of a uum degeneracy , and type IIA/B, s we like, with the r compactification d their potential is us parameters (for non-trivial angles). imensional physics . different kinds of can do perturbation transformations it is r of the Planck mass tive underlying theory ther. The conjecture is or example we perturb canonical layout of 10 s. In the meantime one theory of open strings. tal string theories. The ity. The triumph of the ; these are the massless m that are left over from ) 4 M 32 perturbative contribution to ( then all the neighbouring manifolds are 4 M 5 Five of the labelled points represent the various perturbative regimes (i.e Later in this review I will discuss how phenomenology has taken us to all the dif We now turn to how this idea has been realized in stringy set-ups. For this we fi string theory) that can beall of written which down are in theories 10 of dimensions. closed strings, These and are type Heterotic I which is an SO of this road map.models, starting The with the itinerary most isin conservative case weakly determined of coupled by a heterotic the stringcalled models scale value intermediate down of scale of the to models the orde (type GUT I stringTeV string string and scale scale scale II (in models) in (strongly non-supersymmetric and the coupled models finallyBefore with discussing doing D-branes the so intersecting however, at I willproperties review of the these construction models of are summarized the in 5 the fundamen following table “road-map" of string theory in orderdimensional string to theory orient plus ourselves; supergravity we shown begin in with Figure the 4. In addition the diagram includes2nd a string sixth revolution point was representing to 11Dpossible demonstrate to supergrav that get by from applying any of successivetherefore the duality that 6 the perturbative perturbative points theories on are this simplywhich diagram limits encompasses to of the any some whole o nonperturba of this diagram,can for which consider the the search continue phenomenological possibilitiestheory. for the 6 theories where we 2. On energy scales and model building particles.) In addition we are athope liberty that to our set preferred the choice compactification will to atthe be some moduli as stage potential. large be a So explained by when a itproblem non- comes is to seen lowering as the a fundamental virtue. scale, the vac perturbed and so on, and a signal radiates out at the speed of light in However these parameters correspond to thethe VEVs of higher fields dimensional in metric. the spectru completely These flat fields to turn all orders out in tothe perturbation be theory. compactification massless, (In manifold terms and at of indee a Figure 3, particular if point f in String Phenomenology and shape of aexample particular the compactification various manifold radii), can knownradius collectively be as corresponds specified moduli. by to Choosing vario fixing athey these particula should parameters. of Since course they be determine the the same 4 (i.e. d Figure 3 should look the same) at eve PoS(cargese)001 Steven Abel e classical level d classical. For ical properties. For terotic closed string , on developing the derive the spectrum of n that my approach will 8 allowed 9 allowed , , 6 7 9 allowed , , , ng theory, plus 11D supergravity -branes? 4 5 5 , , , 2 3 1 , , Dp 0 1 = p = = p p ming) 26 dimensional bosonic theory, whose gion is highly nonperturbative and mysterious 6 Closed Closed Closed Closed Open/Closed Open and Closed 8 ) E 32 ( × 8 SO E I IIB IIA Type Heterotic Heterotic Le pays mystérieux. The 5 well-behaved incarnations of stri Many of the concepts that will be important can be understood intuitively at th and indeed many of theexample most the number important of model generations is building givenmodels, issues by or are the number the geometrical of number an fixed of points intersectionsthis in in He reason intersecting I brane shall models, spend bothclassical class longer behaviour than of usual, strings, and inthe then this 5 classical move and string on the theories. to following Ibe consider sections should pragmatic how state and before one brief. beginning can this expositio 3. The classical point particle and geodesic motion Figure 4: String Phenomenology (to outsiders). There isconnections also with a the badly rest behaved of (but the theory very are welco tenuous. are well connected by easy-to-follow routes. The central re PoS(cargese)001 τ = ≡ ] ˙ allel 1 X (3.2) (3.6) (3.3) (3.4) (3.1) . − ˙ so our with s X [ m τ under the fortunately Steven Abel ˙ ct, X d and we need . ˙ 1 X ] i − falling X [ wed the same time ity. By now I hope bles. All observers p le 1 refer to space only. e shorthand: R ≡ . The particle follows − ] is a rather inconvenient − µ 0 D s X X [ ... 1 ≡ t definition is that it is motion = ] i s [ (a set of arbitrary “notches” on the 1 τ − 2 D 1 then τ ν 0 (3.5) ) = dX ˙ X d dX ... = . h ... µ . , ˙ τ X given by ) 0 − = − X dX 2 1 ˙ c ( = X d . ds p ds +++ µν 7 ˙ µν dX X ds g g Z Z Z − − − − Z 1 for the spacetime degrees of freedom. The metric m m 2 0 p δ = ( = − − − dX Z 2 D = = µν world-line parameter δ = ds g S 2 ... 0 1 refer to spacetime whereas latin ones ds − = D ν , ... length of the world-line 0 µ = µ the explicit definitions can vary (you may have seen it defined in terms of par of GR is that particles follow this motion. Unfortunately . ν τ d dX postulate . The only other invariant parameter available to us is the rest mass of the obje µ ] τ d What should we take for the action? Hamilton’s principle suggests We should sort out what this means in terms of invariant physical observa Note that greek indices I will using the labels Let us begin by going back to the classical point particle in general relativ dX 1 µν mass guess for the action takes the form will agree for example that awhen clock it falling passed from B. point This A is to the point B will have sho and in GR we have influence of a background gravitational metric g Geodesic motion is motion that minimizes the proper time, world-line) and use 3.1 The action for geodesic motion geodesic motion; you all have a good idea of what the ideas are. We wish to describe a partic [ and the transport of the velocity 4-vector for example) butthat for minimizes us the the convenien length of the world line. where in special relativity this does not have the right units: in units where playing the role of time as the action, since it is already something that is minimized. Un instead, which is obviously equivalent by eq.3.3. In the above I am using th String Phenomenology parameter, and instead we can define a to make the action dimensionless by multiplying with something that has dimensions of itself is generally a function of the spacetime coordinates which I’ll call PoS(cargese)001 (as (3.8) (3.7) (3.9) (3.13) (3.11) (3.12) gauge Steven Abel fields in a one ) the action now or what we have D will be the fields in µ X That is I can redefine a tion fields this is what I is just a parameter, whereas τ physical gauge ) 2 i ) ˙ ˜ X τ τ , d d 2 0 ) ( ˙ i . X ν ˙ ˜ X τ everywhere in the action. In the − look a bit like a set of ( ( d vdt ˜ τ dX . 2 µ v ∑ 0 (3.10) ) manifestly reparameterization invariant µ i ˜ X τ → ˙ − − µ X = ) d ( ˙ 2 τ 1 dX L τ X ) µ ( ∑ ∂ 0 ∂ √ ˜ τ L µν X ˙ 8 X − m 1 g ∂ ∂ ) ( = = 2 B i t ) A v = ˜ µ t − τ q 0 , P Z , in whatever coordinate system I feel like using. ˙ 1 µ ν t m X ˙ τ P ( − − − d ≡ ( dX p = γ 0 . Note that the = µ S τ m m τ X d L dX = = ) has been removed; the Lagrangian is a function of the “fields” 0 µν µ g P X (i.e. we can indeed just replace s ′ τ X d ) ˜ τ τ d d is the time parameter. ) field theory. This is more than a pedantic observation; when we come later to t τ = ( ˜ , (which for some not very obvious reason is called the τ t d , and = 1 − τ D This is easily achieved because the Lagrangian is It is worth deriving the equations of motion because it will be good practice f Consider special relativity: here we have .. 1 = i it should be since the physicsnew should parameterization of not the depend world-line on by the parameterization). Reparameterization invariance identify string theory as a CFT it will be “field” theory in this sense, and the takes the more comforting form X 3.2 Equations of motion and since dimensional ( we are used to using the actual time, and dots imply differentiation w.r.t. question. I want you to keepmean). this correspondence in mind (whenever I men and the action should look the same: by the chain rule where Note that one of the This gives us where String Phenomenology This is remarkably simple but not quite yet in the usual form since to do later on with strings: the Euler-Lagrange equations are PoS(cargese)001 (3.17) (3.18) (3.16) (3.14) Steven Abel ifferentiation ic differentia- . We can apply 0 ; then invariance ˙ X µ ν / ε hould be familiar to erved currents arise. ) we can work out the + µ µ ν X δ = → µ ν u Λ ) µ 0 (3.19) , ν ρ ε = σν g µν µρ − η γ ε 0 (3.15) ρ / ν ρ , ´ symmetry. The Lorentz symmetry is gener- X m = X ν µ µσ µ σ X L . + g ˙ X X x µ ν 0 v 2 2 ρ . − + Λ ˙ v i x = 9 v γ γ σ , = i µ µ ρσ → P m m ˙ ) = µν P Γ µ ν g = = = ( + X X -coordinate, and replacing µ µ x ¨ 2 ρµ x X H g and we have used the chain rule. The EL equations give us ( ) δ = v . µν v νσ ρ η ; to get the latter we just have to multiply by 1 coordinates of the form − Γ 0 . This has a full Poincar 1 µ X ( µν X p η / 1 ). Consider an infinitessimal transformation = invariant and hence the Lagrangian is invariant even if the metric remains the γ ˙ X . ˙ X and , i 0 X isometry X X . ∂ ∂ X = i not the physical time Derive the general geodesic equation of motion v τ Exercises A: Consider a flat metric As an exercise now read appendices A and B in order to recap how cons 1. reparameterization invariance ated by rotations on just the Following the notes there, (dropping the 3.3 Symmetries and conserved currents Hamiltonian associated with the reparameterization invariance; which is the usual expression for the relativistic energy. Lorentz invariance that leaves of the dot product requires same (it’s an is the Christoffel symbol. Gettingtion. this result is good practice for manipulating metr τ String Phenomenology where where the above to get the geodesicyou. equations Recall of that motion they in are a general setting, which s Note that this isw.r.t. not quite the statement of constant momentum because we have d PoS(cargese)001 = y B (3.26) (3.21) (3.22) (3.23) (3.25) (3.24) (3.20) , 12 B Steven Abel ≡ z B charged particle lude gauge fields are conserved.) itten; i j M m the general formalism, nta τ B A d t t τ ¤ ¶ d µ µ ¢ B A X t t coordinates, and together with the ν X ¤ µν P µ δ µ ε P is constant. The action is invariant as µ − ν X £ ˙ ) µ L X X ν µ k µ ε ) µ ∂ X ∂ ! ε j B X P j µ i v i j ν µ µ ¡ v P = i j E P ε B £ τ τ F τ d d i jk j d d − + d ε . The Lorentz force law is then µν + 2 E ν 0 µ i ε jk ¶ . The boundary term that should vanish is Z Z + X − E i X µ ν 10 B µ ( E ≡ = = X X P q à ( i jk → µ δ ν q ε = ε = = dt µ µ i = ˙ = L X X ¶ ≡ µν i i µν dt ∂ u ∂ dP u B F M dt δ dP δ µ t τ u L d d ∂ ∂ and 0; that is rotations generated by antisymmetric matrices generate , µ Z µ = as the conserved current associated with shift symmetry. d X dt . We wish to write this in a covariant formalism, so we need µ ρµ P B ≡ ε Z is the generalization of the 4D magnetic field which has × u i + is antisymmetric under exchange of any two indices (the Levi-Cevita sym- , v A τ j µρ ≡ i jk ∂ ε or more succinctly ε ≡ k t − B 23 j j v B A i 1 and i jk = ∂ ε x = = B i j 123 B ε and When we reach D-branes much later, it will also be useful to know how to inc This is an additional symmetry of constant shifts of the 13 B in this formalism. In classical electrodynmics thein rate an of em change field of momentum is of given a by the Lorentz force law which in 4 dimensions can be wr the equivalence is the Lorentz symmetry group. Now consult the appendices; converting fro well if the boundary term vanishes; 3.4 Including gauge fields are a set of conserved currents. (Strictly speaking only the angular mome Lorentz rotations the whole forms the Poincaré group. The shift is The Lagrangian is trivially invariant under this shift since identifying the momentum where where Space-shift invariance bol), so that String Phenomenology which is satisfied if so that − PoS(cargese)001 equa- (3.28) (3.27) (3.29) X rmalisms tion previ- Steven Abel introduce a to take advan- 0). For strings, = to? µ j m other gauge. It can be = les ( µν for the particle case is that alent if we use the F ν ∂ . 2 0 (3.30) τ d = em µ 2 ˙ ν X − dl ˙ m µ X . 2 2 A A e ˙ e µν X Z Z . The above is clearly the space parts of + τ qF 11 ) q q 2 d i ˙ v X = − − , Z 1 µ = 1 2 ˙ . The action becomes = = P ) P τ e = ( = S ( em , δ µ e P δ S e ˙ S X indicates the line-integral along the particle’s path. You may have seen this ac Check that the action in eq.3.28 yields the force law eq.3.27 What does the “zero” component of the force law equation correspond Check that Maxwell’s equations correspond to the equation dl Exercises B: String theory has an intimate connection with conformal field theory. In order 3. 2. 1. tage of this it isnew helpful non-propagating to worldsheet field use the so-called einbein formalism. In this formalism we 3.5 The einbein formalism ously in the Aharonov-Bohm effect. where Since the derivatives are the same onshown (see both exercise sides 3 it below) is that now this trivial to equation get results to from any the action Applying the equations of motion for we recover the previous action. The (classical) physics is entirely equiv String Phenomenology Now take the physical gauge, where the equivalent formalism proves vital when it comes to quantization. tions of motion with theare einbein useful equation in of different situations. motion The imposed advantagewe as of a avoid the einbein rather constraint. formalism tricky The square two roots, fo and that we can treat massless partic PoS(cargese)001 . 2 ) b . a ( (4.3) (4.4) (4.2) (4.1) − 2 b 2 a Steven Abel ks and rods . The strings p ] π ew definitions: = 2 , 0) to the other (at θ the world line. In t is invariant. The nder gravity. First 0 lly going to follow [ nature by defining e postulated as sin ∈ 2 ) eodesics that minimize ab σ µ Y (with two endpoints); the 1 2 ∂ ) 1 Y 1 open δ χ the two dimensional area that is ∂ . µ Y Y 0 0 ∂ ∂ ( 0 which I’ll call − between them is τ ) δ χ θ ( ) First consider the world-sheet coordinates. Y i 1 − ) ∂ X world-sheet, dA . , 2 σ | µ µ 0 Y , 1 Z µ Y Y τ ∂ i iX 0 1 Y T 12 1 ∂ ∂ )( 0 1 to counter the dimensions of area. In order to render ∂ − Y 1 with angle = ( = ( 2 0 = ∂ δ χ δ χ µ α . proper area. δ χ S a b | Y χ Y 2 0 mass | ∂ µ ( is a parallelogram with sides given by vectors is something that measures the distance around the string so Y ˆ 0 q A σ ∂ 1 d that is mapped from a small rectangle of parameters with lengths 0 ˆ A δ χ δ χ 0 d | coordinate, the position in space time of the string q δ χ = = has dimensions world-sheet coordinates . That this constant really plays the roll of the tension of the string will be : T ) (i.e. a closed loop with with no endpoints) or dA σ , τ target space = ( closed . The actual element 1 1 , 0 string tension : The = is the proper world-sheet area (as determined by a collection of falling cloc δ χ µ α : A σ T X shown explicitly later and We now want to adapt the previous discussion to the string. We are essentia • • • 0 ) or if the string is a closed loop takes us all the way around it. I’ll now make a f parameter, if you imagine a static string, takes us all the way from one end (at is a time-like parameter whereas π We can insert the above into this expression The area of a parallelogram with two sides 2 it is space-like. In order to get a better picture it helps to go to a Euclidean sig can either be δ χ τ σ String Phenomenology 4. The classical string and Nambu-Goto action Now consider an area element this action in a usableorder form, to we do need this to we parameterize introduce the another world-sheet parameter as as well we as did swept out in space-time by the string as it moves along. The action is therefor the steps of the pointI’ll particle describe generalizing the to a generalization. one Wethe dimensional have world-line. object seen falling that A u string point is particlesgeneralization an follow of the g extended world-line object for that the must particle also is have the an action tha where attached to the string) Next we need to find an expression for the PoS(cargese)001 (4.5) (4.8) (4.9) (4.6) (4.7) (4.11) (4.10) (4.13) (4.12) sheet. It is Steven Abel reparameter- out explicitly Consider a line the square-root. ) . Now we can easily return ′ σ µ X d . Y ′ τ d X induced metric. ) ′ )( ˙ d X X . b . ˜ ′ σ ˙ µ σ X d σ X . ( c X d d ! ν a ˜ )( σ ′ σ τ ′ − ˙ ∂ σ X d 2 X . X dX X . d ) . hd b ′ d ˙ ′ b = µ X dimensions of ) ˙ ˜ X ∂ ( X σ − X . µ X . ′ D ′ ∂σ ∂ b dX ˙ − √ X ˙ in the area in going to the Minkowski signature X X X ∂ c a a X 2 ( . . i . ˜ µν ; ) 13 ∂ Z ˙ σ ′ ˙ X X g X µ q ∂ ∂σ a T X = . X − ∂ à then invariance of the line element means that τ − ab ( ˙ X ab ∂ h = ) ( = − h = 2 − σ given by δτδσ = q , ab = ds µ τ = h NG 2 = ds ( Z ˙ S 2 X ˆ ˜ A σ ds T , ds ) id − σ − , = τ = ( of length ˜ NG τ S dA are indices for the world sheet. This defines what is called the induced metric . This action is quite easily seen to be reparameterization invariant; indeed if I } 1 ab , h 0 det ∈ { b = on the world sheet , h a Before getting the equations of motion we should check that this action also has where I have had to absorb an overall factor of and where for short hand I have defined element ization invariance. In order to do this we first need to define the and now we see that the action is actually remarkably simple; which is the metric requiredalso for known line as elements the that pullbackwe are of have constrained the to space-time lie metric in onto the the world world-sheet. Written We can rewrite this in world sheet elements as where to the Minkowski variables; The nett effect ofFinally having the action a is Minkowski given by signature the integration is over just all the the elements; minus sign inside 4.1 Symmetries and conserved currents Reparameterization (Diffeomorphism or Diff) invariance where redefine the coordinates to String Phenomenology Where the inner product is the Euclidean one over PoS(cargese)001 int e the (4.20) (4.14) (4.17) (4.18) (4.15) (4.16) (4.21) (4.19) Steven Abel for the classical point . ing. 0 π world-sheet = = σ σ ] µ X σ δ d σ µ τ P ) d [ τ ν τ σ ¶ P d d c a 2 µ τ ˜ σ X ¶ Z b d µ . d ∂σ ∂ c a ˜ ε ) σ − T ˜ ¶ ˜ σ σ µ µ τ ∂σ ∂ . ν c a + d − ∂ ∂σ P X ˜ c a 0 ˜ ν τ σ µ L µ ˜ α det µ σ ∂σ ∂ τ X µ ∂ = ∂ X X ∂ ∂σ ˜ ˜ µ ( ν h h d P ( µ 14 a δ µ det ∂ Λ ) σ σ ab − − P a µ a h h d d det = P ∂ p p → a Z Z = = α = µ ∂ µ Z Z ( P ˜ ˜ h X = = σ τ cd T T ˜ d d h µ − − ˜ τ σ µν p d d = = M Z NG T S = NG S δ plays the role of momentum density on the string. The conserved currents ar τ µ P In a Minkowski background the theory has Poincaré invariance just as We now impose the equations of motion and I will be a little more careful than in the po The Euler-Lagrange equations are given by the local contribution on the and the action can be written one finds the following conserved currents; where the canonical momentum is defined as where in the last line I used the Jacobian for the integration Note that String Phenomenology so we have particle. Under infinitessimal transformations of the form particle case, because of the boundary terms; Poincaré invariance corresponding point particle quantities, integrated over the length of the str 4.2 Equations of motion PoS(cargese)001 . 0 R π (4.24) (4.23) (4.22) of some erence of the choice Steven Abel anes. Consider is orthogonal to easily prove the plays the role of σ ν T P 0, or Dirichlet ones, per end NN. ns with period 2 µν = η σ µ behaves pretty much like P 0. The radius of the string , so the = LT t 1 if the string is spinning in the 12 , = X ) ). The constant of proportionality 0 12 2 p ) , M 0 ,... ) p 0 0 ( , R . / 0 t 0 , ( 1 = ( cos 2 t 0 π 0 = = R 2 R σ σ ] RT has an energy X − 15 µ π 1 L 2 X )= t p δ ( σ µ R released from rest at time = P [ 0 µ R p ) of the string at its endpoints. µ X τ ∂ Open string. The lower end has ND boundary conditions, the up along each dimension. Note that there is of course nothing particular about Figure 5: Spinning open strings have an angular momentum (i.e. Strings with only Neumann ends move at the speed of light. A static open string stretched to length The following set-ups tell us a little more about the physics of strings and D-br a circular open string of radius has a time dependence given by plane) that is proportional to the mass-squared (i.e. and the corresponding world sheet is shown below. It executes oscillatio Note that this period is alsothe the string, time so taken for it a is light consistent signal with to causality. go The round momentum the is circumf found to be is known as the Regge-slope. tension. const Using the equations of motion and boundary conditions above, it is fun to think • • • • = µ examples of classical physics to convincea oneself that piece the of classical elastic string (withfollowing zero results: mass when the string is stationary). One can pretty In addition we must impose a boundary condition at the enpoints of the string X This can be satisfied by imposing either Neumman boundary condition, 4.3 Some examples of classical physics the "velocity" (i.e. String Phenomenology of coordinate system. The statement is simply that the momentum vector PoS(cargese)001 (4.25) (4.26) gles, the tion which Steven Abel 12 plane is and writing ) e as one would common to say. σ over the string is , τ ( . the intersection, and tion of the einbein for ab ¶ γ ,... ) ϑ . Now however the momentum , ) cos 0 ab R h − / t 1 ab ( ( t γγ π factor. The worldsheet is shown below: R cos 2 − , 0 γ √ R ϑ σ d sin )= t τ t ( π d R 16 2 R , Z , but this time given Dirichlet boundary conditions on π 2 T 0 ϑ 2 R − µ as shown. Using the formulae above it is not hard to show 2 t ϑ ]= R γ RT , − π 1 X [ 2 P p S = µ p is indeed playing the role of tension. As the string accelerates to the centre, T . Note that initially, the zeroth component is simply the stretching energy dt we of course recover the previous closed loop result. For general an dR π ≡ 2 t , so that R = 1, and its energy diverges with a relativistic T 0 ϑ R → t π Consider now the situation shown in figure.(6). A circular arc of string in the For momentum in the 12directions plane is is now no transferred longer tonot zero the free D-branes and to as is leave the the time string intersection;The dependent. oscillates. the classical string stretching More Momentum "lives energy at provides in a the these potential intersection"when keeping as the the it strings string is at are quantized weintuitively expect, will in find addition a to sector whatever of states intersection exist states in the her "bulk". that the motion is as for the closed string, where is found to be 2 R again released from rest with radius two branes intersecting at angle The square root in the Nambu-Goto action makes it difficult to work with. An ac • does not contain the square root maythe be particle. obtained by For analogy the with string the introduc we introduce an independent world-sheet metric 4.4 The Polyakov action String Phenomenology PoS(cargese)001 (4.33) (4.30) (4.31) (4.32) (4.28) (4.34) (4.29) ory as Steven Abel , the Euler- ab . 0 δγ = ab classical physics is he 12 plane. 0 π γγ time. The diff invari- = = − σ σ ¯ ¯ string. We may choose µ = X δ δγ ¢ . µ µ X X σ , b ∂ ) ∂ ¡ 0 (4.27) µ σ τ , = X d τ a ( equation of motion as a constraint. , ∂ cd , Z h ab γ , ab ab γ γ T . 0 cd ) η ab − ) γ σ − γ , ab τη σ = τ , √ ab µ η ( d τ θ γ µ ( ω X σ 17 φ 2 1 = to be replaced with just one; 2 = X d e δ e a h ¢ − ab ∂ ab Z = γ a γ ab µ − → ab h ∂ 2 X T ) ab h √ a γ σ ∂ − , ≡ a τ ∂ ( ab ¡ ]= T ab X τ [ γ d PC fields to obey the one-dimensional wave equation, σ S d X Z . The Weyl invariance allows this to be further reduced to reads ) T 1 ab + γ , ]= 1 X [ − Arc of open string on D-branes. Momentum is not conserved in t . This is known as the Polyakov action. Using ( PC ab yields S γ µ δ diag X det = Figure 6: = ab γ η The Polyakov formalism allows us to make an important link with conformal field the The first term constrains the follows. Note that (4.28) is unchanged by a Weyl transformation, where In this gauge, the action reads Varying the allowing the Nambu-Goto action to beidentical recovered from if the we Polyakov. use Again the the Polyakov action and impose the whereas vanishing of the second term sets the boundary conditions on the which may be recast as and so Weyl-equivalent metrics correspond toance the allows the same three embedding degrees in of space freedom in String Phenomenology Lagrange equation for where PoS(cargese)001 being (4.38) (4.35) (4.36) (4.39) (4.41) (4.37) (4.40) (4.42) 0. The µ n = ˜ α 2 Steven Abel , X µ n α . ) ft- and right-movers σ − + n σ σ ( in in 2 2 . . − − cos e e n 0 τ µ µ µ n n − . in ˜ ˜ α α α − ¢ 1 1 n n )= , general solutions to the wave e + = 0 0 Z µ σ n . for convenience). Consider branes ∗ 6= 6= σ α 0 ) n n πθ 0 ¡ . ± i µ n 1 n e ∑ ∑ µ µ + τ = πθ ˜ ( ′ ′ 0 α p . X 2 2 . ( α α . π 6= )= ′ ∑ = µ n 2 Im 2 = ) Z + ′ α p ( q q σ ′ 0 end of the open string along ± π ¢ i i | iX α , α − r σ 2 X )= Im τ = + + + 2 ( σ σ Z √ 1 = − + i ¡ σ ∂ µ √ and X µ πθ µ )= σ σ 0 − i X + ˜ = . = µ µ 18 Z e α X = τ n ( ( p p 0 µ 0 ′ ′ µ µ − = Z = )= α p Re Re α α α n σ 0 ′ )= µ ( ( 0 | µ , − α σ σ + + σ = α [5]. τ X α , implies 2 ∂ ∂ ( µ µ ∗ σ τ ∗ x x ; has been replaced by ¢ µ ∂ ( = + ) 2 2 1 1 πθ µ n X µ µ 0 ; ∗ µ π ϑ ) α x X X = = ¡ µ = n = ¢ ¢ α σ ( σ = ( )= − + µ σ σ σ , ¡ ¡ X τ µ µ − + implies ( X X µ ∗ ) X being the centre of mass and momentum of the string and µ µ X p = ( µ and X µ x Neumann boundary conditions, Periodic boundary conditions, Branes intersecting at an angle right- and left-moving Fourier coefficients, where we have defined This choice leads to closed strings. Defining The solution may be written in terms of a Fourier series, with This time, In addition without loss of generality, lie the As discussed above (note that intersecting in the 12 plane, which we can complexify; equation may be written as a superposition of right- and left-moving fields, The reality condition Again, This choice of boundary conditions describes open strings, where the le combine to give a standing wave: boundary conditions are • • • String Phenomenology PoS(cargese)001 . a = χ ª b (4.45) (4.44) (4.49) (4.46) (4.47) (4.48) (4.43) ρ , ngauged a , ρ Steven Abel 1 when the ¶ © b quoted in the χ = a proaches avail- θ ρ b ρ a − . Again, the string is ich introduces space- try on the worldsheet σ 2 χ . ) ing to µ α θ ´ + ψ µ n µ ( i Ψ and e a Ψ ξ 1 ∂ , θ 1 2 θ µ a α + X † n ρ ! + + a i 0 µ α n µ ∂ ´ a i Ψ 0 X a i + b ρ χ i à ∂ requires the addition of a gravitino + µ + µ σ − ) = ) µ Ψ θ Ψ σ 1 X = a , − b − µ ρ n τ µ ρ ∂ ( µ ( ε i b Ψ µ a Ψ e a X , ξ ρ X ρ a δ ∂ a θ b θ i a a ∂ χ − χ ∂ → n ³ ρ − a = 19 2 massless Majorana spinors on the worldsheet: ab µ α ξ n ξ . a + γ ) a ξρ D Ψ µ Z i ³ i , ρ σ ξ δ χ ∈ i 2 ∑ γ Ψ n a n + ! ( ξ ∂ − − ′ i − µ , 2 0 α − X √ = = = = µ sin b i τ a b a µ µ 0 Ψ r ∂ σ e i in ¯ µ ξ X d Ψ Ã δ − . Promoting δ χ τ δ X e δ ξ = a d = . This action is invariant under the global supersymmetry transfor- . Also present is a superconformal symmetry, )= ∂ 0 µ 0 Z σ cd ρ ab , ρ X η T 2 γ T τ d b δ ( ³ e Ψ − Z c a γ e ≡ = − = S Ψ √ are now a suitable independent combination of ab σ γ † d α τ , d α Z 2 T satisfies are two-dimensional gamma-matrices satisfying the usual Clifford algebra a b − e a localized at the intersection point,pedagogical and literature there is is open no strings zero on mode. parallel The branes, case correspond often The mode expansion satisfying these boundary conditions is exponentials combine into where = , ρ We now consider classical supersymmetric string theory. There are two ap L ab S η able; the Ramond-Neveu-Schwarz formalism, which introducesdirectly supersymme and then extends to spacetime,time and supersymmetry the explicitly. Green-Schwarz We formalism work wh in thePolyakov RNS action, formalism. which Our we starting supplement point with is the u and we have defined The 4.5 The RNS superstring mation Then, 2 for an arbitrary Majorana spinor which is invariant under the local supersymmetry transformation where String Phenomenology PoS(cargese)001 L he S The is the (4.52) (4.57) (4.54) (4.55) (4.50) (4.51) (4.53) (4.56) covari- a J . Steven Abel 0 0 : µ r ariation of = = ˜ ugh to select a ψ tion µ ¶ rs, X µ b u-Schwarz, or NS) and ∂ Ψ c µ µ r ∂ + c ψ Ψ σ a ρ ir 2 ρ . µ b r − before selecting this Ψ µ e ρ − i 0; the supercurrent 2 ˜ µ r µ 2 1 ψ ˜ ψ Ψ = , + = ≡ . 0 r a µ ∗ a ∑ 0 J ) ) X ) J = a and c 0 0 µ r = = ∂ , , ∂ µ + ˜ µ ψ τ τ µ µ + 0 π ( ( ( invariance which may be used to select Ψ X = = X Ψ µ µ − + − c σ σ . ] ∂ . ∂ Ψ Ψ + 0 into right- and left-moving fields, ! µ Weyl ± ± Ψ µ µ − + µ = × ab δ reads Ψ Ψ Ψ and · η µ )= )= µ 2 1 à diff 20 + Ψ π π a , , Ψ − Ψ read, in covariant gauge, − = ∂ τ τ σ a ( ( a r µ µ − ir µ ρ χ − 2 µ µ − + i Ψ − Ψ − ) ψ 0 and constrains the Fourier coefficients . Ψ Ψ e a Ψ ) ∗ ∂ µ r and = = δ ¢ σ b · ∗ , ψ µ ρ ± µ − ab ) τ − r γ µ ( Ψ r Ψ + ∑ Ψ ¡ ε + b [ ψ ( ∂ ∂ = = a µ − ρ µ ± ( Ψ µ Ψ , the requirements are identical to those of the previous section. To find Ψ µ i 4 , the Dirac equation for X ) + σ describe right- and left-moving fermionic worldsheet fields respectively. µ ∂ µ + X 0. Finding Euler-Lagrange equations for ± b Ψ ∂ τ = ∂ µ being integer moded in an R sector, and half-integer moded in an NS sector. T a ( X χ 2 1 r and a recovers the one-dimensional wave equation (4.34), plus the Dirac equa . Furthermore, the local super- and superconformal symmetries are eno ∂ µ − = ab Ψ ≡ ± η For closed strings, Periodic (Ramond, or just R) and anti-periodic (Neve giving four sectors in total. The mode expansions are boundary conditions may be chosen independently for right- and left-move with Majorana condition ∂ ab As in the previous section, we have the proviso that the surface terms in the v Just as for the bosonic string, (4.47) has a • = T ab conserved quantity associated with the local symmetry (4.48). ant gauge With boundary conditions on the fermionic fields, split Then, The Euler-Lagrange equations for These are known as super-Virasoro constraints. Note that so that must vanish. For the γ String Phenomenology for an arbitrary Majorana spinor gauge with condition for surface terms to vanish is PoS(cargese)001 9) and (5.5) (5.2) (5.3) (5.1) (5.4) µ ± (4.58) (4.59) Ψ values in Steven Abel elations on the xpansions + σ conditions are ir to − e µ r µν ψ ) , 0 η r ∑ ¢ µν ′ > 2 η σ 1 n µν ¢ √ ( . µν ′ − ) η η σ s = σ σ n ) + , ¡ − r + µ + 0 τ δ δ , m ( µ n µν i σ Ψ δ τ µ ¡ + α ( = η m i = δ n µ + Ψ } i 1 ν s √ Ψ ± in the NS sector. ¢ª ˜ ]= ]= = ′ ψ ) ν m , = σ 2 1 p ˜ ¢¤ , )= )= α µ r 21 ′ µ n , τ , ˜ π π + a ψ σ µ − n ¡ , , , { x ˜ σ Z τ τ α , µ [ ± τ ir ( ( ( should be implicitly understood to be integer valued, µ ¡ = Ψ − µ µ − − ˙ ∈ } ν X e , } n ]=[ Ψ Ψ ) r µ ν m X T r s µ − m , , σ ψ ψ α n α , ) = , { n r , τ σ µ µ ∑ 1 r n ( , √ P α ψ 2 µ τ ± , [ 1 ( { µ ± = √ Ψ µ © † Ψ P µ = n £ iT a iT µ − 1 2 1 2 Ψ in the R sector, and Z ∈ creation/annhiliation operators for right-moving modes. Similarly, inserting (4.5 r should be understood to take integer values in the R sector and half-integer D } s , r { For open strings, left- and right-movers are related. Possible boundary There are now only two sectors; Ramond and Neveu-Schwarz, with mode e Again, In the canonical quantization procedure, one imposes equal-time commutation r In all that follows, subscripts and their canonical momenta • µ X the NS sector. Inserting the mode expansions (4.39) into our relations leads their canonical momenta 5. Canonical quantization with other anticommutators zero. with other anticommutators zero. with other commutators zero. Hence, with other commutators zero. One also imposes equal-time anticommutation relations on gives are a set of whilst String Phenomenology PoS(cargese)001 (5.8) (5.7) (5.9) (5.11) Steven Abel ite the classical onvention is that (R) (NS) (5.10) ) 0 is left to be dealt with 6= a m µ ( ± ψ ± 4.51) applied to physical states . ∂ n r µ ± a + µ ¶ ψ m . 4 1 ψ ) + i δ ) r 2 0 r 0 (5.6) 0 m − − µ µ + m r A > 2 = ψ > r + µ r s ψ + 2 − m r m µ − X µ + J ( ( : n ¶ r . G ± ψ r δ + − Dr D r ∂ = r µ J m ¶ 8 8 1 1 0 − µ B X r L + > ∑ X J ) r ± m = = 0 0 0 + − n ∂ and ± 2 1 r r s 22 + = ∂ µ ± m = = = + B B − n µ r 1 2 i i i Ψ −− µ −− L − m a r −− T T ϕ ϕ ϕ α J µ 2 ∑ )= | | | T − n − r + ) 1 2 m 01 σ µ )= − = σ a = G T 1 L ] = ( ]= im ir α J } + r −− 2 2 n 0 s − ± n e e T L n G > ++ ± 0 G µ ∑ µ n , σ σ σ , T 00 0 L , α m T J ( d d d α r m n + ( ( n L L π π π ¢ G [ − [ µ 1 2 2 1 − 0 0 0 separately, as we have a problem in this case; the raising and low- µ 1 µ { r 0 m Z Z Z 0 α α ψ ′ ′ ′ ≡ ≡ − L µ 0 2 n n ± 1 1 1 ∑ ∑ α πα πα πα J m ±± 1 1 1 4 2 4 2 4 2 ¡ T 3 ≡ = ≡ = = ≡ Dm Dm r 0 m 1 8 8 1 L G L = = m m A A Our operators obey the super-Virasoro algebra, Now, let us begin to examine the physical spectrum of the theory. One may wr are i ϕ ering operators do not commute,the so lowering in operators which go order to should the we right, write and them? the (infinite) The zero-point c energy later. constraint equations (4.51) as Notice that we have treated It is useful to define Fourier components of In terms of these super-Virasoro operators, the| Virasoro constraints ( String Phenomenology 5.1 The super-Virasoro algebra with with sector-dependent anomaly terms PoS(cargese)001 e (5.12) (5.13) (5.14) (5.15) Steven Abel g is moving leaves only the lf as unphysical s for both positive the string is at the the Fourier components we can use the so-called rely bosonic string. First − σ in 2 . − e + − n W . α − . − τ 1 n ) ) + 1 1 V 0 p − − 6= 2 physical transverse degrees of freedom. ′ ∑ − n D D α ′ − − X X 2 2 α D W + − + + 0 0 r i + X X V x 23 ( ( + − 2 2 i − 1 1 2 oscillators on the vacuum. )= √ √ σ W τ i − − , = = p V σ ′ D ( − + = α + X X µ . Hence these operators obey a copy of the algebra (5.9). + X r W ˜ − ψ 2 µ . The nett effect when evaluating physical operators such as the x 1 V − = D . These are exactly similar to equations (5.8), but written in terms of and .. r − 1 n − ˜ G ˜ = X α ). It is now possible to use the string equation of motion to eliminate the i , . Inserting these into the equations of motion we find a linear equation for + α m − + ˜ X L X ; in other words, it is not possible to implement the Virasoro contraints fully at th , which are completely determined. r − n , ˜ α m and coordinate corresponds to the time coordinate seen in a frame in which the strin give operators − n + α This removes all degrees of freedom that are “longitudinal” to the state and For convenience, in order to examine the physical degrees of freedom For open strings, only the above algebra is present; for closed strings, 2 oscillator modes X ++ coordinate as well; i.e. we use the mode expansion T − − Exercise: Use the string equations of motion to determine X with infinite momentum. The light cone gauge is usually expressed by saying X It is not hard to see that a pair of vectors are contracted as The Hamiltonian,is to leave only the contributions of the D each There is no oscillator dependence,same and value it of “time” is ( the frame in which every point on and its equivalent for light-cone gauge. Thisnegative removes norm an states infinite in the over spectrumwe counting called define (which ghosts). light cone manifests Consider coordinates first itse the pu String Phenomenology Because of the anomaly terms (5.10),and it negative is inconsistent to impose these condition 5.2 The light-cone gauge And the spectrum is generated by these the left-moving operators of quantum level. PoS(cargese)001 and ε (5.16) (5.21) (5.17) (5.20) (5.19) (5.18) Steven Abel above, in which eta regularization. the result in (R) (NS) tring phenomenologist und by multiplying the may be found by exam- 2 transverse degrees of ¸ infinite zero-point energy D ) − ε ( D O 2 − 48 D )+ r 1 , µ − (NS) ) ψ − r a ¸ 2 ) µ − θ )= − θ + 24 ( 2 1 ψ D , − 2 r θ n N ) ( − 0 ( 2 1 = ε θ − 2 r ′ > 16 ∑ − r ( r 1 − − 16 α e 1 D − 1 D + = 1 = n ∞ r − 1 ∞ r = n ( ∑ 12 0− (R) 24 µ ∑ · ∑ µ θ = ( 2 α − 2 p ε 2 1 2 n d 2 − 2 2 − d µ = µ 2 1 − in (5.8) and applying (4.40), the mass-shell condition m D ε D p + ′ 0 α ψ − a · − 24 α 0 − − a → 1 0 lim ε 12 > D = ∑ = n → + = = lim ε − 0 − r r 2 X = L 1 2 0 a = ≡ − m = ) comes in because only transverse excitations of the string con- = = N )= ∞ r ∞ r has = n D θ 2 – we will deal with this problem shortly. The first excited state 1 i ∑ ∑ a k ∞ = 2 2 ∑ − n > 2 2 − − 0; n 2 | D D ( D 1 2 − − − ∞ = for the harmonic oscillator by the number of states at each level and the ∑ n D    1 2 = = X 2 (rather than ψ a a − redundancy of the action is elimated. A suitable value for D transforms as a vector under the Lorentz group, and has Weyl i k × 0; | Here, the ground state For open strings, after setting diff 1 2 µ − , which we must regularize. A convenient approach is so-called Riemann z Equivalently one can use the result ining the two sectors of the open-stringare spectrum. the Of states particular which interest to are a massless s at the string level. NS sector the tribute – this may be seen explicitly by going to the light-cone gauge as described where in the intervening steps I have summed the geometric series, expanded where reads number of contributing dimensions, which is tachyonic for all counts the number of states present at eacha level. We still have the issue of thrown away the leading infinity. Theusual normal-ordering constants contribution are of then fo The factor of Therefore, we have ψ String Phenomenology 5.3 The open string spectrum PoS(cargese)001 . If 0, a ) 2 (5.26) (5.27) (5.25) (5.22) (5.23) (5.28) (5.29) = Lorentz − i form the ) ζ D | i 1 ( Steven Abel s , − | 9 a ( b SO g SO are seen to form a ator, µ 0 . In fact, states in the ψ ) 8 ( tates SO in all possible ways with the 4 (5.24) i of ζ | v 8 ,..., 2 1 À 1 , transforming under , 1 2 µ . Therefore, the spinors − = 1 2 ± A µν is an − ] a , , a , η i ± ν 2 1 b 2 ab k 2 9 Γ + δ = Γ ± , − a 0; = 16 | , µ a b , = D s 1 2 ··· 1 2 o Γ [ ¢ µ = 1 − ν o 0 1 − ± i 4 Γ 1 ψ − ψ , + 0 25 ¢ b 1 2 + a 2 − 1 2 1 a Γ b 2 2 Γ , , √ = Γ ± = a i = , + 2 2 , + a µ 0 ± 2 1 0 m b 11 µν M ′ a ψ a Γ Γ n 2 ± δ 2 α M ¯ ¯ ¯ ¯ i ± Γ √ ¡ ¡ = ≡ n 10, in which 1 2 2 1 i a with eigenvalues 32 may be created by acting on s S | = a = = = S in each position. D ± ± 5 a 0 1 2 b b are massless. Furthermore, they obey a Clifford algebra are ten-dimensional gamma matrices. Let us define a set of raising and − µ 0 µ 0 ψ ψ 2 √ = µ Γ is the state with is an eigenvector of i i s ζ | | In the R sector, the The utility of this definition may be seen by noting that the generators of the . These 32 states may be denoted as + a NS sector are spacetime bosons at each mass level. R sector where which obey which may be written in terms of the raising and lowering operators as with other anticommutators zero. Beginning from a lowest weight state satisfyin this constrains us to the value where so-called Dirac representation of the Lorentz algebra, and the ground s representation of dimension 2 b ten-dimensional spacetime fermion. Definining a ten-dimensional chirality oper String Phenomenology freedom. It is therefore a candidate for a spacetime boson lowering operators by this is so, then it ought to be a massless state. Since implying that algebra are PoS(cargese)001 ) 1 . , ) 9 8 ( (5.36) (5.33) (5.34) (5.35) (5.30) ( have a i SO oscillator SO ϕ | Steven Abel ψ of hich removes s consider the C 8 r of , we see that ) 0 or l states ory with space-time S 8 sentations of ,..., 0 so-called modular invari- , 1 ep reasons why the projec- orld-sheet supersymmetry. er an k , 1 k − . 0 (5.32) ¶ ¶ = ( = 0 (5.31) S C 8 i 8 k i , s = , | F 1 2 i 1 2 . N . s ¶ ) r | i − . − 1 2 1 µ ′ Γ ϕ µ µ | · ψ − − r 16 ( k + + µ 0 − + S − GSO ¶ ψ ¶ P 1 µ 0 S C ⇒ 26 h 16 0 8 8 > = ∑ r , , 1 2 Γ = 1 1 2 2 1 0 = k = i −→ + + 2 32 F = ϕ 1; µ µ | is defined as N is given by, = i s ± GSO F i | P s 0 SO | = −→ −→ G G Γ ′ i survive. Now, the two Weyl representations decompose under P · s | 16 as k 2 1 16 ) 11 8 Γ ( =+ survive the physical state conditions (5.11). In particular, 0 SO i s s | × ) 1 , 1 ( SO fermion number operator N −→ ) 1 , As we saw, the lowest-lying state in the NS sector is tachyonic. A prescription w In the NS sector the operator 9 ( the tachyon is the Gliozzi-Scherk-Olive (GSO) projection, in which physica The GSO projection The GSO projection in the NS sector then acts to remove states with an even numbe SO Therefore, surviving physical ground states in the R sector fall into eith The prescription seems at first sight ation little is ad-hoc. necessary. However there At are the de supersymmetry, level which of of the course spectrum, is it not canOn guaranteed be a by seen deeper the level to presence it imbue of can ourance. w be the seen In to the arise following asprescription section a for result we applying of shall the the GSO see requirement projection. this of in detail, but for the moment let u whilst the projection operator applied, String Phenomenology the Dirac representation may be reduced into two inequivalent Weyl repre Not all possibilities for which is the Dirac equation. Choosing the (massless) frame so that only states with depending upon the value of excitations, deleting the tachyon from the spectrum. PoS(cargese)001 2 = (5.39) (5.38) (5.37) N type IIA m which Φ Steven Abel ′ λ λ : nd right-moving dilaton , 11 ction. The physical acts in the opposite Γ µν is irrelevant for open vector supermultiplet B ss level. ual number of degrees constrained to glueing S dilatino dilatino − GSO , 8 ten-dimensional , ± P GSO µ µ ′ P ⊕ oscillator excitations in the . Ψ Ψ C for the left- and right-movers V B-field ψ 8 8 , ± GSO µν P g , . non-chiral ) i gravitino gravitino C F 8 N ) ⊕ 1 V graviton − 8 ( . ( 2 R 11 1 ⊗ m Γ S S ) ⊕ S 27 = ∓ 0 lead to the level-matching requirement that there 56 56 8 1 2 L 28 h ⊕ ⊕ ⊕ = m ⊕ -forms Ramond-Ramond fields S S 1 2 i V rep. Corresponding massless fields p 8 8 8 ϕ ) 35 oscillator excitations in the ( | = 8 = = = ¢ ( = S ψ S a V 8 8 ± 8 V GSO − SO 8 P ⊗ ⊗ 0 ⊗ Massless states of the closed type IIB string. ˜ S L ⊗ V S 8 ¡ 8 8 V 8 = i ϕ | Table 1: ) 1 supersymmetry algebra. In fact, the GSO projection ensures spacetime a now acts to delete states with an odd number of − = 0 L + ( GSO N P , and an even number of 10, ) 8 = ( R-R bosonic NS-R fermionic R-NS fermionic Sector Type NS-NS bosonic D Taking the opposite projection on both sides leads to a spectrum supergravity theory. in which the spinorstheory. have opposite The chiralities spectrum on of either states is side. the same This as is that known of as a a SO When we perform the GSO projection, the relative choice of The closed-string spectrum is obtained by taking tensor products of left- a In the R sector, the definition is modified to include the chirality operator The true importance of the GSO projection lies in its ability to create a string spectru • of S is now important. together only those states with the property be an equal number of excitations of left- and right-movers, so that we are states, each of which is very similarstate in form conditions to the states found in the previous se At each mass level, there are four possible sectors, summarized in table 1. 8 The projection of the 5.4 The closed string spectrum of type II models supersymmetry between NS sector bosons and R sector fermions at each ma String Phenomenology has spacetime supersymmetry. After applying theof projection, freedom there in are an both eq the NS and R sector ground states: these form a strings. fashion, but as there is no absolute definition of chirality the choice of PoS(cargese)001 ws (6.2) (6.1) chiral (5.40) Steven Abel . p bosonic right- 10 dimensions the expression .. 1 out them there is σ e first models that , s in the spectrum.) d fermionization or τ es of freedom. (For e would like to have real bosons, or equiv- kes it consistent, i.e. free satisfies the correct com- J + λ t ely include large GUT groups ) 1 supersymmetry. The first step in ) ) . S − − ) 8 = J J σ σ 2 − 2 − ( ( ⊕ 9 Ψ Ψ 16 .. N V i i with indices for the first 0..9 indices. These .. 0 8 1 = ( + + = µν µ J − 1 1 − ⊗ g − X − X ) Right Movers J J S 2 − 2 − 28 8 Ψ Ψ ), in which the spinors have the same chirality on ) ) ( ) ⊕ + + 2 = C : V 1 σ σ 8 8 √ ( J ( R ( 9 9 ⊕ iX .. .. = 0 e 0 V : = = J 8 − µ theory, and the resulting spectrum of states is that of a + ( µ + λ Left Movers X Ψ ⊗ ’s into complex fermions ) ψ C 8 2 supergravity theory. type IIB ⊕ = V 8 ( N ) and gravity. In this section we shall develop these models, in a way which allo ) 32 ( SO A p-brane theory is based on p-dimensional fundamental objects. Derive , for the Nambu-Goto action using a “world-volume” parameterization Taking the same projection on both sides leads to a spectrum ten-dimensional either side. This is a (or equivalently, 6 Exercises D: The heterotic string is a curious combination of supersymmetric left-movers and So far we have seen how the spectrum of the simplest supersymmetric models in E • , 1. 8 E movers as follows: can be derived.4-dimensions, However these and models also are chiral unrealistic. models, which Most requires obviously w Using the appropriate operator product expansions, one can show tha us to return to the question of consistency of closed string models. this direction was the constructionwere of seriously so-called considered heterotic for models, phenomenology which [6].( were They th perturbativ 6. The heterotic string and modular invariance extra bosons can be regarded as anof extra conformal contribution to anomalies. the theory One thatno ma example Lorentz of invariance many in consistency space-time checks signalled is by that the with lack of massless graviton mutation relations for fermions given above.alently The sixteen right-movers complex have fermions, sixteen thatexample, do in the not spectrum correspond we to will only space find time a degre metric bosonization – of which more in section 12). The relation is given by, In 2 dimensions bosons and complex fermions can be inter-converted (calle String Phenomenology It proves useful to combine the PoS(cargese)001 = ab (6.3) T amely how Steven Abel of freedom lar invariant. persymmetric eory. If we go or the moment wers (for example ld sheet integral is discarded. In fact D models and this ´ tified 10-D models, t was developed in s. J elevant diagram are − In the conformal and λ fermionic or bosonic) gth of the two cycles) a ∂ 8 counts the left-moving ual viewpoint for model nt a ρ ... om thrown in to cancel the leaves a Weyl invariance in important to realize that the J − 1 ) λ i iate constraint equations = ab j + η φ j + e Ψ Minor additional constraints a = .. ∂ 2 a ab ρ γ j + Ψ i + j 29 X b ∂ j X 2 r a ∂ vacuum amplitudes (one loop partition functions) with no ab → η ³ Constrains model Z σ 1 2 dependence). This allows one by a suitable rescaling to go to a flat d 1 r φ Z 2 T − = LC S 16 counts the complex right-moving fermions, and = trivial Z 0 ... Z 1 = J 0 must be the sum of the bosonic contribution from the right movers and the su The technique of constructing the string models with all the additional degrees We now turn to the question that I alluded to at the end of the previous section, n The reason that the one loop diagram is so constraining is that it must be modu = a expressed as world-sheet fermionsrefs.[7, is 8, known 9]. as In the thisconsistent discussion fermionic models I formulation. shall in use 10-D the I are notationused of of to course ref.[8]. derive independent them. It of is the Thein formalism fermionic fact (i.e. formulation was can the also point bebuilding; of it used the disgards to original the geometrical develop papers. interpretation 4- of Thereand the it regards 4-D gives models the as a world-sheet compac slightly fermions unus conformal simply anomaly. as Later extra I degrees shall oflet return freed us to concentrate on the our 4-D task models of in finding this the consistent formalism, models but in6.1 f 10 dimension Modular Invariance - the tool to tell us which models are consiste contribution from the left movers. Consider the one loop diagramof for torus. a First particular recall shape that (i.e. going to given the by conformal the gauge len ( over the region shown in the diagram metric. Now consider the integration region itself: this is now planar, so the wor transverse degrees of freedom. It is not hard to see that the appropr the metric (since there is no G String Phenomenology The full action therefore combines thelight-cone gauges bosonic and supersymmetric actions. where to determine the consistent models.to The complicated trick enough diagrams, is some to putative startmore model doing will than some give one inconsistent perturbation ans answer th we for only the need same to physical go amplitude) as whereupon far it as vacuum can be vertex operators to determine allshown the below. consistent 10 dimensional models. The r PoS(cargese)001 (6.4) (6.7) (6.5) invariance Steven Abel ed with the . The param- 2 τσ + iant, and so in order over-counting. 1 σ = is modular invariant. z 2 τ+1 ) τ 1 (6.6) ( = where Im / ] bc 1 τ π . The total one loop partition function d − 2 ) , τ ) τ 0 τ τ ad ( ( ( 1 m m 1 1 left after moding out the modular transfor- ; r 2 Z ∈ τ π Z τ τ 2 2 Z 2 ) σ + + τ ∈ , τ n 2 1 ( n is still a symmetry of the 2D theory and we can 1 d d σ , τ 30 π z Im 0 c 2 , λ + C b z + and just reorients torus , 2 Z z r → 2 a = z σ = = z 1 b d z Z and + + 1 τ τ c a σ we get a corresponding → 0 τ τ swops . There is an additional invariance under large reparameterizations. Any τ τ 1 redefines torus : / 1 + τ → → − τ τ are integers. Lines with strokes are identified. But we can still use the Weyl is the fundamental region (i.e. the region of m , n C defining the torus is called the Teichmüller parameter: it should not be confus Exercise: using the transformations above show that d For a particular value of τ so that any point is defined by the coordinates world-sheet coordinate to get rid of one of the parameters. i.e. then requires us to integrate over all independent values of this parameter eter where where to make sense our integrand should itself be modular invariant. mations). The measure of the integration renders the integration modular invar reparameterization that describes the same torus has to be moded out to avoid These two transformation generate the modular group, PSL(2,Z) The torus is defined by two complex parameters String Phenomenology reduce it to PoS(cargese)001 (6.8) Steven Abel 1 region. Any into the region > . τ | f the other regions τ | ifferent models) we y. All that we require so that eventually one d into it. All independent ular model is defined by ns as they are transported now show. The calculation ) 1 1 values into the + < τ | ( τ 1 | Z )= 2 τ 1 / σ 31 X C 1 R − J ( 1 λ Z L −1/2 1/2 j λ )= τ ( 1 Z 1 σ be single valued and this leaves some freedom in the phases that the fields 2. The second transformation maps LC / S 1 ≤ ) τ ( Re ≤ The world-sheet fermions do not need to be single valued for consistenc 2 / 1 point outside the fundamental region shown below cantori therefore are be therefore mappe included by integrating over this region. can acquire as they are transported around the world-sheet. Any partic the boundary conditions (phases) ofaround the the world world sheet. sheet bosons and fermio is that the action − Modular invariance is then thewe condition could that have this chosen; region is equivalent to any o String Phenomenology The fundamental region is shown below. The Dehn twist maps all values of It constrains the possible 10-D theoriesis - rather (e.g. intricate, gauge groups) so as at we thesummarize will how end this of happens 1.3.1 and (where then wecan give talk short-circuit a most set about of of how rules the we for interim define model calculation d building, which is included6.2 in Appendix World C sheet boundary conditions, and hamiltonians 6.2.1 What do we mean by ’different models’? PoS(cargese)001 . ). j 8, ± π .. X , 1 (6.9) index (6.11) (6.13) (6.10) (6.12) j = j ordinates ( Steven Abel el indepen- J u J v j avitino which can u elements in the dot j v ve a single-valued , and for that we need or a single left-moving τ + σ r it turns out that all cases only (i.e. phases of 0 ) v 1 2 − n , ( ) ) i ) ) 2 . The left-handed world-sheet 2 2 2 2 J e σ σ σ v σ u , , , , , 1 − 1 , 1 1 † J n ] ] σ σ . v σ σ d ( j ( 20 ( ( ) ) 20 j j + . u + 2 2 u J J v + − − ] j σ σ 8 λ Ψ λ Ψ + v ,.. j , , ) j ,.. J J (we will for convenience drop the 1 σ 5 1 1 5 ) 1 2 iv iu iv iu 4 = u v 1 σ σ ∑ j π π π π ( ( ( 2 2 2 2 − 8 j j 4 4 v σ τ ) − − − − − u v + 0 X e e X e e , , J n ( . For later use I’ll define the inner product as 3 3 ( j u i 32 u v J 4 2 u , , ) v − 2 2 ) = ) = ) = ) = ) = ) = 1 1 2 e v u 2 2 2 π π π = 1 16 , , ∑ J σ σ σ 2 2 2 1 1 − , , , v v u = [( = π π π + + + + it sometimes proves convenient to treat a complex fermion n 2 2 2 2 2 2 V ) functions that is included in Appendix C. For the complex b U = [ = [ 2 1 σ σ σ . + + + 1 , , , , η V 1 1 1 1 1 1 ∞ = ∑ V U n σ σ σ σ σ σ ( ( ( ( ( ( j j and similar for J J j j + + − − )= X X λ λ π t Ψ Ψ , , 0 σ ( = + . We can pair the left-moving Majorana fermions into complex fermions, 4 1 2 λ v , and we can then club the left and right-mover phases into vectors with = + 3 0 or λ v = = j then the boundary conditions can be written 2 ). Hopefully it should be clear when this is the case. u ] v , 1 2 and for the moment these define arbitrary phases. Defining the torus by co j π v ) = 2 , 1 16 0 v [ .. 1 or The partition function is an amplitude for propagation through time 1 ∈ ± = 12 fermions which have space-time indicesbe have to have the same phase as the gr So for example we might have as two real ones in which case we just double the entries (and multiply the real product by It is trivial to see that the action is invariant for arbitrary for the moment) fermions which are of interestparticle here, on we the have world the sheet normal mode labelled expansion canonically f by the hamiltonian for the worlddent sheet product of fields. modular (Dedekind- The bosonic contribution is simply a mod which I’ll denote We can define the other boundary conditions for the fermions with vectors where σ String Phenomenology For the world-sheet to be embedded sensibly in target space-time we must ha In general the phases on thefor complex the fermions 10 can dimensional be models arbitrary, howeve are equivalent to taking entries of 0 6.2.2 Mode expansions for arbitrary boundary condition phases J When the boundary conditions are 0 PoS(cargese)001 ) + σ rd to (6.14) (6.18) (6.15) (6.16) (6.20) (6.19) (6.17) ´ + Steven Abel rticles) leaves σ ) n − m ( tion rules) given i e v ) , and perform the sum. ) whether the mode ) v a v I the left-moving ( − − + otnote. n ( is precisely the raising or n ε ´ ( v i − 0 v e − b ) with one or none fermion ) − n v v † m d a . v d − v ´ − n v † n ( − n d . − ∑ n ) ) d 0 v d 1 6 → v i − . ε − . − 0 + + † | n ab ) , n σ ) v r t d δ ) 2 r ... n d 1 d − − − = + ( m − 2 1 m 1 v † + } ( v r i − b − r b ( b v e v b d ( 2 1 ) + at time ... ( + , i n v 1 n 2 † a 2 2 (in contrast with ref.[7]). As long as the phases n b b − 1 σ ≡ d 1 33 1 † − ] vacuum (with energy 2 1 √ √ { ) d − v − π v v v v v 1 , = n + − = = + + 0 − , so in the Ramond sector † † n [ 1 † n } 0 n 0 1 b n d b b ( d 0 ( j ∈ ) > − b ³ 1 j 1 ’s for normal ordering, and regularizing the infinite contri- 2 n > , 1 = 1 r 2 − = † a d σ v ∑ i Ψ n = b − ∑ Ψ n + φ { . v | m = c b . v + 1 h )= a n t − is needed for the partition function (see Appendix C). It is given by ( ( v + t v + ³ † n λ N b divergence (corresponding to the infinite contribution from all the anti-pa n + ∑ 2 ³ ε λ∂ m / , i ∑ n 2 )= t ( σ σ v d d 1 are the individual excitation numbers. The fermion number operator which H , Z Z as the leading constant term. 0 v = a = ) = i we may always translate the notation to real fermions. To be explicit, it is not ha t n ( } , v i 1 2 H , m 0 The quantization condition for the 2D fermions is given by Exercise: Prove this expression using the procedure described in the fo Again, as a short cut to this result, one can express the sum as Lim dependent 2 v ∈ { i operator for space-time fermions defined in eq.(5.25). Also note for later use that there is no integrating the world sheet hamiltonian over the The hamiltonian at some time Doing the integration, rearranging the the excitations as it should be obvious from the context and the index (i.e. v Throwing away the leading 1 counts the total nett number of excitations is then (by definition and the quantiza see that where by String Phenomenology I will henceforth (for the rest of the heterotic discussion) drop the tilde on operators of each kind (i.e. index and excitation number), is left or right moving. We will take where the properly regularized vacuum energy is given by The particle spectrum is given by exciting the bution gives PoS(cargese)001 (6.23) (6.26) (6.27) (6.21) (6.24) (6.22) (6.25) 24 to the / Steven Abel 1 − osition under the ions, the vacuum sector is when the 0-D graviton. Note e bosons would look and hence preserved ; σ orrespond to the actual as they have no left moving s states in this sector can be R i . 0 ¸ | ) . 24, so that in total we have D † / 1 2 J 1 − 16 d 24 . ] − corresponds to ⊕ 2 , 26 16 † ) , R 1 2 J − † 0 i 24 + 2 1 J ) b D 0 D v 2 | . d ] )( † † 2 1 / − 1 ′ 1 2 I † 16 − − 1 2 J 1 2 . I , b − 15 − n d R λλ 10 ) ψ , . ( i v + † 2 · σ ′ n 2 1 0 ⊕ 1 2 I † | 16 d 2 1 + † J ∑ − † 1 2 − 34 = I j ψ v 1 R d 48 + b + D † V N α ( 1 2 I × (with the left-moving factor providing the transverse 2 ( v =[ a / d 3 L ) verify this result - i.e. that the graviton is massless in N × × i V v L L + − 0 a 15 ]+ 32 | . i i . Each complex fermion above contributes = = = † ( 1 † 2 1 0 0 J ′ 1 2 , 1 2 i | | =[ 16 b Q ) ) 1 2 † SO ψ † † 1 2 = 1 1 2 2 V I i i ≡ a b d d J ( =[ v ⊕ ⊕ ≡ 496 † † = 1 1 2 2 i i † grav j ′ b b 1 2 J ( ( v m ψ † ′ ) adjoint of 1 2 I 32. By the commutation relation of the real fermions, these are seen to ψ 496 ... 1 = 2 = / ′ J , 31 ′ . I subgroup; i.e. ) Exercise: Using the expression for a We can already see the types of massless states that will occur. The NS-NS 16 ( D=10 with 16 internal boson. boundary conditions are all so the graviton mass is Of course here 16 of thecounterpart. bosonic dimensions are regarded as ‘internal’ The extra piece is knowncharges of as the the space vacuum time charge. gauge groups. The charges here will c At this point we canLorentz see invariance, that is getting a masslessenergy good is gravitons check and of gauge D=10 bosons, or 26 dimensions. In D dimens where the two entries are leftbuilt from and excitations right such energies as respectively. Massles where here String Phenomenology Finally we can find the charge which is the fermion density integrated over Lorentz index for theSU gauge boson). The complex notation gives its decomp vacuum energy. The 8 transverse real bosons each contribute form the (32 The former is a gaugethat boson we the could latter equally is have used alike the transverse real field fermion that notation. includes For the example 1 gaug PoS(cargese)001 V ce . (6.28) (6.30) (6.32) (6.29) (6.31) } where V 1 , Steven Abel W U { Let’s now go + 0 e allowed 10D n have different nce. Any model hases and hence that world-sheet W , 1 all possibilities), so ief one first chooses W , riance constraint then 0 ymmetric model. This f allowed survive in the end. For W . The contribution to the , ) iu τ π ( 2 V U e . We must have a sum over all Z V U V U Z C } . ´ V ] , v 8 , to generate the different sectors (the N . For a single fermion with boundary . ] ) ) ∑ ] i when propagated through the complex i 8 u 0 8 allU ) ( ) W − { iv 8 iHt 2 1 1 2 π 1 2 ) e ( ( 2 i × ( 0 h 8 e π a 8 ( ) 2 ) e W 1 2 4 1 2 . We must consider each sector in turn given by v a ) ( ( a H α 2 1 4 q 35 4 ) ) ³ = 1 2 0 = [( V Tr 1 = [( = [( W )= 1 0 + τ mod(1) . For example the two basis vectors ( 0 W W direction gives the phase factor v u 0 W Z σ bosoniccontribution = 2. We then have to consider the sectors 0 a W = )= a 1 τ ( m m 1 Choose a set of basis vectors of Z = 0 m where a m . We leave the detailed discussion of the partition function and modular invarian .. τ the fermion acquires the phase factor i 1 . Propagation in the π v t 2 = , e u a ≡ = α are coefficients that have to be chosen to give the desired modular invaria q πτ V U Defining the model: Exercise: Read Appendix C, and verify the following rules for model building One finds (in Appendix C) that modular invariance constrains the possible p The partition function is given by analogy with C possible sectors in the partition functionthat (by it definition, should since look the like PF sums over boundary conditions). The sectors are given by linear combinations condition time 2 where 6.3 Rules for model building the allowed models. The enda result set is of a basis set vectors ofprojects to out rules generate states for so all model that the building. notexample allowed all In it sectors. the br will states project The that out modular we tachyonsis can inva and write precisely result down the in as a above GSO (10D projectionsupersymmetry space does which time) not we supers automatically earlier mean putstraight a to in space the by time rules hand! and supersymmetric then theory). do (Again, some note example models. partition function is then calculation to Appendix C.models. Here I’ll A just particular set describesectors of and how in boundary it each conditions sector works is we to find called determine a a particular th sector. partition function A model ca is defined by the complete set of possible boundary conditions, i.e. the set o where these phases are mod(1) and we sum over String Phenomenology The integers define a model with PoS(cargese)001 (6.35) (6.34) s match, Steven Abel tor given by n: mmetry, gauge group, # . 2 1 2 ¶ 1 2 or b − V j 0 . W that obey the following conditions . v a a µ = ab W 0 (6.33) 0 W J k 16 ∑ 00 − k 1 2 = = = a 0 + a boundary condition that has resulted in this k ba ab 1 a W k k + . − b a W 1 a , + a m 2 W w β ab 2 1 ¶ 3, so that 36 k + 2 1 = b − = 3 − α 1 a U j 0 apply the modular invariance projection on the states v w ab a W 0 k . µ find the spectrum of states at a given level. The states + 0 W j W 4 a 0 ∑ = a W a α 1 2 k V W a N + = + , the total vacuum energy including the bosonic contribution is given by . α v 1 2 a a V aa 2 and − W = k . " = + V V = 0 a V m a − 0 ˜ L Choose a set of ‘structure’ constants . (It is the sum over the Have = b V ˜ In every sector H = In every sector − V a − so the simplest case is to choose a model with just this vector in the basis. ; 0 0 L Lets use these rules to look at some possible models. All models need the NS sec Using our previous expression for All models can be constructed by applying the rules above in the order give 1) Define the model (i.e. choose the set of basis vectors) 2) Choose the structure constants 3) In each sector find the vacuum energy 4) Then find the massless states that satisfy the modular invariance projection Projections: W 3 = = V Spectrum: Structure constants: Structure constants: Vacuum energies: H must satisfy the level matching condition, that the left- and right-moving Hamiltonian where we sum over mod(1) String Phenomenology projection as in Appendix C.) 6.4 Examples particle content and so on. 6.4.1 A tachyonic SO(32) model with The end result is a massless spectrum which can be examined for supersy V PoS(cargese)001 32. .. 1 There celess (6.41) (6.38) (6.37) (6.39) (6.40) (6.36) . J = J 6= , Steven Abel I I ssless spectrum. 8 and ot SO(8), as is ap- ight movers is odd. .. vitational. They can 1 states appear in repre- = kk and a scalar which is the j f freedom they would be φ , es the required extra longi- i R µν i j i δ B 0 | 2 1 j − 1 − ] anticommute, so that α 1 I R R D i i − ! × ψ 0 0 j V , + | | L ) a 2 1 0,1] 1 1 2 2 i N 1 [ ¶ 0 ( K K 1 − − − | kk [ 8 = . 1 2 ψ ψ ∑ j φ 1 1 2 2 R i − i i j mod J J ] − − − ψ 0 8 δ ] | J ψ ψ , ) 2 8 1 1 2 2 1 1 2 2 ) 1 2 N J − I I 0 1 ( − − R 1 − = ( We have already seen that that massless states are 8 i 37 = ψ 32 , I have written each complex fermion as two real ψ ψ 8 ∑ ) 1 0 J = 0 D ) | 2 1 π × w × × 1 2 0 à ( V ( L − L L J , − . Counting the space time excitations above we find 1 2 i = , . } i i 4 j 0 4 ψ ) 0 0 , | ) i Φ | | N 1 2 = 1 2 µν sector: IJK { . 1 1 2 0 I − 0 A 0 + i j φ − sector N − . ψ W i j α µ 0 ψ = [( = [( 1 2 G × (graviton) an antisymmetric tensor 0 W + i 0 − L ] j + i W , ψ µν i 0 [ i j | G 1 2 φ B i − 496 antisymmetric bosons as above. The indices act on fundamentals of = ≡ ψ = i j φ 2 / 32 × in an effective action. The fermionic excitations . This can be decomposed into irreducible representations (antisymmetric, tra i j µ IJ φ A sector gives only massless states and will not be considered further. At the first excited level we find massive excitations such as The first massless states above are gauge bosons. Massless physical Together these form a physical state which just means that theAll difference the in states above oscillator satisfy numbersHowever this between note so that left none there and are is r projected also a out. tachyon This state completes with the negative mass ma squared 28+8=36, so the physical state is the antisymmetric representation of SO(9) n propriate for a massive state. Thetudonal world-sheet degrees bosonic of excitation freedom. provid But dilaton. States allowed by projections in the W ones: in the above we therefore have the transverse space time indices symmetric and trace) of the transverse rotation group SO(8) as follows; of the form sentations of SO(D-2) as required.written as Restoring the longitudinal degrees o The 0 String Phenomenology Hence we find spin-2 particle where again since we only have phases 0 or are therefore 31 be written 32’s so that this is the 496 of SO(32). The remaining massless states are gra The projection is given by PoS(cargese)001 4. .. 1 (6.42) (6.45) (6.44) (6.43) = i excitations Steven Abel i 0 b f ero modes for the ction, but we have nning from number of right moving ´ ) R L 1 i i ( 0 0 | | 1 ) ) j 4 0 − 1 mod ] b ( α ] 1 3 0 , so that we have four possibilities; 1 0 b ! − + 2 0 j V ! , − b = mod a R , 2 1 ] N 1 0 i 0 0 8 1 b [ 0 − ) 3 4 4 4 8 | [ = W ∑ j . 2 1 1 2 ... 1 ( à J − 8 − + ] ] W ) j ψ 8 1 8 J = , 1 2 ) ) . i 1 2 − b N W J 1 2 1 2 ( I , 6= − 1 0 k ( Initially the spectrum is as before. However now ( 0 , These will turn out to be the fermionic superpart- W sectors are massive. 01 N ψ . 8 8 = 38 b 4 32 k 1 ∑ i ) a J 1 ) W ) = ³ = 2 1 1 2 6= 32 0 ∑ j + W J à W 0 ( ( V × 1 1 b , , 2 2 1 1 + i 0 1 2 4 4 w 0 b ) = [( ) sector: sector: 1 2 = = 0 + 1 = W 0 1 + 2 and i 1 0 sector W 10 W N N W 6= . . w k = j 0 = [( = [( 1 0 2 1 and = = 1 b 1 , 0 W W i 0 = = m 0 b W W 0 0 = = W W + The 0 N N 0 i 0 . . 00 b sector. The right moving excitation can be as before, i.e. space time, or 0 1 m k 0 + W W 1 W and ( Have 2 1 , 0 = 01 k = 10 k Consider adding an additional basis vector Exercise: Using the projection rules verify that the tachyon is projected out. = ’s, so we have to be careful here to use the indices of complex fermions ru 11 we have two projections ners of the states in the antisymmetric internal. However now the left moving side can have any number o and still be massless. Recalld that in the complex fermion notation, there are no z That is the massless excitations are Vacuum energies: States allowed by projections in V String Phenomenology 6.4.2 A supersymmetric SO(32) model with States allowed by projections in V The tachyon state does not satisfyexcitations the so second is condition since projected it out. hasderived Note an it that odd from this the is requirement the of equivalent modular of invariance! the GSO proje k But Structure constants: PoS(cargese)001 (6.48) (6.50) (6.51) (6.46) (6.47) (6.52) (6.53) (6.49) ’s as de- . (c.f. 4 i Γ 1. Steven Abel a | ) as follows; dition on the = χ ced to 8 by this N h supersymmetry explicit, write the , er there are zeroes denote states e as before ymbol of course). Again the ve. It required ) ) 1 ´ 1 ( ( R i L 0 i | ) 0 mod 1 mod 1 | j odd − ) + excitations) gives 16 degrees of j 4 0 α = 2 b 0 0 ( 3 0 j + b b 2 R 2 0 16 (i.e. there can only be one or no i b M moving stu f f 2 1 0 1 0 0 j | j χ = b 1 2 − s ∑ = = ´ 4 − J − ... j R 0 0 ∑ i ψ = + ]= 0 W W 1 2 j . . | 1 + right , I χ 1 0 1 − i 1 (complex) excitations and hence 4 degrees of + 2 j j , i 0 × − 6= ψ W W b 2 0 0 k † α irrelevant stu f f ³ = i 0 39 Γ b − − b i j , 0 = 1 × j × 6= + b i = j 01 11 2 0 4 i L L † k k j j 0 b ∑ 0 i i Γ [ i 0 ) b b a 0 + + | | † b j 1 1 j 0 ) 1 1 1 0 ∑ ³ 4 j = − b ∑ j , + i w w i ( 4 1 i 6= 4 = 6= + + 0 k ∑ j − j 0 b b i 11 01 = i 0 = 6= k k j j b 0 j s b = = N + i 0 j 1 i 0 b ∑ 1 1 4 = b ∑ j W W + N N + i 0 . . b 1 0 1 ( ( W W constraint acts only on the right-movers so projects out exactly the states with 1 W excitations can be written to make the fermionic properties clearer in terms of 0 b As we have seen, space-time fermions appear in Ramond sectors (whenev So, the state constraint requires only odd numbers of left movers so we are left with the oscillator for every index because they anticommute) states, which are redu 0 0 scribed in section 5.3. Now consider the modular invariance projections abo is a single 10D gravitino. Wewe can have. count Here the the number single of gravitino gravitinos implies to that see we how have muc just one supersymmetry But we can show The 6.4.3 Digression on fermions and chirality where We see that the odd numbers of right-moving excitations (there are no massless ones above Counting the space time excitations, we initially had 2 b String Phenomenology In this sector there is no possibility of tachyons. The projections turn out to b in the boundary condition), and these are of the form is the spin-matrix of thespin state, of so the the transverse modularcondition modes, invariance on condition and the is space-time hence actually fermions a (which a chirality here con I’ll projection. denote by To the make generic s that projection. Adding back the longitudinal degrees of freedom ( freedom in each state. These are chiral 10 dimensional fermions which we dimensional fermions which would have just freedom – each chirality then has two elements.) W PoS(cargese)001 2 1 = , 0 (6.58) (6.57) (6.54) (6.55) GSO = P 00 k Steven Abel and e expansion of 2 1 , e fermionic repre- 0 = rm gives χ 01 , so that there are sixteen k L − i    ponds precisely to = ] a | = ] ] ] 1 ] ] 0 0 1 10 χ − , , = 0 0 k V 1 , − , , 1 2 2 1 L 3 4 2 4 4 2 2 2 2 a , 2 1 + 0 0 i j = [ [ 0 − − 4 [ 2 [ [ . −    s [ 11 Γ , χ j k 3 = 2 s − ] b ] , Γ 8 8 2 ) W ) = s and j . ] 2 0 ] , 0 0 (6.56) a ∏ 8 ( 8 2 1 1 χ ( W ) 8 ) s ] , 8 , ] W ] ] ) = 1 2 2 1 , = ) 1 1 8 8 0 8 0 ( ( 2 1 ) 2 1 ) ) + χ s 8 W j χ 8 and their sum satisfies the chirality projection ( 1 2 ( 1 2 , | 0 = ) ) ) ) 2 0 , ( ( , ( sectors are massive. 0 0 2 1 4 8 8 4 Γ 8 2 11 40 π = 2 and ( ( 02 j W ) ) ) ) 1 ) = Γ , k , 2 ± L 1 2 2 1 0 1 2 1 2 4 = 4 W i ( ( Γ ( V sin ) ) + = 0 2 , , , 2 1 | j 0 + 1 1 4 4 4 ) m ∑ = [( ( 0 j ) ) = [( + ) 20 j , 1 2 0 k i 2 0 2 π 2 W 2 = i = [( 6= = [( W Γ [ 1 sector 0 k W = j 2 2 b 2 m = [( i = [( = [( + and W 22 W model with Γ 6= 2 exp 1 1 0 k j 0 = 8 + + b W − W W W 0 1 E 0 i 0 2 m b π + and W W × The 0 0 1 2 8 + W , i 0 E cos 0 ( b ( j = ∏ We have 21 in the representation are k ] i 1 s = . + j ) 2 F 12 N k Γ j ) 2 1 Γ − j ( ∑ 11 π 2 Γ Exercise: Using the Clifford algebra of the gamma matrices, verify the abov Consider adding an additional basis vector to the previous model i [ + 1 Structure constants: exp ( 6.4.4 A supersymmetric Vacuum energies: Taylor expanding the exponential, and using the Clifford algebra term by te or above. It is not hard to see that the projection at general levels corres possibilities; String Phenomenology Using the above relation and the Clifford algebra this becomes giving the 10 dimensional chiralitysentation projection. as As above, we usually denote th where the spins PoS(cargese)001 ’ of a (6.62) (6.60) (6.61) (6.63) (6.59) 120 excitation, Steven Abel . That gives 2 / 1 and a − chiralities. The ) ) ψ ) ) 1 jection. Since the 1 1 16 ( ( ( There are two other ( simply doubling the umber of excitations respectively. Consider SO mod mod mod ) ′ ) ) 1 2 0 0 1 1 16 sectors: ( ( ( = = = 2 2 2 2 ) SO W 1 W W W mod mod ( . . . + 0 1 2 1 0 0 and ! W W W j W mod ) = = R − − − and from our discussion above we see N i + 0 0 0 0 0 1 gauge bosons of 16 2 0 0 | 8 ( = W W W W W ∑ j . . 1 2 . . . W R W 1 2 0 1 2 i J − SO 120 + − a W W W W W = ψ 1 J 1 2 − − − − − N W J J ×| I − 1 Initially the spectrum is as before. However now L 01 02 00 01 02 N N + ψ = i 41 32 k k k k k 1 1 ∑ J 0 0 and V 8 | = = × 32 ∑ ∑ + + + + + J à J 1 2 W 2 L 1 1 1 2 1 0 1 1 1 2 2 contribution from the right mover i i 1 2 2 2 1 1 − W 1 2 / w w 0 w w w ψ | sector: and + 1 2 = = = + + + + + = 0 0 i 2 − 1 0 10 20 02 12 22 W N N N W ψ W . . . w k k k k k 1 1 0 = = + + + + . One of these conditions is already given by the preliminary W W W = = = 0 22 00 10 20 0 0 0 W k k k k W W W N N N = . . . = = = 0 1 2 2 2 2 12 sector. The would-be gauge boson states are W W W W W W k 2 + + + 0 0 0 = W W W W N N N + 02 . . . k 0 0 1 2 (See Appendix E.) W W W W 120 possibilities in each half giving a . ) ′ = come from both from the first 16 real fermions or both from the second 16 16 2 J ( / 6= 15 SO . I It is worth lookingprojections at are the projections briefly to make an observation about the we have three projections this requires that Again the tachyon is projectedin out. the However 1st now or weindices.) also 2nd We require 8. therefore an have even gauge (Again n boson I states am discussing real world sheet fermions and where String Phenomenology States allowed by projections in V us a 16 The chirality projection should beLHS of the the same first for projection these has states an to extra survive 1 the pro for example the second States allowed by projections in V that these will be in the fermionic representations of sectors that give gauge bosons But PoS(cargese)001 can 8 (6.67) (6.66) (6.64) (6.65) E Steven Abel models, I will ist, so we must rge from modular sector which again t moving excitation 1 W ) ) ) 1 1 1 ymmetric models known as ( ( ( 2 =128 bosons in the fermion / 8 mod mod mod 02 model (the gauge bosons of 1 2 0 0 k 1 1 ′ 8 = = = + E = = 1 1 1 21 × k W W W SUSY . 8 . The first of these is the same as the above . . . N N 0 1 2 R E sector to survive the projection, we have the 00 = . i ) k W W W 2 Again these are the fermionic superpartners of No, tachyonic 0 | 12 01 No, no tachyons W 1 − − − k = 16 k j ( − + 00 01 02 01 + = α 42 ) 1 k k k ′ k SO 11 02 × W projection). If this is not satisfied then we have just + + + k 16 k L sectors: 1 0 1 1 1 2 ( 2 ). + i and ′ ) 8 above so that the new condition for these excitations to ) ) = 0 w w w a W ... | E SO 02 above, we have that the middle expression is equal to zero, W 16 32 32 00 + + + ab + k . This gives an ( ( ( k × k × ) 1 = ab ′ 01 11 21 8 ) = k + k k k W SO SO SO E V 16 16 12 ( 01 = ( = = = gauge group k k 1 1 1 SO SO W W W under N N N . . . 0 1 2 and 128 ) W W W model. If it is, then we have additional 2 + 16 ( ) gauge groups rather than ′ 8 120 SO 16 E ( SO × ) 16 ( Having developed the formalism for writing down general modular invariant For the equivalent states in the SO can be as before, i.e. space-time, or antisymmetric internal. Consider the the states in all the sectors above with a single chirality projection, since the righ be decomposed as gives rise to the 10D gravitino and gauginos. The projections are now States allowed by projections in V 6.4.5 Summary of heterotic models representations of condition to have From our preliminary constraints on and hence the new conditions are the same condition (from looking at just the briefly now revisit the type II modelsinvariance to show and firstly also how the the GSO existence projections oftype eme another 0 class models. of related nonsupers survive is Now we require the same chiralityhave projection on the left movers for this state to ex 7. Type II and 0 models, and the link with D-branes String Phenomenology modular invariance constraints on the The would-be gravitino state is as before PoS(cargese)001 . 1 (7.2) (7.4) (7.1) (7.5) (7.3) W and 0 Steven Abel n if we wish W ymmetry in both states ature that both left- ´ entz indices), so that e II models is much j − rmions must be degen- Ψ − ∂ ly. Thus the only two basis − y condition as the 2D world ρ V defined by boundary condition j − . a Ψ i W + , − ] ] only, and one with both a j 8 + 8 0 0 u v k Ψ ] W ] ] + ,.. ,.. 4 ] ] ] 4 4 5 5 ∂ ) 4 4 4 ) )+ ) u v ) ) ) 2 1 + 5 a 1 2 2 1 0 1 1 2 2 ( 4 ρ w ( 4 ( ( ( ( , u j v , , , + 4 , , , , − 4 4 4 ) 4 4 3 3 ) ) ) Ψ 1 a ) ) . Now our entire discussion of modular invariance i 1 2 u v 0 43 1 2 2 1 0 0 , , w ± 2 2 + λ v u 2 , , + ( ) = [( = [( j 1 1 = [( = [( = [( = [( b v u 1 0 0 X α V V V V a W W W ab ∂ = [ = [ k ( ³ = V U σ V 2 d N . a Z W 2 T − = dependence. 5 a LC w S 8 labels the transverse degrees of freedom on each side. Again, we ca .. 1 = j The modular invariance constraints work as before, but with the added fe It is straightforward to adapt the fermionic formalism to incorporate supers In the former model, there are only two sectors, and both can give massless where and right-movers determine the space-time statistics (sincenow they the both projection carry becomes Lor phases for just 4 transverse complex fermions on each side; goes through unaltered, apart from the fact that we now have sectors In addition, as for the lefterate on movers each in side the since heterotic (as case,sheet before) the gravitino. they phases This should of leaves have only the the 4 same fe boundar possible sectors 7.2 The type 0A and 0B models with and there are only two independent models, one given by with an additional which correspond to the labels NS-NS,vectors R-NS, required NS-R are and R-R respective String Phenomenology 7.1 Reinstating supersymmetry on both sides the left- and right-movers oneasier the than the world-sheet, heterotic indeed case. the The construction action of in the typ light cone gauge is complexify the real (Majorana) fermions into PoS(cargese)001 re- 1 2 (7.7) (7.9) (7.6) (7.8) , (7.10) (7.11) 0 = Steven Abel everywhere. 00 k 5 a w e graviton. The − citations from the 1 a w site for e same is customarily gously to type IIB and 8. These contribute the . .. 1 2 1 or = j ) 0 . 1 , replaced by i 0 ( = 1 a ] = ) 00 w 1 2 1 k ] mod ( ! j 0 − V , , 1 2 a R N 2 1 0 , i [ 1 This sector allows a tachyon state; mod 0 0 8 − = | [ ∑ j . . 0 1 2 = R R j − i i − = 0, so that ] ˜ The massless states are fermionic on both sides, j 00 a 0 ψ ] 4 5 0 k ˜ ) 4 = N w ) × 2 1 44 ×| ×| 1 + 0 0 ( = L 8 L L = 5 0 − ( ∑ j i , W (they are like a mesonic bound-state of two fermions i i . , 1 0 4 w 0 V a 0 0 | 4 ) à | | w ) 1 2 1 2 − W 2 1 0 i − 1 0 = ˜ (NS-NS) sector: ψ w = N 0 = [( sector bosons . 0 (R-R) sector: = [( = 0 0 2 and W 0 0 0 N W W . = N . 0 0 0 W m W We have For example projection is given by of the form Note that, these states are space-time and have bosonic statistics). The rules are then as before, with which produces a chiral projection on each side, which is the same or oppo Vacuum energies: where again writing the complex fermion as real ones we have States allowed by projections in the States allowed by projections in the W String Phenomenology Structure constants: same gravitational states to theprojection spectrum is as given by the heterotic string case, including th spectively. The model where the left-mover andcalled right-mover type chiralities 0B, are and th when theyIIA). are Both both models different it have is a a tachyon type and 0A no model space-time (analo fermions. There are also massless states of the form which is trivially satisfied inleft both and cases right since movers; there are the same numbers of ex PoS(cargese)001 (7.14) (7.15) (7.13) (7.12) (7.16) respec- 1 2 , Steven Abel 0 = 00 less states. The k nce to the simple the projection, and . 2 1 site for , ) 0 1 ( = ) . The structure constant con- 1 00 ( mod ! k ] ] ] 1 2 ) ) 1 2 ] 2 1 0 1 1 , mod , 0 − 0 1 0 1 ( ( and V 0 , , − 1 2 a , 2 1 0 1 2 à 2 1 [ − 0 = , [ [ , − R mod mod 0 [ = 0 2 1 i Again this sector has a would-be tachyon 0 b 2 1 0 = | = + . . ] 1 2 W . R R 4 = = j 11 − 00 01 i i a ) k Again the massless states are fermionic on both ] ˜ ] 5 0 5 1 k k a 0 0 ψ ] 4 4 W ( w w ) 4 ) = = = , ) × 2 1 1 2 45 ×| ×| 4 − − 0 5 0 5 1 ( ( = L ) 10 1 L L ( 1 0 1 1 i , , (they are like a bound states of two fermions so have i i k w w 2 1 , 4 0 W 4 V a 0 w w | 4 2 and ) ) | | − − = ) 1 2 1 2 0 1 0 1 1 0 = = i − = = [( 01 (NS-NS) sector: ˜ w w and 0 0 ψ 1 1 k 0 0 = [( W W = [( sector bosons + + (R-R) sector: m = [( W N N 1 0 W . . 0 0 00 01 + = 0 1 W W k k 0 0 W W W = = m 0 0 N N . . 0 1 then selects the chirality on each side.) The model where the left-mover W W 01 We have k Exercise: Work through Appendix D which specializes the modular invaria The type II models have four possible sectors, all of which can give mass type II case. It will help you understand the more general rules. Structure constants: straints leave us with two possibilities, That is only the statesthe with tachyon an is odd projected number out. of excitations on each sideStates allowed survive by projections in the tively. (The value of Vacuum energies: modular invariance can be verified relatively easily. sides, of the form Note that, these states are space-time however now the projections are and massless states of the form state; States allowed by projections in the W String Phenomenology 7.2.1 The type IIA/B models with bosonic statistics). The projection is given by which produces a chiral projection on each side, which is the same or oppo PoS(cargese)001 01 k and 0 = (7.17) (7.20) (7.18) (7.19) (7.21) W 1 forms citation. 00 ojections k + Steven Abel xistence of + p g part is also 01 k As both sectors e both different fermions with a + , again giving the 1 2 00 ) k 1 + ) ( 1 01 ( k mod ral coupling to mod , and subtracting the 2 1 2 1 2 1 = + , 2 2 0 1 2 . 00 1 k + = = = projections are + p µ 1 1 + 01 00 SUSY + X k k N N W p 01 ∂ , ˆ k . + + C 0 R R No, tachyonic No, tachyonic ... i i 1 σ (R-NS) and (NS-R) sectors: W 11 01 ˜ a µ 0 1 k k ). The coupling in question is | 1 1 + X 1 2 + p ×| W = = j p ∂ − d L µ 1 + i 46 ψ .. + 01 00 p 1 0 0 Z k k | µ µ × p 1 2 .. + + C L 1 ρ j − i µ 5 1 5 0 ˜ a ψ C = w w | respectively. The latter can be compared with the chirality and W ) − − = 1 WZ 1 model 1 1 1 0 1 I ( + w w p ˆ + + C mod for this state is given by 11 01 1 2 0A (same chirality) IIA (same chirality) L k k 0B (different chirality) i IIB (different chirality) + a | = = 1 lorentz indices, 11 1 1 k sector. Here the RHS of the + W W 1 = p N N . . . i.e. the gravitinos have the same chirality as the left-moving half of the R-R W 2 1 1 0 1 2 depending on the choice of structure constants. + W W + + 0 01 01 ± W GSO k k P + + 11 00 k k projection is automatically satisfied for this state since there is one right-moving ex + 1 10 There is an important connection between the fields in the R-R sector and the e A second gravitino W k projections on the R-R states shows that the chirality projection on the left-movin 1 where D-branes (for a review see for example [10]). A Dp-brane has a natu The projection on the right-moving half of the R-R states which is given by arises from the same chirality. 7.2.2 Summary of type 0 and II models given by and 7.3 Inferring the existence of D-branes from R-R fields String Phenomenology and right-mover chiralities are theit same is is the our type typecorrespond IIA IIB to model. model, and As when for they ar the heterotic string, it is simple to show that these pr states. States allowed by projections in the W (i.e. tensor fields with The chirality projection of The projection is given by are similar consider the former.lorentz index The (i.e. lowest gravitinos); lying states are massless space time W PoS(cargese)001 ) as σ ( µ µ A e 10D (7.24) (7.23) (7.22) X Steven Abel ings. Now if in auge field site. For type IIB of Wilson lines in ject turns out to be s from fundamental efore we can see the be written as tensors as that for type IIB ees of freedom. The in the R-R sector, then icture is as below, with n R-R flux. µν degrees of freedom. As we C 10 conditions) so we ended up with 1 RR flux W Γ , β ν ) being a ‘charge density’. and X 11 p τ 0 Γ ∂ ρ ( W µ αγ X σ H ∂ D2 brane αβ = 47 µν H space coordinates of the brane plus time, hence the B γβ p τ H d 1 dimensional integral. The world-volume field γα σ ) d + 11 p Z Γ ( 1 form field such as a field strength, it can couple to the world + onto the world volume. p 1 + p ˆ C 10D ‘bulk’ are spinor indices going from 1..32 so we initially have 2 is the same antisymmetric tensor that we derived in the gravitational spectra of th β , µν is the pullback of = 256 states. 1 α B + 4 p 2 Now recast these states as tensors keeping track of the numbers of degr In the case at hand the new bulk R-R fields arose in the R-R sector and can In this equation, the integral is over the The simplest example is the 0 brane (particle). It can couple to the one form g µ .. × 1 µ 4 relative chirality projection removedstrings half the of left and the right degrees movingfor chiralities of example is we freedom, the therefore same have and and for w type IIA it is oppo as follows. The fields we foundconnection were directly. bispinors First so write we the need bispinors to do directly a as little work b in eq.(3.28). This is aQCD well or known the type Aharonov-Bohm of phases in coupling QED. thatstrings. A leads one Here to dimensional the example the coupling come phenomena is of the form string models. The point is now that, if we find a new two form field where defines the position of thethe brane volume volume there in exists space-time some exactly as it did for the str where we can expect it to coupleD1 to brane a (a 1 so-called dimensional D-string object,Dp-branes existing as and in for opposed a R-R to 10D fields bulk, fundamental this having string). ob an R-R The charge p and producing a integral over the world-volume is a 2 String Phenomenology volume in an invariant way as above, with the constant saw there were two projections on chirality (from the C PoS(cargese)001 ) 5 3 1 µ µ + .. Γ 1 2 µ 10 µ terms (7.26) (7.25) (7.27) (7.28) H Γ C . Use it n 3 1 ) µ Γ degrees of µν Steven Abel − η 2 IB is 10 µ 2 Γ 3 = µ We verify this by Γ } in the R-R sector. . 1 ν µ 10 en in type IIA. Γ Γ µ , are different. We can .. − 1 µ 3 + µ Γ n µ Γ { in terms of antisymmetric use the massless bispinors 1 . H µ ] D leaves only the even ispinors as follows Γ 10 µ 10 2 for 4D Dirac fermions) which µ .. µ µ 11 1 .. 5 Γ Γ 1 µ Γ µ .. Γ ] ε − 1 n H ε 5 2 . This has the full 2 + µ ! µ µ n ) 4 4 .. Γ = Γ n µ µ 1 [ 1 .. (c.f. .. 1 − µ µ 1 Γ 1 H 256. Γ µ 0 10 µ H 3 [ 11 ( 0 (7.29) 11 µ Γ . , H Γ Γ = 3 . Γ Γ , 10 0 11 µ = = 2 µ 2 11 + Γ 10 n Γ µ p µ 1 n ... 1 Γ µ . 1 C µ 1 µ µ H .. µ + 4 Γ Γ 1 H ... n 3 H − µ 1 H µ µ + H defined this way has only 512 degrees of freedom , µ , 48 = H Γ H = 2 µ 10 H H ! = µ H ! using the Clifford algebra ( ) which are reduced to 256 by the chirality projection. ) αβ n C H Γ to the above decomposition, we find that only the odd H i n n 2 11 0 H 10 + 0 11 pH n 11 Γ 3 C + − i . = 10 Γ = µ 5 ∑ Γ n Γ 1 2 Γ 10 } 2 10 H ), but the chirality condition above will remove half of these. µ µ = C ( is because the duality above relates the components of + 0 Γ − 0 Γ 1 2 10 1 , 10 = µ 10 αβ C ∑ = n type IIB type IIA Γ 11 C H ( 10 4 Γ 1 6 H = { + + 11 = ] projection. So αβ Γ 10 3 ... µ 0 H C Γ + 3 W .. 1 + 10 µ [ C 10 Γ 2 means antisymmetrized in all the indices C 11 + 2 ] Γ 2 + 10 µ − 0. Applying 256 where the factor C .. 10 0 1 = = C = µ 1 [ } T 11 10 µ + Γ C Γ 5 10 , 2 1 C 11 So that for example Exercise: Derive the rule So we have identified the tensorial equivalent of the bispinors appearing 0 4 + Γ { terms survive the projection in type IIB and applying 10 in type IIA. So our 256 fields are gamma products, yielding An additional duality constraint we have is that inserting it into the sum above and using the 10D identities for to show that the chirality projection leaves only odd tensors in typeBearing IIB in and mind ev that the indices are antisymmetric, the counting of states for type I (i.e. to themselves. For type IIB we have In order to extract the equivalent tensors we can first decompose the b Likewise for type IIA we have freedom (i.e. The tensors that survive this projection in the type IIA and type IIB cases work them out using the usual anticommutation rule of n where the However these are not yetsatisfy the the physical Dirac propagating equation fields. (in both This the is left beca and right moving sectors); This is equivalent to the String Phenomenology or using is C PoS(cargese)001 ) (7.31) 9 ( , 8 7 , Steven Abel , 6 5 , , 4 3 , , field, we should 2 1 btained from the , , 0 now turn to how to s were Note that in µν 1 B = − p fewer dimensions as in = rge distances or, equiva- p rred to ref.[11]. . In this case we can write µ A Dbranes; D-branes; 0 (7.30) ] n µ = .. n 9 2 ν µ µ .. .. 2 C 1 1 µ µ µν [ C 8 , ∂ H 7 µ 1 µ µ .. 1 1 p 1 form electric potentials which naturally couples 1 49 µ µ − C C field can be written locally as a derivative - i.e. like n + = , , ] . This is a 5 brane soliton of the NS-NS sector. For a p 5 6 n 6 H = µ µ ν µ ...... n .. 1 1 1 µ 1 µ ν µ .. µ 1 C C H B µ , , µ 3 4 [ H µ µ p ) . It turns out that (almost) all of these branes can be built as .. .. 1 1 D µ µ µ .. C C 1 , , 1 2 + n µ µ µ 1 supersymmetry. The most geometric formulation is compactification. 1 C µ H , = C 1 , − C C N (i.e. , ) ) 0 n ( ( H H 2 tensor gives rise to a set of + p -brane; type IIB type IIA The 10D models are not much use for describing the everyday world, so I Exercise: Show that these two equations are equivalent p It relies on the fact thatlently, a as higher we dimensional reduce space the can compactificationthe be radius, figure. rolled the up. object appears At to la have get 4 dimensions and ‘lumps of field’, monopole-like solutions of the field equations. These solution addition to the above, as well as the fundamental string which couples to the review of the constructions of these classical solutions the reader is refe 8. Compactification: obtaining D=4 8.1 Background: Kaluza-Klein models to a find an object that couples to its dual So a rank The additional potentials correspondHodge to dual magnetic of potentials which have been o String Phenomenology or equivalently in terms of the tensor fields The first of these relationsthe implies field that strength the in QED it is the covariant derivative of a potential PoS(cargese)001 6 K (8.4) (8.1) (8.2) (8.3) then the theory but R onstrained Steven Abel which only n φ ich I called ht way. This is . These are the s that we see. In mposed as the momentum in 920’s as a way of tes to exist before the h can be written as cation scale, with an t with equally spaced erties give rise to the a 5 dimensional space imensional space. In a ing with its 16 internal y symmetry, gauge groups n R i e n φ ) 2 2 n R . y ) n R − 5 i . The total space is then a product of x µ e R , ) ∂ (by for example using colliders that can µ µ µ . x x R ∂ 6 ( ( ( n K φ φ so that the off-diagonal element plays the role 0 ∑ × ∞ φ 4 )= − ∞ = R ∑ M + 50 = n ) = π 5 5 = 2 µ x A , 10 + )= and from the 4D point of view we find a Kaluza-Klein µ 5 5 + x M x x ( R , , / µν φ 4. If we compactify on a “small” circle of radius µ µ ) n g x .. x 5 ( ( 0 ∂ = φ φ limit. Now consider the Klein-Gordon equation for the 5D state; and the usual 4 dimensional non-compact space. The direct way 5 = ∂ ∞ K ) MN N . + g , rotation symmetry (arrow) of the compactified space that remains in R → µ ) / M ∂ 1 R ( µ ∂ U ( with MN g must obey ) 5 x -modes have an effective mass , n At these scales one would see a typical spectrum of “Kaluza-Klein” states So, the properties of the compactified space determine the low energy physic Broadly speaking the same situation obtains in string theory. We begin with a 10D µ x ( addition one expects a Kaluza-Klein-ish spectruminfinite to spectrum appear of at particles the appearing compactifi sense above we it have as already we seen “open” this the in higher the d spectrum derived in the heterotic str to tell if physics is like this is to probe down to scales Every point in the 4Dobserved space properties has of its the ownetc. internal low 6D energy These space theory, have that such tocompactification. whose as be prop remaining derived super from the 10D string theory which is assumed access equivalent energies 1 We can expand this field in modes in the 5th dimension an internal space which I’ll call residue of the continuous 5Dthe spectrum 5th of dimension momenta. isbecause Before continuous. the compactification, fields After must compactification, be the single momentum valued. values So are for c example a generic field whic The continuous 5D spectrum becomes an infinite but discrete tower of 4D sta in the Introduction. The situation is as in figure.3 with the 10D space being deco reduce it to 4D by compactifying on an internal small 6 dimensional manifold, wh the massless theory even when we take the limit of small The of the electromagnetic field,shown and in the indeed figure couples as to a the charged fields in the rig massless states decompose as becomes continuous in the masses. spectrum, an infinite tower of states whose quantum numbers are the same, bu φ String Phenomenology The idea of extra dimensionsunifying was gravity and first electromagnetism. suggested The idea by workstime, Kaluza as with follows. and metric Consider Klein in the 1 PoS(cargese)001 1 n, = (8.9) (8.8) (8.5) (8.7) j re, so ates. Steven Abel a torus. The . space (indeed s. To see this ) 6 ( 4 supersymmetry vitino we derived massless states in we can imagine, a sless spectrum. e 4 of each chiral- able since the main = SO n states will remain n is accompanied by one gauge. freedom (i.e. the 1 or so are not of relevance . N , ) 4 ( GeV 19 .. 10 R 2 L L i , ≈ i i 0 1 4 (8.6) | 0 0 | | 1 .. ) ) j = 2 − i j , , excitation and pieces that do not; n i 1 α , = 1 1 0 6= j j R 0 × b 6= j i b 0 k L 0 a 1 i b | i 1 SUSY 6= , 0 1 i 0 | + 1 + j 4 j − b ) j 6= j 2 j 0 α , Z i = 51 + n . Hint; use the same identification as earlier, i.e. b 6= iX ) J 1 1 − 0 k = ( 4 6= α b j i 0 + ( N i j b × Z 6= then label the elements of SU × 2 j 0 1 0 + L b X b i 1 i 0 αβ 0 + b 6= | = Γ i 0 , is identified by n j ) of SU b 6 + − ( 4 . Show that the fermionic representations above are a funda- Z T i 0 1 2 ) b × i 4 + − ( ( ˜ 1 4 ψ with coordinated identified under translations. For convenience let’s 6 K , The indices of ¢ 1 = 8 transverse degrees of freedom. However in 4D only the first excitatio + j 2 4 Γ C i 3 + is isomorphic to SU + j is a complex constant. The metric of the compact space is Euclidean everywhe 4 2 ) is the same index we were using before to label the complex space-time coordin j 6 C Γ ( j 1 ¡ a ) each with 4 internal degrees of freedom in the spinor representation of 1 2 1 0 Exercise: count the internal excitations above and confirm that there ar Compactification tends to leave internal gauged symmetries in the effective sub The low energy theory relevant for phenomenology (i.e. the spectrum of Now the 6 dimensional torus, b is related to the 4D space-time fermion, and the rest are just internal excitation = j 1 0 0 decompose the left-moving part into pieces that have a From the 4D point of viewthe we have the two transverse fermionic degrees of b mental and anti-fundamental ( b giving ity. SO String Phenomenology right-moving bosonic degrees of freedom.an infinite For tower example of the states massless such gravito as The extra massive states have masses of order the Planck scale we don’t expect to loseconstraints any states on in the this model compactification. comediagram This from has is a the also one-to-one one-loop reason mapping partitionthe to same. function the which compact is space itself so the projections o collect the compact coordinates together into 3 complex ones which I’ll call where to phenomenology, so for the most part we will be8.2 concerned Toroidal only heterotic with compactifications: the mas this was the original point oftorus, Kaluza is and a Klein). flat The compactified simplest compactifications so that labels the two transverse components of the non-compact space in the light c the string spectrum), are unnaffected, and it is easy to see that this leads to simply by counting the number ofearlier. 4D Recall gravitinos that that are the contained single in chiral the 10D 10D state gra was of the form PoS(cargese)001 y. (8.10) equiva- 2 Z Steven Abel ave a chiral . Consider for 2 R consists of point e the axis to itself oth even and odd interact differently trokes are identified G L ) 2 ( e torus invariant. In the SU which forms a point (where For a review see ref.[14]. the group by a finite group. . An orbifold is the quotient of the notion of manifold, to allow for the 6 Z n 6 T R T
















































































































































GLUE



































































































































































































nt of . The orbifold is defined by the Z and also the origin is a fixed point since it is , Z Z Z − In the case of 52





5 → − =



























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































Z Z that is an “orbifold” of 6 : Orbifolds K FOLD 4 = D of its isometries

























































































































































































































































































































































































































































































































































G SUSY in 1 = by with the single complex coordinate shown in the figure excitations), and this will be true for all 4D fermions that come out of this theor 2 2 1 0 N Z R b / 4 models are not good candidates for low energy physics as they do not h 2



















































































































































































































































































































































































































































































R = N This is the “physicist’s definition”. More correctly an orbifold generalizes To begin with a more simple example consider a cone which is an orbifold of One way to do this is to have 5 a manifold by a subgroup lence relation presence of points whose neighbourhood is diffeomoerphic to the quotie groups (rotations in 6 dimensions)phenomenological and context space they were groups first (shifts) developed that in refs.[12, leave 13]. th leavingthe fundamental domain on the diagram shown insince green. they The are lines mapped with s into each other under We describe String Phenomenology 8.3 Getting example But the Standard Model is chiral, the left-handed particles coupling to from right-handed ones: we need a theory that is chiral. 8.3.1 Origami with 2 dimensional orbifolds - the cone to form a cone. Thethe curvature curvature is is everywhere ill-defined). zero except at the origin mapped into itself - marked with a red circle below. We can fold the sheet and glu spectrum (for one thing):numbers the of 4D gravitinos above had both chiralities (i.e. b PoS(cargese)001 (8.11) Steven Abel ngle. ). uced by half (this is nsported around the n ctors that are parallel is an integer. This divides tated by the deficit angle. n










equivalence maps half the torus 2 where



Z Z



























































































































































n GLUE /



































































































































































































i π 2 e The torus is shown in the first figure below . , = Z 1 i We can define a simple 2D torus by identifying Z − . Z + + 2 . (The deficit angle is the angle of the wedge that = Z Z n Z 53 / / Z 2 = =











































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































π



















































































































































T Z Z − π


but 2 FOLD π




























identical segments. A single segment has the edges identified as above, n moding reduces the fundemantal region by a factor of n Z : the pillowcase 2 projection is again the identification



















































































































































































































































































































































































































































































Z of the plane.) The parallel transported vector is now rotated by the deficit a 2 / 2 Z T Exercise: convince yourself that parallel transported vectors are ro cut out and the is We can make different cones by instead identifying String Phenomenology The effect of the “infinitetransported curvature” at around the it. origin is Weorigin. to show cause When this we a in fold rotation the the in sheet next ve up, figure. it reverses The direction red at the vector join. is tra the complex plane into where lines with equal numbers of strokes are identified. The (the B region) into thegenerally A true, region, i.e. so a the area of the fundamental region is red 8.3.2 A slightly less trivial example is provided by except the deficit angle is now not PoS(cargese)001 → Z (8.12) Steven Abel 1 Z . Folding as shown


















fold along dotted lines fold along dotted














1/2 moding reduces the fundamental 3 A



Z







Z






































































































































































































































































































































































































































































































































































i 3 / i/2 i , π fold along dotted lines and glue remaining lines 1 e −> + + 54 Z Z





operations and translations in the torus (i.e.



2 = = Z We show this in the figure below. The torus is a paral- −> Z Z . A GLUE Z
3

/ 3. As we have seen, the





i / 1 π Z





2 π









e






















































































































































































































































































1





Z =








Z 3 and 2

B /


line stroke single glue


π



are integers) identifies the lines as shown, leaving 4 fixed points shown as a







m , −>









n projection A 3 Z






: the three point cushion 3 , where Z i




















































































































































































































































































































































































































/ im 2 T +



For the final example in 2D we show the torus defined by



































































































































































































































































































































n + region by a factor ofleaves three. a three The pointed cushion. remaining region is shown in the next figure divided by the Next we can show that a combination of String Phenomenology lelogram with angles of Z red dot. Joining the identified lines leaves a ‘pillowcase’8.3.3 with 4 corners. PoS(cargese)001 i 1 Z = ” or (8.15) (8.14) (8.16) (8.13) N Steven Abel ) directions t an i a ree complex satisfy modular heterotic model 8 ns, E × in the action) on the s whose endpoints are 8 , we can add additional E 2 as we shall see shortly. , 1 , 0 W . . ) ) 2 2 3 3 − − , , 1 3 1 3 3 , 3 , 1 2 / Z 1 3 i 1 3 3 Z Z . π / 3 3 i 1 e / / π = ( i i 4 = ( + + π π v 2 2 − i i v 55 e e e ; Z Z ; i i + = = → → → λ Z i i i i 1 2 3 iv iv Z Z Z Z Z π π orbifold (which is simpler to draw). 2 2 2 e e Z identification = = 3 i Z i + (which adds additional phases to the boundary conditions of the heterotic model Z i λ 1 iv π 4 = e , i N iv π 2 e transformation. The states in the spectrum can then be divided into “untwisted 3 as described earlier. The simplest example is to keep the complex ( case is interesting for phenomenology because one can directly construc Z 6 3 : a “realistic” 3 Z T Z acts slightly differently on one of the internal degrees freedom in order to / 6 3 The Now let’s impose this compact structure on 6 space dimensions of the Z T sectors with twists of model as we will shortly see. First complexify the six internal coordinates to th determining the torus orthogonal so that we just repeat the 2D example 3 times; (all at the same time) which is usually written more succinctly as “twisted” as in the next figure for the 3 complex internal space-time fermions).related These by sectors a give “twisted” state In this model, for each of the sectors labelled NS or R generated by defining String Phenomenology 8.4 Then project it out with the single of section 6.4.4.supersymmetric Note side that the fermions (from have to the transform invariance in of the same the way as supercurrent the term boso invariance constraints (i.e. for consistency of one-loop amplitudes again) The PoS(cargese)001 (8.17) vitinos (it Steven Abel so the ends ) projection 16 gravitino 0 ) an be written σ W = ( 1 Z 4 Z amental region is ing. In addition to xed point. This is d the red and blue − different bits of the transformation. The an move throughout , by all our folding and t some of these states 2 i isted state at the origin )= 0 Z |



the GSO ( π 1 us. After the orbifolding,














ˆ µ − + α





fold along dotted lines fold along dotted σ





1 as desired. We return to the ( × 1/2 and translations to map the red Z R = 2 i a Z | N







=





A 4 gravitino multiplet we found on the Z 3) Initially we had 2 L .. i = 0 0









| is the transverse zeroth mode of the 4-D











































































































































































































































































































































































































































i 1 ˆ = 0 0 i/2 µ N − b i α −> × R 4 to 56 i









.. 1 and 0 | , 1 4 0 N = ) = A 3 0 i i GLUE b ( correspond to the 3 complex internal dimensions (6-real), N 3 3 N , 1 Z ) 2 3 indices of the 10D gravitino). (Note for this discussion, the , 2 0


















, 0 1























































































































b 2 b twists for some of the compact dimensions, so we show it exactly ( , ), with the string endpoints separated by a 2 2 2 1 N , Z ) Z 0 1 0 b B ( = 1 N µ )

0 0



b ( −> ’s has changed from i 0 b Untwisted A



i

































































































































































































































































































































































































































twisted here is the 4D space-time index since we are interested in the number of 4D gra ˆ As we have seen untwisted states are the original states before the orbifold µ model and consider the gravitinos. Begin with the 3 and blue parts into thefolded region up, remaining the in all the stringshown second endpoints in figure). rejoin the forming third When a figure,regions the are closed where on fund string the the top. around black Also the regiongluing. shown fi is is Thus an on untwisted the state the twisted which bottom states isthe unnaffected compactified live surface space. at an fixed points, whereas untwisted ones c degrees of freedom which gave us 4 gravitinos in 4 dimensions, but after do not meet before the orbifolding,however, and we these cut states do away not half exist the(which on is torus the a and tor fixed fold point it of up. the The figure shows a tw labelling on the like the 2D example. Twisted states will have boundary condition and The orbifold in the figure has String Phenomenology black, red and blue regionsfundamental show region upon portions orbifolding of (i.e. the use string a that combination are of mapped into where the excitation numbers can be torus coming from the states in the R-NS sector. To recap and simplify a little they c the introduction of new twistedout. sectors, As the we effect shall of now orbifolding see, is this to can projec break the supersymmetry to space time degrees of freedom, Z corresponds to just the PoS(cargese)001 . ′ 8 45 are E (the ). In i 0 = 8 × 2 b (8.23) (8.21) (8.20) (8.24) (8.22) (8.18) 2 6 E W / E 9 + of . projection × 0 Steven Abel ) 3 3 W Z ( 248 , but the 3 Z SU and 0 nstraint on how the ing the 10 → W ′ 8 E ates from the gauge side × 8 riant under the orbifold ac- projection in the gauge side R E 3 i 0 Z | altogether giving † . 1 2 J ) ) ˜ . We then apply the d + i † is 1 2 16 I R λ iX ( ˜ i , 3 b e R 0 Z − i | + SO λ = 1 0 odd | † ˆ , 1 0 (8.19) µ − I I 3 , , 1 2 Z invariant states that remain that in addition .. of = 6= α 6= 0 1 ( 1 1 2 J J − 3 ) 4 ˜ ˜    = v d × i , Z = = − ψ N does not transform under the v † 2 L , 1 2 1 2 I 128 λ 4 0 v ˜ 1 i ( + d I − ˆ − i µ N N 0 − | 3 π 57 ψ ) + 2 α N = 3 0 rotation e † , which came from two sectors × I 1 0 0 0 1 0 0 0 b 3 8 3 + 2 0 L 6= N Z 1 2 E → 2 J i b    ˜ 1 0 b 0 ) N | = † and hence commutes with it. To see it explicitly, consider b 1 2 + I 1 2 2 . However let’s now apply the condition that the states are + ) ˜ b λ i 6 + − 3 1 ( N , K 0 0 ( ψ − N ) b × , so that the only 6 λ ( i so this became 8. We then interpreted this as four 4-D gravitinos L ( , SU iv i s Z π ′ 0 , the latter a spinor ( 2 | SO b ) e 1 2 i operation means that we must leave the first 6 rightmoving fermions as − 16 ( 3 orbifold action. The ψ Z 3 SO Z . The 3 complex right-moving fermions we have singled out must be written as ) 10 ( 16. Now the .. SO 1 1 supersymmetry. = I = When we add an orbifolding, modular invariance provides an additional co N former gave adjoints of the NS-NS sector, the states are of the form adjoint of complex fermions. They can appear in excitations as right-handed (gauge side) behaves (totion). ensure In the fact torus the diagram constraints can is be inva satisfied by again embedding the satisfy the GSO projection have where three complex ones, so only the ten remaining ones can be written this way, giv This projects out some of the gauge bosons and results in the breaking (there are other possibilities).and That fermionize is them we into separate complex out fermions 3 with, bosonic coordin the gauge boson degrees of freedom of the simultaneously on both sides. That is the total action of the of all rotated by the phase factor Indeed this is evident from the fact that the is one of the Cartan generators of also invariant under the String Phenomenology we had to impose a chirality projection, This leaves only the 2 degrees of freedom in the spinor representation of leaving only an odd number of PoS(cargese)001 . t 3 e 1 ) , Z 3 0 any ( pro- gen- ˜ d ) = (8.25) (8.26) ) ˜ 0 b SU 3 derived hen the 8 ( W sector N 2 1 dependent time I will 2 Steven Abel SU ,.. . Verify that − 4 j W X gauge bosons Now we must rom the string ) N v + 1 ( 78 0 + (since the phases n have vanishing since j 3 W ) / i N 3 π ( s contract the path to 2 = e that are homotopically though a gauge rotation j SU that one can use on non- roup that commutes with Q ’s corresponding to the first ) ) of spinor representations of 1 . From the ( X 16 ) 1 + together form the ( possibilities, but the GSO ( ) U 16 5 states are projected out, but for 10 2 ( ˜ ’s transform with the opposite phases to d ˜ × d d to be the charges under U SO µ 3 when constructing the states). Noting that ) µν dx 3 × / on the gauge side where again i µ ∆ and Q X ) the A R π µν ˜ ) b i C under the three U ± 2 F R + ˜ 1 b 0 e λ ( ’s) must be orthogonal to | 1 2 58 2 ) ) ≡ , often called Pe 8 U Q ) 1 − → 3 N ( 1 j ∼ 8 / ( + . i v ˜ b U U 6 1 π U U + ... Q 4 4 j ( 1 theory with chiral fermions, the next task is to break the N − 1 3 4 N ˜ b = = = )( In general, when we construct the path-ordered product of gauge j X 3 0 . ˜ N b 6 so that 2 0 E ˜ b ] however. The fermions all have a phase factor gauge bosons of 1 0 π 3 ˜ b 2 Z 45 , , we have a gauge rotation + 0 C [ + 1 ( 3 labels the three complex world sheet fermions. The states have the correc excitation or not. There are then 2 ∈ , 16 0 2 , b + 1 and so far we haven’t done anything except separate out this factor. for the gauge bosons of E 16 3. Note that without the orbifolding this gives 3+3+9=15 states which is just th = ) 2 1 invariant state, and we have 9 states in all. 8 of these form the adjoint of j .. 6 However, on a small path the Wilson loop becomes (by Stokes’ law) 3 ( 1 − . , Z is the area tensor of the closed path. Thus, on simply connected manifolds, w C 1 = SO , is matrix valued. This quantity is gauge invariant and consquently should be in The 1+ J 0 µν , . I ∆ U ) . The charges of the states can be calculated using the expression ’s under the orbifold action, we find that all Consider the gauge group The situation is precisely the same as when a cosmic string is formed. Far away f Once we have achieved an 6 10 gives a b ( E J . , elements around a loop, of the path on the physical states apply the invariance under Exercise: calculate the charges Q charges. the gauge field strength vanishes. However the gauge fields themselves go adjoint of three complex fermions. Define Q curvature vanishes, we expect Wilsona lines point. to be This unity iscurvature since not tensors, we the but can case Wilson alway onequivalent loops non-simply to that connected non-trivial are cycles. manifolds, not where The equal we remaining to ca gauge unity group is for the paths subg where where gauge group down to the Standardsimply Model connected one. manifolds A is particularly Wilson useful linegive trick breaking. a brief Since outline. it will crop up from time to earlier, where String Phenomenology where SO jection removes half of these, leaving two chiralities (i.e. a The trace combination (i.e. the sum of the they are (by the definition of the mode expansion of the the 8.4.1 Further gauge breaking Wilson lines of I invariance leaves only corresponding to a erators are traceless, hence it is an extra U PoS(cargese)001 . ) 3 ) × = ) 1 1 ( c 3 R ( ) ( ) ) but U 3 ) 3 U (8.29) (8.28) (8.27) ( ( 3 SU ( × 0 for the ) SU SU ch use for 1 SU Steven Abel 6= ( × em to one’s anism for the c ν L U along internal ) A ot continuously 3 × with some other al point of view. a ( bc ) f ld is consequently 1 es from the Green- b ( SU λ arise as follows. The i U p the Standard Model × field along the Wilson avity and gauge fields. c × ∝ ) e the possibility of using L 3 ν a . m ) ( A 2 y    ( . The masses of the broken SU ∂ 1 U − SU gauge group down to δ 0 6 1 × E c − ) γ 3 . Thus ( ) 0 0 δ , are all cube roots of unity, the three fac- b ν b γ 0 0 0 λ SU δ i A , ) 2    γ µ = ´ of the group which contains superfluous , a × c ν b y A β A A ) space, we can identify the higgs fields that are ,    c ab 6 a bc f 2 α 59 f rank K − 0 b − subgroups and we have β (i.e. λ i µ ) ) 0 0 ³ 3 β ∂ 0 ( c When b ( 0 0 0 . δ β µ c ( Y SU A )    y to be a cube root of unity (which is an element of 1 a ( bc f × α b ) U λ i α × e L ) = ( 2 0 )= ( y U ( SU µ a A × c ) which is playing the role of the higgs field. ) b y and its subgroups. We can specify the Wilson line element in terms of the 3 A ( R ) 3 SU ( such that SU y × to be n’th roots of unity we find the gauge group L . If we instead choose only ) 3 At first sight (i.e. perturbatively) only the Heterotic models seemed to be of mu Since the Wilson lines are in the internal ( As an example, let us see how Wilson lines can break the 3 Z ( / β δ 6 , model building so let usThese first models discuss seem those to models be fromD-branes singled a was out more appreciated), for phenomenologic they phenomenology alone because (befor seemed to contain both quantum gr factors. This is a commonmechanism. feature If of for heterotic example theories. they TheySchwarz are must anomalous anomaly be then eliminated cancellation they get mechanism. Stückelbergadvantage mass by Of for course example trying oneYukawa to couplings. might implement also some kind try of to Froggatt-Nielsen use mech th 9. Phenomenology of heterotic models 9.1 Weakly coupled models Note that the Wilson lines have not reduced the responsible for the breaking from the 4-D point of view. Schematically they Yang-Mills terms are functions of the covariant derivatives We choose them to be ofgauge group, this form because at the very least we need to kee which is non-zero for the gauge fields that do not commute with where the indices rundirection over all ten space time6-internal dimensions. indices. The 4-D Let Lagrangian there now gets be a a mass-squared Wilson ter line γ String Phenomenology if we take a test particlenot on simply a path connected, around the the obstruction cosmiccontract being string. the the path The to vacuum cosmic manifo zero string through itself the (i.e. cosmic string). we cann subgroup; SU gauge bosons are proportional toline the (in VEV this case of the component of the gauge tors clearly all commute with the three E PoS(cargese)001 , ′ 8 ′ 8 in E Z , 8 Z orbifold , ompacti- 7 e [16]. In 3 Z actification Steven Abel Z , ′ 6 Z , identification is ave deduced for 6 wn to something e space-time into enough to restrict Z ally, in the simpler , ring amplitudes. As . However the type oup by the compact- 4 Z y) in the effective firld ssibilities are relatively thing about this part of es with some singulari- re the numbers indicate ity degrees of freedom. used for the a high mass scale (much ation of supersymmetric ectively). In addition far ersymmetry breaking by i-Yau. In addition the great ry; hence the gauge groups ely induced contribution to e nett number of generations gauge group. In order to get ) depends on the orbifold in ′ 8 G E hidden breaking) [17]. Early applications × × 1 space-time supersymmetry ((2,0) 8 E = QCD Λ N MSSM −→ contain gravity multiplets and no gauge fields. as derived above – the latter turns out to be dual ′ 8 60 ) E × 32 only ( 6 E are possible. × M Z or SO G ′ 8 × E N −→ Z × ′ 8 8 E E × 8 E factor is already a potential Grand Unified group whereas the second 6 example. As we have seen the 16 additional internal degrees of freedom E 3 Z orbifold of the previous section, the possible orbifoldings are is that it has to be of a certain type (namely Calabi-Yau) [15]. The orbifold c 3 Z 6 K . The prime indicates the same point group but a different compactification lattic 1 supersymmetry in 4 dimensions, the the most general requirement on the comp ′ 12 The modular invariance conditions then require a breaking of the gauge gr Model building in heterotic strings concentrated on the Let us summarize and generalize the phenomenological properties that we h Z = , 12 to string motivated scenarios werethe discussed story in is ref.[18]. that the The effectthe remarkable of superpotential gaugino that can condensation be is determined atheory to non-perturbativ [19]. all (For orders a (thanks review to of holomorph supersymmetry breaking see ref.[20].) example discussed above, namely to embedthe the gauge geometrical degrees orbifold of action freedom. on This th leads to a gauge breaking such as models) can be constructed.resembling The the further Standard symmetry Model breaking by isembeddings Wilson extremely the lines unconstrained. first do Phenomenologic factor forms a hidden sector group.for example The the latter condensing of is the a gauginolike potential of the source some condensation of hidden that sector sup takes group place at in QCD leading to a The precise gauge symmetryquestion. breaking This pattern route became (i.e. known as the therestricted. “standard subgroup embedding” Standard and embedding the generates po thethe so-called supersymmetry (2,2) of the models world-sheet (whe CFTless on restricted the asymmetric left and embeddings right which sides still resp have ification. One attractive route of gauge breaking is to adopt the approach addition product orbifold groups Z fications of the previous section areuseful singular because it limits means of that Calabi-Yau many manifolds. propertiesfor of This example) the can effective be theory derived (th from theadvantage topological of properties of orbifold the models Calab isties, that one because is still they able are to essentiallywell use flat as conformal field the spac theory techniques to calculate scatte N manifold to the SO(32) of the type I models). the bosonic half become gauge degreesin of 10 freedom dimensions in the end effective up theo them being still rank further 16. to be (Indeed either anomaly cancellation alone is our quasi-realistic Heterotic theories are also closed strings,and but bosonic as string they theories are they a contain curious fermions combin and both gauge and grav String Phenomenology Gravity, being a spinII 2 models field, are requires also closed ruled strings out which because rules they out type I PoS(cargese)001 n- tring (9.4) (9.2) (9.1) (9.5) (9.3) gauge 8 E ings there- Steven Abel re possible to all degrees of eory . In terms an intermediate GeV). ..., 16 ergravity compactified s is of order (assuming eas gravity lives in the + 10 vity degrees of freedom 2 the radius of the orbifold trings. All closed strings the Planck scale, but that . i F 4 × Pl , in terms of 4-dimensional 3 Tr M 11 1, imply, in the case that the 6 g R ∼ V √ . 3 x ´ runs over the two 10-dimensional GUT 10 YM i d α M 2 GUT , they read Z α H 3 . ³ λ / 6 2 . / , ) 1 8 s = 6 6 s 2 1), the following relations between the com- 11 V 2 3 H M M / 2 11 λ 6 1 2 6 H ∼ R . πκ 3 V V λ 2 H Pl 61 ( ∼ say. The Planck scale and the gauge couplings can , λ ∼ π M 6 6 / 4 2 Pl 1 V ∼ YM i − i.e. s M α ∑ ) 6 heterotic theory is only tractable thanks to the fact that, M − V ′ 8 E gR GUT × √ α 8 x 2 E groups live. Compactifying down to five dimensions (with a com- 11 d 8 and the radius of the 11-th dimension, = ( E 9 Z 11 / 1 2 11 M − 1 κ ) 2 , and the heterotic string coupling, 2 11 s = M πκ S 4 ( ) and then to four dimensions we can write the fundamental 11-dimensional co 6 π orbifold. Based on anomaly cancellation arguments they argued that an V 2 2 is the 11-dimensional gravitational constant and = Z / 11 11 1 κ S M The 11-dimensional action takes the form One way to address this problem is to go to the strongly coupled limit of heterotic s Let us now turn to the question of the fundamental scale. In heterotic models, The models that arise from the weakly coupled heterotic string with simple embedd is larger than the compactification scale of the 6 extra dimensions. It is therefo 11 consider the compactification of this theory down to5-dimensional 4 model dimensions compactified in two on steps, an with orbifold. group lives on each of11-dimensional the bulk as two sketched 10-dimensional in Fig. orbifoldR 7. fixed In planes the wher case of strong coupling, stant, pact volume heterotic string remains weakly coupled ( where as Horava and Witten showed [22], it is described by 11-dimensional sup on an theory. The strongly coupled These expressions, together with the experimental fact that fixed planes where the two 9.2 Strongly coupled models and freedom in the perturbative model arecan the travel result everywhere of excitations in ofnecessarily the closed feel s compact the space same and compact volume, so both gauge and gra String Phenomenology then be simply computed from the dimensional reduction of the 10-dimensional th of the string scale, fore suffer from the problem thatthe the unification natural scale unification as scale derived for fromthe them the MSSM is RG with running a desert of between the the gauge weak coupling and GUT scales) quantities, pactification, string and Planck scales [21]; PoS(cargese)001 (9.7) (9.6) aningful factor lives, Steven Abel roton decay the Standard 8 terotic string. E e fundamental t for the rather er of the Planck ton decay. There ale while keeping mely the apparent GeV. This accurate g problem; namely heterotic string. The is entails extremely ibilities for orbifold 16 8 E wever such a low unifi- 10 e the problems we have × where each × 8 3 2 E Z ≈ / 1 S , f the GUT 8 M GeV E c N 16 , c 10 e orbifold E c , ∼ T L , , c T , D GUT D , 62 c M H , U which also looks too good to be a coincidence. The ∼ , U 6 ) H Q / 1 : : 10 − ( 6 V 10 16 SO ∼ 8 11 GeV. E ’s of M 15 16 10 − 13 10 ∼ 1 − 11 R are undesirable Higgs triplet superfields. On the other hand any kind of me c Horava-Witten construction for the strong coupling limit o T , T There has in the past couple of years been renewed interest in the poss First I should re-emphasize the apparent success of the minimal MSSM, na At first sight this seems to offer an easy way of uniting the string scale (i.e. th Thus the heterotic string can accommodate both a fundamental scale of the ord where identification of the SM particles is unification of the string, weak and hypercharge forces at about complete and reasonably minimal unification atis that an scale additional serious results problem in for toohow unification, rapid to the pro drive doublet-triplet the mass mass splittin the scale of doublet the components triplet light, higgs anothercomplicated fields form higgs to of sectors. be hierarchy of problem. order the Solving GUT th sc unification looks too precise toModel be fall just promisingly be into coincidence. In addition the multiplets of scale of gravity) with thecation apparently scale successful appears unification to prediction. be Ho minimal too models low that to would be be consistent consistent with with the proton HW decay set-up9.3 limits (see ref.[23]). at New leas orbifold GUT models mass in the weak coupling limit, and of the GUT scale in the strong coupling limit. and therefore model building, mainly focussing(or on the attempts lack to thereof) by solveencountered using the with non-standard the problems embeddings. simplest presented attempts Let to by incorporate us GUT p summariz model building into the he 11-dimensional supergravity propagates in the bulk. It is now possible to have green planes represent the 10-dimensional boundaries of th Figure 7: String Phenomenology PoS(cargese)001 and (9.9) (9.8) ) (9.10) (9.11) 3 ( . In the } ) SU Steven Abel a , 2 this model is ¶ iW G π plane invariant. 4 17 eous phase shift 2 y, and is directly s “GUT” sectors ) ( 3 4 17 can explain why an ( er (unified groups), exp − { , w constructions based SU ist is embedded into the ld models, the spectrum 4 21 uctions in field theory. m). In order to do this we diag − , . = 4 25 a ! U − 5 2 two torus factors are compactified at , , 6 4 19 . ¶ (the suffix indicating the order of the twist leaves the − 2 5 , ¶ 3 , 3 . 2 − 0 v ¶ W 4 Z , 21 such that 3 µ GUT scale. 1 2 1 , , a − = − , 3 1 6 and W , 17 2 4 v 2 4 3 − 25 à , , , 63 W 1 6 − 3 1 ¶ structure after orbifolding. Thus one naturally finds v , , 2 ′ 1 3 8 − 2 3 , 4 E 23 µ = . Specifically, the orbifold action is given by the vector 2 1 compactification lattice is the product of − 6 3 × , = 6 − v 8 0 , ¶µ v T E 3 µ 2 1 planes respectively, given by 0 ) , ! − 5 3 1 2 µ ( , ¶ twists; plane invariant and the 1 2 6 1 = 2 SU , ) Z − 2 1 V 4 ( µ − , and , and 1 2 ) SO 0 , 3 4 , ( Z 1 6 1 2 − − SO µ à = = orbifold of ref.[24]. The 2 3 II W W − Compactification lattice for the model of ref.[26]. The first 6 twist leaves the Z root lattices as shown in figure (8) / 3 6 ) Z taken from ref.[26] with kind permission of the authors These apparent contradictions have been addressed in a variety of ne As an example consider the model of ref.[26]. The compactification lattice for 4 6 T ( Wilson line) in the model of ref.[26] there are two Wilson linaes The Wilson lines are given by a set of 16-vectors on orbifold models [24, 25,retains 26]. a “memory” The of central the observation underlying is that in orbifo a theory that contains “bulk” sectors with larger symmetry together with variou located at the orbifold fixedbut which points are which missing fall troublesome into statesapparently such representations unified as theory of higgs may the triplets. lack larg This theanalogous structure GUT to mass (and states was that inspired mediate by) proton the extra-dimensional deca orbifold constr the string scale. The third torus is compactified at order the The which generates both Figure 8: String Phenomenology SO The orbifolding is embedded into(using the the fermionic gauge formulation degrees for of thebosonize internal freedom the gauge 16 with degrees internal a of right-moving freedo simultan degreesgauge of side freedom. by The the orbifold vector tw the PoS(cargese)001 (9.12) Steven Abel one extends give an hon- imensions. Com- ification, the third ts and shifts act on t preserved symme- ent and get VEVs to eed since those rules rmions. Including the ribed with an effective mechanism. edom on both left- and change in numbers, the ent models are defined by he model of ref.[26] above the GUT . ] ] 32 32 u v ,.. ,.. 11 11 v u factors that are anomalous and hence heavy by ; ; ) 1 10 10 64 ( u v U ,.. ,.. 1 1 symmetry. The fixed points have the symmetry further v u ) 6 ( = [ = [ . The first two tori are compactified at the fundamental scale. SU V U 1 − GUT M theory. This effective orbifold GUT theory is shown in figure 9. 2 Z / 2 T The effective 6-dimensional field theory approximation to t factors. In order to accommodate the successful prediction of gauge un 8 E Before closing this introduction to closed string phenomenology, I would like to The bulk contains an enhanced Again there is a Lorentzianrules convention for for model dot-products. building Apart are from essentially the unchanged from the 10D ones (ind orable mention to the so-calledthe fermionic fermionic formulation formalism of which I the usedpactification heterotic to string. to derive the 4 In 5 dimensions this perturbative leavesright-moving models sides. 6 in 10 These superfluous d can bosonic be degreesdegrees fermionized of of into 12 freedom fre real that (or were 6boundary already vectors complex) there (for fe in complex fermions) 10 of dimensions, the the form differ the Green-Schwarz mechanism. Some additional SM gauge singlets are pres generate the required Yukawa couplings of the SM via the Froggat-Nielsen 9.4 The fermionic construction in 4D projected by the twisting of the third torus. The intersection of all the differen tries is the SM gauge group with some extra torus is compactified at a scale 6-dimensional Thus above the GUT scale and below the string scale the theory can be desc Figure 9: scale. String Phenomenology Note that these models areboth non-standard embeddings in the sense that the twis PoS(cargese)001 ; ) ; 20 (9.13) (9.14) ) u , 20 v 19 , u Steven Abel , 19 v 18 , u 0D world-sheet ( 18 nough to specify , v ) ( , that 17 ) u , 17 . e models should corre- , which consists of five v is fermionized into two ndition vectors are then 16 6 , , so that the supercurrent of freedom of spacetime. u .. , 16 f conformal field theories, X X t. For this it is more con- 1 v ∂ 15 , u = 15 ∼ ( . i , v 2 2 ∀ ) ( + , ψ i 14 ) 1 3 u 14 ψ ψ , i v 1 , 13 + i u 13 3 , v ψ , 12 i 3 u 12 ( ψ v , ( 1 mod(1) ) , 6 = ∑ ) i 11 2 i u 11 65 + , i v + 3 , i 10 v u 10 X , + v ∂ 9 , i 1 in the language of ref.[9]; in order to be consistent with u 9 + ψ ( v i } , ( 2 3 , 3 ) , v 1 8 ) b ∑ = , 8 u i + 2 , v i , 7 b 3 = 7 , u v v 1 , + , 6 b = J 6 , u v 2 ( S and transforms into plus or minus itself on parallel transport around the ( , v , , i ) 1 ) 5 say, we have an SCFT identification = X { 5 u v ∂ 1 2 , , i 4 v . 4 ψ u ψ ] v ] , , 3 64 64 3 = u u v v and ( + ( , 1 , J ,.. ,.. ) ) ψ 2 2 21 21 u v u v , , 1 1 u v A particularly fruitful choice of boundary vectors is the set of ref.[27] = [( = [( V U world-sheet in the R or NS sectors respectively. When a single boson is vectors usually denoted real fermions The model building rules are therefore augmented with the “triplet constraint” String Phenomenology were derived from the requirementthey of could modular hardly invariance be of anything products else).spond o There to is compactified one 10D additional supersymmetric requirement: models.that th In the order choice to of ensure boundaryvenient this to it condition is express leaves e the the theory supercurrentexpressed in rather invarian terms laboriously as of real fermions. The boundary co The two first entries on the left-movingThe side are remaining the two fermions transverse degrees onsupercurrent the is left-moving side are grouped into threes. The 1 PoS(cargese)001 in the (9.15) which [28]. I ) 5 2 ) ) 1 4 10 ¸ ¸ 1 Steven Abel ( ( 1 spacetime ( 16 16 down to one ompactified” ) ) rs lead to 48 ¸ ¸ ¸ ) U = SO 2 2 1 1 16 16 16 ( ( × 10 ) ) ) ( grees of freedom ) N 10 10 on the gauge side 1 1 1 2 2 2 f e.g. s of henomenology see ) ) 2 ( ( ( ( 2 2 1 1 SO 10 10 10 10 ious subgroups. Note ( ( le) Higgs triplets and ; ) ) ) logous to the standard , , SU ) d Model with 3 matter 0 0 0 ) ) ( ( ( lent of 1 2 1 1 2 2 × ; ; ; , , , , , , ) ) ) ) ) ) ) 2 1 2 2 1 1 3 0 0 1 1 2 2 1 1 2 2 , ( , , , , , , , , 1 2 1 2 1 1 2 2 1 2 0 0 2 1 1 2 , , , , SU , )( , ; )( )( 2 1 2 1 1 2 1 2 0 0 ) 1 2 1 1 2 2 0 , )( )( )( , , )( )( )( , 2 1 0 0 1 1 2 2 2 2 1 1 2 2 1 1 0 , , , , , , , , , , 1 2 2 1 0 1 2 0 0 2 1 1 1 2 2 1 2 , , model. The three additional projection , , , , )( b group factor. The gauge group is then )( 2 2 1 1 )( )( 1 1 2 2 0 0 ) 8 0 2 1 1 1 2 2 , E )( )( )( )( )( )( , , , 0 44 0 0 1 1 2 2 , 1 1 2 2 ( 1 2 2 2 1 1 , , , , 0 , , , , , 1 2 1 2 0 0 1 2 2 1 1 2 SO )( 1 1 2 2 , , , , , , 0 66 1 2 1 2 2 1 2 1 )( 0 0 , )( )( 2 1 0 2 2 1 1 )( )( )( )( )( )( , , , , 0 0 gauge group factor. This is broken by the projections 2 2 1 1 1 1 2 2 0 , 1 2 , 1 1 2 2 , , , , 6 , 2 1 )( , , 1 2 0 0 1 2 2 1 E , , 0 2 1 , , 1 1 2 2 , , , 1 2 1 2 2 1 2 1 0 0 0 )( )( )( , )( )( )( 1 2 )( )( )( 2 2 1 1 0 0 , 0 1 1 2 2 , , 1 1 2 2 , , )( , , 1 2 , , 1 1 2 2 2 1 0 1 2 0 , 0 2 1 , , 1 2 , , , , , , 2 1 , 2 2 1 1 0 1 2 2 1 1 2 1 2 0 0 , )( )( )( 0 (16 from each sector) – these will eventually lead to matter. )( )( )( )( )( )( is a supersymmetric 1 2 1 1 2 2 ) 0 0 1 1 2 2 )( 2 2 1 1 , , , , 2 , , , , 0 , , 1 2 1 1 1 2 2 2 10 1 , 2 1 0 0 2 1 1 2 , ( , , , , 0 , , . The untwisted sector gives rise to vector-like pairs of , W , , 2 1 8 1 1 2 2 2 2 1 1 1 1 2 2 0 0 0 ( ( ( ( ( ( SO E ( ( ( ( , , , , , 2 2 2 2 × 2 1 2 0 0 0 3 0 s of ) £ · · · · 6 16 ( = = = = = 1 2 3 4 0 . Finally the last 16 fermions generate an SO ) W W W W W × 10 ) ( The second stage of the model adds at least 3 more vectors to project the 10 ( SO can play the rolemultiplets of in Higgses in the Standard Model. The 3 sets of twisted secto will not show the vectorsare explicitly, complexified but (into suffice 5 to complex say fermions) that and the given 10 boundary fermionic de conditions o SO of its subgroups, a particularly interesting possibility being then to supersymmetry on the gauge side is an vectors cut down the gauge groupthat (without the cutting the space-time rank) side intodegrees is the of the embedded freedom obv are in simply theembedding copied gauge of over side to the the orbifold in right-movers. models. theplay the This sense In same is role addition that ana as 10 the the transverse of fermions 18 on the the “c world-sheet left-movers; the fermions equiva final boundary vector. Thegenerations final remaining models from can the be originalsuitable very 48, Yukawa couplings. close naturally For to heavy more details the (i.e.ref.[29]. and Standar a string complete sca discussion of the p The theory with just String Phenomenology our earlier introduction, I will continue with the notation of ref.[7]; PoS(cargese)001 . 4 M Steven Abel m the 4 di- can be chosen riants of string ttached. If we 9 for type I and 6 ’s point of view , ackground. K space is the same 5 If there are a few e II theories which -branes are added. al theory which as olume giving them , the following way. nd fully dynamical, ivided by the string ticular models, they a single point in e of some consistency ways as in Figure 10. ble feature of D-branes erator which represents s [30, 31]. These can be es of freedom are included space that we observe in order 4 gauge group. In order to reach a M ) 9 for type IIB, 1 , 2 ( 7 , U 5 but have 4 internal degrees of freedom , 4 3 , M 1 = p 67 4 theory with = 1 configuration, the compactified space 3 the branes appear as points in the compactified space.) N = = p N D-brane realization of a U(2) gauge group. 3. (If ≥ p Figure 10: 8 supersymmetry in 4 dimensions (if the compactified space is toroidal) ends up = N 4. We thus end up with an 8 for type IIA. The interesting feature of D-branes from a model builder = , dimensions on their world volume where 6 N , p The arrival of the large extra dimension idea stimulated interest in the other va It also turns out that the strings have to have an excitation from the brane v What do we see when we observe this from 4 dimensions? Remember that fro Before we start throwing branes together at random, we need to take car 4 , 2 , 10. Open string models: string at singularities String Phenomenology 0 theory as model building tools. In particularhave attention in turned their to the nonperturbative type spectrum Ibuilt and objects like typ known monopoles as from Dirichlet the brane with effective a field typical theory, and surface are tensioncoupling). membrane-like and a As a we width argued ofhave earlier order based the on fundamental the scale types (d of R-R fields in the par in the perturbation theory by addingthe “Chan-Paton emission indices” of on such the a vertex state. op corresponding to the adjoint of U(2). As we shall see later, these degre is that open stringsAssociated can with end an on open thembranes string and together, end this the point can is index generateconsider an simply two gauge index, labels branes groups for the the example, in Chan-Paton branes the index. endpoints to can which be attached the in open onemensional string of point 4 is of a view weeverywhere. need In to particular arrange the things brane such must that be the lying compactified in the large Given this, the open strings may freely propagate in for the open string to be able to travel along it (otherwise it would be stuck at So the branes must have a Lorentz (gauge boson) or internalis (matter that field) they index. break Finally onlywe a half saw remarka the has supersymmetry (i.e. they are BPS). The origin being more phenomenogically interesting in such a way thatThe the simplest supersymmetry (i.e. is most calculable) already way partially to broken achieve this before is the to D use orbifolds as the b PoS(cargese)001 (10.1) Steven Abel with all the 6 ted by gravity K In a large extra as “bottom-up” me suppression nd form part of ancelled locally. dditional branes lar the details of R fields that are nd the untwisted string states they 1 supersymmetry of each other at a d-Ramond tadpole ellation are usually present for example = tent. One has to be ed space is rather di- of Ramond-Ramond s, for example when M configuration. We luded consisting of a ups, particle content, N point of rge, and couples to the interesect (for example es around, for example, tual properties are inter- rsymmetry breaking and locally supersymmetry is ). We then need to satisfy (as of course we require if 4 int of view of phenomenol- oidal compactification they 4 M ve to do with supersymmetry iated with objects far away in M the flux lines are all absorbed. . GeV 11 10 ∼ Pl 68 M W M p ∼ I M fixed point far away in the compactified space. The communica- other , but with their world volumes filling the whole of 6 K The bottom-up approach begins therefore by focussing on the local MSS These requirements led to an approach to model building which became known the open strings on their world volumes are able to travel anywhere in the requirements of local(D7 RR-tadpole branes cancellation. for example) That such thatpreserved. is the we “twisted” This need RR-tadpoles puts to cancel a but add constraint in on a the angles at which the branes can assume an intermediate fundamental scale of necessary elements to make up the standard model gauge group and leave in the visible sector.single This point can in for example be a set of D3-branes lying on top This scale is familiar fromand had the been hidden suggested sector earlier supersymmetrymediation on by breaking more gravity communica general [33]. grounds First a to set do of with D-branes supe is included at some fixed tion to the visible sector thenas has that to felt be by through gravity. the This bulk,D3 and is branes will shown localized get schematically at the some in point same Figure in volu These 11. the are The compactified space points chosen with re in twisted RR suchRR flux fields a c live way in that the theuntwisted bulk RR visible of flux the sector cancellation compactified are is space. less the well These MSSM. determined. details Gravity and a in particu [32]. Consider what are the important featuresogy. of The any model leading from factors the po arenumber those of generations things and that so have on. tobreaking, Secondary the do factors are cosmological with things constant the that etc. gauge ha twined gro The with latter gravity. are As things such whosedimension their even set-up, influence the on correspondence phenomenology with is therect. less configuration in The important. the primary compactifi factors have tosome do orbifold with fixed the point, local whereas the arrangementsthe secondary of factors bulk D-bran are of all the assoc compactified space.collection of For branes example at a some “hidden” sector can be inc String Phenomenology conditions. The most importantconditions. of As these we for saw D-branes everyRamond-Ramond are D-brane fields has the of a famous the “Ramond-Ramond” Ramon do closed (RR) not cha string care spectrum. about thepropagate Since presence throughout these or the are otherwise entire closed of compactifiedthe the volume. compactified D-branes. Curvature space singularitie In is aconfined tor an to orbifold, the introduce fixedthe point. a gravitational second spectrum. The type However RR they offields fields differ must “twisted” in behave the be R like respect absorbed gravitons thatcareful in and flux therefore a lines dilatons to compact choose a space theOnce otherwise arrangements of the this D-branes theory requirement such is is that satisfied inconsis satisfied, as other well. requirements such as anomaly canc PoS(cargese)001 (10.2) (10.3) effective , since the Steven Abel ct volume 3 y therefore, − p . The precise I V e clear. The ad- hoose the funda- M absorbed by anti- hidden sector, and ∼ effect is the same as s ponsible for the large M t. A consistent set-up is lly break supersymmetry. ctor without affecting the anes elsewhere in the bulk and make sure those fluxes ctive 10 dimensional type I are of the local consistency points represent the local config- en blob represent the global structure, less , . 2 2 s P 3 I 6 M M − p λ 2 I V λ 69 ∼ = p 6 K α V just K just lives here gravity -dimensional brane. The gauge interactions are proportional to the string p on a p α Schematic picture of the bottom-up approach. The small blue is the string coupling. To get an idea of what this has to be, we can look at the I λ The reason for the particular choice of the intermediate scale can now be mad ditional ingredients required to ensure globalSince tadpole it cancellation genera is only thehidden global sector configuration supersymmetry that breaking breaks communicatedmental supersymmetry, by scale the gravity accordingly. net and In we other mustPlanck words, c scale the and volume of the the dilution bulk of can be supersymmetry res breaking effects only if dependences on volumes can be derivedaction from to 4 the dimensions reduction [34]. of We the begin effe with the Planck mass relation to the total compa gauge bosons are free to roam anywhere in this volume. Hence uration of D-branes leading toimportant the from MSSM a phenomenological whereas viewpoint. the large gre that the D7 branes intersectconditions, at however one right should angles). also take This care arrangementcancel of as the takes global well. c RR-tadpoles This can beor done by may adding be other D-branes done andthe in anti-D particular br some way other in which way.consequently the the global soft From supersymmetry tadpoles the breaking areshown point and cancelled schematically of cosmological affects in constan view only Figure the of 12.branes, 4D This but phenomenolog figure the shows set-up the canMSSM global be set-up RR entirely directly. flux different being away from the visible se where coupling but are diluted by the volume of the branes in the compactified space, gauge coupling Figure 11: String Phenomenology PoS(cargese)001 . 6 K use (10.4) (10.5) Steven Abel same factor acts like a loop I λ ndence s-squareds communi- e dimensionality of the 4 M so that from the above, and W brane). Any process we care to M ts of 3-branes at a fixed point in p hidden ∼ are one-loop and enhancement factor arises from the 3 − SUSY p SUSY . to cancel local RR-tadpoles. Global absence , V M p 3 3 p M − − − − P p 9 p 9 V V M V V in the bulk, or possibly something else entirely. 2 s 2 s W M M M 70 6 ∼ = ∼ 2 K visible P 2 s , we also turn that dimension from a brane dimension i M M R p 2 SUSY / is obvious. The α m 1 p −









































































































































































→ 9 i V R contribution as there is in the tree level Yang-Mills terms (hence the 1 we need I λ hidden ∼ / p α . Essentially this is like a phase space factor. (As a rule-of-thumb, one can p α / 1. Hence the volumes must appear as the ratio of brane volume to co-volume, ) because the diagrams that contribute to p ± α p is the co-volume (i.e. the volume orthogonal to the .) There is no 1 Set-up for the bottom up approach. The visible sector consis p p → − − 9 p 9 V V / Now, for reasonable phenomenology we would like 3 − p into a dimension orthogonal tobrane, the brane or vice-versa, and also change th equation for assuming that we have The dilution due to the co-volume sum over Kaluza-Klein (momentum) modes inas the that brane arising in volume 1 and is essentially the the fact that if we invert a radius, expansion parameter. D7 branes have to be includedof passing tadpoles through requires this additional fixed branes point and/or anti-branes Substituting Eq. (10.3) into Eq. (10.2) gives us Figure 12: String Phenomenology calculate that breaks supersymmetry, such ascated a via contribution closed to string the modes scalar from mas an anti-brane, feels the same volume depe where V PoS(cargese)001 e- (10.6) branes so that naively ) 6 1 K ( Steven Abel U s the , between the ere should be L . ) and their pres- and Q 6 ) n to assume a nice Thus we can iden- K sector particle con- 3 bal validity of these This crucial stringy ting for a number of ( arts of the construc- outlined in Refs.[32, this effect may make and fixed in U 4 matically in figure namely NS-NS tadpoles. nes translates into a mass d (i.e. etween different stacks of M ure shows the arrangement ks of branes corresponding where we can still calculate, factors, and the final reduction ) 1 ( U U(1) P W M M L,H stacks with left handed quarks, ∼ ) 71 U(2) 2 p 3 Q ( − − p 9 U . The branes are extended in U 6 V V K and the dimension orthogonal to the branes should be in and 4 . The gauge states are those strings with ends attached on a ) ) M 3 D 1 ( ( U(3) X U U × ) 2 ( U Local arrangement of states on D3-branes leading to the MSSM of hypercharge comes about because there is only one linear combination × Y ) ) 3 1 ( ( massive, and one expects that the anomalous combinations will be broken. R U s U ′ ) branes with left handed leptons and higgses, and between the 1 Figure 13: ) ( 1 ( U that is anomaly free. Of course string theory is a consistent theory, and th U s ′ ) 1 and ( Let us turn briefly to the local arrangement of branes that yields the visible The bottom up approach has a number of advantages, many of which were 3 branes at a particular fixed point in ) 2 U D . In addition the branes are on top of each other. (Any separation of bra ( 6 tent and gauge group. This isof often represented as in Figure 13. The Fig branes and consequently appear (in thistify simple strings example) stretched in between the the bifundamental. single stack of branes. The matter states correspond to strings stretched b K for the relevant states due toto the a stretching energy.) gauge There group are three stac The beauty of the bottom-up approachtion that is are that not is vital allows to usmodels phenomenology. to due For to disregard example the those there fact is p that a thereThese question are however of uncancelled can glo tadpoles be of absorbed another dynamically kind, by adjusting the backgroun markably the states turn outanomaly to cancellation have (the the Green-Schwarz mechanism) hypercharge is assignments represented of sche the SM. 36]. For example the prediction of an intermediate fundamental scale is interes String Phenomenology no anomalies at all.anomalous But the way in which string theory cancels the anomalies make ence does not automatically renderthe theory the intractible theory on inconsistent a global [35].(tractable) scale, flat it Although or may still orbifold be background afor near reasonable example, the approximatio interactions. visible sector branes, of down to a single two of the dimensions shown are in with right handed quarks. The gauge groups contain too many as expected, and consequently a volume ratio U PoS(cargese)001 Steven Abel nergy spectra. nological impli- o branes. If one is consistent with n axion solution to therefore the possi- e global configuration e bottom-up approach mbles the MSSM and, king communicated by grav- cellation mechanism which is re- mmetries, however D-branes in- propagation of a closed string. right lower diagram which is “stringy”. The built out of the left lower diagram (which has photon, to closed string modes which then emit tree-level ) supersymmetries, and so one may hope to get 1 ( U 72 anomalies. The upper diagram is the usual field theoretic different ) 1 ( U 1 spectra, and even break all the supersymmetry this way. Schematic representation of the Green-Schwarz anomaly can = N As we saw earlier, classical strings can be trapped at the intersection of tw Figure 14: String Phenomenology quired to understand the cancellation of imagines D-branes of some dimensonality wrappingbility a that compact upon space, quantization there these is In intersection particular, states D-branes could are lead (BPS) to andtersecting interesting so at low preserve different e half angles the will supersy preserve latter corresponds to the coupling of an open string diagram. In the stringa theory field the theory anomaly limit contributions equivalent are to the upper diagram), and the chiral 11. Intersecting branes two open string gauge bosons. Note that this process is a reasons. It is a natural realization ofity. hidden The sector model supersymmetry provides brea axions withthe just string the CP right Peccei-Quinn problem. scale In toa addition allow fundamental a the intermediate see-saw scale, mechanism and so foris on. neutrino that, masses One by of its the verycations. disadvantages nature of This th it is is because difficult theby to approach construction, make aspects begins such concrete with as a predictions supersymmetryover visible of breaking which sector phenome we have that to assume rese do very little with control. th PoS(cargese)001 of Z (11.1) (11.3) (11.2) ’, i.e. we Steven Abel ce to introduce rather than the 2 orbifold twist field, is 2 . N similar to twisted states iX hat happens at the inter- X Z + πϑ . An open string stretched 1 τ X . and , 0 , 0 σ , , 0 0 1 1 )= )= − − < ≥ ϑ ϑ ) ) πϑ − + z z , πϑ ( n n ( ( ( 0 − − z z cot Im Im , as depicted in figure 15, has the boundary cot ϑ ϑ ) )= ) ) ) + − ¯ π z z 0 k n π πϑ ( ( ( ( ¯ ( 1 α α 73 1 ¯ 2 X X n n X X ¯ . Hence, we see that there is a natural correspondence ∂ ∂ X ∑ ∑ σ σ τ 1 N ∂ ∂ ∂ ( = − )= )= ) z z )+ )= ϑ )= ( ( π z 0 π ¯ ( X X ( ( ( πϑ 2 as the worldsheet coordinate with domain the upper-half com- 2 1 ∂ ∂ X X X X σ ∂ i A ‘twisted’ open string state - the angle is τ τ σ to stand for the complex coordinate − ∂ ∂ ∂ τ 1 e X − X = z . ) Figure 15: z ( ¯ X ∂ Let us recap and extend what we saw in section 4.4. This will give me a chan This domain is often extended to the entire complex plane using the ‘doubling trick In order to carry out this program it is first necessary to understand w Now the mode expansion of a closed string state in the presence of a a more convenient complex worldsheet coordinate to replace identical to (11.2) with the replacement define, section of two branes a littleon better. orbifolds. It turns out that the states here are very section 4.4, to avoid confusion.) conditions, plex plane. (Note I am using 11.1 Quantizing the intersection between two D-branes intersecting at an angle Thus the correct holomorphic solutions to the string equation of motion are, and similarly for String Phenomenology where I have intriduced PoS(cargese)001 or is the NS (11.4) (11.5) (11.6) L for Steven Abel 1 2 es. A simi- e later). This + can be written which are sewn to the boundary Z , 1 2 e understood geo- y can be separated Z p to form a twisted ∈ tretched between two i losed string state on an r te is the stretched string , what’s called a twist field ¶ ϑ ϑ a − in these coordinates is then, n + nteraction. This configuration has − X 2 1 ¯ z 3 ϑ ϑ − f closed string relation. However, we + 2 n + ) √ ˜ , α n ϑ ) i ∼ 2 4 + f ϑ + r ϑ − ( + iple there could be three independent angles. n 1 to be those of eq.(11.1), where the intersection ( + − | i z X . ϑ ′ ϑ ϑ | 74 − 1 2 n − bosonic πα i α n 2 N ∑ / µ + L = with 4 entries representing 4 complex coordinates (three ′ n ∑ i ϑ .) The mode expansion for = α ′ Πϕ 2 ) a ϑ 2 2 ¯ L α 2 w π f , LT 4 w r ( = Πϑ X 2 )= ¯ z M , ′ z 1 α ( f X represents the obvious contribution from bosonic oscillators. Here bosonic N A set of D-branes which can lead to a nonperturbative 4 point i into the vertex operator which represents the emission of the open string (se ) ¯ w , Quantization then proceeds in the usual manner. In particular the spectrum The correspondence with the spectrum of twisted states on orbifolds can b w ( sectors respectively. Then the GSO projected spectrum of open string s ϑ point of the two D-branes is at displacements between the branes (ofbut course at in angles). more than This twowhich is dimensions, already the the has classical classical energy stretching energy; i.e. the groundsta with the right andlar left mode moving expansion modes is being obtained mapped for into the upper fermions and with lower the half obvious plan addition of field’s job is to change the boundary conditions of metrically as in figure 17.together at This their figure edges. showsclosed two An string. identical open As three string a point living result diagrams at we the expect intersection to is find doubled a u open string where conditions for NS sectors. and where internal and one transverse space time). Introduce a lattice of excitations R two independent angles (i.e. two parallel branes). In princ between open strings stretched between intersectingorbifold. branes and (To a take twisted account c of this correspondence, we must introduce σ Figure 16: String Phenomenology branes is given by as follows. Introduce a twist vector PoS(cargese)001 ow a class Steven Abel many of the tersections that im in this review breaking), have a lanation for family mong many others. ther restrictive ones upersymmetries (su- ted set of multiplets and, especially, their TeV is required. The or an earlier application ∼ ce four-dimensional chi- ly because the branes are ry constrained and minimal f this kind of models and the 75 Identifying open strings to form closed strings Figure 17: TeV: Branes at angles ∼ s M Having understood the open string spectrum, at least a little, we will discuss n In particular, configurations with branes at angles typically break all the s Models with D-branes intersecting at non-trivial angles [5] (see [37] f also note that the intersection angles in this case are more general than the ra found in supersymmetric orbifolds of closed strings. 11.2 persymmetric configurations have been constructed [38] butmodels they are are ve very difficult to obtain) and therefore a very low string scale String Phenomenology of the same idea, innumber of the very dual appealing version phenomenological ofrality or features branes a such with reduced as fluxes, amount for toOne of instan supersymmetry symmetries particularly (both important gauge feature and thatreplication. supersymmetries) these a Specifically models the have matter isare fields an stretched between correspond attractive two exp to branes. thewrapped There string so are states then that at three each generations thestretched type simp in between of the branes intersection at appears the three intersections. times, with a repea of models that represent, withinfeatures of a the bottom-up SM, approach, allowing realisticis in string principle to for models account a with for veryflavour low their structure, string which, phenomenological scale. as features, Our it main theirmost turns a stringent out, realistic constraints provides on structure the the deepest string probe scale o as well. PoS(cargese)001 3, , 2 . Of right gauge , 7 1 ) 2 ( = (b), I SU Steven Abel of the three , 3 left ∼ is a reflection + ms of models ) l [48] that ex- I 2 2 st the SM were R ( X (a), ymmetries of the 6-brane that will al gauge symme- i e models, from the II theory with four ram. view some of these D (see Table 3) auge bosons corre- + times the horizontal 2 nd [41] (see [42] for d , GUT or realistic SM I k ) + 0] and its profound ex- n I in Table 3 and a subset nsions. This mouthfull 1 2 the study of the consis- ( X -branes) corresponding in ing identified. Recall that U baryonic p = × D I wraps . The compactified space is a c Z 6 ) N k 1 K ( . The nett effect of the orientifold ise directly to a USp 2 (in the plane running horizontally). U T 2 × T a ) 1 the stack ( , 2 U is the world-sheet parity and . I i.e. T × ¯ live in the world volume of the corresponding Ω Z , ) × ) ) I k 2 2 = 1 ( 76 m ( I T , Z U I k SU × where n R 2 ( × × R ) T ) wrapping numbers. The number of branes in each stack, N Ω 3 ( ( ) I K SU m times the vertical direction. We have to include for consistency I k − ∼ , m ) I k n N ( so that there are only three dimensions of each ( 4 due to the orientifold projection [31]. M ) 1 ( th torus and − I (d). Three of the dimensions of each D6-brane wrap a 1-cycle on each -dimensional gauge bosons (for the case of a stack of . The branes therefore appear as just lines in each ) 6 1 K + leptonic p branes. In oursponding particular to configuration, the gauge we group have SU seven-dimensional g ( general to the group U This particular model contains at low energies just the particle content and s Note that the left stack of branes consists of just one brane that gives r For the sake of clarity we will concentrate here on one very particular mode The open string light spectrum in these models consists of the following fields: 7 • projection is to introduce mirror images of the branes in each The images do not add any new states so we have not included them in the diag appear in and the orientifold projection is given by emplifies most of the interestingwith branes properties intersecting as at well angles.stacks as It of some is D6-branes of an wrapping the orientifold compactification factorizableis possible of displayed 3-cycles proble type in on Fig. the 18 compact which dime shows just the compactified space, compact factorizable 6-Torus so the three boxes in thethe figure 6 represent branes each 2 must torus, lie with in the edges be group instead of the usual U String Phenomenology first semi-realistic models were constructedsome in related [39] technical and developments). soontries These after or initial in matter [40] models content a beyond presentedpresented the addition in ones [43]. in the Sincetency SM. and then, The stability first a [44] models and great phenomenological containingconstruction deal implications of ju of of supersymmetric intersecting models effort bran [38], has gaugeconstructions symmetry gone [46] breaking to into cosmological [45] implications into [47]. Indevelopments the paying following particular we attention will to re theirperimental flavour structure implications. [48, 49, 5 We have denoted the coordinates of the tori by complex coordinates about the horizontal axis of each of the three 2-tori, of them, together with some of the relevant moduli, are displayed in Fig. 18. their wrapping numbers and the gauge groups they give rise to are shown MSSM. In order to get that we include four stacks of D6-branes, called their orientifold images with dimension of the (c), and 2-tori, with wrapping numbers denoted by PoS(cargese)001 1 j=¼ & s =½ £ M j ¢ espond to ¾»¿ j=½ Steven Abel branes and -form fields esponding to =¹½ rizable cycles omenology of £ in massless or j ¢ nt feature of this even in low scale ve level as unbro- ½»¿ j=¹½ 1), up type singlets , 0 =¼ heir number depend s a mass through the , £ j 1 ´¿µ ¢ − ¯ ¼ = ifferent stacks in the models i ´¿µ ½»6 ½»¿ ~ ¯ ptonic sector is not represented the corresponding radius (except R . ) d with Q R + π c uark doublets ( (1,0);(1,3);(1,-3) (0,1);(1,0);(0,-1) (0,1);(0,-1);(1,0) (1,0);(1,3);(1,-3) wrapping numbers Q right are respectively the dark solid, faint solid, ( s of 2 1 2 a ) ¾»¿ i=½ − 1 a ( 77 ) c d Q U ) 2 ) 1) are denoted by an empty circle, full circle and a cross, 1 6 ( , 1 1 × ( 0 ( ½»¿ , ) i=¹½ = U U SU 1 3 ¢ ( Y − ´¾µ Q Gauge group ¯ = SU ∗ i=¼ ¼ j ). R k 1 1 3 1 π N s has been studied in [51] finding a bound on the string scale ′ ) 1 ( c a d b Stack Brane configuration in the model discussed in the text. The le 1) and down type singlets ( , Number of branes, gauge groups and wrapping numbers for the d ¢ 0 , 1 Four-dimensional chiral massless fermions living on the intersections of two transforming as bi-fundamentals of theon corresponding a gauge topological groups. invariant, the T intersection number, which in the case of facto TeV. Interestingly enough, these gauge symmetries remainken at the global perturbati symmetries [43].baryon, lepton, Quite or generally Peccei-Quinn like these symmetries,models. new preventing In proton global our decay particular symmetries example, corr thethe anomaly hypercharge free is massless combination corr Green-Schwartz mechanism, whereas anomaly-freenot, combinations depending can on the rema particularclass brane of configuration. models This that allow is non-anomalous indeedacquiring gauge a bosons a salie to mass couple of to the thethese RR order extra two massive of U the string scale in this form [43]. The phen the several abelian groups, every anomalous linear combination receive Ö − which is measured in units of 6 • ) = 3 ( j ˜ while the baryonic, left, rightdashed and and orientifold dotted. image of The the intersections corresponding to the q ε Figure 18: Table 3: String Phenomenology ( discussed in the text. respectively. All distance parameters are measured in unit PoS(cargese)001 all ssibility ere that in o exist. In Steven Abel ce of modi- ay from the t of massive and therefore des and string rsecting branes ns thereof). As to get it [41]. r in four. As we bservable levels. at depend on the symmetry is pre- ot automatic and ne structure [53], tersections. In re- s transverse to s, the simplest one rally massive when t these branes wrap at is stopping us are y of dimensionalities, ession seen in Eq.10.3 rs of the corresponding al target space, contains eutral Currents in these h to account for the large tates. , natural family replication s that relatively light vector- why in such a large volume manifold, , ) I b y or another always present in the large extra n I a ed, the vacuum degeneracy problem makes this m − I b m I a n ( 1 78 3 = ∏ I [41], localized near the intersections and with angle- = ab I gonions . This approach is in spirit quite similar to the bottom-up approach. A second po 8 Four-dimensional scalars, also localized at the intersections, with masses th particular configurations of theSUSY branes. is broken They by can the intersection) bealistic superpartners models, seen of scalars the as fermions with the at the the quantum (gene the in numbers example we of are the considering (MS)SM the Higgs configurationserved is boson such at als that each the of same super thefermions intersections completing and the massless matter scalars, spectrum superpartne of the MSSM live at the intersections. with different signs corresponding tocompact different dimensions chiralities. naturally provide The intersectionreplication fact numbers of tha fermions greater with than the one samethe quantum case numbers. of It should lower-dimensional be branes, mentionedlocating like h the D5 whole configuration or at D4-branes, orbifold chirality singularities is is required n in order on a factorizable torus is simply The massive spectrum comprises, apart from the usual winding and KK mo In our particular example, as can be seen in Fig. 18, there are no dimension There is a conceptual difficulty in this construction that can be phrased as We have therefore seen that at the level of the light spectrum, models with inte 8 • the branes and therefore noeffective transverse four-dimensional volume Planck can mass be with a madeof small large course string enoug the scale. gauge The couplings thingand which th become would extremely receive small. the This same problemis volume can to suppr be connect circumvented our in small several torusfor way to a instance, large cutting volume manifold a withoutbranes affecting hole the and bra sewing and large volume manifold in a region aw the relevant physics occurs in such adimensions tiny approach region. to This the difficulty hierarchy is problem inpossibility but, one at as wa least we conceivable have in emphasiz a stringy set-up. excitations not related tovector-like the fermions, intersections the so-called normally present in string models, a se dependent masses. Although a purely effectivelike field theory fermions, study especially show when theyfications mix of with trilinear the couplings top [52],models quark, overcomes the in are presence general the of any phenomenological most Flavour relevance likely Changing of sour these N s have a number of nice features, namelyand four-dimensional chiral local fermions and global symmetrieswe and have matter seen, content the closed of string the sector,among SM which other (or lives in fields simple the the extensio full graviton. ten-dimension with gravity These propagating models in thus ten have dimensions,sketched a in gauge natural the interactions introduction, hierarch in this seven will and allow matte us to reduce the string scale down to o String Phenomenology PoS(cargese)001 1 ∼ s (11.7) (11.8) M p and the ang-Mills Steven Abel is preferred to ut the superpo- ing scale, but the transverse actions in the su- istency conditions suppressed by the adly behaved. The on the latter. e obtained [56] (see In these models the ucture we have dis- n yield finite results t the observed values. g the presence of more s to all branes do exist. roken supersymmetries. is the string coupling and . Contrary to the original e models). 3 TeV have been constructed II λ TeV few ∼ 3 P M , 2 , ∼ V a 4 a group. Reasonable values for the couplings V V V ) II a √ λ D6-branes wrapping 3-cycles on the compact 3 N s 4 s 4 ( π M M 79 II i.e. 16 2 λ = = 2 a 1 P g M [4]. mm ∼ stand for the volume of the four-dimensional manifold where the branes wra 2 , 4 V We have not yet elaborated on the details of the construction and their cons Models with intersecting branes therefore allow in principle for a very low str Gauge couplings can be simply computed from a dimensional reduction of the Y In order to do much meaningful phenomenology, one needs information abo is the string scale. In this situation it is possible to have all scales of order TeV s TeV, while keeping the Planck massNotice (11.7) and as the well gauge that couplings inavoid (11.8) large a the corrections case to the of Higgs non-supersymmetric vev. models, a low string11.3 scale Globally consistent models tential. In string modelsperpotential in can flat be backgrounds computed it withfor is standard amplitudes rather CFT that satisfying techniques. in that the thetechniques Often equivalent are inter extra these common dimensional ca to field both theory closed would and be open strings b and I will concentrate dimensions then have to be M 12. Interactions, esp. Yukawa couplings such as the absence ofThese Ramond-Ramond conditions greatly tadpoles restrict or the number the ofcomplicated presence possibilities, spaces usually of requirin by unb furthercussed. orbifolding Nice and reviews are orientifolding given the in refs. toroidal [58, str 59]. expectation, under certain mild assumptions, gaugealso coupling [57] unification for a can study b of gauge threshold corrections in intersecting bran theory living on thevolume world-volume of of the the compact dimensions stack of of the brane, branes. As expected, it is where we have considered the case at hand, in [54]. (See alsoeffective [55] four-dimensional Planck for mass other reads examples with extra vector-like fermions.) String Phenomenology is to consider lower-dimensional branes,Realistic for examples which with transverse D5-branes dimension and a string scale as low as are obtained if the relevant volume for the brane is space and considered the gauge coupling of an SU volume of the two-dimensional one transverse to all the branes and where PoS(cargese)001 c (12.3) (12.1) (12.4) (12.2) Steven Abel ficient than single string is the extrinsic e ordered topo- k nsion parameter is asics, and develop r at a well-defined rvers will agree that rdered topologically: heets with boundaries eory, we may define a stly, since the different oint in string diagrams ne closed string can be , and ect the spectrum found n theory with strings is only on the topology of ular we need to be able the same as making two ling strength of open and M d. escribed in sec. 5.3 which ∂ hole in the worldsheet, and . , dsk M ∂ Z is the number of boundaries it has and , π 1 b σ c 2 with boundary − − + Euler number . b π M R C γ − g h − −→ is the 80 2 ∼ √ σ 2 O χ τ − g d 2 σ : = d Ω χ M Z π 1 4 = χ respectively. C g to the action (4.44), where and O g λ χ is the Ricci scalar for a given worldsheet is the number of handles a given worldsheet has, h R The Euler number may also be expressed as In order to discuss the calculation of couplings, let us briefly return to the b In field theory, terms in the perturbation expansion of a scattering amplitude ar Cross-caps occur only in unoriented theories, inare which preserved only under those the states d worldsheet parity operation is the number of crosscaps present.and Some handles example are diagrams showing presented worlds inreplaced figure by 19. two In open the strings.open third strings, diagram, Therefore, so we making that see a the that couplings closed o have string the ‘costs’ relation where Here, the formalism for doing perturbationto theory describe in the flat emission backgrounds. and absorptionperturbation In of series partic physical expansion states. for Just stringpotentially as scattering superior in amplitudes. to field perturbation th theory Perturbatio with particleselements for of two reasons. a Fir string worldsheetdiagram contain contains a many multiple Feynman of diagrams:field string string theory. states theory simultaneously, Second, is a there potentiallya is much no string more unique ef interaction point takes ininteraction place. spacetime point at lead which In to all ultraviolet fieldis obse divergences, in theory, but this since propagators sense the ‘smeared coming interaction out’ togethe over p spacetime, UV divergences are avoide logically according to the number of loopstaken in to the be Feynman the diagram, coupling and the strengthwe expa of add the a field. term In string theory, terms are also o are retained, as described earlier.then To make glue a together all cross-cap, diametrically we opposed cut edges. a small curvature of the worldsheet.above: This instead, term its is effect not is dynamical, tothe weight and worldsheet. the does The action not perturbation by expansion aff a parametersclosed factor strings, are which taken depends to coup String Phenomenology 12.1 Perturbation theory PoS(cargese)001 Steven Abel . The general worldsheet, as ) σ overcounting, we r topology, insert , τ he Weyl invariance , k sly occuring, and sum ( V nd row, the external states have been tors. vertex operators 81 -matrix for string theory, one should imagine that the incoming and outgoing S Example diagrams in string perturbation theory. In the seco To obtain the There are three complications associated with this procedure. First, to avoid (asymptotic) states are taken off to infinity, just as one does in field theory. T (4.29) may then beshown used in to the map figure, the which external are states then to replaced by local local disturbances on the conformally mapped to points, to be replaced by vertex opera The Polyakov path-integral procedure for calculating scatteringvertex amplitudes operators onto is it, then calculate the toover probability all consider of physically distinct the a cases. diagram particula spontaneou Figure 19: String Phenomenology PoS(cargese)001 c , b one (12.7) (12.6) (12.5) n theory, fields ccount, so tors on the Steven Abel , which are 6 CKVs are to spacetime. = αβ sistent with the the positions of ghost κ time, and as such are points on the can be thought of ically unchanged. i is. Here, one first pologies may have y conditions for the ghost for each CKV. iy tor methods. , e fundamental region c + ) i i bra involved is tiresome. z x , i path-integral approach, as k = nd ‘oval-ness’ – and indeed, ( , the Teichmüller parameter. i i 3 by eq. (12.5). As the vertex z τ spin-structures V = g κ √ z 2 d are related to the Euler number by the and also Z κ ) 1 x χ . n 1 and so , = i ∏ χ iy = 3 λ χ − χ ( − − E S = 82 − and CKVs −→ κ e ) g µ − σ D , µ τ Weyl ψ ( × 0. There are three possible topologies to consider, none of D diff X > V D χ 2. The Riemann-Roch result (12.5) tells us that Z ghost is introduced for each modulus, and one = αβ (CKVs) associated with them. These isometries lead to worldsheets , b ∑ χ of the vertex operators, and integrates over the positions of those that χ Weyl gauge invariance of the action. Second, some of the topologies have Weyl space is then fixed by a Faddeev-Popov procedure [2], in which κ × = × S , with 2 , which has one boundary. Here, S 2 D on a particular topology. is the Euclideanized version of the action (4.44) and the ψ E associated with them, describing different embeddings of the worldsheet in S The disk operators must be on thethree vertex worldsheet operators. boundary, this is again enough to fix The sphere present. We may use theseworldsheet. to completely fix the positions of three vertex opera We now discuss some of the specific topologies which play a role in perturbatio A gauge in the diff There are two general approaches to calculating amplitudes in a manner con • • In practice, the contributions of the ghosts can be simply determined by opera beginning with tree level where after which one may write down a well-defined path integral, fixes the coordinates of Euclideanized worldsheet. The sum is over topologies remain. The moduli and CKVs are accounted for by introducing anticommuting all possible ways in whichfermions we may choose periodic and anti-periodic boundar All values of thewhatever moduli the diagram, associated one with wouldof a always some expect particular moduli to topology space, have as must toConformal we integrate be Killing did over taken Vectors for th the into one-loop a partition function. Third, to we should divide out by them.as the Taking the two torus ways as in anFor which example a a again, given (regular) topology, the torus the CKVs may numberRiemann-Roch be of theorem, rotated moduli whilst leaving it phys above: the operator approach,typified by as [2]. typified The operator by approach is [1],Therefore, not and without we its the merits, generally but (Polyakov) make the alge Euclideanizes use the of worldsheet, the path-integral formalism in this thes which are mathematically distinct but have the same physical embedding in space and on the worldsheet: one String Phenomenology must account for the diff where For the torus, for instance, one mayas imagine we tori saw of different in ‘fatness’ section a 6, the torus is defined by a complex modulus moduli which have any moduli associated with them: PoS(cargese)001 , z (12.9) (12.8) (12.12) (12.13) (12.10) (12.11) in space- ) as , the fields + Steven Abel z X σ i of course the , 2 3. 1 e . , − = ) ) n = 0 z section 6.1, where − κ ( ( r point z g the tools of con- z e expansions (4.36) µ µ + µ 1, n X X ˜ perator: the complex plane and α n = ∂ ∂ n n ∑ z χ ! excited state of the closed ′ ) z π arbitrary points 2 1 α d . 2 1 ) − . C z r n I τ i fields, ( ( ) fields are written in terms of ′ . These expressions invert to X − ′ z 2 · X α 2 ik α e . r , )= ν r z ) i − X ( z )) ( z ∂ µ + = , X µ · z X −→ µ ( n ik X ∂ ˜ n holomorphic α ∂ X µ − z e z ˜ 2 α − 2 d d 83 X ( Z Z 1. 10 = δ 1. 1 −→ ) fields are in terms of −→ 2. κ z ) − i 2 z = n i ( k d = = k − ) κ z µ − 0 0; κ Z µ 0; µ | ( n | X = µ and hence, on the Euclidianized worldsheet, ¯ 1 α ∂ = 2. ν µ n − X − n z ˜ µ n ∑ σ = α i , which has one cross-cap and hence also 1 ′ π ∂ , with 2 2 κ dz µ 2 2 ! 2 − e α ) C α C RP 1 = , with = I r 2 , with 1 i z ′ antiholomorphic µ − 2 ) has the vertex operator M 2 − α 0. There are four possible topologies, n K ( . Notice that the left-moving ( µν ′ z = r g 2 ∂ taken to enclose the origin of the complex plane anti-clockwise. Applying the )= α χ , with z = 2 ( ≡ C r T µ n µ i − ∂ α X , z ∂ ∂ −→ n ≡ µ − ∂ α The torus The projective plane The cylinder/annulus The Klein bottle The Möbius strip Mathematically, the state-operator correspondence may be described usin • • • • • Excited states are then constructed usingstring eq.12.10; (the for graviton, instance, the first are simply translated. Now, an operator which localizes the string to a particula This result is valid for operators inserted at the origin; for operators at time is where integrate over just the fundamental domainfundamental in region the of amplitude. the one-loop The partitionthe classic function moduli example are which is represented we by already the met complex in Teichmüller parameter The simplest way to construct each of these is to identify various regions of String Phenomenology At one-loop level, whilst the right-moving ( residue theorem gives the state-operator correspondence for the with the contour formal field theory. After themay Euclideanization be eq.12.6, written (defining the closed-string mod and the (tachyonic) ground state is the spacetime Fourier transform of this o Vertex operators PoS(cargese)001 . µ n α (12.16) (12.21) (12.15) (12.14) (12.19) (12.17) (12.20) (12.18) Steven Abel or the closed . Here, the mode . z . µ n 2 a set of five bosonic ) α d , is e identification. The ctor states. R-sector 0 heet is determined by i ( . k . µ 1 2 ¢ boundary, and a factor of ˜ 0; − 2 Ψ | r 2 1 1 2 ve one set of operators − Ψ i − solution lies in bosonization: µ z − r µ r ) ψ − ∂ z ˜ ( ! ψ 1 . ¢ iH r Ψ 1 2 ) ∑ ¡ z − 1 ( ) e − 2 z X 1 r · )= . √ ¡ − ik z )= z ( e z w = H µ ( µ · ( 1 − s i ˜ −→ Ψ Ψ Ψ Ψ w φ log µ r − dze ˜ ∼ e ψ a ) Z z 84 ∼ − λ ( O ) z Ψ g ( −→ ) obeying ) 1 2 i ¢ z w H s 2 ( − ( )= | ) H r z iH Ψ ) w − , Ψ i e ( z k 0 ( ( µ + r H 1 is added, and the integration is taken over µ may be treated in a similar fashion to the 1 µ ψ )= − (OPE): ˜ z Ψ Ψ µ H r r V ( 2 1 ¡ ∑ ψ − 2 Ψ r has been explicitly inserted. Two other points about this expression 1 may be bosonized in an analogous manner. Bosonising the ten dimen- ∂ √ )= ) O ! z z ( g ¢ ( = into complex pairs as 2 1 , are potentially more complicated since the expansion in eq.12.14 has a µ ˜ i Ψ 1 µ Ψ s Ψ − | Ψ r ¡ −→ µ r ψ is the vector (5.26) and the integration is over the worldsheet boundary. F , the open string R-sector ground state is then identified as s H operator product expansion As an example, the vertex operator for the open string photon, The fermionic oscillators The behaviour of these fieldstheir as they come together at a point on the worlds then the identification first group the fields branch-cut, and the state-operator correspondence is not simple. The This is all thestates, information built required up to from construct vertex operators of NS-se And the state-operator correspondence is is consistent with the OPEantiholomorphic fields (12.17); as such, all physics is unchanged by th If we introduce a complex bosonic field expansions (4.56) become String Phenomenology For the open string, the procedure is analagous, except that we only ha string, an symmetric operator in where deserve further comment. This operator should be understood tothe be open integrated string over coupling the worldsheet sions of the string into fivefields complex pairs of the form (12.16) and introducing PoS(cargese)001 (12.22) (12.23) included. φ Steven Abel the general − e ro constraints (4.51) to [3], so that the composite ) 2 which appears together with . + n traces of Chan-Paton factors. dz ) a z ( ( , of our vertex operators. Such a ... i a 1 2 V ) )+ − z w ( ( picture µ 0, so we have a problem. The solution is V ψ 2 = ) µ onto the worldsheet, which may be bosonized z 2 X γ k h − ∂ , φ 85 w of a particular topology. In general then, we will β e ( z χ -charge, or → fields on the worldsheet. In general, the OPE of this lim w φ )= then has weight z Ψ fields ( φ . To offset the factor of a )= has been introduced into (12.21). This is a non-dynamical e z V V , and ) CFT, it is necessary that the total superghost charge in a par- a 1 has been attached to eq.12.21, and an operator k w ( λ − of X ( 1 respectively and βγ + T takes the form i 2 V k must have a total conformal weight of one. The conformal weights ′ V 4 factor α V and . The operator 2 1 φ − superconformal ghost conformal weight Chan-Paton are Scattering amplitudes with orientated worldsheets contai X · ik e is the h and To avoid an anomaly in the Second, a First, notice that the subscript Figure 20: Ψ The argument for its presencedefine a goes stress-energy as tensors follows; for firstly, we use the Viraso tensor with a vertex operator in terms of the field ticular amplitude sums to the Euler number operator (12.21) correctly has unit weight. String Phenomenology quantity which may be associated with the endpoints of strings. The idea is to write to add commuting prescription is the picture-changing operation, where of Chan-Paton factors eq.12.21, it turns out that need some prescription for changing the Picture-changing PoS(cargese)001 , ) s, N runs ( U m (12.27) (12.25) (12.26) (12.28) (12.24) ics is one of Steven Abel N lar, each living ukawa triangle. . Therefore, the is the Euler Beta tructure of these iption of Yukawa N B e dy of Yukawa cou- hould depend on the , eld theory techniques symmetry ts are typically able to ) ′ m ntain a factor ( d as we will see shortly ver each sub-torus. The πα I branes wrap factorizable 2 mion contact interactions A re. Nonetheless, since both − e an think of this as the classical m ∑ . ) I ¢ representation, which supports our θ 4 λ − N 3 ) I I ⊗ λ ν . ν 2 a i j N − λ , − † λ ′ 1 1 1 i α , U of the string transforms in the λ , I / k field and Wilson lines and a I ¡ ; i ν ijk ν λ ( B tr A i j ( | F U − B j ) = e 86 , I π i ∑ 4 li θ 4 are the angles at the fermionic intersections, ∼ , λ −→ I = I 3 kl ν a θ i i jk ( λ k λ Y ; B 2 jk a | λ and s 1 i j I 1 λ ν 3 = l ∑ , I k , j π ∑ transforms in the adjoint , 2 i 1 II µ − λ V 2 √ = in the basis i Y k ; a . Under this symmetry, one end | ) N ( is the area of the minimal area worldsheet with vertices at the three intersection runs over the three tori, U I ∈ i jk A U (with one end on each brane) and travelling to the opposite corners of the Y The leading contribution to Yukawa couplings between two fermions and a sca Among the many phenomenological implications of low scale models, flavour phys whilst the other end (due to the relative orientation reversal) transforms in th function, at a different intersection, is due to world-sheetaction instantons for [41]. a One stretched c string leaving an intersection bounded by the correspondingplings, branes. using calibrated (See geometry [48], Fig.[60], and showed 21.) that confirmed when later the A by compactcycles, conformal more space the fi is detailed relevant a area stu factorizable is torus thefinal and result, sum the including of the the quantum projected part areas reads of the triangle o of the most pressing, so itprobe is mass to scales flavour much that higher wethis now than turn. is the Flavour energy particularly experimen of true currentmodels in experiments is an not the restricted case to Yukawa couplings of butare intersecting flavour also violating present brane four-fer at the models. classical level,sources giving The them of flavour a flavour uniquely s rich violation structu arecouplings. intimately related we shall start with the descr where Since the trace is cyclic, scattering amplitudes are invariant under the gauge when where we have neglected the presence of non-zero identification of it as a gauge boson. 12.2 Yukawas and flavour in open strings String Phenomenology open string state The action for a stringarea is the string the sweeps worldsheet out; area, and therefore the amplitude s Then, as figure 20 demonstrates, open-string scattering amplitudes must co open string vertex operator PoS(cargese)001 ) 3 ( ˜ θ , ) 3 (12.29) (12.32) (12.31) (12.30) (12.33) ( θ , th triangle ) Steven Abel 2 − ( m θ yed in detail in ence of non-zero lead to a realistic ses, the dynamics models since it is nvalues. There are nded quarks. This Yukawa couplings. shall see, in practice th torus, e tori (there is an infinite − . ] k t tori and the property that torus while they live at the φ ) l + , , ´ δ s. At each intersection a fermion or a ´ ( ) i ) 3 ′ ( 3 ′ π ( α J 2 , α J 3 j , d 3 + ³ b ´ is the projected area of the i ³ ) # ) κ a 2 ′ # ( 2 ) m ) ) 3 α J = ( ( 3 l ) 3 I ( ) ˜ ε 3 d 3 ˜ i j ε ( A + ³ ( Y ˜ ˜ − θ θ # δ + ) ( ) ) 87 3 i − + ( 2 3 , ( ( ) j ) π u ) ε 3 [ 2 ε ε 3 b ( ( ( i + θ θ a + + θ ∗ exp 3 j j i 3 3 Z = ε " " " ∑ l u i j is the complex theta function with characteristics, defined as Y ϑ ϑ ϑ )= ϑ ≡ ≡ ≡ κ i j j ( u d a b b # denotes the complex Kähler structure of the δ φ ) " k ( ϑ J 1, , 0 , 1 − = ∗ j World-sheet instanton contribution to the Yukawa coupling th torus. , j − , I i This exponential dependence has been claimed as a nice feature of these This factorizable form of the Yukawa couplings, Eq. (12.29), is too simple to field and Wilson lines. The coefficients are − expected to naturally give athis hierarchical does pattern not hold, of at fermion leastof masses. in left-handed the and As simplest right-handed we models. fermionsonly The turns reason the out is to projected that occur triangles in inAn are many differen ca example relevant is translates the into very aFig. model 18. factorization we of Left-handed have quarks the been livesame discussing at unique in different this points intersection only section in in andresults the the displa in third second the one. following factorizable form The of opposite the Yukawa happens couplings for right-ha where we have only explicitly written the classicalB part, including this time the pres parameterize the Wilson lines and on the where Figure 21: String Phenomenology scalar is localized. over all possible triangles connecting thenumber three of vertices them on due each to of the the toroidal thre periodicity) and fermion spectrum. It is a rank one matrix with one massive and two massless eige PoS(cargese)001 . 3 9 − 2 10 (12.34) & dels with C Steven Abel 2 M ing angles / 1 pression goes | le of the com- lti-dimensional m esent it: ace-time where 1 J | ing theory greatly 3 m antum corrections anes. A string cal- simply worldsheets ∏ ], is the appearance FCNC are therefore t ′ 0] using an extension rely stringy source of ver another feature of the different fermions hich four-dimensional α re a three Higgs model tion scale.) The origin ory investigations, par- − ation scale uge bosons then induce ht dynamics occur at the le in a different way to olds [16], indeed repro- re allowed to propagate, 1 source to increase as the on-factorizable) compact awa couplings, providing − ging neutral couplings that ) ifferent intersections in the x − 1 ke the bounds in that case milder [64]). ( s ′ CNC also affect the model in [50] as well, α − 1 − dx x 1 0 . R ) x ) ( 1 cl λ S . (See also [65] for a model with light vector-like 88 2 − 10 e λ 3 ∑ λ ¤ 4 ) 3 λ ( u + µ 4 γ ) λ 4 3 ( u λ ) 2 1 ( λ u 1 µ λ γ ( ) ′ 2 ( α u s £ g − × ) = 4 , 3 , 2 , 1 ( The full amplitude is a bit of a beast to work out but for completeness I will pr We have emphasized in this review that, after the second string revolution, str The particular localization properties of KK modes in warped scenarios ma Although not necessary for the generation of fermion masses, these F A 9 10 fermions, relevant for phenomenology despite this very large compactifica of these FCNC can be simplygauge traced bosons, to the having fact a thatthe non-trivial Kaluza-Klein fermions modes profile localized of in the at the mu theFCNC extra different in dimensions, positions. the fermion coup Family mass non-universal eigenstateexpected ga basis from [66]. a purely Gauge field boson theoryculation KK of viewpoint generated the in tree models level with four-fermion intersectingof amplitude, D-br the which conformal can field be performed theoryduces [6 techniques the developed field for theory the expectation. heteroticflavour violation orbif In in these addition, models though, mediatedthat it by string directly reveals instantons connect a [49]. four new These fermions pu are same of way different that generations living theroughly at Yukawas as d connected the the area, higgs socompactification that to length one two and would hence fermions. expect worldsheet the area FCNC decrease. Again effect the from this sup TeV in the case of flat extra dimensions [63] influenced (and in turn receivedticularly some in degree the of area inspiration of from)plementarity models field with between the extra string dimensions. and Weintersecting shall field D-branes see are theory a a in salient stringy examp fermions extra realization live of dimensions in the the in brane boundaries this world ofthese idea, extra section. latter dimensions in where w dimensions gauge Mo being bosons agravity a lives further [4, restriction 61]. to alive One in submanifold well separate of known points the property of fullof of the flavour sp brane extra changing dimensions, worlds neutral currents the in that split which tightly fermion constraint scenario the [62 compactific 12.3 Flavour Changing Neutral Currents and therefore similar bounds on the string scale apply. String Phenomenology of course different ways out of this,manifold either or by by looking using for a configurations more of complicatedsame branes (n in torus. which An the example left of and the rig with latter has democratic been rather provided than recently in hierarchical [50],these Yukawas whe is very studied. simple models There that isare makes howe taken the into naive account. assertion This above newpropagate invalid through feature when quantum is qu loops the to presence the ofthem otherwise flavour trivial with chan structure enough of complexity Yuk to give rise to a realistic set of fermion masses and mix PoS(cargese)001 0 th ′ tori = m 2 ’s are cl ´ T S λ ] (12.35) (12.36) x ] ; x one of the 2 one brane, ; Steven Abel ϑ with 4 are the usual ϑ + 2 ) 1 + 3 ϑ k 3 , ϑ 3 + , ϑ 1 3 world-sheet areas are k sub-torus. The ϑ ( − , reas in the three 2 pen string states. These all of the necessary pole 1 1 − T , ϑ gs exchange (which I shall 1 = ϑ − [ , u U(1) 1 ) 1 , [ F 2 1 dL 2 ) is the contribution from the F ] 3 ψ 2 x k | ] µ m x SU(2) γ − + J 2 cL − k ;1 ¯ ( ψ 4 lepton ;1 ϑ e − 3 )( ϑ + bL = 1 t ψ + , ϑ µ 2 l 2 , γ ϑ ) 3 SU(3) , 2 aL 89 ϑ 3 (which is of course the world-sheet area in the full k ¯ ψ ϑ − cl ( + , S 1 1 1 , k ϑ 1 ( abcd 2 n ϑ ) − − [ q ) M 1 1 n ~ [ ( LL = F 1 c 2 Higgs exchange as a “double instanton” s ( F ) 2 4 ) = ϑ ) 3 3 ) + ϑ ) n ϑ 1 ~ 3 ( LL + ϑ − ϑ 2 ( 1 O ( Γ ϑ ( Γ u,d ) ( Γ ) 2 Γ ) Figure 22: 1 ) ϑ 1 ϑ 4 ϑ + ϑ ( − 1 Γ 1 + ϑ ( 3 ( Γ Γ ϑ ( 3 Γ are; ϑ 3 2 − | ϑ 2 ϑ J ϑ 2 − − | 2 ϑ 1 − 1 ϑ ϑ − ) 1 − − x 1 1 ϑ are the standard hypergeometric functions. Each | ) x − − x 1 1 1 x ( − = F 1 2 | ( J + ³ | The point of displaying this lengthy expression is that it allows me to demonstrate The KK mediated flavour violating four fermion interactions come from diagrams beauties of string theory: grotesque asstructure to it generate may the be, correct this field expression theorydiscuss contains behaviour. presently) For can example Hig be extracted from the situation shown in fig.22: The – i.e. two fermionswhich propagates annihilate, to produce a an differentcontributions open intersection are string where of it the KK form, produces mode two with new both o ends on internal complex dimension and one must use the relevant angles for that when, as in this case, the compactification manifold is factorizable. where two Yukawa couplings and the Higgs field is the intersection state in middle. the famous Chan-Paton factors, and 6D internal space) turns out to be the sum of the projected world-sheet a Mandlestam variables. The classical action String Phenomenology The functions PoS(cargese)001 , s is L s 100 / c M & / R (12.37) (12.39) (12.38) s 1 M = Steven Abel s L K modes, the and the string | pend very much K hich is generally s eigenstates and er of independent ε . K modes typically r all LH or all RH) | cale, independently i 3) and ontribution, the case ) . , j / L d ) ) everal differences with y c brane configurations ibutions on the ratio of ~ string smoothing arises not aligned (so that the e larger the ratio 2 c 2 erefore unable to excite e factorization property n in the case of Yukawa + R − i lization of this. Note that a occurs in one torus and 2 ( ed at separate points of the separate points in order to L i ) π y ~ g theory automatically cuts- L ( 4 , · ( U ) ( n ~ ) dL ∼ 1 ~ M ψ + h i ( µ c γ ) cos † L cL ¯ jd U ψ ( ) b )( L ) 1 U bL ( + i ψ ( 1). Secondly, the exponential dependence on c j ) µ ) L γ † L = U 90 aL U ( ( ¯ ψ ai ib ( ) ) † L L i.e. U U U ( abcd 2 ( s ) i ) ai M ∑ n ) ~ ( LL † 2 s L c L A ( π U 2 and relatively small mixing angles, leads to the bound ( − = i j µ e ∑ 2 s str LL ee ∼ M O / the larger the area in string units) the stronger the suppression. Notice 2 n 0). The exponential smoothing provided by the string dynamics, which ~ → M abcd τ 6= − ) i.e. j δ str LL y ,( c S ∼ ( − L i / y c abcd R ) ) n ~ ( LL c ( is the area of the corresponding parallelogram (which is A On the other hand, string instanton flavour changing neutral couplings de are the corresponding unitary matrices rotating current eigenstates into mas is an order one (but always larger) number that depends on the specifi L is crucial in the case ofdiverge, has more to than be one introduced extra by dimensionsoff hand the where in contribution the a of sums KK field-theory over modes approach. K heavier Strin than the string scale. Therefore th where the string scale. Already in thisrespect chirality preserving to interaction the we field observe s theorycouplings case. aligned The with first gauge one couplings is ( that there are FCNC eve the bigger the number of KK modes that contribute and the largeron the the effect chiralities is ofangles). the Four-fermion external interactions fermions with (through all fermionscorrespond the of to difference the a in same parallelogram chirality the with (eithe of numb only the one model independent we angle. arethe Given discussing, result th the is of only the non-vanishing form world-sheet are and represents the stringdivergent smoothing in the of field the theory calculation KK ofbecause contribution the the same at branes effect. have high (Essentially, a the energies finitemodes width w of of a order shorter the wavelength than string this.)of length, Right-Right We and have and are only th Left-Right written the contributions Left-Leftin is c order a to straight-forward have genera FCNC itrotation matrices is are essential non-trivial) that and the current different andextra generations dimension mass are ( localiz eigenstates are the ratio of string andlarger compactification the scales ratio is opposite to that coming from the K with the following dependence of the coefficient instanton contribution to however that it is stillhave necessary FCNC. to The have opposite different dependence generations ofcompactification living the and at KK string and scales allows string us instanton toof contr put this a lower ratio. bound on An the string estimation s of this bound [49], using the KK contribution to δ String Phenomenology with the following dependence of the coefficient TeV as shown in Fig. 23. U PoS(cargese)001 and tion | K ε (12.41) (12.40) | lated to Steven Abel except in some . 24. The Higgs ed and two right- ntribution goes, in g from the vertices the final expressions case if all the flavour rn of fermion masses tricacies of the calcu- ch of the quadrangles tributions in more than e three-point amplitude , this introduces enough 5 n the four vertices 1,2,3,4 d for a low string scale and 4.5 , | K from the KK contribution to 2 H ε 14 | s 4 100 TeV is found. M Y L / . 23 c & − Y L t s 14 2 Y s M 3.5 / ∼ M 23 2 s L Y s π /l 3 c 2 L ∼ / H 2 s 91 14 L A π 2 H 2 invariance whereas in the four-point one we have to − 2.5 / 2 s e M ) 2 s M L R 1234 − , π . A global bound A t 2 2 2 µ − / ( µ e H ee 23 SL A -> ee τ → − 1.5 e τ 1 0

400 300 200 100 700 600 500

s (TeV) M is the Higgs mass. On the other hand there is another, purely stringy contribu Bound on the string scale as a function of the ratio H M channel, like Another nice feature with possible important phenomenological implications is re The chirality changing four-fermion interactions, connecting two left-hand t Higgs-mediated like processes.mediated Let process us can be consider obtained as the2 the and situation field 3 theory displayed down limit to in of thethe a Fig Higgs string vertex propagatin and then back to the vertices 1 and 4. This co integrate over the position of the fourth vertex, see.) As we shall see soon flavour violation to generate, throughand loop mixing corrections, angles. a semi-realistic patte handed fermions, is a bit morehere involved but and far outline more the interesting. reasons We for willlation. the give The new main features new without feature enteringlimiting is into cases). the the absence in The of L-R reasonone factorization is in 2-torus that the and now amplitude the in ( classical(incidentally, general this action there does is are not no non-zero happen longer for con we the the can sum Yukawa couplings fix of because all the in three areas th vertices of using ea (not expected on field theorycorresponds grounds) to that a can string be sweeping verywithout much out going enhance the through area the of Higgs thedynamics vertex quadrangle happens (shaded on betwee area a in single torus the the Figure). amplitude goes In as this the string instanton contribution to Figure 23: String Phenomenology where PoS(cargese)001 ) dL q cR ¯ q )( are the (12.42) bR q 14 aL v Steven Abel ¯ q , ( ) 23 dL v statement about en Schofield for q mion masses and feature of models l form. In order to cR the relevant angles ¯ q , de alluded to above, 2 )( ) ld who kindly allowed pends on the particular bR n 14 sted on the ar that only in the trivial q v to be [60] aL − ed (as opposite to the usual ¯ q ( n 23 v ( n ∑ 2 ) H m 14 v − m 23 v ( m ∑ 92 r 4 ) 3 3 πϑ πϑ + sin 2 2 πϑ 3 2 1 πϑ ( sin sin ′ 1 ) or in the degenerate case (when distances in all sub-tori are equal) the πα c 4 factorizes. Higgs vs string instanton mediation of the process = = ) b cl cl S S or − ( d = exp Figure 24: a are the (independent) angles at the corresponding intersections and ∼ 3 , 2 θ It is a pleasure to thank Oleg Lebedev, Manel Masip, Anthony Owen and B Let us now concentrate on the relevant amplitude for the generation of fer their comments, discussion and collaboration. Specialthe thanks incorporation to of Ben some Schofie of his notes. Acknowledgements distances between the relevant intersections. Fromcase this (when expression it is cle amplitude where String Phenomenology If the flavour dynamics happens in moreconfiguration than due one torus to the the detailed non-factorization resultbut de property is of roughly this the four same. pointthe amplitu phenomenological A implications more of detailed thiswith study property intersecting is branes but necessary is it before the seems presence makingexpected that of suppression) any Higgs-like a by processes general light enhanc Yukawas. amplitude. The full expressions are intricategive and some do feeling not of admit what a happens simpleare we analytica the will same consider on a each simplified sub-torus. case In in this which case the classical action turns out mixing angles. In particular we will consider the quark sector and are intere PoS(cargese)001 (12.49) (12.47) (12.43) (12.45) (12.44) is always Steven Abel σ em here. I’ll use a harge at the bound- ) field theory to do this. u although there µ X dxdt ≡ ) u , ) ,... dx u σ xxx x dx X . u ≡ ∂ 0 ∂ , cos t ∞ T x ∞ − ∂ xx ∞ such that 0 (12.46) , ∂ = − ∞ ] u − τ x 1 , u ∞ X = ∞ Z [ ( x − T dx c ≡ u u u x − 0 (12.48) − Z X , − sin ∞ t is playing the role of the potential. Assume that ∞ ∞ t ∂ ∂ − 2 x → u 93 2 + u = = = = Z , u + X u xx ( = − → ± u t cos T 2 x t ∂ Q L 2 ∂ T dx − u − plus time, with a generic field ∞ ∞ tt ∞ 1 x ∞ u − = Z Z = 2 t 1 t d L dt Z P ]= u [ S can be assembled from the remains constant at T X and X Conserved currents are an important concept so it is worth recapping th (Note that in string theory the obvious analogy is two dimensional (1 noncompact space so compact). We will use the lagrangian formalism; Then eq.12.46 means that For example the Sine-Gordon equation has String Phenomenology Appendix A: Conserved currents in 1+1D Assume further that two functions with the Euler-Lagrange equation giving Here for later reference note that is a conserved quantity. Notearies. that the RHS of eq.(12.48) is the net inflow of c PoS(cargese)001 . x u , t u , (12.54) (12.56) (12.50) (12.52) u and con- Steven Abel dtdx nds only on symmetries ) famous examples that ome “boundary terms” x rinciple: that if we let but do not differentiate coordinate and insisting u ) , x t t f motion which are obeyed , u ¶ ¶ x , ; (12.53) x ( t u x u L u ( L u ∂ ∂ ∂ ∂ dxdt L µ µ x u dtdx − t and ¶ 0 (12.51) dtdx u u ) δ = u ) x x δ t δ = δ u u ¶ , dt t + x x δ δ x + u ( L u u L x L ¶ ¶ u 0 (12.55) ∂ ¶ + ∂ u ∂ ∂ L ∂ . ∂ u x x t ∂ ∂ u u µ δ u u = L L d d dx δ , dx t ∂ ∂ ∂ B A ∂ t + t t d u t dt L u t ¸ − u µ − ∂ µ L ∂ u u δ + ∂ 94 t t ∂ δ d δ d µ u u dx dt L + L t ¶ t ∂ t u u ∂ = ∂ u u ∂ u d L L dt u δ δ ∂ δ ∂ we immediately find a conserved current ∂ , ∂ Q d d dt dt x u u · Z = = u + L δ − δ − ∂ u ∂ = ¶ ¶ + δ u u u u µ → L L u S ∂ ∂ ( δ δ u ∂ u ∂ δ L d t x dx ∂ , then things get trivial since we drop the ∂ u L µ u L L t ∂ ∂ ∂ ∂ boundary terms µ µ + Z Z Z Z Z Z Z Z d d dt dx means use the chain rule with = = = 0 implies a set of locally obeyed equations of motion. One thing which is 0 dt = / = d S δ S .) Setting everything in brackets to zero gives the E-L equations δ x then u 0 gives δ = + S u Emmy Noether was the first to elucidate the deep connection between If the theory has only time, δ → servation laws. Theu Euler-Lagrange equations derive from Hamilton’s p and if there is an invariance under important though are the boundary terms. Assume that the Lagrangian depe that and the additional boundary terms are (This will suffice fordepend perturbative on string the higher theory, derivatives.) although Making the there variation are we actually many get String Phenomenology Appendix B: Symmetries and conservation laws in 1+1D However to get the last two(i.e. terms I complete integrated derivatives); in by fact parts I once and used generated s since they are total derivatives they make nolocally. difference to the equations o (Here for example with respect to PoS(cargese)001 ) t → u ely t ∂ / (12.58) (12.60) (12.61) L ∂ such that Steven Abel ≡ T only and does p ) x ariance, and the and u of the action, and , X t u k cally, all we have left , u ( ) u L = y terms. We see it as follows: cos ) 0 (12.59) ¶ xt tt t u ¶ u − u L u t u = ε ε u t 1 δ cos u ε ¶ L t ∂ u )+ ∂ + ( L − )+ Now look at the shift in the action, or t t L ∂ , )+ ∂ 2 x 1 µ , we have the relation 2 t x u x L , − ( µ ( ) d x t x dt etc. t + ( t L dt ( − + u u u d d ε dt t 2 x u 0 0 (12.57) t 2 t xx 2 or u u 2 time-translation symmetry u u u + L + t x → = = ∂ δ ∂ u ) = + ) = L ¶ − ¶ , ) = t t ε ∂ ε ∞ 2 t x ∂ µ 2 t ε L u T ε u 2 95 u ± u δ u + 0 with δ x + d + = + δ = dt t t u x L x , x t , , | = = = u ε L ∂ , t ∂ x x Consider the infinitessimally small constant shift X + t x X ( u ∂ ( ∂ H T . ( x t + µ δ ¶ 0 u u u t t L µ , + ≡ For the SG equation u d u dx x = − d x T dx → → → → δ ε t X u L u t ) ) ) dx ∂ ∂ t t t t = = x 0 limit to get , , , u L µ X x x x ∂ ( ( ( ∂ R t → x L u d dx u u δ ε − = = H dx t T So the hamiltonian is an expression of time-translation invariance. Convers R . or very importantly explicitly on = xx H u and taking the ε Tdx is invariant under this shift it is called a is a conserved current; you may recognize this as the Hamiltonian (with R S t dx d . If Now let’s find the current conserved under time translation; recall we see H ε R + an explicit time dependence incurrent would the no Lagrangian longer would be break conserved. time translation inv dividing by so hence the name The extra bits on the RHS I will call there is a conserved current associatedunder with the that given shift by we the have boundar by Taylor expanding that rather everything inside the integral.are Since the the boundary E-L terms. equations These are are satisfied (assuming lo for the moment that String Phenomenology The Hamiltonian density: time translation symmetry not depend on This relation is in precisely the form which is clearly the kinetic plus potential energy (densities) of the system. so that t PoS(cargese)001 (12.63) Steven Abel term. In this dx / d is invariant under this S . If that ε 0 (12.62) went in the xx tx + u u x = x ε ε u L ¶ ε − x → )+ u )+ t x t t , )+ , t u x L x , ( ( ∂ x ∂ x t ( u u . µ u x u d dt t ) = ) = u ) = L ε ε + ε ∂ 96 ∂ + ¶ + of the action, and again there is a conserved current + t t = , t , ε , L x x P x ( ( + x ( t − u u u x x u x → → → → u L ) ) ) x ∂ ∂ t t t , , , µ x x x ( ( ( t x u d u dx u space-translation symmetry Now consider the infinitessimally small constant shift case the conserved current is related to the momentum density To get the conserved current we can read of from the boundary terms The only difference from the time-translation case is that the shift it is called a associated with that given by the boundary terms. Under the shift we have String Phenomenology Space translation symmetry PoS(cargese)001 . t ≡ (12.64) (12.67) (12.65) (12.66) (12.68) (12.69) πτ ... Steven Abel se there are + . This means rmion excita- i u 0 , | v † v s for each fermion. b contributions since v t is we sum over all iu + ic excitation) π † 1 2 . e ) ˆ . Ob ) ossible sets of boundary ) v u τ + − ( 1 1 2 ( V U b i v Z π b (always allowed) when we add 2 V operators which then annihilate | U − 0 C v V U e h α v N C f ) # ... − ) ´ τ n v 1 2 v . 1 ( ) ) q + N j j and ) n τ v − − u i ... − u ( q ( + v 0 Z 2 1 u − | 4 + η 1 1 1 2 H v − − " i ( = 16 + i ) )( ∏ 1 J 0 † 1 ) π θ τ | ( ) u 2 ( v 1 ) τ e ˆ 1 2 − ∞ − = Ob v ( 1 2 † ∏ 2 n j j v − Im ( H i v u u + q 97 24 π ˆ 16 1 )( Z Od / 2 ³ 1 2 − b 1 v 1 e when propagated through the complex time 2 | | 4 1 = q − − ) 0 j ∏ v Tr 2 iv h τ − ( i d v π ( | π 2 + + )= η 2 0 e n | )= )= i h e τ q τ 0 τ } ( | ( ( + V = † v η , + v u V U i v u Z 1 0 Z ∑ ˆ ( | Z Ob 1 for states that are space-time fermions. As we will now see, the ) is then 1 v allU v i { ∞ = − b − ∏ † | n 1 iHt v 0 a h e ˆ )= Od h q v τ + ( − = i 1 1 0 v u Z d | | Z ˆ can be commuted left through the O 0 | h v 0 − is a factor of h + † n can be chosen to give modular invariance. α and where the final factor includes a phase from every world sheet fe d f V U τ ) i C ) = 1 π ˆ 2 and O − e ( ( 1 − = v Tr q + † n First consider a single complex world sheet fermion with boundary condition The total contribution is then trivially given by the product of the individual For the complete one loop partition function we must include bosons and in this ca b the contribution from different sectors The partition function (c.f. where they commute; Conventionally this is expressed in terms of Jacobi theta functions where the Dedekind eta function is 8 real bosons for theconditions, left and and include right a movers. (very In important) addition overall we phase sum factor over all p tion in a particular physical state.possible excitations To (i.e. take all the possibilities trace one we or sum zero over numbers all of states; each tha fermion Propagation in the other direction on the torus must give the phase factor that the fermion acquires a phase factor where the String Phenomenology Appendix C: Modular invariance in detail C.1 The partition function for the complex fermions phase factors The on the vacuum. The end result is PoS(cargese)001 (12.73) (12.72) (12.74) (12.75) (12.70) (12.71) Steven Abel such that U V C . ) e τ ( well known V that all phases appearing U − ) Z τ ( V U # . C ) ) ) 1 2 )) τ τ τ ( ( V ) ( − + V # ) τ 1 requires that ) V ) ( U − U τ 1 β U − α ( v τ α ( . V + Z U ( β Z 0 u − V U V + v + − C , τ )) V W Z ] C − − τ )) ) ) V U − 8 " − α u u U ( 1 2 U ) V V + → C U 1 . Z θ Z . " − + 0 1 2 ) U } Z ) V v τ ( 1 6 ( ( . W U θ i V ) V ( )( 0 αβ , ) 8 . . ) i − + π 1 2 τ 0 1 ) 0 v W π τ 2 V ( ∑ . )= W − 2 W − ( 2 1 − e − V u 2 e τ allU α − ( − U ( } 1 2 η v ( { . i / V ( τη τ ( U . V i 4 V i 98 . π i i , 1 12 ( V ) π 2 i V π πα / ( i ( i i 2 − − e e π i − 1 2 ∑ 3.) )= − π 2 π e ( π allU / τ e e √ √ e e 2 1 { ( 2 . This sends the fermionic part of the p.f. to e = Z V τ U ) = ) = = [( V ≡ = / Z 1 τ 0 )= ) = ) = ) = ) = ) = − ) = 1 / V 3 U )= V τ 1 1 τ τ 1 τ V + W U 1 / τ C U − / / / / C ( τ 1 + + } + − 1 1 1 1 C ( 1 ( V v u − τ τ → − τ , Z v − u − − − ( ( ( Z ( ( ( ( Z τ ∑ V U η # V U V U η # allU Z Z Z { β α β α V U " " C θ } θ V , ∑ allU { are to be taken mod(1) - e.g. Remember that the point was to now apply the modular invariance condition. Now consider the effect of Z C.2 Modular properties of the partition function Comparing the two expressions, the partition function is invariant if we choos Using these it is straightforward to show that or collecting all the contributions together where But then since we sum over all boundary conditions, we can trivially write for all sectors in the model. Likewise invariance under String Phenomenology for all sectors. The individual transformation properties of the eta and theta functions are defines the NS sector which asin we have seen gives us the graviton. (Note PoS(cargese)001 nted (12.79) (12.76) (12.77) (12.78) se rules. Steven Abel ss the term in the ex- e trace) , model building rules in . The modular invariance ´ l V N . c and writing W ) a c β W . ) ) − V . 0 τ c c ( δ for integer W ( i V U − 0 π a l c Z 2 0 δ k α β e W a + V C = 1 c β H α w c f l q m + ) = α c β b 1 β α i C U − π cb ³ k 2 ( ; e } 99 )( a that is giving us the projection in the text simply a c Tr a 1 β m β , , c α = a f a W − c ∑ c β a 0 α ) m m β c satisfies the conditions for modular invariance derived { 1 δ α ( ∑ i V U − = π = ( C 2 } a e a V β m , , = a a ’s and ∑ c α m f ermion α β { β Z C = f ermion ’s is zero, thereby enforcing the projection on states that we have prese Z . It is the sum over β c is summed over. The fermionic part of the PF can be written a To show modular invariance for the models outlined in the text, we now adopt tho C.3 Proof of modular invariance above. The more masochistic reader may like to do this. To repeat the argument, the contribution toponent the summed partition over function vanishes unle where projection in the rules means that we must have That is we define the sum over sectors by using a basis of vectors Consider where this expression came from (i.e. write it before evaluating th because we are summing over all in the text. It isthe a main straighforward text, (but tedious) that exercise this to expression show, for using the where we sum over String Phenomenology PoS(cargese)001 for , then c (12.83) (12.80) (12.81) (12.82) ounting integer Steven Abel = c t we define the s there are only φ 2 for any / + ce. First let us infer V , odd N ´ . aluating the traces; V c . = ) N . W c τ c ( φ W ) V U c + and show it is modular invari- β Z − α V β 1 for states that are space-time α β 0 c N ) C . C δ ± V ( α c . i f c π ) is W W 2 1 e − α ) ) ) a c f V − 0 01 01 01 ) k ( H W k k k 1 4 a q + + + + 1 c − β α β − 11 00 00 w ) ( k k k C τ + ( ( ( = α b ( 2 2 2 f ) ) ) α ) U 1 1 1 cb Im 1 k and we have ; − − − and writing 16 )( − c 1 . Assume that the fermionic part of the PF can be 100 that is giving us the projection in the text for the ( ( 1 ( c ( φ a , − a β ) | c − 1 − 1 − 1 − 1 c ³ 0 W ) − W β β a 0 τ c − ======( Tr δ α 0 and that particular state cannot contribute to the partition ( c } a i , η δ 1 1 1 1 0 0 0 0 1 0 1 0 1 0 1 0 a | , , , , , , , , π , , , , , , , , ( = β 0 β i 1 1 1 0 0 1 1 0 0 2 0 1 0 1 0 1 0 1 , } , , π a a e 1 a ∑ C C C C C C C C V 2 = β 1 α , , α e { a 1 = ∑ 1 α 4 1 { = α β − simplifies. Indeed we can substitiute in the right hand side of the 1 4 C α β C α )= β τ C ( )= τ ( 1 Z . It is the sum over f ermion c Z 1 gives a factor 1 , 0 1 is summed over, and , = 0 c = β a contribute 1 to the partition function. The partition function is then doing its job of c Now we just need to evaluate the partition function with this The type II theories allow a particularly simple proof of modular invariance, a c since this reproduces the right projection. where 4 sectors. Here we willfrom verify the that projection the rules rules whatsum in the over the sectors form text by of give using the modular a invarian basis partition of function vectors is. As in the tex following reason. If we write physical states, and we can interpret the rules in the text as implying projections we worked out in the text, to find ant. For the type II models all where the first piece is the bosonic contribution and bosons/fermions. Consider the fermionic contribution written out before ev written function and isn’t in the spectrum. On the other hand states that satisfy String Phenomenology Appendix D: Simple modular invariance for type II where we sum over the sum PoS(cargese)001 , ¸ 4 ) 1 2 1 2 Z + ( 4 ) 2 1 0 (12.85) (12.86) (12.84) (12.87) (12.88) Z ( − Steven Abel 4 cribed at the ) 1 2 0 Z deed give ( e to prove modular − 4 ) oked up. They are utions since they com- 0 0 Z ( · × ¸ prefactors, gives an expression 4 ) α β 1 2 1 2 ) C Z ) # α 1 f τ v 1 2 . ) ( + ) ) ) j j + ( 1 n τ v − u τ − 4 01 ) ( q ( k − ) Z 2 1 u 1 ( 00 1 2 0 + η 1 k − Z " 4 Z 11 = + 11 01 ∏ 1 k J ( k ( k θ ( 01 ) 2 2 2 01 00 1 ) k − ) ) ) τ k ( k )= 1 2 ∞ = 2 2 2 ( 4 1 1 1 τ ∏ n j j − ) ) ) ) / v u u − 1 − 1 − 1 1 2 0 24 1 101 ( 1 ( ( )( Z / Z − − − 1 2 1 1 − ( − − − 1 − 4 ( = q − j ∏ v 1 − ( i = ( = ( = = = = = ( = Z 4 π ) 1 1 1 0 1 0 0 0 )= 2 0 0 1 1 1 0 0 1 , , , , , , , , , , , , , , , , 0 )= 0 e τ for type IIB,A respectively. This expression should be 1 0 1 1 0 1 0 0 0 0 1 1 1 0 0 1 τ Z ( )= ( C C C C C C C C 2 1 ( τ = η ’s and including the , V ( U 00 v 0. u k 1 0 β Z 2 Z Z ) = = 1 11 00 − k ( k ,s and just give an overall definition of chirality and without loss of · = α 4 01 k − 01 ) k τ ( Im 4 16 − | ) τ ( η | )= We now need to evaluate the partition function. By performing the trace as des τ ( 1 generality we can take beginning of Appendix C, we express it in terms of Jacobi theta functions The different choices of The total contribution is then givenmute; by the product of the individual contrib that can be factorized into left moving and right moving parts; Z Performing the sums over where the Dedekind eta function is modular invariant where as we saw in the text, String Phenomenology The modular properties of jacobi thetawritten and in dedekind Appendix eta C.2 functions and can substitutinginvariance. be them lo into the above expressions does in Exercise: substitute the modular transformations ininvariance. Appendix C.2 into the abov PoS(cargese)001 + ; the 45 ) celess + 12 1 ( giving 248 Steven Abel → SO 7 , that satisfy U × ) ories, and hence 1 ( 64 and the adjoint U consist of the adjoint = 8 6 E 2 identity matrix. They are = k × rred to in the text are ... k . T T in exactly the same way fermions = is the ) ) 1 2 independent entries in such a matrix, k I can be decomposed into representations / − 10 12 ) ( 8 ( 1 E , SO − SO 7 n was made evident in the text, so if nothing else MTM where × E ( , 8 ) n 1 independent elements, and hence there are the 6 1 E ! matrices of determinant 1. The generators are an- ( E k 102 − 0 n I U 2 k n × I 0 n − Ã , that satisfy i T = M we can decompose it into representation of 7 1, both chiralities of 66, giving 133 generators in total. Finally we can decompose the plus a single chirality of the fermionic representation 2 E ) = matrices, ) = 16 2 where k ( 1 / 1 ( 2 . For − into representations of 11 SO U 8 matrices, which have × . ) 6 E T k n E 12 U 2 of × / n ) )= = ( 1 1 12 : the group of unitary matrices with determinant one. The generators are tra − − : the group of real orthogonal ( : The symplectic groups are the groups consisting of unitary matrices, 78 generators. The decomposition of ) ) ) n n k 16 ( ( = SO ( ( SU Sp SO and hence the same number of generators same number of generators. hermitian MUM The generators of the exceptional groups tisymmetric Hermitian matrices. There are generated by 2 of the corresponding maximal subgroups. For example generators of 16 you can use the heterotic string to help you decompose large groups! generators for of generators of single boson of 32 The Lie algebra determines the local structure of the gauge group in our the • • • • the number and structure of generators and so forth. 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