aa r X i v : . [ h e p - t h ] A ug Semi-realisti heteroti Z × Z orbifold modelsThesis submitted in a ordan e with the requirements ofthe University of Liverpool for the degree of Do tor in PhilosophybyElisa MannoJune 2009bstra tSuperstring phenomenology explores lasses of va ua whi h an reprodu e the low en-ergy data provided by the Standard Model. We onsider the heteroti E × E stringtheory, whi h gives rise to four-dimensional Standard-like Models and allows for their SO (10) embedding. The exploration of realisti va ua onsists of (cid:28)nding ompa ti(cid:28) a-tions of the heteroti string from ten to four dimensions. We investigate two di(cid:27)erents hemes of ompa ti(cid:28) ation: the free fermioni formulation and the orbifold onstru -tion. The relation of free fermion models to Z × Z orbifold ompa ti(cid:28) ations impliesthat they produ e three pairs of untwisted Higgs multiplets. In the examples presentedin this dissertation we explore the removal of the extra Higgs representations by usingthe free fermion boundary onditions dire tly at the string level, rather than in thee(cid:27)e tive low energy (cid:28)eld theory. Moreover, by employing the standard analysis of (cid:29)atdire tions we present a quasi(cid:21)realisti three generation string model in whi h stringent F (cid:21) and D (cid:21) (cid:29)at solutions do not appear to exist to all orders in the superpotential. Wespe ulate that this result is indi ative of the non(cid:21)existen e of supersymmetri F (cid:21) and D (cid:21) (cid:29)at solutions in this model and dis uss its potential impli ations. By ontinuingour sear h of semi-realisti models in di(cid:27)erent string ompa ti(cid:28) ations we present asimple, yet ri h, set up: the orbifold. The simplest examples of orbifold ompa ti(cid:28) a-tions generally produ e a large number of families, whi h are learly unappealing forexperimental reasons. We show that, by hoosing a non-fa torisable ompa ti(cid:28) ationlatti e, de(cid:28)ned by skewing its standard simple roots, we de rease the total number ofgenerations. Although we do not provide a semi-realisti model in this framework, themethod represents an intermediate step to the (cid:28)nal realisation of phenomenologi allyviable three generation models. Moreover, we mention other possible tools whi h maybe applied in the sear h of Standard Model-like solutions. Finally, the onstru tionof modular invariant partition fun tions for E × E orbifold ompa ti(cid:28) ations is pre-sented. Several interesting examples are derived with this formalism, su h as the aseof a Z × Z shift orbifold model, in order to provide a more te hni al approa h in the onstru tion of onsistent string models. iontentsAbstra t iContents ivList of Figures vA knowledgement vi1 Introdu tion 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 String theory as a theory of uni(cid:28) ation . . . . . . . . . . . . . . . . . . . 31.3 Organisation of the hapters . . . . . . . . . . . . . . . . . . . . . . . . . 72 Ba kground notions on onsistent perturbative superstring theories 82.1 Bosoni strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Vertex operators and string intera tions . . . . . . . . . . . . . . . . . . 132.3 The superstring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 One loop amplitude and modular invarian e . . . . . . . . . . . . . . . . 172.5 Spin stru tures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6 Partition fun tions of 10D superstrings . . . . . . . . . . . . . . . . . . . 212.7 Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.7.1 Produ t expansion operator . . . . . . . . . . . . . . . . . . . . . 222.7.2 Free bosons and free fermions . . . . . . . . . . . . . . . . . . . . 232.8 The heteroti string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.9 Toroidal ompa ti(cid:28) ations . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Free Fermioni Models 283.1 The free fermioni formulation . . . . . . . . . . . . . . . . . . . . . . . 293.1.1 Model building rules and physi al spe trum . . . . . . . . . . . . 313.1.2 Constru tion of semi-realisti models . . . . . . . . . . . . . . . . 333.2 Minimal Standard Heteroti String Models . . . . . . . . . . . . . . . . . 363.2.1 Yukawa Sele tion Me hanism . . . . . . . . . . . . . . . . . . . . 383.2.2 Higgs Doublet(cid:21)Triplet Splitting . . . . . . . . . . . . . . . . . . . 39ii.3 Models with redu ed untwisted Higgs spe trum . . . . . . . . . . . . . . 403.3.1 Flat dire tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Stringent (cid:29)at dire tions . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.5 The string model with no stringent (cid:29)at-dire tions . . . . . . . . . . . . . 473.5.1 Third and Fourth Order Superpotential . . . . . . . . . . . . . . 503.5.2 Flat dire tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 Z × Z orbifold onstru tions 574.1 Heteroti string and toroidal ompa ti(cid:28) ation . . . . . . . . . . . . . . . 584.2 Orbifold onstru tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2.1 Consisten y onditions . . . . . . . . . . . . . . . . . . . . . . . . 624.2.2 Generalities on the spe trum . . . . . . . . . . . . . . . . . . . . 634.3 Z × Z orbifold with SO (4) ompa ti(cid:28) ation latti e . . . . . . . . . . . 654.3.1 Analysis of the latti e . . . . . . . . . . . . . . . . . . . . . . . . 664.3.2 Introdu tion of Wilson lines . . . . . . . . . . . . . . . . . . . . . 724.3.3 Massless Spe trum . . . . . . . . . . . . . . . . . . . . . . . . . . 745 Constru tion of partition fun tions in heteroti E × E models 815.1 Shift orbifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Partition fun tion of the heteroti E × E shift orbifold superstring with Z a tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2.1 The bosoni ontribution . . . . . . . . . . . . . . . . . . . . . . 885.2.2 Spe trum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.3 A string model with no gravity . . . . . . . . . . . . . . . . . . . . . . . 915.4 Supersymmetri Z × Z shift orbifold model . . . . . . . . . . . . . . . 945.4.1 Untwisted spe trum . . . . . . . . . . . . . . . . . . . . . . . . . 965.4.2 Twisted se tor h . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.4.3 Torus amplitudes for the right and for the left se tor . . . . . . . 985.4.4 Massless spe trum . . . . . . . . . . . . . . . . . . . . . . . . . . 986 Con lusions 101A 103A.1 η and θ -fun tions and modular transformations . . . . . . . . . . . . . . 103A.1.1 SO (2 n ) hara ters in terms of θ -fun tions . . . . . . . . . . . . . 104A.1.2 Modular transformations for SO (2 n ) hara ters . . . . . . . . . . 105A.2 De(cid:28)nition of latti e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.2.1 De(cid:28)nition of shifted latti es . . . . . . . . . . . . . . . . . . . . . 105A.2.2 Proof for the transformation properties (A.13) . . . . . . . . . . . 106A.3 Expansion of SO (2 n ) hara ters in powers of q . . . . . . . . . . . . . . 108iii 109B.1 Tables for two models with redu ed Higgs spe trum . . . . . . . . . . . . 110C 128C.1 Weight roots of E representations in the twisted se tor θ of the SO (4) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128D 130D.1 Total amplitude ontributions of the shift orbifold in eq.(5.16) . . . . . . 130D.2 Left amplitudes of Z × Z orbifold model in eq.(5.43) . . . . . . . . . . 130D.3 Right amplitudes of Z × Z orbifold model in eq.(5.43) . . . . . . . . . 134Bibliography 141ivist of Figures1.1 Supersymmetri perturbative onsistent string theories in 10 dimensions. 42.1 a) losed string worldsheet. b)open string worldsheet. . . . . . . . . . . . 92.2 Mapping of the worldsheet ylinder into the omplex plane. The dottedlines of onstant τ are on entri ir les while the lines of onstant σ follow radial dire tions from the origin. . . . . . . . . . . . . . . . . . . . 132.3 1)Torus diagram. 2)The (cid:29)at torus as a two dimensional latti e. a and b represent the two non- ontra tible y les of the Riemann surfa e. . . . . 182.4 Fundamental domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.1 Modular transformations for a generi amplitude in orbifold models,where a , b ∈ { , g, h, f } for a Z × Z orbifold. . . . . . . . . . . . . . . . 945.2 Modular orbits in Z × Z orbifolds. . . . . . . . . . . . . . . . . . . . . 95v knowledgementNon basta guardare, o orre guardare on gli o hi he vogliono vedere, he redono in quello he vedono. Galileo Galilei.I would like to express my gratitude to my supervisor, Prof. Alon Faraggi, for hisvaluable suggestions and onstru tive advises through this resear h work. His guidan einto the ompli ated and at the same time fas inating world of String Phenomenologyenabled me to omplete my work su essfully. I am immensely grateful to Dr. CristinaTimirgaziu for her pre ious instru tions in the study of string models and her will-ingness to answer all my questions without hesitation. Her warm en ouragement andthoughtful guidan e have been ru ial in the ompletion of this proje t. I wish to a -knowledge the String Phenomenology Group whi h provided numerous ideas and usefuldis ussions during the weekly meetings, as well as the useful seminars whi h presentednew aspe ts and outlooks on the progress of string theory in the rest of the s ienti(cid:28) world. In parti ular, I would like to thank Dr. Thomas Mohaupt for helpful suggestionsin several o asions and Dr. Mirian Tsulaia for the interesting ollaboration of the lastfew months, whose help and knowledge has been signi(cid:28) ant in the solutions of numerousquestions. My great appre iation goes to Prof. Carlo Angelantonj, whose kindness andavailability allowed me to answer riti al doubts in the analysis of the partition fun -tion onstru tions. I wish to thank Prof. Gerald Cleaver for the ollaboration whi hprodu ed the se ond paper presented in this thesis. I am thankful to Prof. Ian Ja kfor his on rete, and moral, support during these years of study at the University ofLiverpool. I wish to thank Prof. Claudio Coriano who persuaded me to apply for thisPhD and (cid:28)rst got me interested in the resear h of Theoreti al Physi s. My PhD wouldnot have been so ex iting and enjoyable without my great friends Ben, Cathy and Chris,who shared with me an o(cid:30) e and the di(cid:30) ulties of the s ienti(cid:28) resear h. Among mymany friends in Liverpool, I want to thank Paola, Linda and Laura for the pleasantevenings spent together. Their pre ious friendship and advises during these years havebeen fundamental to me. A spe ial thanks to Adriano and Andrea for their deli iousitalian dinners, that were a ni e break from the writing up. In the last year of my PhDthe onstant and loving support of my boyfriend Gary has been de isive to over omemoments of frustration and tiredness. The great time we have spent together gave methe enthusiasm and drive to a omplish my work. I will herish all the memories ofvihe last days for the rest of my life. I am forever grateful to my family, my motherFilomena, my father Lu iano and my sister Paola, sin e they taught me the way torea h my goals in life. viihapter 1Introdu tion1.1 MotivationThe Standard Model (SM) of Parti le Physi s des ribes orre tly the physi s of theelementary parti les and their intera tions, as on(cid:28)rmed by the experiments up to theele troweak s ale M W = 246 GeV. It ombines three of the four fundamental for es innature, the weak, the strong and the ele tromagneti intera tion, into a unique theo-reti al framework, whi h is a Yang-Mills gauge theory based on the symmetry group SU (3) C × SU (2) L × U (1) Y ( C , L and Y denote the olour, the weak isospin and thehyper harge quantum number respe tively). In parti ular, the weak and the ele tro-magneti intera tions are des ribed by the SU (2) L × U (1) Y gauge symmetry, whi h isspontaneously broken to a U (1) em by the Higgs me hanism [1℄. The resulting massivegauge bosons, W ± and Z , mediate the weak intera tions, while the massless boson γ ,the photon, is the arrier of the ele tromagneti for e. The Quantum Chromodynami sis des ribed by the SU (3) C se tor, whi h remains unbroken, where the messengers ofthe strong intera tion are eight massless gluons. The Standard Model ontent onsistsof three generations of leptons and three generations of quarks, in agreement with theobserved experiments. The predi tability of the Standard Model is a onsequen e of itsrenormalizability, whi h assures a onsistent perturbative analysis of quantities relatedto the parti le physi s (in(cid:28)nities that may appear in the al ulations are onsistentlyabsorbed into a (cid:28)nite number of physi al parameters). Despite the a hievements a - omplished in this set up, several issues have not been resolved yet. We list below someamong the most important short omings of the Standard Model [2℄. • Absen e of gravity: the Standard Model does not in lude in its des ription theNewtonian for e, whi h is orders of magnitude smaller than the nu lear for es. Al-though General Relativity des ribes its infrared properties onsistently, gravity is har-a terised by non-renormalizable operators whi h produ e ultraviolet divergen es. • The hierar hy problem: the Higgs boson, responsible for the ele troweak symmetrybreaking and for the generations of the masses for the elementary parti les, has a mass ofthe order of
GeV (if orre tly predi ted by the Standard Model). This mass re eives1adiative orre tions whi h an make the Higgs very heavy ( ∼ GeV), while itsva uum expe tation value is of the order of the ele troweak s ale. The hierar hy betweenthe two energy s ales in the physi s of the Higgs boson appears very unnatural, and ertainly unappealing for a fundamental theory. The introdu tion of supersymmetry (asymmetry between fermioni and bosoni degrees of freedom in the theory) solves thisproblem by preventing the s alar parti le to a quire the dangerous ontributions fromthe perturbation theory, thus stabilising its mass. • The grand uni(cid:28) ation: the oupling onstants for the ele tromagneti and nu learfor es are parameters whi h depend on the energy s ale. If their behaviour is extrapo-lated at high energy, roughly GeV, these values approa h to one point but do not oin ide. If supersymmetry is in luded, the (cid:28)nal theory provides a uni(cid:28)ed des riptionof the for es of the Standard Model at high energy. • The arbitrariness: more than twenty free parameters des ribe the physi s of theStandard Model and their values are ompletely arbitrary. For instan e, the fermionmasses, the gauge and Yukawa ouplings, the Kobayashi-Maskawa parameters and manyothers have to be (cid:28)xed by the experiments and put by hand into the theory.There are many other open questions related to the physi s of the Standard Model,su h as the problem of the osmologi al onstant, whose small value annot be explainedin this set up. Also, the number of families does not (cid:28)nd a reasonable explanation.Moreover, we mention the non-zero neutrino masses, due to their os illations, whi hdoes not (cid:28)t into the des ription of the leptoni physi s of the Standard Model. Theattempts of surmounting all these in onsisten ies lead to several di(cid:27)erent theoreti alsolutions in the physi s beyond the Standard Model, for instan e the introdu tion ofgrand uni(cid:28) ation theories (GUTs) and supersymmetry. The main target of GUTs the-ories [2, 3℄ is solving the uni(cid:28) ation problem previously mentioned, by extending thegauge symmetry group of the SM to a G GUT hara terised by only one gauge oupling.In prin iple, the strong, the weak and the ele tromagneti intera tion merge togetherat some higher energy s ale M GUT where the theory has the larger gauge symmetry G GUT . When the energy de reases below M GUT then the GUT symmetry breaks tothe SM gauge group SU (3) × SU (2) × U (1) and the ouplings asso iated with di(cid:27)erentfa tors evolve at di(cid:27)erent rate. The smallest simple group whi h a ommodates the SMis the SU (5) with M GUT ≃ GeV [4℄. A typi al feature of grand uni(cid:28)ed theories isthe mixing of quarks and leptons into the same group representation. Thus, in the aseof SU (5) gauge group, a matter generation is ontained into the two irredu ible rep-resentations { , ¯5 } ∈ SU (5) . By onsidering a larger G GUT , for example an SO (10) symmetry [5℄, it is possible to ombine one generation into only one irredu ible repre-sentation, pre isely the of SO (10) . In the last ase, the presen e of a singlet state,the right-handed neutrino, and the absen e of exoti parti les makes the model verypredi tive. Unfortunately, there are several unsolved questions appearing in grand uni-2ed theories, most of whi h originated from the quark-lepton mixing. A (cid:28)rst exampleis given by the existen e of new intera tions that violate lepton and baryon number,whi h are responsible for the instability of the proton. Another typi al problem is thepresen e of olour-triplet Higgs states whi h we do not expe t to see in the low energyspe trum (the so- alled doublet-triplet splitting problem). Additionally, the hierar hyproblem, whi h a(cid:27)e ts already the physi s of the SM, does not (cid:28)nd a solution in GUTstheories. Finally, they still su(cid:27)er from the la k of gravity.Several answers to the previous problems are presented by supersymmetri theories.In parti ular the hierar hy problem is solved with the introdu tion of supersymmetry(SUSY), as anti ipated earlier, whi h asso iates to ea h boson of the theory a fermioni superpartner with the same quantum numbers (sin e any internal symmetry ommuteswith SUSY). This symmetry is an extension of the Poin aré algebra whi h in ludesthe fermioni generators Q i , i = 1 , ..N , satisfying anti ommutation relations. The waysupersymmetry over omes the hierar hy problem is by "doubling" the spe trum, whereea h s alar oexists with its fermioni partner. Basi ally, the radiative orre tions of thes alar Higgs at one-loop in lude a divergent s alar self-energy term. In supersymmet-ri theories a quadrati ally divergent term from the bosoni superpartner arises, givingexa tly an opposite ontribution. Hen e, we assist to a an ellation of terms whi h sta-bilises the s alar masses of the theory. At low energies there is no experimental eviden eof supersymmetri parti les, implying that SUSY has to be broken at a relatively lows ale, while being an exa t symmetry at high energies.1.2 String theory as a theory of uni(cid:28) ationAs mentioned before, the non-renormalizability of General Relativity makes a onsistentdes ription of quantum gravity problemati . Therefore, the formulation of a quantumtheory that in ludes gravity and the other for es is very important. String theoryseems to be the most su essful andidate for a uni(cid:28)ed theory of all for es in nature,as we explain in the following. The regularization of the gravitational intera tionsis realised thanks to the introdu tion of an extended obje t, the string. The knownparti les are identi(cid:28)ed with massless ex itations of the string. Beside these parti lesthere is an in(cid:28)nite tower of (cid:28)elds with in reasing masses and spins [6, 7℄ with typi almass of the order of the Plank s ale M P ∼ GeV. Among all ex itation modes thegraviton, the quantum of the gravitational (cid:28)eld, arises in the spe trum, and suggests theinterpretation of string theory as a quantum theory of gravity. Moreover, the presen e ofonly one parameter (the string oupling g s ) used in the des ription of all phenomena, is onsidered a key feature in the prospe tive of an unifying pi ture. From a more te hni alpoint of view, string theory ontains gauge symmetries whi h may in orporate the SMsymmetry. Finally, supersymmetry arises in a natural way in this set up, despite theexisten e of onsistent modular invariant string theories whi h are not supersymmetri .3n the quantization pro edure, the onsisten y of the string theory requires spa etimeto have the riti al dimension, whi h orresponds to D = 10 for supersymmetri strings.In the table below we present the (cid:28)ve 10-dimensional perturbative superstring theoriesand some of their most important properties.Type N susy String Massless bosoni ontent H E × E g µν , ϕ, B µν , A µ of E × E H SO (32) g µν , ϕ, B µν , A µ of SO (32) I − SO (32) g µν , ϕ, A µν , A µ of SO (32) IIA g µν , ϕ, B µν , C µνρ , A µ of U (1) IIB g µν , ϕ, B Nµν , ϕ ′ , B Rµν , D † µνρσ In the table above, g µν , ϕ, B µν , A µ represent the graviton, the dilaton, the antisym-metri tensor and the gauge bosons respe tively. The bosons A µ belong to the adjointrepresentation of E × E or SO (32) for the (cid:28)rst three ases, while they are bosonsof U (1) symmetries for the type IIA ase. C µνρ , ϕ ′ , B Rµν and D † µνρσ are respe tively athree-index tensor potential, a zero-form, a two-form and a four-form potential, the lat-ter with self-dual (cid:28)eld strength. The (cid:28)ve superstring models are onsidered as di(cid:27)erentmanifestations (in di(cid:27)erent regimes), of an unique theory, known as M -theory, andthey are onne ted by some kind of equivalen es, the so- alled string dualities [8℄. Theunderlying fundamental theory, whose low energy limit is 11 dimensional SUGRA [9℄,is unfortunately still poorly understood. M−TheoryI IIA IIB
SO(32) E x E
S−Duality T−Duality onT−Duality onT−Duality on T−Duality on g g s s o ooo
S R S R x x
S R x I R x Figure 1.1: Supersymmetri perturbative onsistent string theories in 10 dimensions.As we an see from (cid:28)g.1.1, the duality transformations relate the superstring theo-ries in nine and ten dimensions. T duality inverts the radius R of the ir le S , alongwhi h a spa e dire tion is ompa ti(cid:28)ed, R → R . In parti ular, this duality relates4he weak- oupling limit of a theory ompa ti(cid:28)ed on a spa e with large volume to the orrespondent weak- oupling limit of another theory ompa ti(cid:28)ed on a small volume. S duality instead provides the quantum equivalen e of two theories whi h are pertur-batively distin t. In fa t, it inverts the string oupling g s → g s . The perturbativeex itations of a theory are mapped to non-perturbative ex itations of the dual theoryand vi eversa. Fig.1.1 summarises the relevant information of the perturbative stringtheories and their web of dualities.In order to make onta t with the real world, the ompa ti(cid:28) ation of the six extradimensions is needed. This pro edure follows the Kaluza-Klein dimensional redu tion ofquantum (cid:28)eld theory and is generalised to the ase where a ertain number of spa etimedimensions give rise to a ompa t manifold, invisible at low-energy [10, 11℄. Demandingfour-dimensional N = 1 supersymmetri models leads us to a spe ial hoi e of inter-nal manifolds, the so- alled Calabi-Yau manifolds [12℄. Compa ti(cid:28) ations of this kindare hara terised by some free parameters, the moduli, generally related to the sizeand shape of the extra dimensions. The low energy parameters often depend on thesefree values whi h spoil the predi tivity of the theory. The moduli des ribe possibledeformations of the theory and their ontinuous hanges allow to go from one va uumto another. So far, the problem of (cid:28)xing the moduli has not been solved yet, sin eno fundamental prin iple is able to single out a unique physi al va uum. The studyof Calabi-Yau manifolds is, unfortunately, fairly ompli ated sin e the omputation ofproperties whi h are not of topologi al nature is very di(cid:30) ult. A simpler lass of om-pa t manifolds is given by the toroidal ompa ti(cid:28) ation, although the resulting theory isnot hiral. Hen e, ombining the desirable pi tures of Calabi-Yau manifolds and toroidal ompa ti(cid:28) ations, we (cid:28)nally arrive to the orbifold onstru tion. The orbifold seems toprovide a simple framework for the realisation of N = 1 supersymmetri models in fourdimensions, with hiral parti les in the spe trum.In this thesis we dis uss two main ompa ti(cid:28) ation s hemes whi h o(cid:27)er omplemen-tary advantages in the understanding of semi-realisti heteroti string models. The (cid:28)rstapproa h is the free fermioni onstru tion, whi h is based on an algebrai method tobuild onsistent string va ua dire tly in four dimensions. In the fermioni formalism allthe worldsheet degrees of freedom, required to an el the onformal anomaly, are givenby free fermions on the string worldsheet. This set up o(cid:27)ers a very onvenient settingfor experimentation of models, allowing a systemati lassi(cid:28) ation of free fermion va uaand their phenomenologi al properties. Moreover, this set up provided the most semi-realisti models to date. On the other hand, the orbifold ompa ti(cid:28) ation, previouslymentioned, leads to the analysis of other interesting features of heteroti models. Forinstan e, the geometri pi ture provided by the orbifold onstru tion may be instru-mental for examining other questions of interest, su h as the dynami al stabilisation ofthe moduli (cid:28)elds and the moduli dependen e of the Yukawa ouplings. The orrespon-5en e of free fermioni models [13, 14, 15℄ to Z × Z orbifold ompa ti(cid:28) ation is a keypoint of this thesis. In fa t, the phenomenologi ally appealing properties of the freefermioni models and their relation to Z × Z orbifolds provide the lue that we mightgain further insight into the properties of this lass of quasi(cid:21)realisti string ompa t-i(cid:28) ations by onstru ting Z × Z orbifolds on enhan ed non-fa torisable latti es (thepoint at whi h the internal dimensions are realised as free fermions on the worldsheetis a maximally symmetri point with an enhan ed SO (12) latti e, whi h is in prin iplenon-fa torisable).In this thesis we produ ed the following results. We presented two semi-realisti models in the free fermioni formulation with a redu ed Higgs spe trum. The trun a-tion of the Higgs ontent is realised for the (cid:28)rst time in this set up at the level of thestring s ale, by the assignment of asymmetri boundary onditions to the internal right-and left-moving fermions of the theory. Moreover, the analysis of (cid:29)at dire tions, per-formed with the standard methods, leads to an unexpe ted result. The Fayet-IliopoulosD-term whi h breaks supersymmetry perturbatively in our models is not ompensateby the existen e of D- and F- (cid:29)at solutions, whi h would restore supersymmetry. TheBose-Fermi degenera y of the spe trum implies that the models are supersymmetri attree level. Thus, the models presented may provide a new interpretation of the super-symmetry breaking in string theory. In the framework of the orbifold onstru tion, webuilt a Z × Z orbifold with a skewed SO (4) ompa ti(cid:28) ation latti e and analysedits spe trum and symmetry group. Our main goal initially was reprodu ing a threegeneration free fermioni model [16℄ with gauge symmetry E × U (1) × SO (8) H . Un-fortunately we ould not obtain the wished features, not even after the introdu tion ofWilson lines. Nevertheless, several interesting properties are dis ussed on erning the ompa ti(cid:28) ation latti e and its possible tools to realise semi-realisti four dimensionalmodels in the onstru tion of orbifold models. Finally, we on luded this thesis withthe onstru tion of modular invariant partition fun tions for heteroti shift orbifolds.In this ontext we presented di(cid:27)erent examples of onsistent va ua with the derivationof the full perturbative spe trum. In parti ular, we dis ussed the details of a Z × Z shift orbifold model whi h ontains some te hni al subtleties due to the elements of theorbifold group, and presented in detail its massless spe trum.6.3 Organisation of the haptersThe topi s of this thesis are organised as follows. • Chapter 2A general introdu tion on the bosoni and fermioni string is presented in orderto provide perturbative superstring onstru tions. A brief overview on the partitionfun tion whi h en odes the modular invariant properties of the theory is dis ussed. Weexplain the bosonization pro edure ne essary for the orresponden e between fermioni and bosoni onformal (cid:28)eld theories. We lose the hapter with some generalities onthe heteroti string, whi h will be analysed in great detail in the next hapters. • Chapter 3We present the main features of four-dimensional semi-realisti models in the freefermioni onstru tion and show the advantages of using this ompa ti(cid:28) ation s heme.We (cid:28)x the formalism to provide the onsisten y onstraints and the model building rulesfor this framework and explain the general derivation of the spe trum. In the se ondpart of the hapter we present two very pe uliar examples of semi-realisti free fermioni models, where the redu tion of the Higgs ontent is, for the (cid:28)rst time, realised at thestring s ale. Moreover, the standard analysis of (cid:29)at dire tions is in both ases unable torestore supersymmetry perturbatively, although the models are supersymmetri at the lassi al level. This point opens new interpretations for the supersymmetry breakingme hanism in string theory. • Chapter 4We start by introdu ing the heteroti string in its bosoni formulation, followed bythe des ription of the toroidal ompa ti(cid:28) ation. We pro eed by providing the generali-ties of orbifold onstru tions. The dis ussion of the spe trum is initially performed atan abstra t level to (cid:28)nd in the last part of the hapter a on rete appli ation, in the ase of a Z × Z orbifold with SO (4) ompa ti(cid:28) ation latti e. In our example we seizethe opportunity to present the expli it derivation of the (cid:28)xed tori for a non-fa torisablelatti e and investigate possible ways to ontrol the number of families, for example by onsidering Wilson lines. • Chapter 5Some interesting examples of heteroti strings ompa ti(cid:28)ed on shift orbifolds arepresented, providing the te hni al details on the derivation of Z and Z × Z orbifoldpartition fun tions. As an example is obtained, a onsistent modular invariant stringva uum with no graviton. This model is in a way reminis ent of string va ua withoutgravity - "little string" models. • Chapter 6We on lude this thesis underlining the main aim of our resear h, the semi-realisti heteroti string onstru tions in di(cid:27)erent ompa ti(cid:28) ation s hemes. We present themain results obtained and (cid:28)nally provide possible interesting outlooks.7hapter 2Ba kground notions on onsistentperturbative superstring theoriesIn this hapter we brie(cid:29)y present aspe ts of the perturbative formulation of string theoryand introdu e the ne essary tools for the onstru tion of semi-realisti four dimensionalsuperstring models. The sour es of the introdu tory part are given by [17, 18, 19, 20,21, 22, 23, 24℄.We start by presenting the bosoni string, whi h is the simplest instan e of a stringtheory. This two-dimensional onformal theory at the lassi al level is onsistent onlyat the riti al dimension D=26. In its low energy spe trum, provided by the masslessex itation modes, the presen e of a symmetri metri tensor g µν , the andidate of thegraviton (cid:28)eld, gives the main motivation for interpreting string theory as a quantumtheory of gravity. Two main reasons make the bosoni string inadequate for a ompletedes ription of the fundamental intera tions, su h as the existen e of ta hyoni states,a sign of instability for the theory, and the absen e of fermioni ex itations in theperturbative spe trum. The solution to these problems leads to the introdu tion of thesuperstring, a super onformal theory with riti al dimension D=10. After presentingthe lassi al a tion for the bosoni and fermioni string, we will dis uss the quantizationpro edure of the theory. The on epts of onformal invarian e and modular invarian eare explained in detail. We on isely mention how to al ulate string intera tions whilstgiving a detailed overview on the partition fun tion for the losed bosoni string, thetorus amplitude, sin e this quantity represents one of the main topi s treated in thefollowing hapters [25, 26℄. In the last se tion we introdu e the on ept of toroidal ompa ti(cid:28) ation whi h will be onsidered extensively in hapter 4, before the orbifold onstru tions of semirealisti models. In most ases we restri t our dis ussion to the losed strings sin e our target is the onstru tion of the heteroti string.8.1 Bosoni stringsStrings are one dimensional (cid:28)nite obje ts whose propagation in a D dimensional spa e-time gives rise to a two dimensional worldsheet X µ ( σ, τ ) , µ = 0 , ..D − . In (cid:28)g. 2.1 thissurfa e is shown in both ases of free losed and free open strings. The worldsheet isparametrized by the two real independent oordinates, τ and σ , where the (cid:28)rst variableis a time-like parameter while the se ond is spa e-like and belongs to the interval [0 , π ] . σττσ σσ = π= 0 (a)(b) Figure 2.1: a) losed string worldsheet. b)open string worldsheet.The physi s of the string1 is des ribed by the Polyakov a tion that, in a (cid:29)atMinkowski D dimensional spa etime, assumes the form [27, 28℄ S = − T Z d σ √− hh αβ η µν ∂ α X µ ∂ β X ν , (2.1)where T is the string tension, h αβ is the worldsheet metri and h = det ( h αβ ) , while d σ implies the equivalent notation σ = ( σ , σ ) = ( τ, σ ) .For a general ba kground we an simply repla e the (cid:29)at metri η µν by g µν ( X ) andeq.(2.1) be omes the worldsheet a tion of D dimensional s alar (cid:28)elds X µ oupled to thedynami al two-dimensional metri (theory of quantum gravity oupled to matter).The Polyakov a tion has three symmetries:1) Poin aré invarian e in the target spa e X µ .2) Lo al reparametrization invarian e.3) Conformal (Weyl) invarian e.1To be more pre ise, the simplest a tion whi h des ribes the motion of the string is the Nambu-Gotoa tion, S NG = − T R d σ √− γ , where γ is the determinant of the indu ed metri on the worldsheet, γ αβ = ∂ α X µ ∂ β X ν g µν . This a tion is proportional to the area swept from the worldsheet, thus itprovides a more geometri and intuitive meaning of the string a tion. The Polyakov a tion, whi hsupplies in a simpler way the equations of motion, is equivalent to the Nambu-Goto a tion and an beobtained by introdu ing the independent metri on the worldsheet h αβ .9he last two properties are lo al symmetries whi h an be used to (cid:28)x the worldsheetmetri in the onformal gauge, h αβ = e φ ( τ,σ ) η αβ , obtaining a (cid:29)at metri up to a s alingfun tion. The equations of motion (e.o.m.) for the bosoni (cid:28)elds X µ and for the metri h αβ are obtained in the usual pro edure, as the variation of the a tion with respe t toea h of these (cid:28)elds respe tively. At this point it is onvenient to introdu e the two-dimensional stress tensor T αβ whi h provides the onstraints for the string theory. Wede(cid:28)ne T αβ (also known as energy-momentum tensor) as the variation of the Polyakova tion with respe t to the world-sheet metri T αβ = − T √− h δSδh αβ = ∂ α X µ ∂ β X µ − h αβ h ργ ∂ ρ X µ ∂ γ X µ , (2.2)then the request that the energy-momentum tensor vanishes, T αβ = 0 , (2.3) orresponds exa tly to the e.o.m. for h αβ . This ondition is alled the Virasoro on-straint and represents a very important ingredient when onsidering the physi al statesof the model under onsideration. The stress tensor is symmetri , tra eless ( T αα = 0 ),as onsequen e of the Weyl invarian e and onserved.It is very onvenient to rewrite the Virasoro onditions in the light- one oordinates σ + = τ + σ , σ − = τ − σ , where ∂ ± = ( ∂ τ ± ∂ σ ) . Then eq.(2.3) would simply be ome T −− = 12 ( ∂ − X ) = 0 ; T ++ = 12 ( ∂ + X ) = 0 ; T ±∓ = 0 . (2.4)The equations of motion for the (cid:28)elds X µ take the form ∂ + ∂ − X µ = 0 , whose generalsolution an be written as the sum of a (cid:16)right-moving(cid:17) solution plus a (cid:16)left-moving(cid:17)solution, X µ ( τ, σ ) = X µR ( τ − σ ) + X µL ( τ + σ ) . (2.5)Together with the periodi ity onstraint X µ ( σ, τ ) = X µ ( σ + 2 π, τ ) , eq.(2.5) leads to themode expansion X µR ( τ − σ ) = 12 x µ + α ′ p µ ( τ − σ ) + i r α ′ X n =0 n α µn e − in ( τ − σ ) ,X µL ( τ + σ ) = 12 x µ + α ′ p µ ( τ + σ ) + i r α ′ X n =0 n ˜ α µn e − in ( τ + σ ) , (2.6)where the Regge slope parameter α ′ is de(cid:28)ned in terms of the string tension as α ′ =1 / πT . From (2.6) we see that the lassi al motion of the string is des ribed by the entre of mass position x µ , the momentum p µ and the os illator modes.For later onvenien e we de(cid:28)ne the Virasoro operators as Fourier modes of the stresstensor, that in the right-moving se tor be ome L m = T Z π dσe im ( τ − σ ) T −− = 12 ∞ X n = −∞ α µm − n · α µn ( m = 0) . L m = 0 , ∀ n ∈ Z and for the ase n = 0 we obtain the mass equation for the right os illation modes, dis ussed in the followingse tion. Moreover α µ = q α ′ p µ . The orrespondent left-moving expression ˜ L m is givenby the substitutions T −− → T ++ , σ − → σ + and the omplex onjugate os illators andsimilar onditions to the right se tor hold in the left se tor as well.Quantization of the bosoni stringThe os illators, the entre of mass position and the momentum presented in eq.(2.6)satisfy the standard ommutation relations, while the Virasoro operators form the so- alled Virasoro algebra. In the ovariant anoni al quantization pro edure the previous onditions are translated into the following ommutators [ x µ , p µ ] = iη µν , [ α µm , α νn ] = mδ m + n η µν , [ ˜ α µm , ˜ α νn ] = mδ m + n η µν , [ L m , L n ] = ( m − n ) L m + n + D m ( m − δ m + n . (2.7)The other ommutators between di(cid:27)erent ombination of operators are zero. The Her-miti ity of X µ gives ( α µn ) † = α µ − n ; ( ˜ α µn ) † = ˜ α µ − n . D represents the entral harge andfor the bosoni string D = η µν η µν . The same algebra holds for the left operator ˜ L m .From now on, when de(cid:28)ning properties of operators in the right se tor, we will assumeimpli itly that analogous relations hold in the left se tor. In the quantization of a las-si al system an ambiguity is introdu ed in the de(cid:28)nition of the operators. This anbe solved if we onsider the orresponding normal-ordered expressions. In the ase ofthe Virasoro operators the orre t de(cid:28)nition is given by L m = P ∞ n = −∞ : α µm − n α µn : .The only term sensitive to normal ordering is L where a normal ordering onstant a isintrodu ed.In the ovariant quantization we obtain states with negative norm whi h destroy theunitarity of the theory, but we an dis harge those by imposing the following onstraints L m> | phys i = 0 , ( L − a ) | phys i = 0 . (2.8)It has been shown that the subset of positive norm states exists only for D ≤ and a ≤ [29℄.It is easier to solve the Virasoro onstraints in the light- one quantization (we havealready de(cid:28)ned the operators in terms of light- one oordinates) where the states, ob-tained by solving the mass-shell equation, are always positive. But if unitarity is guar-anteed in this pro edure, we will need to verify the Lorentz invarian e, whi h is notmanifest. We have already mentioned that for D = 26 and a = 1 Lorentz invarian eis preserved. D = 26 is thus a very spe ial hoi e of spa etime dimensions, alled the riti al dimension of the bosoni string. 11e use now a residual invarian e, leftover after imposing the onformal gauge, whi his a reparametrization invarian e up to s aling, generally de(cid:28)ned as σ ′ + → f ( σ + ) , σ ′− → f ( σ − ) . This invarian e allows to (cid:28)x the value of X + as follows, leading to the light one gauge, X + = x + + 2 α ′ p + τ. The light- one oordinates are given by X ± = ( X ± X D − ) / √ and by using theVirasoro onstraints we an express X − in terms of the transverse oordinates X i ,where i takes values in the transverse dire tions. This means that we are left only withthe transverse os illators, while the light- one ones are given by α − n = 1 √ α ′ p + { X m ∈ Z : α in − m α im : − aδ n } ,α + n = r α ′ p + δ n , (2.9)and analogous expressions hold for ˜ α ± n . The Virasoro onstraints in the light- one gaugede(cid:28)ne the mass-shell ondition for the physi al states p + p − = 2 α ′ ( L + ˜ L − D − L = ˜ L . (2.10)In the (cid:28)rst equation of (2.10) the Riemann ζ fun tion2 ζ ( −
1) = − / has been used,as a result of the divergent sums of zero-points energies due to the normal ordering a of L and ˜ L [30℄. The se ond equation in (2.10) is the level mat hing ondition, arelation whi h onne ts the left with the right ex itation modes of the losed string.This onstraint has to be imposed for the onsisten y of every losed string model and ontains an important information on erning the physi al states of the model, the rightand the left modes provide the same ontribution to the mass of the physi al states.The masses of the string ex itations are obtained by the ontributions of the transversemomenta, whi h for the right se tor are provided by the formula L = α ′ p i p i + N . Themass operator is M = 2 α ′ ( N + ˜ N − D −
212 ) (2.11)and N = P m> α − m · α m . In the ase at hand D = 26 , thus the (cid:28)rst state obtained fromeq.(2.11) is the ground state | p µ i , with N = ˜ N = 0 . Its mass is given by M = − a/α ′ ,where a takes the value for onsisten y, as we said before. This state is the ta hyon.The (cid:28)rst ex ited state is the tensor α i − ˜ α j − | p µ i . If we de ompose it into irredu iblerepresentations of the group SO (24) we obtain a symmetri tensor g µν (a spin-2 parti le,the graviton), the antisymmetri tensor B µν and a s alar ϕ , the dilaton.At the next level we obtain states whi h are organised in representations of SO (25) and whi h are massive.2The in(cid:28)nite sum due to the zero-point energy is al ulated by a regularisation pro edure introdu ingthe Riemann ζ fun tion: ζ ( s ) = P ∞ k =1 k − s . It provides the value of a in terms of the spa e-timedimension D , whi h is exa tly a = D − , as shown in formula (2.11) for ζ ( −
1) = − / [30℄.12.2 Vertex operators and string intera tionsA lo al unitary quantum (cid:28)eld theory has an operator-state orresponden e whi h asso- iates to ea h (cid:28)eld a quantum state reated from the va uum. In string theory the same orresponden e is realised by mapping the worldsheet ylinder to the omplex plane. phys>| = − τ ττσσ π00 < σ < 2 τ i τ τ −i σ V phys (0) τπ oo z = e z Figure 2.2: Mapping of the worldsheet ylinder into the omplex plane. The dotted linesof onstant τ are on entri ir les while the lines of onstant σ follow radial dire tionsfrom the origin.In this ontext it is possible to build the so- alled vertex operators whi h give riseto a spe trum generating algebra. By using this formalism, for instan e, an in omingphysi al state | phys i in the in(cid:28)nite worldsheet past ( τ = −∞ ) is given by the insertionof a vertex operator V ( z ) at the origin z = 0 , see (cid:28)g.(2.2).In this thesis we will not go into further details on erning the vertex operators, butit is important to stress their role in the onstru tion of string amplitudes and in thedes ription of strings intera tions.In quantum (cid:28)eld theory the perturbative expansion of Feynman diagrams des ribesthe intera ting parti les at well de(cid:28)ned points. The worldline of parti les in spa etimeis des ribed by propagators that meet in a vertex, singular point whi h is responsiblefor ultraviolet divergen es in loop amplitudes. The string Polyakov perturbation theoryis given by the sum of two-dimensional surfa es whi h orrespond to the worldsheets.When onsidering all ontributions of the in(cid:28)nite tower of massive parti les of the stringspe trum, the ultraviolet divergen es of quantum gravity loop amplitudes an el out.The reason why the non-renormalizability of quantum (cid:28)eld theory is solved in stringtheory is be ause its intera tions are des ribed by smooth surfa es with no singular13oints. The main onsequen e of this property is that string intera tions are ompletelydetermined by the worldsheet topology. In oriented losed strings the perturbativeexpansion is given by only one ontribution at ea h order of perturbation theory. This ontribution orresponds to losed orientable Riemann surfa es with in reasing numberof handles h and the perturbative series is hen e weighted by g − χs , where χ is the Euler hara ter, de(cid:28)ned as χ = 2 − h , while the string oupling g s is dynami ally determinedby the va uum expe tation value of the dilaton (cid:28)eld ϕ , g s = e <ϕ> .A generi string s attering amplitude is given by a path integral of the form A = Z D h αβ D X µ e − S P n Y i =1 Z M d σ i V α i , (2.12)where h αβ is the metri on the worldsheet M , S P is the Polyakov a tion and V α is thevertex operator that des ribes the emission or absorption of a losed string state of type α i from the worldsheet. The onformal invarian e redu es these expressions to integralson non-equivalent worldsheets whi h are des ribed by some omplex parameters, themoduli. The amplitudes in eq.(2.12) are then (cid:28)nite dimensional integrals over themoduli spa e of M .2.3 The superstringAs we have mentioned at the beginning, the bosoni string su(cid:27)ers of two main problems:the absen e of spa etime fermions (ne essary for a realisti des ription of nature) andthe presen e of ta hyons (sign of an in orre t identi(cid:28) ation of the va uum). The solutionto these problems leads us to the onstru tion of the superstring. The new theory is onstru ted by the introdu tion of worldsheet supersymmetry, realised by in luding D two-dimensional Majorana fermions Ψ µ = ( ψ µ − , ψ µ + ) , µ = 0 , ..D − , on the worldsheet.These (cid:28)elds are ve tors from the spa etime point of view but when ombined withappropriate boundary onditions will provide spa etime fermions. In the following wewill work in the RNS (Ramond-Nevew-S hwarz) formalism [31, 32℄, where the GSO(Gliozzi-S her k-Olive) proje tions are introdu ed in order to obtain supersymmetry[33℄. The generalised a tion S T in the onformal gauge S T = − T Z d σ ( ∂ α X µ ∂ α X µ − iψ µ ρ α ∂ α ψ µ ) (2.13)is invariant under worldsheet global supersymmetri transformations δ ǫ X µ = ¯ ǫψ µ , δ ǫ ψ µ = − iρ α ∂ α X µ ǫ, with ǫ onstant spinor and ρ α , α = 0 , , Dira matri es whi h an be hosen as follows ρ = (cid:18) − ii (cid:19) , ρ = (cid:18) ii (cid:19) .
14n the light- one oordinates the fermioni ontribution of eq.(2.13) is simply ψ − ∂ + ψ − + ψ + ∂ − ψ + , (2.14)where the spa e-time index µ has been suppressed.The equations of motion are simply the Dira equations ∂ ± ψ ∓ = 0 . Their solutionsare of the form ψ − = ψ − ( σ + ) and ψ + = ψ + ( σ − ) , hen e we an say that ψ − representsthe right-moving (cid:28)eld while ψ + is the left-moving one. The boundary onditions ariseby requiring that ( ψ + δψ + + ψ − δψ − ) | σ = πσ =0 = 0 . (2.15)Equation (2.15) is satis(cid:28)ed if ψ + and ψ − are periodi or anti-periodi ψ µ + ( σ + π, τ ) = ± ψ µ + ( σ, τ ) ,ψ µ − ( σ + π, τ ) = ± ψ µ − ( σ, τ ) . (2.16)The periodi ase is alled Ramond (R) boundary ondition while the anti-periodi isknown as Neveu S hwarz (NS). The general solution in terms of mode expansion is givenby ψ µ − = X r b µr e − iπ ( σ − ) , (2.17)for the right-moving states and an analogous expression applies for the left-movers ψ µ + (by repla ing σ − by σ + and b µr by ˜ b µr ). As a result of the boundary onditions, thefrequen y r is integer for R boundary onditions and half-integer for the NS ase.The Ramond boundary onditions and the integer modes will des ribe string statesthat are spa etime fermions. In fa t, if we onsider the fundamental state b i | p µ > , wesee that it is massless and degenerate, as b satis(cid:28)es the Cli(cid:27)ord algebra { b i , b j } = δ ij .This means that the Ramond va uum is a spinor of SO (8) and all the states obtainedfrom the va uum with the reation operators are fermioni as well. Instead the NSboundary onditions with the half-integer ex itations give bosons. The fundamentalstate | p µ > has negative mass (ta hyon) and is a s alar. The (cid:28)rst ex ited masslessstate b i − | p µ > is a ve tor of SO (8) and all the states in this se tor, reated byhalf-integer modes, provide bosons.Sin e the superstring is an extension of the bosoni ase, it is ne essary to enlargethe algebra whi h des ribes the theory. Thus, the lassi al Virasoro onstraints are nowgeneralised to J ± = 0 , T ±± = 0 , (2.18)where the super urrents and the energy-momentum tensors are given in their light- onegauge oordinates J + = ψ µ + ∂ + X µ , T ++ = ∂ + X µ ∂ + X µ + i ψ µ + ∂ + ψ + µ , − = ψ µ − ∂ − X µ , T −− = ∂ − X µ ∂ − X µ + i ψ µ − ∂ − ψ − µ . Quantization of the superstringThe quantization of the fermioni (cid:28)elds is obtained by imposing the anti ommutationrelations { b µr , b νs } = η µν δ r + s , { ˜ b µr , ˜ b νs } = η µν δ r + s . The anti ommutator of left and right os illators vanishes. For r < ( r > ) b r denotes reation (annihilation) operators. The omplete spe trum is provided by the a tion ofthe reation operators on the va uum.The mass-shell ondition in eq.(2.11) is now generalised by rede(cid:28)ning N as thenumber of right bosoni plus right fermioni os illators a ting on the va uum. Samerede(cid:28)nition applies to ˜ N . We have to take into a ount that fermions an assume R orNS boundary onditions and this will hange the ontribution to the zero point energy a . Ea h fermioni oordinate ontributes with a − / in the NS se tor and / in theR se tor, while ea h boson gives a ontribution of − / . In D dimensions, if we are inthe light- one gauge, we have D − transverse bosons and D − transverse fermionswhi h give a = 0 in the Ramond se tor while a = − / D − in the Neveu-S hwarz.After quantizing the supersymmetri theory, the Virasoro onstraints be ome [ L m , L n ] = ( m − n ) L m + n + D m ( m − δ m + n , [ L m , G r ] = ( m − r ) G m + r , { G r , G s } = 2 L r + s + D r − a δ r + s , (2.19)where the operators are de(cid:28)ned by their normal ordered expressions L m = L a ′ m + L b ′ m ,L a ′ m = 12 X n ∈ Z : α − n · α m + n : ,L b ′ m = 12 X n ∈ Z + a : ( r − m b m − r · b r : ,G r = X n ∈ Z : b r − n · α n : . (2.20)For ompleteness with respe t to the bosoni ase, we shall provide the light- one quan-tization for the superstring ase. The theory is ghost-free but not expli itly ovariant,but we an assure Lorentz invarian e if D = 10 and a = 1 / [17℄.The gauge is (cid:28)xed with the relation ψ + = 0 and X + = α ′ p + τ and sin e we are(cid:28)xing the longitudinal os illator modes, the only independent degrees of freedom arethe transverse ones.A supersymmetri non-ta hyoni theory is obtained when the spe trum is trun atedby some GSO (Gliozzi, S herk and Olive) proje tions [34℄. We will explain this trun- ation separately in the NS and in the R se tor. In the Neveu-S hwarz se tor the GSO16roje tions P GSO is de(cid:28)ned by keeping states with an odd number of b i − r os illator ex i-tations and removing those with even number. We de(cid:28)ne below the proje tion operatorin the NS se tor and the fermion number, P NSGSO = 12 (1 − ( − F ) , F = ∞ X r =1 / b i − r · b ir . Thus, the bosoni ground state is now massless and the spe trum no longer ontains ata hyon (whi h has fermion number F = 0 ). In the Ramond se tor, the fundamentalstate (a Majorana spinor) lives in the spinorial representation of SO (8) , as mentionedbefore. If we introdu e the proje tor operator P RGSO = 12 (1 + ( − F Γ ) , where Γ = b ·· b is the hiral operator in the transverse dimensions, then the fundamen-tal state be omes a Majorana-Weyl spinor of de(cid:28)nite hirality. P RGSO , while proje tingonto spinors of opposite hirality, guarantees spa etime supersymmetry of the physi alsuperstring spe trum (we note that the hoi e of sign of ( − F Γ = ± , orrespondingto di(cid:27)erent hirality proje tions on the spinors, is a matter of onvention).The general pro edure to obtain the massless spe trum is to solve the masslessequations for right and left se tor, apply level mat hing ondition and the parti ularGSO proje tions depending on the perturbative superstring model onsidered, (cid:28)nallytensor the left with the right states. If we want to pro eed with the expli it al ulation ofthe spe trum we need to spe ify the string theory we want to analyse. Supersymmetri theories with only losed strings are type IIA, type IIB and heteroti models. For typeIIA and type IIB (where supersymmetry is realised in the left and right se tor), bytaking the tensor produ ts of right and left movers we get four distin t se tors: NS-NS,R-R, NS-R, RN-R, where the (cid:28)rst two sets give bosons and the last two se tors providefermion (cid:28)elds in the target spa e. The features and di(cid:27)eren es among these two modelshave been given in the introdu tion. In this thesis we are interested in the heteroti string hen e we will fo us on the te hni alities on erning the heteroti ase startingfrom se tion 2.8.2.4 One loop amplitude and modular invarian eThe one loop va uum amplitude, also known as genus-one partition fun tion, representsa fundamental quantity of the theory sin e it en odes the full perturbative spe trum.Di(cid:27)erently from quantum (cid:28)eld theory, in the string theory this is a (cid:28)nite quantitythat makes the theory modular invariant. The modular invariant onstraints are in fa tderived from the al ulation of the one-loop va uum amplitude. The Feynman diagram,whi h des ribes a losed string propagating in time and returning to its initial state, isa donut-shaped surfa e, equivalent to a two-dimensional torus. We an parametrize the17 τ a ba b Figure 2.3: 1)Torus diagram. 2)The (cid:29)at torus as a two dimensional latti e. a and b represent the two non- ontra tible y les of the Riemann surfa e.torus by a omplex parameter τ = τ + iτ , τ > . If we de(cid:28)ne in the omplex planea latti e by identifying z = z + 1 , z = z + τ , then the torus is obtained by identifyingthe opposite sides of this parallelogram (see (cid:28)g.2.3).The full family of equivalent tori is obtained by the transformations S : τ → − τ , T : τ → τ + 1 , (2.21)that are the generators of the modular invariant group, whose most general transfor-mation is given by τ → aτ + bcτ + d ad − bc = 1 a, b, c, d ∈ Z . (2.22)The formula (2.22) generates the modular group P SL (2 , Z ) . The non-equivalent toriare ontained in the so- alled fundamental region F = C /P SL (2 , Z ) = {| τ | ≥ , − ≤ τ < , τ > } (see (cid:28)g.2.4). Any point outside the modular domain an be mapped by a modulartransformation inside F . We al ulate now the va uum amplitude for the bosoni F SL (2 , Z ) −1 1 −1/2 1/2 iIm( τ ) Re( τ ) τ Figure 2.4: Fundamental domain.string in analogy with the quantum (cid:28)eld theory approa h. In the ase of a single s alarparti le the va uum energy Γ is de(cid:28)ned by the path integral e − Γ = Z D φe − S , (2.23)where S is the a tion of the boson in D dimensions. If we want to make expli it thedependen e of the integral on the parti le mass M we an rewrite it in terms of the18 hwinger parameter t and eq.(2.23) assumes the form Γ = − V Z ∞ ǫ dtt e − tM Z d D p (2 π ) D e − tp , (2.24)where V is the volume of the spa etime and p the momentum of the parti le. Theparameter ǫ is an ultraviolet uto(cid:27) that will disappear when we restri t the integrationregion to the fundamental region of the torus. If we al ulate the Gaussian momentumintegral and generalise formula (2.24) for bosoni and/or fermioni (cid:28)elds then we obtain Γ = − V π ) D/ Z ∞ ǫ dtt D/ Str ( e − tM ) , (2.25)where the Supertra e Str takes into a ount the Bose-Fermi statisti s.Let us now onsider the ase of the bosoni string for whi h we want to derive theone-loop amplitude. For the bosoni theory we have D = 26 and M = α ′ ( L o + ˜ L o − .At this point we need to take into a ount the level mat hing ondition that an beimplemented by a onstraint given in terms of a real variable s . Subsequently, werearrange the t and s parameters in the new omplex "S hwinger" parameter τ = τ + iτ = s + i tα ′ π . Sin e the losed string sweeps a torus at one loop then we identify τ as the Tei hmuller parameter parametrizing the torus (see for example [25℄).De(cid:28)ning q = e iπτ and q = e − iπτ and al ulating the integral in the fundamentaldomain gives the partition fun tion of the torus amplitude T = Z F d ττ τ trq L − q ˜ L − . (2.26)The same expression an be obtained by some geometri onsiderations. A point onthe string propagates in the time dire tion as πτ and in spa e as πτ . The timetranslation is given by the Hamiltonian H = L + ˜ L − and the shift along the stringis given by the momentum operator P = L − ˜ L . The path integral is then T ∝ tr ( e − πτ H e iπτ P ) ∼ tr ( q L − q ˜ L − ) . The expansion of the operator L and the al ulation of the tra e will transformequation (2.26) into T = Z F d ττ τ | η ( τ ) | , (2.27)where the Dedekind η fun tion is de(cid:28)ned in Appendix A, as well as its properties undermodular transformations. Ea h bosoni mode then gives a ontribution to the partitionfun tion equal to | η | . The integrand of eq.(2.27) is modular invariant, as we an proveby using the formulae in Appendix A. 19.5 Spin stru turesWhen we onsider the parallel transport properties of spinors on a two dimensionalsurfa e, for example on the torus, we need to introdu e the so- alled spin stru tures.They provide the fermioni ontributions to the partition fun tion and have to be de(cid:28)nedin both Ramond and Neveu-S hwarz se tors. Some kind of GSO proje tions enter inthe game to ensure the onsisten y of the theory.A fermion moving around the two non- ontra tible loops of the torus gives rise tofour possible spin stru tures, indi ated as following: A (++) ( τ ) , A (+ − ) ( τ ) , A ( − +) ( τ ) and A ( −− ) ( τ ) . The (cid:28)rst entry in the exponent represents the boundary ondition in the σ dire tion while the se ond gives the boundary ondition in time dire tion σ . The ” + ” and ” − ” signs label the Ramond and Nevew-S hwarz boundary onditions respe tively.For brevity we fo us our dis ussion on the spin stru tures of the right se tor of the string.The NS se tor provides states with anti-periodi boundary onditions in the σ dire -tion and if we implement the periodi ity in the time dire tion we will need to introdu ethe Klein operator ( − F in the tra e. The fermioni ontributions to the path integralare given, in the R and NS se tor Hilbert spa e, by the following expressions A (+ − ) ∼ T r R ( e − πτ H ) , A (++) ∼ T r R (( − F e − πτ H ) ,A ( −− ) ∼ T r NS ( e − πτ H ) , A ( − +) ∼ T r NS (( − F e − πτ H ) . (2.28)The modular transformations hange the boundary onditions, thus it is possible toobtain a spin stru ture from another by applying T and S transformations. We notethat A (++) is modular invariant while for the other expressions the following relationshold A (+ − ) S −→ A ( − +) T −→ A ( −− ) . Ea h of these ontributions is multiplied by a phasewhi h an be derived by imposing modular invarian e of the total partition fun tion ofthe model under onsideration. A detailed explanation on the derivation of the phases an be found in [20℄.The one-loop modular invariant partition fun tion for the right-moving se tor isgiven by Z = 12 T r NS [(1 − ( − F ) q L − ] + 12 T r R [(1 + ( − F ) q L ] . (2.29)The total superstring amplitude is obtained by ombining eq.(2.29) with the left-movingfermioni ontribution and multiply the whole expression by the bosoni part.If we al ulate the tra es in eqs.(2.28) we an rewrite the spin stru tures in terms20f the Ja obi θ -fun tions A (++) = 16 q / ∞ Y n =1 (1 − q n ) = θ (0 | τ ) η ( τ ) ,A (+ − ) = 16 q / ∞ Y n =1 (1 + q n ) = θ (0 | τ ) η ( τ ) ,A ( −− ) = q − / ∞ Y n =1 (1 + q n +1 / ) = θ (0 | τ ) η ( τ ) ,A ( − +) = q − / ∞ Y n =1 (1 − q n − / ) = θ (0 | τ ) η ( τ ) . (2.30)Eq.(2.29) orresponds to the famous Ja obi identity θ − θ − θ = 0 , whi h tells us that the superstring amplitude vanishes. The meaning of the previousresult is that the ontribution of NS spa etime bosons and R fermions is the same(but the two ontributions have opposite statisti s). This is onsidered an indi ation ofsupersymmetry. The general de(cid:28)nition of θ -fun tions as Gaussian sums and in prod-u t representations are given in Appendix A, along with their modular transformationproperties.2.6 Partition fun tions of 10D superstringsIn this se tion we present the partition fun tion for the (cid:28)ve perturbative superstringtheories and the ase of the heteroti E × E string with orbifold a tions will bedis ussed widely in the hapter 5. A very onvenient and ompa t way of writing thefermioni ontributions (in the previous se tion they were given in terms of θ -fun tions)is by de(cid:28)ning the hara ters O n , V n , S n and C n , representations of the SO (2 n ) group.Their general de(cid:28)nitions and modular transformations are presented in Appendix A.Here we give as an example the hara ters of the little group SO (8) O = θ + θ η , V = θ − θ η ,S = θ + θ η , C = θ − θ η . (2.31)Ea h de(cid:28)nition in eqs.(2.31) represents a onjuga y lass of the SO (8) group, in par-ti ular, O is the s alar representation, V the ve torial, S and C are spinors withopposite hirality. The hara ters V and O provide a de omposition of the NS se tor,while the C and S give the R spe trum. We are (cid:28)nally ready to present the partitionfun tions for the D spe tra of type II and type 0 T IIA = ( V − C ) | ( V − S ) | , T A = | O | + | V | + C | S | + S | C | , T IIB = | V − S | , T B = | O | + | V | + | S | + | C | . (2.32)21he spe trum an be read by expanding the hara ters in powers of q and ¯ q , as indi atedin Appendix B.For the heteroti ase we need to introdu e SO (16) and SO (32) hara ters in thepartition fun tion, in order to in lude the gauge degrees of freedom of the theory. Theonly two supersymmetri modular invariant heteroti models in 10 dimensions are thosewhere the E × E and the Spin (32) symmetries are realised and their torus amplitudeis respe tively T E × E = ( V − S )( O + S )( O + S ) T S = ( V − S )( O + S ) . (2.33)2.7 BosonizationIn this se tion we present the equivalen e between fermioni and bosoni onformal (cid:28)eldtheories in two dimensions, a orresponden e whi h allows the onsistent onstru tionof free fermioni models.Before entering into the details we will give the de(cid:28)nition of operator produ t ex-pansions (OPEs) in onformal theories in two dimensions.2.7.1 Produ t expansion operatorIn quantum (cid:28)eld theory, the in(cid:28)nitesimal onformal transformations z → z + ǫ ( z ) , ¯ z → ¯ z + ¯ ǫ (¯ z ) produ e a variation of a (cid:28)eld Φ( z, ¯ z ) given by the equal time ommutator with the onserved harge Q = πi H ( dzT ( z ) ǫ ( z ) + d ¯ z ¯ T (¯ z )¯ ǫ (¯ z )) , where T and ¯ T are the stress-energy tensors in omplex oordinates. The produ ts of the operators is well de(cid:28)nedonly if time-ordered. The radial quantization introdu ed in se tion 2.2 is an example ofthe onstru tion of a quantum theory of onformal (cid:28)elds on the omplex plane. In thisset up the time-ordered produ t is repla ed by the so alled radial-ordering3, realisedby the operator R . A omplete treatment of the omplex tensor analysis an be foundin [23, 24℄. Here we only mention the main results whi h will be useful for our purpose.The ommutator of an operator A with a spa ial integral of an operator B orre-sponds to h Z dσB, A i = I dzR ( B ( z ) A ( z )) . (2.34)3The radial ordering operator R for two (cid:28)elds A and B is given by R ( A ( z ) B ( w )) = A ( z ) B ( w ) | z | > | w | B ( w ) A ( z ) | z | < | w | , where a minus sign appears if we inter hange two fermions.22his result leads [24℄ to the operator produ t expansions (OPEs) of the stress energytensors T ( z ) and ¯ T (¯ z ) with the (cid:28)eld Φ( w, ¯ w ) R ( T ( z )Φ( w, ¯ w )) = h ( z − w ) Φ + 1 z − w ∂ w Φ + ...,R ( ¯ T (¯ z )Φ( w, ¯ w )) = ¯ h (¯ z − ¯ w ) Φ + 1¯ z − ¯ w ∂ ¯ w Φ + ... . (2.35)Eqs.(2.35) ontain the onformal transformation properties of the (cid:28)eld Φ , hen e they an be used as a de(cid:28)nition of primary (cid:28)eld4 for Φ with onformal weight ( h, ¯ h ) . Weobserve that the above produ ts are given by the expansion of poles (singularities that ontribute to integrals of the type (2.34)) plus regular terms, whi h we an omit. Fromnow on we assume that the operator produ t expansion is always radially ordered.2.7.2 Free bosons and free fermionsWe start by onsidering a massless free boson X ( z, ¯ z ) , where we an split the holomor-phi and anti-holomorphi omponents into X L ( z ) and X R (¯ z ) . For our purpose it issu(cid:30) ient to onsider the holomorphi part only. The propagator of the left omponent orresponds to < X L ( z ) X L ( w ) > = − log ( z − w ) , whi h says that it is not a onformal(cid:28)eld, but its derivative ∂X L ( z ) is a (1,0) onformal (cid:28)eld. This is showed by taking theOPE with the stress tensor, that is de(cid:28)ned as T = − : ∂X L : , and omparing witheq.(2.35) one obtains T ( z ) ∂X L ( w ) ∼ z − w ) ∂X L ( w ) + 1 z − w ∂ X L ( w ) + ... . (2.36)We now onsider two Majorana-Weyl fermions ψ i ( z ) , i = 1 , , where a hange ofbasis rearranges the fermions into the omplex form ψ = 1 √ ψ + iψ ) , ¯ ψ = 1 √ ψ − iψ ) . The theory ontains a U (1) urrent algebra (see following se tion) generated by the(1,0) urrent J ( z ) =: ψ ¯ ψ : . The OPE for ψ ¯ ψ and the holomorphi energy tensor arede(cid:28)ned as ψ ( z ) ¯ ψ ( w ) = − z − w , T ( z ) = 12 : ψ ( z ) ∂ψ ( z ) : . (2.37)If we al ulate the produ t expansion T ( z ) ψ ( w ) with the above de(cid:28)nitions, we see that ψ is an a(cid:30)ne primary (cid:28)eld5 of onformal weight (1 / , .4Its de(cid:28)nition is given in .5The formal de(cid:28)nition of primary (cid:28)eld is the following: Φ is primary of onformal weight ( h , ¯ h ) if itsatis(cid:28)es the transformation law Φ( z, ¯ z ) → ` ∂f∂z ´ h “ ∂ ¯ f∂ ¯ z ” ¯ h Φ( f ( z ) , ¯ f (¯ z )) , where h and ¯ h are real values.23e present the boson-fermion orresponden e by showing that the same operatoralgebra is produ ed by two Majorana-Weyl fermions on one hand and a hiral boson onthe other hand. In fa t, in the fermioni ase T ( z ) = 12 : J : , formula that says that the stress tensor has entral harge c = 1 . We an produ e thesame operator algebra by using a single hiral boson X ( z ) , whose urrent is providedby J ( z ) = i∂X ( z ) , where is the stress-energy tensor T = − : ∂X : , as presented at the beginning of these tion. The de(cid:28)nitions below thus ontain expli itly the boson-fermion equivalen e ψ =: e iX ( z ) : , ¯ ψ =: e − iX ( z ) : . (2.38)Further details an be found in [19, 24, 35℄.2.8 The heteroti stringThe heteroti string [36℄ was onstru ted after the famous work of Green and S hwarz[37℄ had shown that the onsisten y of an N = 1 supersymmetri string theory requiresthe presen e of an E × E or Spin (32) gauge symmetry. 10 dimensions supergravitywith these gauge groups is free of gravitational and gauge anomalies. This observationfuelled an in reased a tivity in heteroti models. Before this dis overy, the standardpro edure to introdu e gauge groups in string theory onsisted of atta hing the Chan-Paton harges at the endpoints of open strings [38℄. This pres ription does not produ ethe ex eptional E × E [39, 40℄, a non-abelian GUT gauge group whi h allows a morenatural embedding of the Standard Model spe trum at low energy.In this se tion we des ribe the basi s of the heteroti superstring, an orientable losed-string theory in ten dimensions with N = 1 supersymmetry and with gaugegroup E × E or Spin (32) /Z [17℄. Its low-energy limit is supergravity oupled withYang-Mills theory. This theory is an hybrid of the D = 10 fermioni string and the D = 26 bosoni string and the resulting spe trum is supersymmetri , ta hyon free,Lorentz invariant and unitary. The absen e of gauge and gravitational anomalies isobtained by the ompa ti(cid:28) ation of the extra sixteen bosoni oordinates on a maximaltorus of determined radius. All these properties make the heteroti string one of themost appealing andidates for an uni(cid:28)ed (cid:28)eld theory.Current algebra on the string worldsheetIn heteroti models the gauge symmetries are introdu ed by distributing symmetry harges on the losed strings. These harges are not lo alised, so we obtain a ontinuous harge distribution throughout the string. A way to des ribe their urrents is to intro-du e, on the worldsheet, fermions with internal quantum number, whi h are singlets24nder the Lorentz group. If we take n real Majorana fermions λ a , a = 1 , ..n , and wesplit them into right- and left-moving modes ( λ a ± ), then we an write the bosoni a tionon the worldsheet, in luding the new internal symmetries, as S = − T Z d σ ( ∂ α X µ ∂ α X µ − λ a − ∂ + λ a − − λ a + ∂ − λ a + ) . (2.39)The equivalen e of bosons and fermions in two dimensions (see eq.(2.38)) allows usto onvert two Majorana fermions on the worldsheet into a real boson. We an thenobtain n bosons φ i in the pla e of n fermions λ a . With this substitution the theory ontains D + n/ free bosons and has a SO ( D − , Lorentz symmetry plus an internal SO ( n ) × SO ( n ) symmetry. Its onsisten y requires D + n/ , and in the ase ofa supersymmetri theory ( D = 10 ) it means that n = 32 . Let us go ba k to eq.(2.39)and onsider for our purposes only a SO ( n ) R symmetry. The right-fermion urrents aregiven by J α + ( σ ) = 12 π T αab λ a + ( σ ) λ b + ( σ ) . (2.40)The T α generators satisfy the algebra [ T α , T β ] = if αβγ T γ and this relation (cid:28)xes the ommutation relation for the urrents [ J α + ( σ ) , J β + ( σ ′ )] = if αβγ J γ + ( σ ) δ ( σ − σ ′ ) + ik π δ αβ δ ′ ( σ − σ ′ ) . (2.41)The previous formula des ribes the a(cid:30)ne Lie algebra ˆ SO ( n ) with entral extensionrepresented by the se ond term (anomaly ontribution). If this algebra is built up from n fermions in the fundamental representation of SO ( n ) then k = 1 . If the fermions arenot in the fundamental representation we would obtain a di(cid:27)erent (quantized) value of k . We are interested in obtaining the extended algebra for the ex eptional group E but it turns out that the task is unrealisable in terms of free fermions with a minimalvalue of k . It has been shown [17℄ that this realisation is possible by using eight freebosons.We are now ready to des ribe the heteroti string as it was (cid:28)rst formulated by Gross,Harvey, Martine and Rohm. As we said already, the left moving modes are des ribedin a bosoni string theory (D=26) while the right movers are supersymmetri (D=10).Spe i(cid:28) GSO proje tions ensure supersymmetry for our model. The gauge degrees offreedom are in luded in the left se tor with an appropriate urrent algebra.The general a tion of this theory is S = − T Z d σ X µ ( ∂ α X µ ∂ α X µ − ψ µ + ∂ − ψ + µ ) − n X a =1 λ a − ∂ + λ a − ! . (2.42)We observe here that the spa etime fermions ψ µ have only right-moving omponents,superpartners of X µR . The ontent therefore di(cid:27)ers from the type IIB, where supersym-metry is realised in both left and right se tors. The left-moving se tor ontains thespa e-time (cid:28)elds X µL and the internal Majorana fermions λ a − .25f the boundary onditions for λ a − are all the same, we obtain the Spin (32) heteroti theory; hoosing di(cid:27)erent boundary onditions for the internal fermions will provide the E × E heteroti string. In this thesis we want to analyse the se ond possibility. It anbe shown that the two theories are ontinuously related [41℄. In fa t an equal numberof states at every mass level appear in the two heteroti string theories.The E × E heteroti theory is obtained when we split the internal fermions intotwo groups and assign di(cid:27)erent boundary onditions to ea h set. In this ase the gaugegroup would be SO ( n ) × SO (32 − n ) . The interesting ase for us is when n is a multipleof 8 and in parti ular n = 16 . The massless left-moving states are of the form λ i / λ j / | Ω > i, j = 1 , .. . These ombinations give rise to the ve tor and the adjoint representations for ea h SO (16) present in the urrent algebra of the theory. We also obtain the spinorialrepresentation of SO (16) . The introdu tion of appropriate GSO proje tions produ esthe (cid:28)nal ontent given by the adjoint and the spinorial representations of SO (16) . Thissum enhan es the Lie algebra of SO (16) to the ex eptional group E . Sin e we startedwith an SO (16) × SO (16) symmetry we on lude that the enlarged urrent algebraobtained is E × E . In the next se tion we onsider the toroidal ompa ti(cid:28) ation,fundamental in the des ription of the bosoni formalism.2.9 Toroidal ompa ti(cid:28) ationsThe urrent algebra an be realised in the bosoni formulation by introdu ing a toroidal ompa ti(cid:28) ation. We an start with a bosoni theory in 26 dimensions and ompa tifyone dimension on a ir le. In this simple ase we only get one toroidal boson while ifthe ompa ti(cid:28) ation in ludes d of these bosons the spa e-time is redu ed from 26 to − d dimensions.In this se tion we des ribes the simple ompa ti(cid:28) ation on a ir le, leaving the expla-nation on how gauge groups are reated in this setup for the ase of higher dimensional ompa ti(cid:28) ations in hapter 4.The oordinate ompa ti(cid:28)ed on the ir le satis(cid:28)es the ondition x ≡ x + 2 πRn . R is the radius of the ir le and n an integer whi h de(cid:28)nes the winding number, aquantity that gives the number of times the string wraps around the ir le. The windingrepresents a stringy new feature whi h arises in the ompa ti(cid:28) ation pro edure.The general expansion for the ompa t boson be omes X = x + 2 α ′ mR τ + 2 nRσ + ( oscillators ) . (2.43)The expression (2.43) an be rewritten in terms of the hiral omponents p L and p R ofthe ompa t oordinate as X L,R = 12 x + α ′ p L,R ( τ ∓ σ ) + ( oscillators ) L,R , (2.44)26here the hiral momenta are de(cid:28)ned as p L,R = mR ± nRα ′ . (2.45)The invarian e under x → x + 2 πR requires m to be integer. The presen e of a n = 0 des ribes a soliton state that does not exist in the un ompa ti(cid:28)ed theory, sin e its energywould diverge for R → ∞ . This means that the spe trum of a ompa ti(cid:28)ed theory anin general be larger than the non ompa t orresponding ase. When a non- ompa tboson is ompa ti(cid:28)ed, its ontribution to the partition fun tion be omes a dis rete sum,given below √ τ ηη → X m,n q α ′ p L / q α ′ p R / ηη . (2.46)We an underline here the presen e of a symmetry whi h relates m and n quantumnumbers, the so- alled T-duality, one of the symmetries relating the (cid:28)ve perturbativestring models [42, 43℄. n ↔ m R ↔ α ′ /R. (2.47)The previous formula tells us that the losed bosoni string ompa ti(cid:28)ed on a radius R is equivalent to the theory with radius α ′ /R . T duality is an exa t symmetry of theperturbative theory for the losed bosoni string and it relates type 0A with type 0B,type IIA with type IIB, as mentioned in the introdu tion. As we announ ed before,the generalisation to higher dimensional tori will be onsidered in hapter 4. We willintrodu e the ompa ti(cid:28) ation on a 16 dimensional tori that leads to the E × E symmetry, as expe ted. 27hapter 3Free Fermioni ModelsIn this hapter we des ribe the free fermioni formulation of the heteroti superstringand mainly fo us on a subset of these models whi h are alled semi-realisti free fermioni models. Moreover, we provide some indi ative examples among this lass of string om-pa ti(cid:28) ations, whose results are published in [44, 45℄.In the (cid:28)rst part of our dis ussion we will des ribe the onsisten y rules ne essaryfor the onstru tion of the theory. The interested reader an (cid:28)nd further details in theoriginal papers [46, 47, 48, 49, 50℄.In the se ond part of this hapter we present some examples of semi-realisti mod-els in the free fermioni formulation produ ed in the past, in whi h the only StandardModel harged states are the MSSM states [51, 52℄. Therefore we revisit some of theirproperties. The presen e of three Higgs doublets in the untwisted spe trum is anotherfeature of semi-realisti free fermioni models and the general pro edure to redu e themto one pair is given by the analysis of the supersymmetri (cid:29)at dire tions. This method onsists in giving heavy masses to some of the Higgs doublets in the low energy (cid:28)eldtheory [53, 54℄. The two models largely dis ussed in this hapter introdu e instead anew me hanism that a hieves the same redu tion by an appropriate hoi e of boundary onditions, in parti ular, asymmetri boundary onditions among left and right internalfermions. An additional e(cid:27)e t related to this hoi e is the redu tion of the supersym-metri moduli spa e. The pro edure, explained in detail later on, represents a sele tionme hanism useful to pi k phenomenologi ally interesting string va ua. We will presentsome generalities on the analysis of (cid:29)at dire tions and introdu e the on ept of strin-gent (cid:29)at dire tions, sin e this will allow the investigation of the low energy propertiesof free fermioni models. The (cid:29)at dire tion analysis is needed be ause of an anomalous U (1) whi h generally appears in this set up. Its presen e gives rise to a Fayet-IlliopulosD-term whi h breaks supersymmetry but, by looking at supersymmetri (cid:29)at dire tionsand imposing F and D (cid:29)atness on the va uum, supersymmetry an be restored. Inthe last example presented in this hapter an extensive sear h ould not provide any(cid:29)at solution, raising the question on the perturbatively broken supersymmetry. At the28ree level the Bose-Fermi degenera y of the spe trum implies that the theory is insteadsupersymmetri , yielding a vanishing osmologi al onstant. Therefore, this un on-ventional result may lead to an interesting new interpretation of the supersymmetrybreaking me hanism in string theory.3.1 The free fermioni formulationIn ontrast with the ten dimensional superstrings, where the ompa ti(cid:28) ation of the"extra-dimensions" is needed to redu e the spa etime to four dimensions, the freefermioni formulation provides dire tly a four-dimensional theory with a ertain numberof internal degrees of freedom. In fa t, an internal se tor of two-dimensional onformal(cid:28)eld theories is required in order to ful(cid:28)l • onformal invarian e, • worldsheet supersymmetry, • modular invarian e.In this approa h all internal degrees of freedom are fermionised, thus produ ing world-sheet fermions. Requiring anomaly an ellation (cid:28)xes the number of (cid:28)elds in the leftand right se tor, obtaining 18 left-moving Majorana fermions χ a , ( a = 1 , .. , and 44right-moving Majorana fermions ¯Φ I , ( I = 1 , .. . The spa etime is des ribed by theleft-moving oordinates ( X µ , ψ µ ) and the right-moving bosons X µ . Sin e the heteroti string is N = 1 spa etime supersymmetri (we hoose here a di(cid:27)erent onvention w.r.t.the bosoni approa h by (cid:28)xing the supersymmetry in the left se tor), then we requireleft-moving lo al supersymmetry. This is realised non-linearly [47℄ among all the (cid:28)eldsin the left se tor, spa etime and internal ones, by the super urrent T F = ψ µ ∂X µ + f abc χ a χ b χ c , (3.1)where f abc are the stru ture onstants of a semi-simple Lie group G of dimension .The χ a transform in the adjoint representation of G . In [55℄ it is shown that N = 1 spa etime supersymmetry an be obtained in four dimensions when the Lie algebra G = SU (2) . In this ase it is onvenient to group the χ a into six triplets ( χ i , y i , w i ) , ( i = 1 , .. . Ea h of them transforms as the adjoint representation of SU (2) . Sofar we have ensured super onformal invarian e of the theory. We still need to verify itsmodular invarian e to get a onsistent theory. The target is a hieved by investigating theproperties of the partition fun tion. In this pres ription, a modular invariant partitionfun tion must be the sum over all di(cid:27)erent boundary onditions for the worldsheetfermions, with appropriate weights. For a genus- g worldsheet Σ g , fermions moving29round a non trivial loop α ∈ π (Σ g ) transform as Φ I → R g ( α ) IJ Φ J ,ψ µ → − δ α ψ µ ,χ a → L g ( α ) ab χ b , (3.2)where the (cid:28)rst transformation refers to the right-moving (cid:28)elds, L aga ′ L bgb ′ L cgc ′ f abc = − δ α f a ′ b ′ c ′ and δ α = ± . The spin stru ture of ea h fermion is a representation ofthe (cid:28)rst homotopy group π (Σ g ) [56℄. The transformations (3.2) ensure the invari-an e of the super urrent. We need to require the orthogonality of R g ( α ) to leave theenergy-tensor invariant in the right se tor. In order to keep the theory tra table, om-mutativity of the boundary onditions has been assumed [46℄, implying the followingrestri tions on L g ( α ) and R g ( α ) : they have to be abelian matrix representations of π (Σ g ) ; it is assumed ommutativity between the boundary onditions on surfa es ofdi(cid:27)erent genus. The previous onstraints allow the diagonalization of the matri es R ( α ) and L ( α ) , simplifying the equations (3.2) into f → − e iπα ( f ) f, (3.3)where f is any fermion ( ψ µ , χ a , Φ I ) and α ( f ) is the phase a quired by f when movingaround the non ontra tible loop α .Thus, the spin stru ture for a non ontra tible loop an be expressed as a ve tor α = { α ( f r ) , ..α ( f rk ); ˆ α ( f c ) , .. ˆ α ( f ck ′ ) } , (3.4)where α ( f r ) is the phase for a real fermion while ˆ α ( f c ) orresponds to a omplexone. By onvention, α ( f ) ∈ ( − , . Obviously for the omplex onjugate fermion α ( f ∗ ) ∈ [ − , . We set the notation δ α = (cid:26) if α ( ψ µ ) = 0 − if α ( ψ µ ) = 1 where, a ording to eq.(3.3), the entry represents a periodi boundary ondition and is the anti-periodi boundary ondition. Sin e there are g non- ontra tible loops fora genus g Riemann surfa e, we have to spe ify two sets of phases α , ..α g , β , ..β g toobtain the full partition fun tion. In its general form it an be written as Z = X genus g X i,j =1 c (cid:18) α i β j (cid:19) z (cid:18) α i β j (cid:19) , (3.5)where z (cid:18) α i β j (cid:19) an be expressed in terms of θ -fun tions. The modular invarian eimposes onstraints onto the oe(cid:30) ients c (cid:18) α i β j (cid:19) . It was shown [57℄ that modular30nvarian e and unitarity imply that these oe(cid:30) ients for higher genus surfa es fa toriseinto the form c (cid:18) α , ..α g β , ..β g (cid:19) = c (cid:18) α β (cid:19) c (cid:18) α β (cid:19) ...c (cid:18) α g β g (cid:19) . For this reason it is su(cid:30) ient to onsider only the one-loop oe(cid:30) ients.3.1.1 Model building rules and physi al spe trumIn the free fermioni framework, the onstru tion of onsistent string va ua in fourdimensions is a hieved by applying two sets of rules, namely, the onstraints for theboundary ondition ve tors (we restri t to the ase of rational spin stru ture [46℄) andthe rules for the one-loop phases.A set of onsistent boundary ondition ve tors form an additive group Ξ ∼ Z N ⊗ ... ⊗ Z N k , generated by the basis B = { b , ..b k } , where ea h b i is in the form of eq.(3.4).This basis has to satisfy the following onditions • P m i b i = 0 ⇐⇒ m i = 0 (mod N i ), ∀ i, • N ij b i · b j = 0 mod 4 , • N i b i · b i = 0 mod 8 , • b = 1 , • the number of periodi real fermions must be even in ea h b i ,where N i is the smallest integer for whi h N i b i = 0 (mod2) and N ij is the least ommonmultiplier between N i and N j . The inner Lorentz produ t is de(cid:28)ned by b i · b j = X real left + X complex left − X real right − X complex right b i ( f ) b j ( f ) . For a onsistent basis B there are several di(cid:27)erent modular invariant hoi es of phases,ea h one leading to a onsistent string theory. The phases under onsideration have tosatisfy the requirements, whi h provide the se ond group of onstraints below • c (cid:18) b i b j (cid:19) = δ b i e πiniNj = δ b j e πimiNi e iπbi · bj , • c (cid:18) b i b i (cid:19) = − e iπbi · bi c (cid:18) b i (cid:19) , • c (cid:18) b i b j (cid:19) = e iπbi · bj c ∗ (cid:18) b j b i (cid:19) , c (cid:18) b i b j + b k (cid:19) = δ b i c (cid:18) b i b j (cid:19) c (cid:18) b i b k (cid:19) , where < n i < N j and < m i < N i . Moreover, there is some freedom for the phase c (cid:18) b b (cid:19) = ± e iπb · b , while by onvention c (cid:18) (cid:19) = 1 and c (cid:18) α (cid:19) = δ α , onditionwhi h assures the presen e of the graviton in the spe trum.If we indi ate by1 α a generi se tor in Ξ , the orresponding Hilbert spa e H α on-tributes to the partition fun tion of the model. We adopt the notation α = { α L | α R } to separate the left and the right phases. The states in H α have to satisfy the Virasoro onditions and the level mat hing ondition, that, in our formulation, appear as M L = −
12 + α L · α L N L = − α R · α R N R = M R , (3.6)where N L and N R are respe tively the total left and the total right os illator numbera ting on the va uum | > α . The frequen ies are given respe tively for a fermion f andits onjugate f ∗ by ν f = 1 + α ( f )2 , ν f ∗ = 1 − α ( f )2 . The physi al states ontributing to the partition fun tion are those satisfying the GSO onditions e iπb i · F α | s > α = δ α c (cid:18) αb i (cid:19) ∗ | s > α , (3.7)where | s > α is a generi state in the se tor α , given by bosoni and fermioni os illatorsa ting on the va uum. The operator ( b i · F α ) is given by b i · F α = n X left − X right o b i ( f ) F α ( f ) , (3.8)where F is the fermion number operator. F gets the following values F ( f ) = (cid:26) → for f − → for f ∗ . If the se tor α ontains periodi fermions, then the va uum is degenerate and transformsin the representation of a SO (2 n ) Cli(cid:27)ord algebra. Hen e, if f is su h a periodi fermion,it will be indi ated as |± > and F assumes the value below F ( f ) = (cid:26) → for | + > − → for |− > . The U (1) harges for the physi al states orrespond to the urrents f ∗ f and are al u-lated by the following expression Q ( f ) = 12 α ( f ) + F ( f ) . α a generi boundary ondition ve tor andat the same time the generi se tor in the Hilbert spa e. We assure that from the ontext it is always lear to understand whi h quantity we are referring to.32.1.2 Constru tion of semi-realisti modelsThe onstru tion of semi-realisti free fermioni models is related to a parti ular hoi eof boundary ondition basis ve tors and the general pro edure of the onstru tion isbased on two prin ipal steps. The (cid:28)rst stage is onsidering the NAHE (Nanopoulos-Antoniadis-Hagelin-Ellis) set [58, 59, 60℄ of boundary ondition basis ve tors B = { , S, b , b , b } , whi h orresponds to Z × Z ompa ti(cid:28) ation with the standard em-bedding of the gauge onne tion [13, 61℄. The basis B is expli itly given below { ψ , , χ ,.. y ,.. , w ,.. | ¯ y ,.. , ¯ w ,.. , ¯ ψ ,.. , ¯ η , , , ¯ φ ,.. } ,S = { ψ , , χ ,.. } ,b = { ψ , , χ , , y ,.. | ¯ y ,.. , ¯ ψ ,.. , ¯ η } ,b = { ψ , , χ , , y , , ω , | ¯ y , , ¯ ω , , ¯ ψ ,.. , ¯ η } ,b = { ψ , , χ , , ω ,.. | ¯ ω ,.. , ¯ ψ ,.. , ¯ η } , (3.9)where the notation means that only periodi fermions are listed in the ve tors. Theleft-moving internal oordinates are fermionised by the relation e iX i = 1 / √ y i + iw i ) ,as explained in se tion 2.7 and a similar pres ription holds for the right-moving internal oordinates. The superpartners of the left-moving bosons are indi ated by χ i . Theextra 16 degrees of freedom ¯ ψ ,.. , ¯ η , , , ¯ φ ,.. are omplex fermions. The GSO one-loopphases for the NAHE set are given below c (cid:18) b i b j (cid:19) = − , c (cid:18) S (cid:19) = 1 , c (cid:18) b i , S (cid:19) = − . The gauge group indu ed by the NAHE set is SO (10) × SO (6) × E and N = 1 supersymmetry. The spa etime ve tor bosons generating the symmetry group arise inthe Neveu-S hwarz se tor and in the se tor ξ = 1 + b + b + b . In parti ular, the ¯ ψ ,.. are responsible for the SO (10) symmetry, the ¯ φ ,.. generate the hidden E andthe internal fermions { ¯ y , ··· , , ¯ η } , { ¯ y , ¯ y , ¯ ω , ¯ ω , ¯ η } , { ¯ ω , ··· , , ¯ η } generate the threehorizontal SO(6) symmetries. In the untwisted se tor we note the presen e of states inthe ve torial representation of SO (10) , that represent the best andidates for theHiggs doublets. The three twisted se tors b , b and b produ e 48 multiplets in the representation of SO (10) , whi h arry SO (6) harges but are singlets under the hiddengauge group.In the se ond stage of the onstru tion we onsider additional basis ve tors (generallyindi ated by α, β, γ ) whi h redu e the number of generations to three and simultane-ously break the four dimensional gauge group. This breaking is implemented by theassignment of boundary onditions, in the new ve tors, orresponding to the generatorsof the subgroup onsidered. For instan e, the breaking of SO (10) is due to the bound-ary onditions of ¯ ψ ,.. in α, β, γ , whi h an provide SU (5) × U (1) [62℄, SO (6) × SO (4) SU (3) × SU (2) × U (1) gauge groups [64, 65, 59, 53℄. Further attempts in the on-stru tion of realisti models an be found in [66, 67℄. The SO (6) symmetries are alsobroken to (cid:29)avour U (1) symmetries. The worldsheet urrents η i ¯ η i , i = 1 , , , produ e U (1) harges in the visible se tor and further U (1) n symmetries arise by the pairing ofreal fermions among the right internal se tor. If a left moving real fermion is paired witha right real fermion then the right gauge group has rank redu ed by one. The pairing ofthe left and right movers is a key point in the phenomenology of free fermioni models,for example it is stri tly related to the redu tion of the untwisted Higgs states, as wewill dis uss widely in the following.The orresponden e of the free fermioni models with the orbifold onstru tion is il-lustrated by extending the NAHE set, { , S, b , b , b } , by at least one additional bound-ary ondition basis ve tor [13, 14, 15℄ ξ = { ¯ ψ , ··· , , ¯ η , , } . (3.10)With a suitable hoi e of the GSO proje tion oe(cid:30) ients the model possesses an SO(4) × E × U(1) × E gauge group and N = 1 spa e-time supersymmetry. The matter (cid:28)eldsin lude 24 generations in the 27 representation of E , eight from ea h of the se tors b ⊕ b + ξ , b ⊕ b + ξ and b ⊕ b + ξ . Three additional and pairs are obtainedfrom the Neveu-S hwarz ⊕ ξ se tor.To onstru t the model in the orbifold formulation one starts with the ompa ti(cid:28)- ation on a torus with nontrivial ba kground (cid:28)elds [68, 69℄. The subset of basis ve tors { , S, ξ , ξ } , (3.11)where ξ = { ¯ φ , ··· , } , generates a toroidally- ompa ti(cid:28)ed model with N = 4 spa etimesupersymmetry and SO(12) × E × E gauge group. The same model is obtained in thegeometri (bosoni ) language by tuning the ba kground (cid:28)elds to the values orrespond-ing to the SO(12) latti e. The metri of the six-dimensional ompa ti(cid:28)ed manifold isthen the Cartan matrix of SO(12), while the antisymmetri tensor is given by b ij = g ij ; i > j, i = j, − g ij ; i < j. (3.12)When all the radii of the six-dimensional ompa ti(cid:28)ed manifold are (cid:28)xed at R I = √ ,it is seen that the left- and right-moving momenta P IR,L = [ m i −
12 ( B ij ± G ij ) n j ] e Ii ∗ (3.13)reprodu e the massless root ve tors in the latti e of SO(12). Here e i = { e Ii } are sixlinearly-independent vielbeins normalised so that ( e i ) = 2 . The e Ii ∗ are dual to the e i ,with e ∗ i · e j = δ ij . 34dding the two basis ve tors b and b to the set (3.11) orresponds to the Z × Z orbifold model with standard embedding. Starting from the N = 4 model with SO(12) × E × E symmetry, and applying the Z × Z twist on the internal oordinates,reprodu es the spe trum of the free-fermion model with the six-dimensional basis set { , S, ξ , ξ , b , b } [13, 14, 15℄. The Euler hara teristi of this model is 48 with h = 27 and h = 3 .It is noted that the e(cid:27)e t of the additional basis ve tor ξ of eq. (3.10) is to separatethe gauge degrees of freedom, spanned by the world-sheet fermions { ¯ ψ , ··· , , ¯ η , , , ¯ φ , ··· , } ,from the internal ompa ti(cid:28)ed degrees of freedom { y, ω | ¯ y, ¯ ω } , ··· , . In the realisti freefermioni models this is a hieved by the ve tor γ [13, 14, 15℄, with γ = { ¯ ψ , ··· , , ¯ η , , , ¯ φ , ··· , } , (3.14)whi h breaks the E × E symmetry to SO(16) × SO(16) . The Z × Z twist indu edby b and b breaks the gauge symmetry to SO(4) × SO(10) × U(1) × SO(16) . Theorbifold still yields a model with 24 generations, eight from ea h twisted se tor, but nowthe generations are in the hiral 16 representation of SO(10), rather than in the of E . The same model an be realised [70℄ with the set { , S, ξ , ξ , b , b } , by proje tingout the ⊕ from the ξ -se tor taking c (cid:18) ξ ξ (cid:19) → − c (cid:18) ξ ξ (cid:19) . (3.15)This hoi e also proje ts out the massless ve tor bosons in the 128 of SO(16) in thehidden-se tor E gauge group, thereby breaking the E × E symmetry to SO(10) × U(1) × SO(16) . We an de(cid:28)ne two N = 4 models generated by the set (3.11), Z + and Z − , depending on the sign in eq. (3.15). The (cid:28)rst, say Z + , produ es the E × E model, whereas the se ond, say Z − , produ es the SO(16) × SO(16) model. However,the Z × Z twist a ts identi ally in the two models, and their physi al hara teristi sdi(cid:27)er only due to the dis rete torsion eq. (3.15).The free fermioni formalism provides useful means to lassify and analyse Z × Z heteroti orbifolds at spe ial points in the moduli spa e. The drawba ks of this approa his that the geometri view of the underlying ompa ti(cid:28) ations is lost. On the other hand,the geometri pi ture may be instrumental for examining other questions of interest,su h as the dynami al stabilisation of the moduli (cid:28)elds and the moduli dependen e ofthe Yukawa ouplings. In hapter 4 we will analyse Z × Z orbifolds on non-fa torisabletoroidal manifolds.On e we extra t the massless spe trum of a parti ular free fermion model, the nextstep is the analysis of its superpotential. We postpone the explanation of this topi sin e it will be treated in the next se tions. Further details on erning the onstru tionof free fermioni models arried on step by step an be found in [71℄.35.2 Minimal Standard Heteroti String ModelsAfter providing the main tools on the onstru tion of the theory, we would like to revisitsome of the properties of semi(cid:21)realisti Standard Model(cid:21)like free fermioni models. Oneof their remarkable su esses has been the fa t that they an a ommodate the righttop quark mass [72, 73, 74, 75℄. The models o(cid:27)ered an explanation why only the topquark mass is hara terised by the ele troweak s ale, whereas the masses of the lighterquarks and leptons are suppressed [65, 76, 77, 78℄. The reason is that only the topquark Yukawa oupling is obtained at the ubi level of the superpotential, whereas theYukawa ouplings of the lighter quarks and leptons are obtained from nonrenormaliz-able terms whi h are suppressed relative to the leading order term. As we explainedbefore, the three generations arise from the three twisted se tors, whereas the Higgsdoublets, to whi h they ouple in leading order, arise from the untwisted se tor. Atleading order ea h twisted generation ouples to a separate pair of untwisted Higgsdoublets. Analysis of supersymmetri (cid:29)at dire tions implied that at low energies onlyone pair of Higgs doublets remains light and other Higgs doublets obtain heavy massfrom VEVs of Standard Model singlet (cid:28)elds. Hen e, in the low energy e(cid:27)e tive (cid:28)eldtheory, only the oupling of the twisted generation that ouples to the light Higgs re-mains at leading order. The onsequen e is that only the top quark mass is obtainedat leading order, whereas the masses of the remaining quarks and leptons are obtainedat subleading orders. Evolution of the al ulated Yukawa ouplings from the string toele troweak s ale then yields a predi tion for the top quark mass. The analysis of thetop quark mass therefore relies on the analysis of supersymmetri (cid:29)at dire tions andthe de oupling of the additional untwisted ele troweak Higgs doublets, that ouple tothe twisted generations at leading order. In the examples presented in the followingan alternative onstru tion is given, where only one pair of untwisted Higgs doubletsremains in the massless spe trum after the appli ation of the Generalised GSO (GGSO)proje tions. Therefore, the massless string spe trum ontains a single ele troweak Higgsdoublet pair, without relying on analysis of supersymmetri (cid:29)at dire tions in the e(cid:27)e -tive low energy (cid:28)eld theory. Although the Higgs redu tion is obtained by applying thenew pro edure, the (cid:29)at dire tion analysis is still ne essary to investigate the supersym-metri properties of the model. The existen e of an (cid:16)anomalous(cid:17) U (1) symmetry isa ommon feature of free fermioni models [79℄. The anomalous U (1) A is broken bythe Green(cid:21)S hwarz(cid:21)Dine(cid:21)Seiberg(cid:21)Witten me hanism [80℄ in whi h a potentially largeFayet(cid:21)Iliopoulos D (cid:21)term ξ is generated by the VEV of the dilaton (cid:28)eld. Su h a D (cid:21)term would, in general, break supersymmetry, unless there is a dire tion ˆ φ = P α i φ i in the s alar potential for whi h P Q iA | α i | is of opposite sign to ξ and that is D (cid:21)(cid:29)atwith respe t to all the non(cid:21)anomalous gauge symmetries, as well as F (cid:21)(cid:29)at. If su h adire tion exists, it will a quire a VEV, an elling the Fayet(cid:21)Iliopoulos ξ (cid:21)term, restoringsupersymmetry and stabilising the va uum. The set of D - and F -(cid:29)at onstraints is36iven by h D A i = h D α i = 0 ; h F i ≡ ∂W∂η i i = 0 ; (3.16) D A = h K A + X Q kA | χ k | + ξ i ; (3.17) D α = h K α + X Q kα | χ k | i , α = A ; (3.18) ξ = g (Tr Q A )192 π M ; (3.19)where χ k are the (cid:28)elds whi h a quire VEVs of order √ ξ , while the K (cid:21)terms ontain (cid:28)eldslike squarks, sleptons and Higgs bosons whose VEVs vanish at this s ale. Q kA and Q kα denote the anomalous and non(cid:21)anomalous harges, and M Pl ≈ × GeV denotes theredu ed Plan k mass. The solution (i.e. the hoi e of (cid:28)elds with non(cid:21)vanishing VEVs)to the set of eqs.(3.16)(cid:21)(3.18), though nontrivial, is not unique. Therefore in a typi almodel there exist a moduli spa e of solutions to the F and D (cid:29)atness onstraints, whi hare supersymmetri and degenerate in energy [81℄. Mu h of the study of the superstringmodels phenomenology (as well as non(cid:21)string supersymmetri models) involves theanalysis and lassi(cid:28) ation of these (cid:29)at dire tions. The methods for this analysis instring models have been systematised in [82, 83, 54, 84, 79℄.In general it has been assumed in the past that in a given string model there shouldexist a supersymmetri solution to the F and D (cid:29)atness onstraints. The simpler typeof solutions utilise only (cid:28)elds that are singlets of all the non(cid:21)Abelian groups in a givenmodel (type I solutions). More involved solutions (type II solutions), that utilise alsonon(cid:21)abelian (cid:28)elds, have also been onsidered [79℄, as well as in lusion of non(cid:21)abelian(cid:28)elds in systemati methods of analysis [79℄. The general expe tation that a givenmodel admits a supersymmetri solution arises from analysis of supersymmetri pointquantum (cid:28)eld theories. In these ases it is known that if supersymmetry is preservedat the lassi al level, then there exist index theorems that forbid supersymmetry break-ing at the perturbative quantum level [85℄. Therefore in point quantum (cid:28)eld theoriessupersymmetry breaking may only be indu ed by non(cid:21)perturbative e(cid:27)e ts [86℄.In the model of table 3.23 the redu tion of the Higgs states is obtained by impos-ing asymmetri boundary onditions in a boundary ondition basis ve tor that doesnot break the SO (10) symmetry. Another onsequen e of the Higgs redu tion me h-anism is the simultaneous proje tion of untwisted SO (10) singlet (cid:28)elds, provoking avast redu tion of the moduli spa e of supersymmetri (cid:29)at solutions. The model underinvestigation does not ontain supersymmetri (cid:29)at dire tions that do not break some ofthe Standard Model symmetries. Thus, by ontinuing the sear h of semirealisti modelswith redu ed Higgs spe trum we are lead to the se ond model proposed in table 3.35,where the Higgs redu tion me hanism utilises boundary onditions that are both sym-metri and asymmetri in the basis ve tors that break SO (10) to SO (6) × SO (4) , with37espe t to two of the twisted se tors of the Z × Z orbifold. The onsequen e is thattwo of the untwisted Higgs multiplets, asso iated with two of the twisted se tors, areproje ted entirely from the massless spe trum. As a result, the string model ontains asingle pair of untwisted ele troweak Higgs doublets.In the pro ess of seeking supersymmetri (cid:29)at dire tion, we arrive to the unexpe ted on lusion that the model may not ontain any supersymmetri (cid:29)at dire tions at all. Inthe least, this model appears to have no D -(cid:29)at dire tions that an be proved to be F -(cid:29)atto all order, other than through order-by-order analysis. That is, there does not appearto be any D -(cid:29)at dire tions with stringent F -(cid:29)atness (as de(cid:28)ned in [87, 88℄). In theanalysis of the (cid:29)at dire tions we in lude all the (cid:28)elds in the string model, i.e. StandardModel singlet states as well as Standard Model harged states. The model thereforedoes not ontain a D (cid:21)(cid:29)at dire tion that is also stringently F (cid:21)(cid:29)at to all order of non(cid:21)renormalizable terms. The model may of ourse still admit non-stringent (cid:29)at dire tionsthat rely on an ellations between superpotential terms. However, past experien esuggests that non(cid:21)stringent (cid:29)at dire tions an only hold order by order, and are notmaintained to all orders [66℄. We therefore spe ulate that in this ase supersymmetryis not exa t, but is in general broken at some order. If this (cid:28)nding remains true afterthe entire parameter spa e of possible all-order non(cid:21)stringent (cid:29)at dire tions has beenexamined, we must ask what are the impli ations. If a model without all-order F -(cid:29)atnesswere to be found, then supersymmetry would remain broken by the Fayet(cid:21)Iliopoulosterm at a (cid:28)nite order, whi h is generated at the one(cid:21)loop level in string perturbationtheory, rather than be an elled by a D -(cid:29)at dire tion with anomalous harge. If so,then this would imply, although supersymmetry is unbroken at the lassi al level andthe string spe trum is Bose(cid:21)Fermi degenerate, that supersymmetry may be brokenat the perturbative quantum level. Nevertheless, sin e the spe trum is Bose(cid:21)Fermidegenerate, the one(cid:21)loop osmologi al onstant still vanishes. The details of this modelare given in se tion 3.5.Below we provide the details of the Yukawa me hanism and the Higgs doublet-tripletsplitting whi h are realised in the examples proposed in the next se tions.3.2.1 Yukawa Sele tion Me hanismAt the ubi level of the superpotential the boundary ondition basis ve tors (cid:28)x theYukawa ouplings for the quarks and leptons [75℄. These Yukawa ouplings are (cid:28)xedby the ve tor γ whi h breaks the SO (10) symmetry to SU (5) × U (1) . Ea h se tor b i gives rise to an up(cid:21)like or down(cid:21)like ubi level Yukawa oupling. We an de(cid:28)ne threequantities ∆ i , i = 1 , , , in the ve tor γ , whi h measures the di(cid:27)eren e of the left(cid:21)and right(cid:21)moving boundary onditions assigned to the internal fermions from the set { y, w | ¯ y, ¯ ω } and whi h are periodi in the ve tor b i , ∆ i = | γ L (internal) − γ R (internal) | = 0 , i = 1 , , . (3.20)38f ∆ i = 0 then the se tor b i gives rise to a down(cid:21)like Yukawa oupling while the up(cid:21)type Yukawa oupling vanishes. The opposite o urs if ∆ i = 1 . In models that produ e ∆ i = 1 for i = 1 , , the down(cid:21)quark type ubi (cid:21)level Yukawa ouplings vanish andthe models produ e only up(cid:21)quark type Yukawa ouplings at the ubi level of thesuperpotential. Models with these hara teristi s were presented in refs. [59, 75℄.3.2.2 Higgs Doublet(cid:21)Triplet SplittingThe Higgs doublet(cid:21)triplet splitting operates as follows [89, 90℄. The Neveu(cid:21)S hwarzse tor gives rise to three (cid:28)elds in the 10 representation of SO (10) . These ontainthe Higgs ele troweak doublets and olour triplets when breaking the gauge group tothe SM symmetry. Ea h of those is harged with respe t to one of the horizontal U (1) symmetries U (1) , , generated by ¯ η , ¯ η and ¯ η . Ea h one of these multiplets isasso iated, by the horizontal symmetries, with one of the twisted se tors, b , b and b .The doublet(cid:21)triplet splitting results from the boundary ondition basis ve tors whi hbreak the SO (10) symmetry to SO (6) × SO (4) . We an de(cid:28)ne a quantity ∆ i in thesebasis ve tors whi h measures the di(cid:27)eren e between the boundary onditions assignedto the internal fermions from the set { y, ω | ¯ y, ¯ ω } and whi h are periodi in the ve tor b i , ∆ i = | α L (internal) − α R (internal) | = 0 , i = 1 , , . (3.21)If ∆ i = 0 then the Higgs triplets, D i and ¯ D i , remain in the massless spe trum while theHiggs doublets, h i and ¯ h i are proje ted out and the opposite o urs for ∆ i = 1 . Therule in eq.(3.21) is a generi rule that operates in NAHE(cid:21)based free fermioni models.Another relevant question with regard to the Higgs doublet(cid:21)triplet splitting me ha-nism is whether it is possible to onstru t models in whi h both the Higgs olour tripletsand ele troweak doublets asso iated to a given twisted se tor b j from the Neveu(cid:21)S hwarzse tor are proje ted out by the GSO proje tions. This is a viable possibility as we an hoose for example ∆ ( α ) j = 1 and ∆ ( β ) j = 0 , where ∆ ( α,β ) are the proje tions due to the basis ve tors α and β respe tively. This isa relevant question as the number of Higgs representations, whi h generi ally appear inthe massless spe trum, is larger than what is allowed by the low energy phenomenology.Attempts to onstru t su h models were dis ussed in ref. [91℄. In se tion 3.3 we presentthree generation models with redu ed untwisted Higgs spe trum, without resorting toanalysis of supersymmetri (cid:29)at dire tions.39.3 Models with redu ed untwisted Higgs spe trumAs an illustration of the Higgs redu tion me hanism we onsider the model in table3.22. ψ µ χ χ χ ¯ ψ ,..., ¯ η ¯ η ¯ η ¯ φ ,..., α β γ
12 12 12 12 12 12 12 12 12
12 12 12 y y y ¯ y y ¯ y ¯ y ¯ y y ω y ¯ y ω ¯ ω ¯ y ¯ ω ω ω ω ¯ ω ω ¯ ω ¯ ω ¯ ω α β γ c (cid:18) α, βα (cid:19) = c (cid:18) β, γβ (cid:19) = − c (cid:18) γ , α (cid:19) = c (cid:18) αb (cid:19) = c (cid:18) γb (cid:19) = − c (cid:18) βb j (cid:19) = − c (cid:18) αb , b (cid:19) = − c (cid:18) γb , b (cid:19) = 1 (j=1,2,3), with the others spe i(cid:28)ed by modular invarian e and spa etime supersymme-try. As noted from the table, in this model the boundary onditions with respe t to b and b in the basis ve tor α are asymmetri and symmetri , respe tively, while theopposite o urs for the basis ve tor β . At the same time, the boundary onditions withrespe t to the se tor b are asymmetri in both α and β . Therefore, in this model ∆ ( α )1 = ∆ ( β )1 = 1 ; ∆ ( α )2 = 1 , ∆ ( β )2 = 0 and ∆ ( α )3 = 0 , ∆ ( β )3 = 1 . Consequently, irrespe -tive of the hoi e of the generalised GSO proje tion oe(cid:30) ients, both the Higgs olourtriplets and ele troweak doublets asso iated with b and b are proje ted out by theGSO proje tions, whereas the ele troweak Higgs doublets that are asso iated with these tor b remain in the spe trum. However, the se tor α produ es hiral fra tionally harged exoti s, and is therefore not viable. We also note that in this model the non(cid:21)vanishing ubi level Yukawa ouplings produ e a down(cid:21)quark type mass term, and nota potential top(cid:21)quark mass term.An alternative model is presented in table 3.23. ψ µ χ χ χ ¯ ψ ,..., ¯ η ¯ η ¯ η ¯ φ ,..., b β γ
12 12 12 12 12 12 12 12
12 12
12 12 y y y ¯ y y ¯ y ¯ y ¯ y y ω y ¯ y ω ¯ ω ¯ y ¯ ω ω ω ω ¯ ω ω ¯ ω ¯ ω ¯ ω b β γ c (cid:18) b b , β, γ (cid:19) = c (cid:18) ββ, γ (cid:19) = c (cid:18) b , γb j (cid:19) = − c (cid:18) γ (cid:19) = − c (cid:18) βb j (cid:19) = 1 , (j=1,2,3), with the others spe i(cid:28)ed by modular invarian e and spa etime supersymme-try. In this model the basis ve tor2 b preserves the SO (10) symmetry, whi h is brokenby the basis ve tors β and γ to SU (3) × SU (2) × U (1) . The b proje tion is asymmetri with respe t to the internal fermions that are periodi in the se tors b and b and, there-fore, proje ts out the entire untwisted ve torial representations of SO (10) , that ouple tothe se tors b and b , irrespe tive of the β proje tion. On the other hand, it is symmetri with respe t to b , while the basis ve tor β , that breaks SO (10) → SO (6) × SO (4) , isasymmetri with respe t to b . Therefore, the Higgs doublets that ouple to b remainin the massless spe trum. We note also that the boundary onditions in the ve tor γ , that breaks SO (10) → SU (5) × U (1) , are asymmetri with respe t to the internalfermions that are periodi in the se tor b . Therefore, this model will sele t an up(cid:21)quarktype Yukawa ouplings at the ubi level of the superpotential. The gauge group of thismodel is generated entirely from the untwisted ve tor bosons and there is no gaugesymmetry enhan ement from additional se tors. The four dimensional gauge group is SU (3) C × SU (2) L × U (1) B − L × U (1) T R × U (1) , ··· , × SU (2) H , ··· , × U (1) H , .The spe trum of the model is detailed in the Table .a in Appendix B. The u-bi level superpotential, in luding states from the observable and hidden se tors, isstraightforwardly al ulated following the rules given in [92℄ and reads: W = N cL L ¯ h + u cL Q ¯ h + C − ++ D − ¯ h + C + −− D + h ++ ( φ φ ′ + φ ′ φ ) φ + ( C − ++ C + −− + C − + − C + − + ) φ ′ + ( D + D − + C + C − + T + T − + D (6)+ − D (6) − + + D (6) −− D (6)++ ) φ + ( D (3 , − D (3 , − + + D (5)+ D (5) − + D (3)++ D (3) −− + D (3)+ − D (3) − + ) φ + A + A − φ ′ . As expe ted, we obtain a Yukawa oupling for the top quark, but also ouplingsof the Higgs with exoti states. One an also see that not all the fra tionally harged3states in the spe trum appear in the ubi level superpotential, whi h means that theyremain massless at the trilinear level. However, this does not ex lude the possibility ofgiving them masses at higher orders.2We use a di(cid:27)erent notation here for the boundary ondition ve tor α , whi h is now alled b , sin ein the literature α breaks the SO (10) gauge group while in this ase the boundary onditions w.r.t. ¯ ψ ,.. leave inta t the SO (10) symmetry.3The hyper harge is de(cid:28)ned as Q Y = 1 / Q C + 1 / Q L and the ele tri harge is given by Q e = T L + Q Y , with T L the ele troweak isospin. 41.3.1 Flat dire tionsIn this se tion we investigate the (cid:29)at dire tions of the model of table (3.23). The model ontains 6 anomalous U (1) 's with Tr Q = Tr Q = − Tr Q = Tr Q = − , Tr Q = − Tr Q = 12 . (3.24)The total anomaly an be rotated into a single U (1) A and the new basis reads Q ′ = Q − Q ,Q ′ = Q + Q ,Q ′ = Q + Q ,Q ′ = Q + Q + Q − Q ,Q ′ = Q + Q − Q + Q + 4( Q − Q ) ,Q A = 2( Q + Q − Q + Q ) − Q + Q . (3.25)In the following we will all Q ′ i , i=1,...,5, simply Q i .To sear h for (cid:29)at dire tions we use the methodology developed in [93℄. We start by onstru ting a basis of D-(cid:29)at dire tions under Q ... and then we investigate the exis-ten e of D-(cid:29)at dire tions in the anomalous U (1) A . Subsequently we will have to imposeD-(cid:29)atness under the remaining gauge groups and F-(cid:29)atness. To generate the basis of(cid:29)at dire tions under Q ... we start by forming a basis of gauge invariant monomialsunder U (1) , then we use these invariants to onstru t a basis of invariant monomialsunder U (1) and so forth.We in lude in the analysis only the (cid:28)elds with vanishing hyper harge and whi h aresinglets under the Standard Model gauge group. The Q ... ,A harges of these (cid:28)elds aredetailed in table 3.27, where, following the notation of [93℄, we signal by ( ′ ) ( ( ′′ ) ) thepresen e in the spe trum of a se ond (third) (cid:28)eld with the same U (1) ... ,A harges andby √ the presen e of a (cid:28)eld with opposite U (1) ... ,A harges. For instan e, the (cid:28)eld φ stands for φ , while φ ′ stands for φ and the two (cid:28)elds with opposite harges are φ ′ and φ ′ . The (cid:28)elds with opposite harges to A + and A − are D (5) − and D (5)+ , respe tively,while the (cid:28)eld with opposite harges to D is D (3 , − and ˜ D ′′ stands for D (3 , − + , in thenotation of the Table 3.a in Appendix B. We did not in lude in table 3.27 the (cid:28)elds ˜ φ , φ and ˜ φ , whi h have vanishing harges. These (cid:28)elds are trivially (cid:29)at dire tions in the U (1) ... , but they are not (cid:29)at under the anomalous U (1) .For simpli ity we res aled the harges Q , Q and Q A by a fa tor 2 and the harges Q , Q and Q by a fa tor 4. The seventh olumn is given by ˆ Q = 118 ( Q A − Q + 9 Q ) (3.26)42nd, as explained in [93℄, it will be useful for the sear h of (cid:29)at dire tions in the anomalous U (1) . Q Q Q Q Q Q A ˆ Qφ ( ′ ) √ ( ′ ) S ( ′ )1 , D ˜ S ( ′ )1 , ˜ D ( ′ )1 S ( ′ )2 , D √ -1 4 0 -2 -2 -2 0 ˜ S ( ′ )2 , ˜ D ( ′ )( ′′ )2 -1 0 0 2 -6 -6 0 S ( ′ )3 , D ˜ S ( ′ )3 , ˜ D ( ′ )3 N -1 0 -1 -2 -10 -1 0 N N A + √ -1 0 0 -6 2 2 0 A − √ F ( ′ ) ˜ F ( ′ ) -1 1 0 1 15 -3 -1 F F F F φ ( ′ ) √ ( ′ ) , S ( ′ )1 , ˜ S ( ′ )1 , S ( ′ )2 , ˜ S ( ′ )2 , S ( ′ )3 , ˜ S ( ′ )3 , N , N and N .Bearing in mind the equivalen e in the harges for some (cid:28)elds, these ount as 11 (cid:28)eldsand so, given the fa t that we have to impose 5 onstraints, the basis of (cid:29)at dire tionsshould ontain 6 elements. But a simple Mathemati a program an show that it isimpossible to in orporate the (cid:28)elds S ( ′ )1 , S ( ′ )3 , N , N and N into the (cid:29)at dire tions.This leave us with 6 (cid:28)elds, so we expe t a basis with just one element. It turns out that,in respe t with the harges of the remaining (cid:28)elds, Q and Q are a linear ombinationof the previous U (1) 's, so there are a tually only 3 independent onstraints and, hen e,we obtain three basis elements φ ¯ φ, ¯ φ ˜ S ˜ S ˜ S , ¯ φ ˜ S S ˜ S , (3.28)where we expressed the (cid:29)at dire tions as gauge invariant monomials. For example, themonomial ¯ φ ˜ S ˜ S ˜ S orresponds to the following hoi e of VEVs | ¯ φ | = | ψ | , | ˜ S | = 2 | ψ | , | ˜ S | = 2 | ψ | , | ˜ S | = 2 | ψ | , (3.29)for an arbitrary | ψ | .Note that in the pre edent basis any (cid:28)eld A an be repla ed with its opy A ′ . Any(cid:29)at dire tion, P , an be obtained from the elements of the basis as43 n = Y α M n α α , (3.30)where M α stand for the elements of the basis, n is a positive integer and n α are integers[93℄.In order to obtain D-(cid:29)at dire tions in the anomalous U (1) we need to onstru tinvariant monomials ontaining the (cid:28)eld S ( ′ )3 , sin e this is the only (cid:28)eld with a positive ˆ Q harge4, ne essary to an el the negative Fayet-Iliopoulos term generated by theanomalous U (1)
5. And, sin e none of the elements of the basis ontains this (cid:28)eld, we on lude that there are no (cid:29)at dire tions involving only VEVs of the singlets.Therefore, we pro eed with the analysis in luding also non-abelian (cid:28)elds under thehidden gauge group. This amounts to in luding all the (cid:28)elds in table (3.27), whi h ontains 22 (cid:28)elds with non-equivalent harges. Again, we look for a basis of gaugeinvariant monomials under Q ... . Su h a basis is given by φ ¯ φ, D ¯ D , A + ¯ A + , A − ¯ A − , ¯ φ ˜ S ˜ S ˜ S , ¯ φ ˜ S S ˜ S , ¯ φA + A − , ¯ φS N F ˜ F F , ¯ φS S N ˜ F F , ¯ φS N N N F ˜ F , ¯ φS N N N F F , ¯ φS N N N F F , S ˜ S S ˜ S ¯ A + ˜ F F ,S ˜ S S ˜ S N N ¯ A ˜ F F F , ˜ S S ˜ S ¯ A F F , ¯ φS ˜ S S ˜ F F ,S ˜ S S N N ¯ A + ˜ F F , (3.31)where, again, any (cid:28)eld an be repla ed with one of its opies with equal Q ... harges.All the elements of the basis have negative or vanishing ˆ Q harges, but, sin e some ofthe elements ontain the (cid:28)elds S and F , whi h have positive ˆ Q harge, and, sin e (cid:29)atdire tions an be obtained as a ombination of the basis elements with negative powers,we annot on lude immediately that there are no D-(cid:29)at dire tions under the anomalous U (1) . Nevertheless, a simple Mathemati a program shows that it is impossible to obtainviable invariant monomials with positive ˆ Q harge, by viable meaning that the (cid:28)eldsthat do not have a partner (cid:28)eld with opposite harges should appear with positivepowers in the monomials. We on lude that there are no (cid:29)at dire tions involving onlysinglets of the visible gauge group.Therefore, the only possibility to obtain (cid:29)at dire tions whi h do not break ele tri harge is to onsider the option of giving a VEV also to the neutral omponent of the4The ˆ Q harge of an invariant monomial is equal, up to positive fa tors, with his Q A harge, sin ethe di(cid:27)eren e between the two is a linear ombination of Q ... , under whi h the invariant monomialshave zero harge by onstru tion.5In our model Tr Q A < . 44iggs (cid:28)eld, in whi h ase the (cid:29)at dire tions would break the ele troweak symmetry.The Higgs doublets in our model have the following harges: Q Q Q Q Q Q A ˆ Qh √ ¯ φhS S ˜ S ˜ F F to the basis (3.31). The new basis element also has a negative ˆ Q harge and, again,it turns out to be impossible to onstru t (cid:29)at dire tions with positive ˆ Q harge. Thismeans that the only stable va uum solutions of our model are the ones that break theStandard Model gauge group.Interested in the analysis of (cid:29)at dire tions in free fermioni models with redu edHiggs spe trum we performed an extensive sear h in a similar ase, where we ould not(cid:28)nd any solutions. Before providing the details of this model, the de(cid:28)nition of stringent(cid:29)at dire tions is introdu ed.3.4 Stringent (cid:29)at dire tionsIn general, systemati analysis of simultaneously D - and F -(cid:29)at dire tions in anomalousmodels is a ompli ated, non-linear pro ess [94, 95℄. In weakly oupled heteroti string(WCHS) model-building, F -(cid:29)atness of a spe i(cid:28) VEV dire tion in the low energy e(cid:27)e -tive (cid:28)eld theory may be proved to a given order by an ellation of F -term omponents,only to be lost a mere one order higher at whi h an ellation is not found. An ex ep-tion is dire tions with stringent F -(cid:29)atness [52, 88℄. Rather than allowing an ellationbetween two or more omponents in an F -term, stringent F -(cid:29)atness requires that ea hpossible omponent in an F -term have zero va uum expe tation value.When only non-Abelian singlet (cid:28)elds a quire VEVs, stringent (cid:29)atness implies thattwo or more singlet (cid:28)elds in a given F -term annot take on VEVs. For example, inse tion 3.5.1, whi h presents the third and forth order superpotential for the modelunder onsideration, the omponents of the F -term for Φ are (through third order): F Φ = ¯Φ ¯Φ ′ + ¯Φ ′ ¯Φ . (3.33)For stringent F -(cid:29)atness we require not just that < F Φ > = 0 , but that ea h omponentwithin is zero, i.e., < ¯Φ ¯Φ ′ > = 0 , < ¯Φ ′ ¯Φ > = 0 . (3.34)Thus, by not allowing an ellation between omponents in a given F -term, stringent F -(cid:29)atness imposes stronger onstraints than generi F -(cid:29)atness, but requires signi(cid:28) antlyless (cid:28)ne-tuning between the VEVs of (cid:28)elds.The net e(cid:27)e t of all stringent F - onstraints on a given superpotential term is that atleast two (cid:28)elds in the term must not take on VEVs. This ondition an be relaxed when45on-abelian (cid:28)elds a quire VEVs. Self- an ellation of a single omponent in a given F -term is possible between various VEVs within a given non-abelian representation.Self- an ellation was dis ussed in [96℄ for SU (2) and SO (2 n ) states.A given set of stringent F (cid:29)atness onstraints are not independent and solutions toa set an be expressed in the language of Boolean algebra (logi ) and applied as on-straints to linear ombinations of D -(cid:29)at basis dire tions.The Boolean algebra languagemakes lear that the e(cid:27)e t of stringent F -(cid:29)at onstraints is strongest for low ordersuperpotential terms and lessens with in reasing order. In parti ular, for the modelpresented in the following, stringent (cid:29)atness is extremely onstraining on VEVs of theredu ed number of (untwisted) singlet (cid:28)elds appearing in the third through (cid:28)fth ordersuperpotential, in omparison to its onstraints on the larger number of singlets in themodel of table 3.23 [44℄.One might imagine that stringent F -(cid:29)atness onstraints requires order-by-order test-ing of superpotential terms. This is, in fa t, not ne essary. All-order stringent F -(cid:29)atness an a tually be proved or disproved by examining only a small (cid:28)nite set of possibledangerous (i.e., F -(cid:29)atness breaking) superpotential terms. Through a pro ess su h asmatrix singular value de omposition (SVD)6, a (cid:28)nite set of superpotential terms anbe onstru ted that generates all possible dangerous superpotential terms for a spe i(cid:28) D -(cid:29)at dire tion. This basis of gauge-invariants an always be formed with parti ularattributes: (1) ea h basis element term ontains at most one unVEVed (cid:28)eld (sin e tothreaten F -(cid:29)atness, a gauge-invariant term, ne essarily without anomalous harge, an ontain no more than one unVEVed (cid:28)eld); (2) there is at most one basis term for ea hunVEVed (cid:28)eld in the model; and (3) when an unVEVed (cid:28)eld appears in a basis term, itappears only to the (cid:28)rst power. The SVD pro ess generated a possibly threading basisof superpotential terms for several models (see for example [52, 66, 98℄).To appear in a string-based superpotential, a gauge invariant term must also fol-low Ramond-Neveu-S hwarz worldsheet harge onservation rules. For free fermioni models these rules were generalised from (cid:28)nite order in [92, 99℄ to all-order in [54℄.The generi all order rules an be applied to systemati ally determine if any produ t ofSVD-generated F -(cid:29)atness threatening superpotential basis elements survive in the or-responding string-generated superpotential. If none survive, then F -(cid:29)atness is provedto all (cid:28)nite order. This te hnique has been used to prove F -(cid:29)atness to all (cid:28)nite orderfor various dire tions in several models [52, 66, 98℄. Alternately, if any terms do survive,the lowest order is determined at whi h stringent F -(cid:29)atness is broken.How should stringent (espe ially all-order) (cid:29)at dire tions be interpreted in ompar-ison to general (perhaps (cid:28)nite order) (cid:29)at dire tions? All-order stringent (cid:29)at dire tions ontain a minimum number of VEVs and appear in models as the roots of more (cid:28)ne-tuned (generally (cid:28)nite-order) (cid:29)at dire tions that require spe i(cid:28) an ellations between6A SVD FORTRAN subroutine is provided in [97℄.46 -term omponents. The latter may involve an ellations between sets of omponentsof di(cid:27)erent orders in the superpotential.All-order stringent (cid:29)at dire tions have indeed been dis overed to be su h roots inall prior free fermioni heteroti models for whi h we have performed systemati (cid:29)atdire tion lassi(cid:28) ations. However, the model presented in the next se tion appears tola k any stringent (cid:29)at dire tions, at least within the expe ted range of VEV param-eter spa e. We have rea hed this on lusion after employing our standard systemati methodology for D - and F -(cid:29)at dire tion analysis.3.5 The string model with no stringent (cid:29)at-dire tionsThe string model that we present here ontains three hiral generations, harged un-der the Standard Model gauge group and with the anoni al SO (10) embedding ofthe weak(cid:21)hyper harge; one pair of untwisted ele troweak Higgs doublets; a ubi leveltop(cid:21)quark Yukawa oupling. The string model therefore shares some of the phenomeno-logi al hara teristi s of the quasi(cid:21)realisti free fermioni string models. The boundary ondition basis ve tors beyond the NAHE(cid:21)set and the one(cid:21)loop GSO proje tion oe(cid:30)- ients are shown in table 3.35 and in table 3.36, respe tively. ψ µ χ χ χ ¯ ψ ,..., ¯ η ¯ η ¯ η ¯ φ ,..., α β γ
12 12 12 12 12 12 12 12
12 12 12 12 y y y ¯ y y ¯ y ¯ y ¯ y y ω y ¯ y ω ¯ ω ¯ y ¯ ω ω ω ω ¯ ω ω ¯ ω ¯ ω ¯ ω α β γ S b b b α β γ − − − − − iS − − − b − − − − − − − ib − − − − − − ib − − − − − − α − − − − β − − − − − γ − − − − − − i (3.36)47oth the basis ve tors α and β break the SO (10) symmetry to SO (6) × SO (4) andthe basis ve tor γ breaks it further to SU (3) × U (1) C × SU (2) × U (1) L . The basisve tor α is symmetri with respe t to the se tor b and asymmetri with respe t tothe se tors b and b , whereas the basis ve tor β is symmetri with respe t to b andasymmetri with respe t to b and b . As a onsequen e of these assignments and of thestring doublet(cid:21)triplet splitting me hanism [90℄, both the untwisted Higgs olour tripletsand ele troweak doublets, with leading oupling to the matter states from the se tors b and b , are proje ted out by the generalised GSO proje tions. At the same timethe untwisted olour Higgs triplets that ouple at leading order to the states from these tor b are proje ted out, whereas the untwisted ele troweak Higgs doublets remainin the massless spe trum. Due to the asymmetri boundary onditions in the se tor γ with respe t to the se tor b , the leading Yukawa oupling is that of the up(cid:21)typequark from the se tor b to the untwisted ele troweak Higgs doublet [75℄. Hen e, theleading Yukawa term is that of the top quark and only its mass is hara terised bythe ele troweak VEV. The lighter quarks and leptons ouple to the light Higgs doubletthrough higher order nonrenormalizable operators that be ome e(cid:27)e tive renormalizableoperators by the VEVs that are used to an el the anomalous U (1) A D (cid:21)term equation[75℄. We remind on e again that the novelty in the onstru tion of the model in [44℄,and in the model of table 3.35, is that the redu tion of the untwisted Higgs spe trumis obtained by the hoi e of the boundary ondition basis ve tors in table 3.35, whereasin previous models it was obtained by the hoi e of (cid:29)at dire tions and analysis of thesuperpotential [79℄.The (cid:28)nal gauge group of the string model arises as follows: in the observable se torthe NS boundary onditions produ e gauge group generators for SU (3) C × SU (2) L × U (1) C × U (1) L × U (1) , , × U (1) , , . (3.37)Thus, the SO (10) symmetry is broken to SU (3) × SU (2) L × U (1) C × U (1) L , where, U (1) C ⇒ Q C = X i =1 Q ( ¯ ψ i ) , (3.38) U (1) L ⇒ Q L = X i =4 Q ( ¯ ψ i ) . (3.39)The (cid:29)avour SO (6) symmetries are broken to U (1) n with ( n = 0 , · · · , . The (cid:28)rstthree, denoted by U (1) j ( j = 1 , , , arise from the worldsheet urrents ¯ η j ¯ η j ∗ , asmentioned previously. The additional horizontal U (1) symmetries, denoted by U (1) j ( j = 4 , , ... ) , arise by pairing two real fermions from the sets { ¯ y , ··· , } , { ¯ y , , ¯ ω , } and { ¯ ω , ··· , } . The (cid:28)nal observable gauge group depends on the number of su h pairings. Inthis model there are the pairings ¯ y ¯ y , ¯ y ¯ ω and ¯ ω ¯ ω , whi h generate three additional U (1) symmetries, denoted by U (1) , , . 48t is important to note that the existen e of these three additional U (1) urrentsis orrelated with the assignment of asymmetri boundary onditions with respe t tothe set of internal worldsheet fermions { y, ω | ¯ y, ¯ ω } , ··· , , in the basis ve tors that extendthe NAHE(cid:21)set, { α, β, γ } . This assignment of asymmetri boundary onditions in thebasis ve tor that breaks the SO (10) symmetry to SO (6) × SO (4) results in the proje -tion of the untwisted Higgs olour(cid:21)triplet (cid:28)elds and preservation of the orrespondingele troweak(cid:21)doublet Higgs representations [90℄.In the hidden se tor, whi h arises from the omplex worldsheet fermions ¯ φ ··· , theNS boundary onditions produ e the generators of SU (2) , , , × SU (4) H × U (1) H . (3.40) U (1) H orresponds to the ombinations of the worldsheet harges Q H = X i =5 Q ( ¯ φ i ) . (3.41)The model ontains several additional se tors that may a priori produ e spa etimeve tor bosons and enhan e the gauge symmetry, whi h in lude the se tors + b + b + b and + S + α + β + γ . Additional spa etime ve tor bosons from these se tors wouldenhan e the gauge symmetry that arise from the spa etime ve tor bosons produ ed inthe Neveu(cid:21)S hwarz se tor. However, with the hoi e of generalised GSO proje tion oe(cid:30) ients given in table 3.36 all of the extra gauge bosons from these se tors areproje ted out and the four dimensional gauge group is given by eqs. (3.37) and (3.40).In addition to the graviton, dilaton, antisymmetri se tor and spin(cid:21)1 gauge bosons,the Neveu(cid:21)S hwarz se tor gives one pair of ele troweak Higgs doublets h and ¯ h ; sixpairs of SO (10) singlets, whi h are harged with respe t to U (1) , , ; three singlets ofthe entire four dimensional gauge group. A notable di(cid:27)eren e as ompared to modelswith unredu ed untwisted Higgs spe trum, like the model of ref. [65℄, is that the SO (10) singlet (cid:28)elds, whi h are harged under U (1) , , , are proje ted out from the masslessspe trum. The three generations are obtained from the se tors b , b and b , as usual.The model ontains states that are ve tor(cid:21)like with respe t to the Standard Model andall non(cid:21)abelian group fa tors, but may be hiral with respe t to the U (1) symmetriesthat are orthogonal to the SO (10) group. The full massless spe trum of the model isdetailed in Table 3.b in Appendix B.As a (cid:28)nal note we remark that the boundary onditions with respe t to the internalworldsheet fermions of the set { y, ω | ¯ y, ¯ ω } , ··· , in the basis ve tors α , β and γ , thatextend the NAHE(cid:21)set, are similar to those in the basis ve tors that generate the stringmodel of ref. [65℄, with the repla ements α (¯ y ¯ y ) ←→ γ (¯ y ¯ y ) β (¯ y ¯ ω ) ←→ γ (¯ y ¯ ω ) . (3.42)49he worldsheet fermions { y, ω | ¯ y, ¯ ω } , ··· , orrespond to the ompa ti(cid:28)ed dimensions ina orresponding bosoni formulation. The substitutions in eqs.(3.42) are augmentedwith suitable modi(cid:28) ations of the boundary onditions of the worldsheet fermions { ¯ ψ , ··· , , ¯ η , ··· , , ¯ φ , ··· , } , whi h orrespond to the gauge degrees of freedom. The ef-fe t of these additional modi(cid:28) ations is to alter the hidden se tor gauge group. Whilethe substitutions in eqs.(3.42) look inno uous enough, they in fa t produ e substantial hanges in the massless spe trum and, as a onsequen e, in the physi al hara teristi sof the models. With regard to the (cid:29)at dire tions of the superpotential, the e(cid:27)e t ofthese hanges on the untwisted states will be parti ularly noted.3.5.1 Third and Fourth Order SuperpotentialThe three singlets of the entire four dimensional gauge group are obtained from: ξ = χ ∗ ¯ ω ¯ ω | > ,ξ = χ ∗ ¯ ω ¯ y | > ,ξ = χ ∗ ¯ y ¯ y | > . We show below the ubi and fourth order superpotential terms.Trilinear superpotential: W = N c L ¯ h + u c Q ¯ h + H ¯ H h + ¯ H H ¯ h ++ ξ ( H ¯ H + H ¯ H + H ¯ H )+ ξ ( H ¯ H + H ¯ H + H ¯ H )+ ξ ( H ¯ H + H ¯ H + H ¯ H + H ¯ H + H ¯ H )+ ξ (Φ αβ ¯Φ αβ + Φ αβ ¯Φ αβ )+ Φ ( ¯Φ ¯Φ ′ + ¯Φ ′ ¯Φ ) + ¯Φ (Φ Φ ′ + Φ ′ Φ )+ Φ ′ ( ¯Φ Φ + ¯Φ ′ Φ ′ ) + ¯Φ ′ (Φ ¯Φ + Φ ′ ¯Φ ′ )+ Φ ′ (cid:16) (Φ αβ ) + (Φ αβ ) (cid:17) + ¯Φ ′ (cid:16) ( ¯Φ αβ ) + ( ¯Φ αβ ) (cid:17) + ¯Φ ′ H H + Φ H H + ¯Φ ′ H H + Φ ′ ( H ) + ¯Φ ′ ( ¯ H ) + ¯Φ ′ ( H ) + Φ ′ ( ¯ H ) + Φ αβ H H + ¯Φ αβ ( ¯ H ¯ H + ¯ H ¯ H ) + ¯ H ¯ H H + H ¯ H ¯ H . (3.43)Quarti superpotential: W = Q u H ¯ H + Q u H ¯ H + L N c H ¯ H + L N c H ¯ H . (3.44)We provide the expression of the quinti order superpotential in (B.1) in Appendix B.50.5.2 Flat dire tionsThe model in table 3.35 possesses nine lo al U (1) symmetries, eight in the observablepart and one in the hidden part. Six of these are anomalous: Tr U = Tr U = − Tr U = 2Tr U = − U = 2Tr U = − . (3.45) U (1) L and U (1) C of the SO (10) subgroup are anomaly free. Consequently, the weakhyper harge and the orthogonal ombination, U (1) Z ′ , are anomaly free. The hiddense tor U (1) H is also anomaly free.Of the six anomalous U (1) s, (cid:28)ve an be rotated by an orthogonal transformationto be ome anomaly free. The unique ombination that remains anomalous is: U A = k P j [Tr U (1) j ] U (1) j , where j runs over all the anomalous U (1) s and k is a normalisation onstant. For onvenien e, we take k = and therefore the anomalous ombination isgiven by: U A = − U − U + 2 U − U + U − U , Tr Q A = 180 . (3.46)The (cid:28)ve rotated non-anomalous orthogonal ombinations are not unique, with dif-ferent hoi es related by orthogonal transformations. One hoi e is given by: U ′ = U − U , U ′ = U + U + 2 U , (3.47) U ′ = U + U , U ′ = U − U − U , (3.48) U ′ = U + U − U − U + 2 U − U . (3.49)Thus, after this rotation there are a total of eight U (1) s free from gauge and gravitationalanomalies. In the following we use a di(cid:27)erent method to al ulate D- and F- (cid:29)atness,whi h is suitable for the implementation of a FORTRAN program. A basis set of (norm-squares of) VEVs of s alar (cid:28)elds satisfying the non-anomalous D -(cid:29)atness onstraints(3.19) an be reated en masse [84, 54℄. The basis dire tions an have positive, negative,or zero anomalous harge. In the maximally orthogonal basis used in the singular valuede omposition approa h of [84, 54℄, ea h basis dire tion is uniquely identi(cid:28)ed with aparti ular VEV. That is, although ea h basis dire tion generally ontains many VEVs,ea h basis dire tion ontains at least one parti ular VEV that only appears in it.A physi al D -(cid:29)at dire tion D phys , with anomalous harge of sign opposite that ofthe FI term ξ , is formed from linear ombinations of the basis dire tions, D phys = . X i=1 a i D i , (3.50)where the integer oe(cid:30) ients a i are normalised to have no non-trivial ommon fa tor.In our notation, a physi al (cid:29)at dire tion (3.50) may have a negative norm-squarefor a ve tor-like (cid:28)eld. This denotes that it is the oppositely harged ve tor-partner (cid:28)eldthat a quires the VEV, rather than the (cid:28)eld. Basis dire tions themselves may have51e tor-like partner dire tions if all asso iated (cid:28)elds are ve tor-like. On the other hand,if in parti ular, the (cid:28)eld generating the VEV uniquely asso iated with a basis dire tiondoes not have a ve tor-like partner, that basis dire tion annot have a ve tor-like partnerdire tion.In pursuit of physi al all-order (cid:29)at dire tions for this model, we (cid:28)rst examineddire tions formed solely from the VEVs of non-abelian singlet (cid:28)elds. An asso iatedmaximally orthogonal basis set, denoted by {D ′ i =1 to 13 } , ontaining only non-abeliansinglet VEVs is shown in Table 3. in Appendix B. The respe tive unique VEV (cid:28)elds ofthese basis dire tions are identi(cid:28)ed in Table 3.d in the same Appendix. Examination ofTables 3. and 3.d reveals that no physi al D -(cid:29)at dire tions an be formed solely fromVEVs of non-abelian singlet (cid:28)elds. Sin e the FI term ξ in eq.(3.19) is positive for thismodel, with Tr Q A = 180 , a physi al (cid:29)at dire tion must arry a negative anomalous harge. However, of the 13 singlet D -(cid:29)at basis dire tions, three arry anomalous hargeof +15 , +30 , +30 while the remaining ten do not arry anomalous harge. Further,the unique VEVed (cid:28)elds for the 3 basis dire tions with positive anomalous harge donot have orresponding ve tor-like partner (cid:28)elds. Hen e, there are no ve tor-like pairedbasis dire tions with negative anomalous harge. Thus, Tables 3. and 3.d imply thatone or more (cid:28)elds arrying non-abelian harges must also a quire VEVs in physi al D -(cid:29)at dire tions. This result is, in itself, not ne essarily unexpe ted, as non-abelianVEVs have been required for physi al (all-order) (cid:29)at dire tions in other quasi-realisti free fermioni heteroti models in the past, for example [66℄.Thus, we expanded our (cid:29)at dire tion sear h to in lude VEVs of both non-Abeliansinglet (cid:28)elds and non-abelian harged (cid:28)elds. Our hosen set of 50 maximally orthogo-nal D -(cid:29)at basis dire tions for both non-abelian singlet VEVs and non-abelian hargedVEVs, denoted by {D i =1 to 50 } , is presented in Table 3.e. The respe tive unique (cid:28)eldVEVs identi(cid:28)ed with these basis dire tions are given in Table 3.f. In this enlarged basisthe anomalous harges are given in units of ( Q ( A ) ) and the dire tions ontaining onlysinglet VEVs are rotations of those in Table 3. .Nine of the 50 dire tions, denoted D i =1 ,..., , arry one or two units of negativeanomalous harge. Twenty basis dire tions, denoted D through D , arry no anoma-lous harge. Twenty-one basis dire tions, denoted D through D , arry one or twounits of positive anomalous harge. All basis dire tions possessing negative anomalous harge ontain SU (3) C ⊗ SU (2) L harges or hidden se tor SU (4) ⊗ Q j =1 SU (2) j harges.(Thus, this basis set also reveals that anomaly an ellation will ne essarily break oneor more non-abelian lo al symmetries.) All of the Φ (cid:28)elds, the H (cid:28)elds and h have ve tor-like pairs. Thus, physi al (cid:29)at dire tions an have negative omponents forany of these. A subset of these (cid:28)elds, spe i(cid:28) ally Φ , Φ ′ , ¯Φ ′ , and H , , , , has VEVsappearing in multiple basis dire tions. The only non-ve tor-like (cid:28)eld with a VEV thatappears in multiple dire tions is e c . 52 through D and D are omposed solely of varying ombinations of the ve tor-like (cid:28)elds. Hen e, all of these basis dire tions have orresponding ve tor-like partnerbasis dire tions, ¯ D i ≡ − D i , for whi h the VEV of ea h (cid:28)eld is repla ed by the VEVof the ve tor-like partner (cid:28)eld. Thus, in a physi al (cid:29)at dire tion in eq.(3.50), ea h ofthe respe tive integer oe(cid:30) ients a through a and a , may be negative, positive,or zero.Note that D , D , D and D are ve tor-like ex ept for their e c omponents.Thus, ea h of a , a , a and a may be negative, positive, or zero in a physi al D -(cid:29)atdire tion, so long as the net norm-square VEV of e c is non-negative.7 The remainingbasis dire tions ontain at least one unique non-ve tor-like (cid:28)eld VEV. Thus, in a physi al(cid:29)at dire tion, the oe(cid:30) ients of the remaining basis dire tions must be non-negative.What does this mean for a physi al D -(cid:29)at dire tion formed as a linear ombination ofthe basis dire tions? For a physi al (cid:29)at dire tion there are, thus, two spe i(cid:28) onstraintson the a i oe(cid:30) ients and one general set of non-negative norm-square onstraints on asubset of the a i . First, negative anomalous harge for a (cid:29)at dire tion requires − X i =1 a i − X i =3 a i + X i =30 a i + 2 X i =45 a i < . (3.51)Se ond, a non-negative norm-square VEV for e c requires − X i =1 a i − a − X i =4 a i − a − X i =8 a i − X i =18 a i − a + a − X i =23 a i − a − a + 2 a − a + 2 a + 6 a + 6 a + a +3 X i =39 a i + 6 a + 6 X i =45 a i + 2 X i =48 a i + 6 a ≥ . (3.52)Last, for the set of non-ve tor-like (cid:28)elds that are ea h identi(cid:28)ed with a respe tive unique D -(cid:29)at dire tion, the general set of non-negative norm-square VEV onstraints is a i ≥ i = 1 to 6 , , , ,
23 to 50 . (3.53)At low orders, ea h individual superpotential term also indu es several stringent F -term onstraints on the a i oe(cid:30) ients of physi al (cid:29)at dire tions. As stated prior, theset of onstraints from superpotential terms with only singlet (cid:28)elds translate into the7Note that non-ve tor-like (cid:28)elds, su h as e c , that appear in multiple dire tions with some basisdire tions having positive and some having negative norm-square omponents, are ommon in thispro ess. Further, some models explored in the past have had (at least) one basis dire tion with two (ormore) (cid:28)eld VEVs unique to it and with norm-square VEVs with di(cid:27)ering signs. This latter type of basisdire tion an never appear in a physi al dire tion and, hen e, implies that the (cid:28)elds unique to it annever appear in a D -(cid:29)at dire tion. (If all of the norm-squares of the (cid:28)elds unique to a basis dire tionwere initially negative, then these signs, along with those of the norm-squares of any ve tor-like (cid:28)eldVEVs in that basis dire tion, ould all be hanged together to allow the basis dire tion to appear in aphysi al dire tion.) 53equirement that two or more singlet (cid:28)elds in a given superpotential term annot take onVEVs. For the model under investigation, onstraints from third order superpotentialterms are espe ially severe. For this model, all six Φ singlet (cid:28)elds and their ve tor-likepartners appear in third order superpotential terms (spe i(cid:28) ally, the sixth and seventhlines) of eq.(3.43). Stringent F -(cid:29)atness from these terms forbids at least 8 of the 12singlet (cid:28)elds from a quiring VEVs.For example, when solely third order stringent F -(cid:29)atness onstraints are applied tothe six pairs of Φ ve tor-like singlets (and no F -(cid:29)atness onstraints are applied to thenon-abelian states), there are just nine solution lasses that allow the maximum of 4singlet VEVs. (Flat dire tions in any of these nine lasses are de(cid:28)ned by their respe tivenon-abelian VEVs.)For three of these nine singlet third order (cid:29)atness lasses, the VEVs are of two (cid:28)eldsand their respe tive ve tor-like partners: either, < Φ >, < Φ ′ >, < ¯Φ >, < ¯Φ ′ > = 0 , or (3.54) < ¯Φ >, < ¯Φ ′ >, < Φ >, < Φ ′ > = 0 , or (3.55) < ¯Φ ′ >, < ¯Φ >, < Φ ′ >, < Φ > = 0 . (3.56)Higher order stringent (cid:29)atness onstraints an further redu e the allowed number ofsinglet VEVs of ea h of these solutions. Further, a omponent of a D -(cid:29)at basis dire tionin Table 3.a in Appendix B only spe i(cid:28)es the di(cid:27)eren e between the norm-squares of theVEV of a given (cid:28)eld and of the given ve tor-like partner (cid:28)eld (if it exists). Completely hargeless VEVs solely involving a (cid:28)eld Φ i and its ve tor-like partner ¯Φ i su h that | < Φ i > | = | < ¯Φ i > | an always be added to a physi al D -(cid:29)at dire tion. However,it is preferable for higher order F -(cid:29)atness to impose that a (cid:28)eld and its ve tor-partnerdo not simultaneously a quire VEVs. Hen e, these three solutions e(cid:27)e tively allow onlytwo unique singlet (cid:28)elds to a quire VEVs.The next three lasses of singlet solutions do allow up to four distin t singlet (cid:28)eldsto a quire VEVs: either, < Φ >, < Φ ′ >, < Φ >, < Φ ′ > = 0 , or , (3.57) < Φ >, < Φ ′ >, < Φ > < ¯Φ ′ > = 0 , or , (3.58) < ¯Φ >, < ¯Φ >, < Φ ′ >, < Φ ′ > = 0 . (3.59)For the three remaining solution lasses, the (cid:28)elds in (3.57), (3.58) and (3.59), arerespe tively repla ed with their ve tor-like partner (cid:28)elds. For any of these nine stringent F -(cid:29)at hoi es, no other Φ singlet (cid:28)elds an a quire VEVs.Any of the onstraints on allowed and disallowed VEVs, su h as the above, an bere-expressed in terms of onstraints on the a i oe(cid:30) ients spe ifying the basis dire tions ontributions to a physi al D -(cid:29)at dire tion. For example, setting < Φ > = 0 would54equire a + a + 2 X i =3 a i + 8 a + 2 a + a − a − a + a + a − a + 2 a + a − a + a − a − a + a + 4 a + a − a + X i =33 a i − X i =36 a i − a + a − X i =41 a i + a − a + 2 a − a − a − a = 0 . (3.60)To systemati ally investigate physi al D -(cid:29)at dire tions with non-abelian VEVs, overa ourse of several months we generated and examined physi al D -(cid:29)at dire tions om-posed of from 1 to 6 basis dire tions. Under the assumption that all VEVs of physi al(cid:29)at dire tions are nearly of the same order of magnitude, we allowed oe(cid:30) ients of 0 to20 for the non-ve tor-like basis dire tions and oe(cid:30) ients of -20 to 20 for the ve tor-likebasis dire tions.To be lassi(cid:28)ed as a physi al D -(cid:29)at dire tion, a linear ombinations of basis dire -tions needed to obey eqs.(3.51-3.53) and was, of ourse, also required to have non-abelian D -(cid:29)atness. (The general pro ess by whi h we enfor ed non-abelian D -(cid:29)atness followedthat presented in [84, 54℄.) Ea h resulting physi al D -(cid:29)at dire tion was then tested forstringent F -(cid:29)atness from all third order through (cid:28)fth order superpotential terms andadditionally for some key sixth order superpotential terms.8Following the SVD method dis ussed earlier in se tion 3.4 and des ribed in [51, 88℄,we had planned to then test for possible all-order stringent F -(cid:29)atness, the subset ofphysi al D -(cid:29)at dire tions that had proved stringently F -(cid:29)at to at least (cid:28)fth or sixthorder. Based on all of the prior models we had investigated, we had expe ted to (cid:28)ndaround four to six physi al D -(cid:29)at dire tions that were, in fa t, stringently F -(cid:29)at to all(cid:28)nite order. However, in ontrast we dis overed that no physi al D -(cid:29)at dire tions thatwe had generated even kept stringent F -(cid:29)atness through sixth order. So there wereno physi al D -(cid:29)at dire tions to examine for all-order testing. For this model, with itsredu ed set of singlet (cid:28)elds from the untwisted se tor, not even self- an ellation of non-abelian terms ould provide stringent F -(cid:29)atness through sixth order for any of thesephysi al D -(cid:29)at dire tions.We will ontinue a sear h for F -(cid:29)atness past sixth order for physi al D -(cid:29)at dire tionsin this model that are omprised of seven or more basis dire tions. However, a ontinuednull result is likely: sin e ea h of our basis dire tions ontains a unique (cid:28)eld VEV,in reasing the number of non-zero a i oe(cid:30) ients linearly in reases the minimum numberof unique (cid:28)eld VEVs. With ea h in rease in number of basis dire tions omposing aphysi al D -(cid:29)at dire tion, the probability of obtaining stringent F -(cid:29)atness mu h beyondsixth order further de reases.8While only the third through (cid:28)fth order superpotential is given in se tion (3.5.1), we have generatedthe omplete superpotential to eighth order and an generate it to any required order.55n this model no physi al D-(cid:29)at dire tion that we generated kept F-(cid:29)atness throughsix order. We said that only stringent (cid:29)at dire tions an be (cid:29)at to all orders of non-renormalizable terms. This would indi ate that this model has no D-(cid:29)at dire tions that an be proved to be F-(cid:29)at to all order. If a non-vanishing F-term does exist, then su-persymmetry remains unbroken at (cid:28)nite order. The Fayet-Iliopoulos term that breakssupersymmetry is generated at one-loop level in the perturbative string expansion. Onthe other hand the string spe trum is Bose-Fermi degenerate and possesses N = 1 spa etime supersymmetry at the lassi al level. This would suggest that, ontrary tothe expe tation from supersymmetri quantum (cid:28)eld theories, perturbative supersymme-try breaking may ensue in string theory. Futhermore, the modular invariant one-looppartition fun tion vanishes, giving a vanishing one-loop osmologi al onstant. Thismodel may therefore represent an example of a quasi-realisti string va uum with van-ishing one-loop osmologi al onstant and perturbatively broken supersymmetry.56hapter 4 Z × Z orbifold onstru tionsIn the previous hapter we have largely dis ussed the free fermioni models, whi h or-respond to Z × Z orbifolds at spe ial points of the moduli spa e (see se tion 3.1.2). Inthis hapter we want to present the orbifold onstru tion as it provides omplementaryinformation on heteroti models away from the spe ial points.We (cid:28)rst onsider the heteroti superstring ompa ti(cid:28)ed on a (cid:29)at torus, where thephysi al dimensions are redu ed from ten to four. In order to obtain models withappealing phenomenology, for instan e with N = 1 supersymmetry, the initial toroidal ompa ti(cid:28) ation is modi(cid:28)ed by modding-out a dis rete symmetry des ribed by a pointgroup P and giving rise to an orbifold [100, 101℄, for review see [102℄. We brie(cid:29)ypresent the orbifold onstru tion rules, the derivation of the massless spe trum and theproje tion onditions required for modular invarian e. We mainly follow [103, 104, 105,106, 107, 108, 109℄, where an extensive treatment of the topi an be found.An indi ative example of orbifold ompa ti(cid:28) ation is presented in the se ond part ofthis hapter. Our model is a six dimensional torus de(cid:28)ned by the SO (4) root latti e,with Z × Z dis rete symmetry. The derivation of its (cid:28)xed tori, their entralisersand the introdu tion of the Wilson lines is explained in details. The main motivationfor onsidering the skewed model analysis was the attempt of reprodu ing the threegeneration free fermioni model [16℄, with E × U (1) × SO (8) H gauge symmetry. Amodel with these properties was not found in the lassi(cid:28) ation by Donagi and Wendland[110℄, whi h extended the analysis of Donagi and Faraggi [15℄. The aim of the skewedmodel analysis is to try to build an orbifold model with similar hara teristi s to thefree fermioni model. While unsu essful, the in lusion of this analysis in the thesisaims to provide details of the omplementary orbifold onstru tion. In parti ular, weexplain the impli ations of using fa torisable or non-fa torisable latti es and how thepresen e of Wilson lines may hange the phenomenology of the model. A detailed study on erning Z × Z orbifolds with di(cid:27)erent ompa ti(cid:28) ation latti es is given in [111℄.Several semi-realisti orbifold models have been presented in the literature with di(cid:27)erentdis rete symmetry [112, 113℄, although we are mainly interested in the Z × Z ase.57his is mainly be ause we believe that the orresponden e with free fermioni models an provide some intuition in the sele tion of phenomenologi al interesting va ua, sin ethe number of modular invariant orbifold models is huge and a omplete lassi(cid:28) ationunder general physi al properties represents an in redible feat. This hapter des ribesthe general pro edure of the orbifold onstru tion and suggests some te hni al tri ks in hoosing the most favourable latti es to onstru t possible semi-realisti orbifold va ua.4.1 Heteroti string and toroidal ompa ti(cid:28) ationThe ten dimensional heteroti superstring an provide a realisti four-dimensional the-ory if six of the nine spatial dimensions are ompa ti(cid:28)ed to a (cid:16)su(cid:30) iently small(cid:17) s ale,unobservable in nowadays experiments. The simplest ompa ti(cid:28) ation s heme is on atorus that, being a (cid:29)at surfa e, assures no modi(cid:28) ations in the equations of motion.We start this se tion by revisiting the ontent of the heteroti string in ten dimensionsin the bosoni onstru tion, sin e in hapter 2 we have presented the orrespondentfermioni des ription, where the ompa t bosons are substituted by internal degrees offreedom (32 real left-moving fermions with a pre ise hoi e of boundary onditions).In the bosoni formalism, the heteroti string is a right-moving superstring ombinedwith a bosoni left-moving string. In the light- one gauge the eight fermioni and eightbosoni right oordinates are given respe tively by Ψ iR and X iR , i = 1 , .. . The indi es i = 1 , denote the two transverse spa etime dimensions, while the other six refer to the ompa t spatial dimensions. The left movers are given by the bosoni X iL and sixteenfurther bosons X IL , I = 1 , .. , ompa ti(cid:28)ed on a 16-torus. The anomaly an ellationrequirement imposes that the 16-torus is either the root latti e of E × E or the oneof Spin (32) / Z [37℄. In this thesis we are interested in the E × E symmetry, then theequations given below will refer to the (cid:28)rst ase. The ompa ti(cid:28) ation pro edure doesnot a(cid:27)e t the mode expansion of the (cid:28)elds, whose expressions have been provided in hapter 2. We spe ify here the expansion of the gauge degrees of freedom X IL ( τ + σ ) = x IL + p IL ( τ + σ ) + i X n =0 ˜ α In n e − in ( τ + σ ) , (4.1)where we (cid:28)xed α ′ = 1 / and the momenta p IL lay on the E × E ′ latti e. In the anoni albasis, any element of the E latti e an be written as eight-dimensional ve tors ( n , ..., n ) , ( n + 1 / , ..., n + 1 / , where P n i = SO (16) ,while the se ond ve tor represents the spinorial of the same symmetry group.The ompa ti(cid:28) ation of the internal oordinates on the 6-torus, namely X i = X iL + X iR with i = 3 , .. , identi(cid:28)es the entre of mass oordinates x i with points that are58eparated by latti e ve tors of the torus x i = x i + 2 πL i , where ~L = ( L , ..., L ) belongs to a six-dimensional latti e Λ = { P t =3 r t ~e t | r t ∈ Z } and ~e t are the basis ve tors of the latti e. This implies that the boundary onditionsfor the ompa t spatial bosons are also satis(cid:28)ed if X i ( τ, π ) = X i ( τ,
0) + 2 πL i , whi h orrespond to winding states around the torus. The ompa ti(cid:28) ation also requires thequantization of the momenta p i and this result is a hieved by imposing the ondition P i =3 p i L i ∈ Z . Thus, the momenta are quantised on the dual latti e Λ ∗ , de(cid:28)ned as Λ ∗ = { X t =3 m t ~e t ∗ | m t ∈ Z } , where the basis ve tors ~e t ′ ∗ satisfy the relation ~e t ′ ∗ · ~e t = δ t ′ t .After the ompa ti(cid:28) ation to four dimensions, the mass formula for the right moverstakes the form m R = N R + 12 p iR p iR − a R , (4.2)where N R is the number operator whi h ounts the bosoni and fermioni (both R andNS) os illators. The onstants a R,L are the normal ordering for the Virasoro operators ˜ L and L , introdu ed in hapter 2. There we have showed that they get di(cid:27)erentvalues when onsidering the Ramond or the Neveu-S hwarz se tor (we noti e that thesevalues were determined for the non- ompa ti(cid:28)ed theory, while di(cid:27)erent values will be al ulated in the next se tion for twisted states arising in orbifold onstru tions).For the left movers in four dimensions the mass formula is given by m L = ˜ N L + 12 p IL p IL − a L , (4.3)where the left number operator ˜ N L in ludes the spatial os illators ˜ α i − n ˜ α in and the leftgauge ontributions ˜ α I − n ˜ α In .In eq.(4.2) and (4.3) the ontribution from the momenta p iL,R an give rise to mass-less states for parti ular values of the parameters of the latti e Λ , su h as the lengthof the basis ve tors, the angles between them, a s ale fa tor. Apart from these isolatedvalues, massless states arise when momenta and winding numbers are zero p iR = p iL = 0 , as we an see from their de(cid:28)nition in eq.(2.45). The toroidal ompa ti(cid:28) ation des ribedso far provides a N = 4 supersymmetri theory in four dimensions.In fa t, let us show expli itly how four gravitinos are generated in this set up. Inthe massless spe trum, we noti e the presen e of the states b i − / | > R ⊗ ˜ α j − | > L , b i | > R ⊗ ˜ α j − | > L , (4.4)59here i = 1 , and j takes values in the ompa t spa e. The (cid:28)rst ombination providesspa etime ve tors, the se ond state is in the Ramond groundstate and transforms asan SO (8) hiral spinor, the opposite hirality spinor being deleted by GSO proje tionsused in the superstring onstru tion. The weight ve tor notation for su h a spinor isgiven by q = ( ± ± ± ± ) , with an even number of (cid:16) + (cid:17). The SO (8) hiral spinor an be de omposed into representations of SO (2) × SO (6) ∈ SO (8) , with the SO (2) orresponding to the two transverse spa etime oordinates and the SO (6) referringto the six ompa ti(cid:28)ed oordinates. Hen e, there are four spa etime spinors of ea h hirality, providing four gravitinos. The analogous notation is used for the NS rightmoving state, orresponding to the (cid:28)rst entry in eq.(4.4) and indi ated by q = (1 , , , (the unders ore denotes that all permutations are in luded).For ompleteness we provide the massless physi al states of the heteroti string in D = 10 .Spe trum of the heteroti string | q > R × ˜ α i − | > L : i = 1 , .. supergravity multiplet , | q > R × ˜ α I − | > L : I = 1 , .. un harged gauge bosons of E × E , | q > R ×| p > IL : 240 + 240 harged gauge bosons of E × E , (4.5)where | q > R indi ates both R and NS solutions, meaning that the bosoni and its orrespondent fermioni state are present in the spe trum at the same time (susy su-perpartners).4.2 Orbifold onstru tionSo far we have shown that the toroidal ompa ti(cid:28) ation redu es our ten dimensionalheteroti string to four dimensions, but the theory is not hiral. In order to obtain aphenomenologi al interesting N = 1 supersymmetri theory, we onsider the orbifold onstru tion by starting with the toroidal ase. A torus is reated by the identi(cid:28) ationof points ~x of the underline spa e that di(cid:27)ers by a latti e ve tor ~l ∈ Γ = 2 π Λ ~x ∼ ~x + ~l. (4.6)In the toroidal ompa ti(cid:28) ation six spatial internal dimensions are ompa ti(cid:28)ed on thetorus T and the sixteen left-moving oordinates, orresponding to the gauge degreesof freedom, are ompa ti(cid:28)ed on the self-dual latti e T E × E ′ . T is generated by thelatti e Λ de(cid:28)ned in the previous se tion, while T E × E ′ is given by the root latti e of thegroup E × E ′ . An orbifold is obtained when we identify points on the torus whi h arerelated by the a tion of an isometry θ , more pre isely, an automorphism of the latti e( θ~l ∈ π Λ ) that preserves the s alar produ ts among the basis ve tors ~e a ∈ Λ , a = 1 , .. ,60here the ve tor ~l = ~e a n a . In the following we indi ate the latti e roots simply as e a ,spe ifying the entries of the ve tor with a new label i when ne essary. The orbifold isde(cid:28)ned as Ω = T /P × T E × E /G, (4.7)where P is the point (isometry) group, G its embedding in the gauge degrees of free-dom. The onstru tion of an orbifold depends on the hoi e of the point group P, itsembedding G and the latti e T . In parti ular, the requirement of N = 1 spa etimesupersymmetry is a hieved by imposing P ⊂ SU (3) . We restri t our dis ussion to anabelian P . In this ase the point group is dis rete and there are two possible hoi es : • P ≡ Z N = { θ k | k = 0 , ..N − }• P ≡ Z N × Z M = { θ k ◦ θ l | k = 0 , ..N − and l = 0 , ..M − } (4.8)where θ an be seen as a rotation of π/N , with N being the order of the twist. Thegauge twisting group G is an automorphism of the E × E Lie algebra and its a tionis required in order to satisfy modular invarian e. The six-dimensional torus an bewritten in the equivalent notation T = R / Γ when onsidering the identi(cid:28) ation ~x ∼ θ~x + ~l. The previous expression is useful when we de(cid:28)ne the spa e group, given by the set ofelements S = { ( θ,~l ) | θ ∈ P, ~l ∈ π Λ } . By using the previous de(cid:28)nition the following equivalen e holds: T /P ≡ R /S .The inner automorphism of the E × E algebra an be realised [114℄ by a shift V I in the root latti e and the embedding of a generi element of S is implemented by ( θ, n a e a ) → ( σ V I , n a σ A Ia ) , where σ A Ia orresponds to the a tion of the shifts A a in the gauge latti e. These shiftsare the gauge transformations asso iated with the non- ontra tible loops given by e a and they are alled Wilson lines.We an (cid:28)nally present the orbifold a tion on the spatial ompa t oordinates andon the gauge degrees of freedom X i → ( θX ) i + n a e ia , X IL → X IL + V I + n a A Ia . (4.9)In parti ular, if we use the omplex notation Z a , a = 1 , , , for the ompa t dimensions X ,.. , the a tion of θ is simply θ k : Z a → e πikv a Z a (4.10)61here the ve tor ~v = ( v , v , v ) orresponds to the twist a tion θ k → kv. Sin e the number of independent y les on a six-torus is six, we ould initially thinkthat there are six independent Wilson lines. However, the latti e ve tors de(cid:28)ning thetorus are generally related by the point group symmetry, thus some of the Wilson linesare identi(cid:28)ed. We will larify this statement when we onsider our example in se tion(4.3). Di(cid:27)erently from the toroidal ompa ti(cid:28) ation, in orbifold ba kgrounds there aresingular points, the so- alled (cid:28)xed points, where the metri is not isomorphi to R .This is a ru ial feature of orbifold models, related to the presen e of twisted se tors inthe spe trum. A (cid:28)xed point is de(cid:28)ned by X if = ( θ k X f ) i + n a e ia , for i = 3 , .. . We willshow the expli it derivation of the (cid:28)xed points for the Z × Z orbifold with a given ompa ti(cid:28) ation latti e in the se ond part of this hapter.At this point we show how the twist ve tor v a has to be (cid:28)xed to a hieve N = 1 supersymmetry. Sin e P is abelian, it must belong to the Cartan subalgebra of SO (6) asso iated with the oordinates X ,.. . If the generators of this subalgebra are indi atedas M , M and M , then the a tion of the point group element a ts on the omplexbasis Z a as θ = exp [2 πi ( v M + v M + v M )] , (4.11)where | v a | < , a = 1 , , . The ondition P ⊂ SU (3) thus requires ± v ± v ± v = 0 . (4.12)The ondition (4.12) and the fa t that the twist is a symmetry of the torus restri tthe hoi es of P to the following possibilities: it has to be a Z N symmetry with N =3 , , , , , or a Z N × Z M symmetry, with N multiple of M and N = 2 , , , [100,101℄. In general there an be several latti es for a given P . The massless spe trumand the gauge symmetries are determined by the point group and not by the hoi e ofthe latti e. We point out that when the spa e group is taken into a ount, then theembedding into the gauge latti e E × E provides properties depending on the latti e.A omplete list of point group generators for Z N and Z N × Z M ⊂ SU (3) orbifolds anbe found in [102℄.4.2.1 Consisten y onditionsThe embedding of the point group P into the twist gauge group G is an homomorphismof the latti e, thus for a N order twist θ the a tion of N V I orresponds to the identityon the root latti e. The same prin iple holds for the Wilson lines and these onditionsare translated into the equations below N V ∈ T E × E , N A a ∈ T E × E . (4.13)62odular invarian e has to be required in order to guarantee anomaly freedom and inthe orbifold onstru tion this requirement is implemented by the following onditions N ( V − v ) = 0 (mod2) ,N V · A a = 0 (mod1) ,N A a · A b = 0 (mod1) ,N A a = 0 (mod2) . (4.14)For Z N × Z M orbifolds the previous relations an be generalised. For instan e, in these ond part of this hapter, when the Z × Z orbifold is introdu ed, it will be de(cid:28)nedby two independent twist ve tors ~v and ~v , while the standard embedding is realisedby the shifts V I and V I . The (cid:28)rst two formulae in eqs.(4.14) must hold for both ofthese ve tors. Moreover, the Wilson lines onditions in eqs.(4.14) must be ful(cid:28)lled byboth these ve tors as well.4.2.2 Generalities on the spe trumThere are di(cid:27)erent ways in whi h the losed boundary onditions an be satis(cid:28)ed on anorbifold. This leads to the on lusion that there are two types of strings, the untwistedstring losed on the torus before the identi(cid:28) ation of points by the twist, and the twistedstring whi h is losed on the torus after imposing the point group symmetry. This issimply resumed in the following expression X ,.. ( τ, σ ) = θ k X ,.. ( τ,
0) + n a e a , (4.15)where the untwisted se tor ( k = 0 ) orresponds to the toroidal ompa ti(cid:28) ation, whilethe additional twisted se tors generate all new string states, lo alised at the points left(cid:28)xed under the a tion of the elements ( θ k , n a e a ) of the spa e group S. A generi element h ∈ S ⊗ G has a orrespondent operator ˆ h whi h implements the a tion of h on theHilbert spa e. We all ˆ h a onstru ting element and denote the states lo alised at the orresponding (cid:28)xed point by H h . Hen e, sin e the orbifold is de(cid:28)ned by modding outthe a tion of S ⊗ G , then physi al states must be invariant under the proje tion S ⊗ G .We will explain this on ept on an expli it example in se tion 4.3.UNTWISTED SECTORThe untwisted states are those obtained by the heteroti string ompa ti(cid:28)ed on atorus whi h survive the S ⊗ G proje tions. Below we rewrite eqs.(4.2(cid:21)4.3) demandingthe level mat hing ondition and using the weight ve tor notation for the right movers,as introdu ed previously, q −
12 = 14 m R = 14 m L = 12 p + N L − . (4.16)Under the a tion of S ⊗ G the left and the right states transform respe tively as | p > = e (2 πip · V ) | p > ; | q > = e (2 πiq · v ) | q > . p · V = 0 (mod 1) , p · A a = (0 mod 1) , (4.17) ombined with right movers whi h are invariant under S . When onsidering right moverstransforming non trivially, we get the se ond set of solutions. In this ase, in fa t, theonly possible surviving states are those tensored with left states transforming as p · V = k/N (mod 1) , k = 1 , ..N − , p · A a = 0 (mod 1) . (4.18)The most important result in the untwisted spe trum is that three of the four gravitinospresent in the toroidal ompa ti(cid:28) ation are proje ted out, giving a four-dimensional N = 1 supergravity theory.TWISTED SECTORSThe boundary onditions in eq.(4.15) for k = 0 provide the massless states of thetwisted se tors. Ea h twisted se tor orresponds to a onstru ting element, previously alled h . Obviously, the new boundary onditions hange the mode expansions of thebosoni and fermioni os illators, while the weight latti e has been shifted. In parti ular,we obtain | q > twR = | q + kv > R and shifted momenta | p I > tw = | p I + kV I + n a A a > . Themass formula in ea h twisted se tor reads as
12 ( q + v i ) −
12 + δ c = 14 m R = 14 m L = 12 ( p I + V I + n a A a ) + N L − δ c = 0 . (4.19)In the formula above the quantity δ c is the zero point energy due to the moded os il-lators. It an be al ulated by δ c = P a =0 η a (1 − η a ) , where η a = kv a (mod 1) and ≤ η a < . We anti ipate here that for the ase of the Z × Z orbifold δ c = 1 / . As mentioned already, in the twisted se tor the os illators are moded if they orrespondto a omplex dimension a where the twist a ts non trivially, giving for example ˜ α am − η a for a bosoni os illator. In this ase the number operator ˜ N an be fra tional. Thephysi al spe trum is obtained after the proje tions under ea h element of S ⊗ G . If weindi ate with h = ( θ, n a e a ; V, n a A a ) a onstru ting element of this group, the invariantstates under h de(cid:28)ne the Hilbert spa e H h , as we stated at the start of this se tion.Now we onsider a di(cid:27)erent element of the group, that we all g = ( θ, n a e a ; V, n a A a ) .If g ommutes with h , then, by using the de(cid:28)nition of twisted boundary onditions,we an see that the states invariant under g belong to H h . Moreover, all states in H h whi h transform non trivially under g have to be proje ted out. This reasoning has tobe applied for all ommuting elements of S ⊗ G and the whole set that ontains theseelements is alled entraliser Z h = { g ∈ S ⊗ G su h that [ h, g ] = 0 } . (4.20)64equiring that non invariant states are proje ted out means that all the elements in the entraliser a t as the identity on H h . For ea h non ommuting element ˜ g , [˜ g, h ] = 0 , thepro edure to apply onsists in building linear ombinations of states of Hilbert spa es H h , H ˜ gh ˜ g − , ... H ˜ g n h ˜ g − n , with ˜ g n = 1 . In the Z × Z ase it is always possible torestri t the previous pro edure to a reasonable (cid:28)nite number of elements of S ⊗ G .4.3 Z × Z orbifold with SO (4) ompa ti(cid:28) ation latti eThere are several latti es with Z × Z symmetry that an be onsidered to des ribe the T torus. One of the simplest instan es [107, 108℄ is the fa torisable T = T × T × T ,with orthogonal roots e i = (0 , , .., i, .., , i = 1 , .., . The a tion of the point groupin the Z × Z orbifold is given by P Z × Z = { , θ , θ , θ } , where the trivial element generates the untwisted spe trum and θ k , k = 1 , , , generate the twisted se tors. θ is the ombination of the two independent twists θ and θ . We present expli itly thetwist ve tors asso iated to ea h twisted se tor → (0 , , , , θ → v = (0 , / , − / , ,θ → v = (0 , , / , − / , θ → v = (0 , / , , − / , where the four entries in the ve tors v k refer to the spatial dimensions in omplex oor-dinates. The spa e group is de(cid:28)ned by S = { ( kv + lv , n a e a | k, l = 0 , , n a ∈ Z ) } andthe twisted se tors are obtained by the ombinations of k, l = 0 , . It has been shown[107, 108℄ that the fa torisable T = T × T × T latti e needs proper Wilson lines toprovide three standard model generations, although it still does not realise the standardembedding of the hyper harge. For this reason we an on lude that the fa torisablelatti e an be onsidered a toy model in the lass of orbifold onstru tions. We needto introdu e more hallenging ases to implement some interesting phenomenologi alproperties. The next step is to onsider di(cid:27)erent ompa ti(cid:28) ation latti es for the six-dimensional torus that for instan e generate an inferior number of generations beforeeven adding Wilson lines. Hen e, we rely on two me hanisms for the generation re-du tion, su h as the introdu tion of Wilson lines and the hoi e of the latti e. Beforeentering into the details, we spe ify the fa t that for the Z × Z orbifold the (cid:28)xedpoints are a tually two dimensional obje ts, thus providing (cid:28)xed tori. They give riseto generations or anti-generations (representations of the symmetry group of the modelunder onsideration in terms of multiplets whi h provide the Standard Model families,eventually after the breaking of the gauge group). The number of (cid:28)xed tori dependson the ompa ti(cid:28) ation latti e and for this reason and appropriate hoi e of the lat-ti e provides the options to de rease the net number of generations, often too many inorbifold ompa ti(cid:28) ations. In the standard embedding the net number of generationsis a tually given by the Euler number, hen e we ompare the result obtained by theexpli it al ulation of the (cid:28)xed tori for our model with its the Euler number.65s we have mentioned in the introdu tion, the SO (4) orbifold example is far frombeing a semi-realisti model. Our point is showing how the presen e of Wilson lines,that in general hange drasti ally the out ome of a model, in this parti ular ase donot modify the number of generations, for any hoi e of Wilson lines. The proof of thisstatement is shown at the end of the hapter.We introdu e now our example, where T is obtained by ompa tifying R on an SO (4) root latti e, whose basis ve tors are given by the simple roots e = (1 , , , − , , ,e = (1 , , , , , ,e = (0 , , , , − , ,e = (0 , , , , , ,e = (0 , , , , , − ,e = (0 , , , , , . (4.21)We remark here that the a tion of the orbifold on the SO (4) ompa ti(cid:28) ation latti e isnon-fa torisable, as it is obvious from the displa ement of the entries in the roots (4.21).This hoi e produ es interesting onsequen es for the spe trum of the model. In fa t,the number of (cid:28)xed tori is redu ed from 48 in the standard SO (4) to 12 for the asewith skewed a tion on the ompa ti(cid:28) ation latti e, resulting into a drasti redu tion ofthe number of generations. The derivation of the massless spe trum follows the rulesgiven in the previous se tions. We (cid:28)nd onvenient to obtain at this point some relevantinformation whi h will be used in the al ulation of the twisted states. In fa t, from theanalysis of the SO (4) skew latti e, we obtain the (cid:28)xed tori and the entralisers whi hare ne essary for the dis ussion of the massless twisted states.4.3.1 Analysis of the latti eThe study of a latti e onsists of the following steps: • (cid:28)nd the generators of the latti e, • look at the symmetries of the roots under the orbifold a tion, • al ulate the (cid:28)xed tori and the entraliser, • analyse the onsisten y onditions for the Wilson lines.We remind that we removed the ve tor symbol on the roots and any general ve torlatti e to simplify the notation. 66enerators and symmetriesThe generators of the latti e are de(cid:28)ned as the minimal shifts that, added to a (cid:28)xedtorus, provide exa tly the equivalent torus. In order to make this on ept more under-standable, we will illustrate the pro edure to obtain the generators in the ase of thetrivial torus1 of the θ twisted se tor (cid:8) ( x , x , , , , (cid:12)(cid:12) x , x ∈ R / Λ (cid:9) . (4.22)The ompa ti(cid:28) ation latti e Λ is generated by the ve tors (2 , and (0 , ; in fa t weneed to satisfy the ondition ( x , x , , , ,
0) = ( x + a, x + b, , , ,
0) + X a i e i , (4.23)where the e i are the SO (4) roots and a and b are the minimal shifts on the ( x , x ) oordinates of the 2-torus. The onstants a i have to be integer sin e we are looking forequivalent tori, meaning that they an di(cid:27)er only by SO (4) latti e shifts. Eq. (4.23) an be written as x e + e ) + x e + e ) = x + a e + e ) + x + b e + e ) + X a i e i . Requiring a i ∈ Z implies ( a, b ) = (0 (mod2) , (mod2) ) . Hen e, we are lead to the on lusion that the minimal shift is determined by the points (0 , , (0 , , (2 , , (2 , and we an hoose the two independent generators to be a = (2 , , b = (0 , .The symmetry of the roots (4.21) determines analogous results for the (cid:28)xed toriin the θ and the θ twisted se tors, although this is not a general property2. Thesymmetries of the latti e are derived by looking at the transformation properties of theroots under the elements θ i . θ θ θ e → - e e → e e → - e e → - e e → e e → - e e → - e e → e e → - e e → - e e → e e → - e e → - e e → - e e → e e → - e e → - e e → e (4.24)We observe that there are three sets of roots { e , e } , { e , e } and { e , e } , whi h behaveanalogously under the twists. This means that the Wilson lines asso iated to ea h groupmust be equal, in parti ular A = A , A = A , A = A θ se tor are al ulated in the next se tion.2The SO (6) non fa torisable latti e is an example where the (cid:28)xed tori in the three twisted se torshave di(cid:27)erent generating elements [111℄.3This result gives rise to the onsisten y onditions for the Wilson lines that we an possibly intro-du e in the ase of the SO (4) skew latti e. 67ixed tori and entraliserIn this se tion we present the (cid:28)xed tori for ea h twisted se tor of the model. The elementin parenthesis on the right-hand-side of a (cid:28)xed torus represents the onstru ting elementfor its orrespondent entraliser. We have also spe i(cid:28)ed if the torus provides a generationor an anti-generation to the twisted spe trum, a on ept that will be explained lateron. The (cid:28)xed tori for the se tor θ : (cid:8) (0 , , , , x , x ) (cid:12)(cid:12) x , x ∈ R / Λ (cid:9) , , (4.25) (cid:8) (1 , , , , x , x ) (cid:12)(cid:12) x , x ∈ R / Λ (cid:9) , ( e + e ) 1 generation , (4.26) (cid:8) (1 / , , , / , x , x ) (cid:12)(cid:12) x , x ∈ R / Λ (cid:9) , ( e ) 1 generation , (4.27) (cid:8) (1 / , , , − / , x , x ) (cid:12)(cid:12) x , x ∈ R / Λ (cid:9) , ( e ) 1 anti-generation . (4.28)The (cid:28)xed tori for se tor θ : (cid:8) ( x , x , , , , (cid:12)(cid:12) x , x ∈ R / Λ (cid:9) , , (4.29) (cid:8) ( x , x , , , , (cid:12)(cid:12) x , x ∈ R / Λ (cid:9) , ( e + e ) 1 generation , (4.30) (cid:8) ( x , x , / , , , / (cid:12)(cid:12) x , x ∈ R / Λ (cid:9) , ( e ) 1 generation , (4.31) { ( x , x , / , , , − / (cid:12)(cid:12) x, y ∈ R / Λ (cid:9) , ( e ) 1 anti-generation . (4.32)The (cid:28)xed tori for se tor θ : (cid:8) (0 , , x , x , , (cid:12)(cid:12) x, y ∈ R / Λ (cid:9) , , (4.33) (cid:8) (0 , , x , x , , (cid:12)(cid:12) x , x ∈ R / Λ (cid:9) , ( e + e ) 1 generation , (4.34) (cid:8) (0 , / , x , x , / , (cid:12)(cid:12) x, y ∈ R / Λ (cid:9) , ( e ) 1 generation , (4.35) (cid:8) (0 , / , x , x , − / , (cid:12)(cid:12) x, y ∈ R / Λ (cid:9) , ( e ) 1 anti-generation . (4.36)Derivation of the (cid:28)xed tori (4.29)-(4.32)Only the al ulation of the (cid:28)xed tori in θ twisted se tor is presented in detail, sin e thetreatment for the other twisted se tors is similar. The mathemati al ondition whi hprovides (cid:28)xed tori in the θ se tor is the following θ T = T + X a i e i , (4.37)where we indi ate the generi torus as T = ( x , x , x , x , x , x ) . Equation (4.37) gives ( x , x , − x , − x , − x , − x ) = ( x , x , x , x , x , x ) + ( a , , , − a , , a , , , a , ,
0) + (0 , a , , , − a , , a , , , a ,
0) + (0 , , a , , , − a )+(0 , , a , , , a ) , (4.38)68r equivalently x = x + a + a x = x + a + a − x = x + a + a − x = x − a + a − x = x − a + a − x = x − a + a . (4.39)The (cid:28)rst two equations restri t some of the oe(cid:30) ients by requiring the equivalen e of(cid:28)xed points (see eq.(4.23)) a + a = a + a = 0 (mod2). We an distinguish several ases whi h give di(cid:27)erent solutions for the x i oordinates • a + a = 0; a + a = 0 , (4.40) • a + a = 2; a + a = 0 , (4.41) • a + a = 0; a + a = 2 , (4.42) • a + a = 2; a + a = 2 . (4.43)If we take the ase (4.40), for example, we would get x = x x = x − x = a + a − x = 2 a − x = 2 a − x = − a + a . Let us onsider initially the ase a = a = a = a =0, whi h implies ( x , x ) = (0 , . Weare left with the equations (cid:26) − x = a + a − x = − a + a (4.44)whi h means looking for all ( x , x ) ∈ [0 , su h that a , a are integers. The possibleoptions are ( x , x ) ∈ { , / , , / } . We note here that / ∼ − / (mod2). It is easyto verify that the omplete set of solutions is given by ( x , x ) = (0 , , (0 , , (1 / , / , (1 , , (1 / , / , (3 / , / , where the underline s ript indi ates any solution obtained by swapping the entries. We an (cid:28)nally olle t the results orresponding to the (cid:28)rst ase analysed and write downthe (cid:28)xed tori T = ( x , x , , , , T = ( x , x , , , , T = ( x , x , , , , T = ( x , x , , , , T = ( x , x , / , , , / T = ( x , x , / , , , / T = ( x , x , / , , , / T = ( x , x , / , , , / . (4.45)Among these solutions we have to sele t only the independent ones, sin e there areidenti(cid:28) ations up to shift latti es: T = T + e , T = T + e ,T = T + e , T = T + e . (4.46)69he total independent (cid:28)xed tori are then T , T , T , T as shown in eqs.(4.29-4.32).If we apply the same pro edure for the other ases in eqs.(4.41(cid:21)4.43) we noti ethat the solutions are redundant, reprodu ing equivalent (cid:28)xed tori. For instan e, it isstraightforward to he k that eq.(4.43) may (cid:28)x the onstants a = a = a = a =1 whi hprovides the solutions in eqs.(4.45). Furthermore, we noti e that for ea h ase in (4.40-4.43) there are several hoi es to (cid:28)x the onstants a i . For example, eq.(4.40) an(cid:28)x a =1, a = − , a =1, a = − . This hoi e provides ( x , x ) = (1 , = (0 ,
0) + e + e , yielding exa tly the same solutions obtained for the hoi e a = a = a = a =0. Ananalogous al ulation has been performed for the twisted se tors θ and θ , whose resultsare shown in eqs.(4.25(cid:21)4.28, 4.33(cid:21)4.36).In the analysis of the twisted se tors, the string states arising at the (cid:28)xed points ingeneral do provide a generation (or anti-generation) of fermions of the Standard Model,after the breaking of the gauge symmetry group into the Standard Model gauge group.For instan e, if the gauge bosons of the model provide an E symmetry, a generationis identi(cid:28)ed by the supermultiplet whi h falls into the representation of E , while a would indi ate the anti-generation (the hoi e of generation/anti-generation w.r.t.the representation is a matter of onvention). We are interested in the net number ofgenerations for our model, thus we need to know what ea h (cid:28)xed torus gives rise to.Let us onsider a (cid:28)xed torus under θ i . If the (cid:28)xed points of this torus under the a tionof θ j , where i = j , are mapped onto points of the same torus, then that torus providesa generation. In the ase where this torus is mapped onto a di(cid:27)erent (cid:28)xed torus in thesame twisted se tor, then these two tori give a generation and an anti-generation. Byapplying this reasoning to ea h (cid:28)xed torus, we get a total number of nine generationsand three anti-generations (hen e a net number of six generations) in the twisted se tor.Cal ulation for the entraliserThe analysis of the ompa ti(cid:28) ation latti e pro eeds with the al ulation of the en-traliser. This information will provide the proje tions under whi h the twisted stateshave to be invariant. For brevity we give the details only for se tor θ , sin e the al ula-tion for the other Z × Z non-trivial elements θ and θ is a straightforward modi(cid:28) ationof the following derivation.The (cid:28)rst step is to (cid:28)nd the onstru ting element for ea h torus, whi h we all now g = ( θ , e inv ) . As mentioned in the introdu tory part, the entraliser is the set of allelements h = ( θ i , P a i e i ) of the orbifold group that ommute with g . This ondition inthe θ se tor is translated by the formula X a i e i − θ ( X a i e i ) = e inv − θ j ( e inv ) , j = 1 , , . (4.47)70he invariant ve tor e inv is determined for ea h torus by the transformation T = ( x , x , , , , θ −→ ( x , x , , , ,
0) + e inv. ( e inv. = 0) ,T = ( x , x , , , , θ −→ ( x , x , , , ,
0) + e inv. ( e inv. = e + e ) ,T = ( x , x , , , ,
12 ) θ −→ ( x , x , , , ,
12 ) + e inv. ( e inv. = e ) ,T = ( x , x , , , , −
12 ) θ −→ ( x , x , , , , −
12 ) + e inv. ( e inv. = e ) , obtaining for ea h of the (cid:28)xed four tori above the respe tive onstru ting elements g = ( θ , , g = ( θ , e + e ) , g = ( θ , e ) , g = ( θ , e ) . Let us see expli itly howwe get the entraliser for the torus T , for instan e, by applying eq.(4.47). X a i e i − θ ( X a i e i ) = e + e − θ j ( e + e ) , j = 1 , , , (4.48)will give the solutions for θ and θ : a = a = 1 ; a = a ; a = a and for the θ theset of solutions : a = a ; a = a . The entraliser is then determined by all possiblelinear ombinations of the previous onstants w.r.t. the orrespondent twisted se tor.The (cid:28)nal result is shown below Z g =( θ ,e + e ) = (cid:8) h = ( θ , e + e ) , h = ( θ , e + e + e + e ) ,h = ( θ , e + e + e + e ) , h = ( θ , e + e + e + e + e + e ) ,h = ( θ , e + e ) , h = ( θ , e + e + e + e ) ,h = ( θ , e + e + e + e ) , h = ( θ , e + e + e + e + e + e ) ,h = ( θ , e + e ) , h = ( θ , e + e ) ,h = ( θ , e + e + , e + e ) (cid:9) . (4.49)None of these elements indu e a proje tion on the states from the T torus be auseof the onsisten y onditions in se tion 4.3.1. For the trivial torus T there are obvi-ously no proje tions at all indu ed by the Wilson lines and this ondition implies thatthe transformation laws of the massless states of T (and T for the same reason) aredetermined under θ only. By analysing the entralisers of T and T we see that thetransformations under the θ se tor are not de(cid:28)ned, meaning that it is impossible to reate invariant states by tensoring with the twisted right movers in θ se tor (in fa twe will show later on that these transform as e ± iπ under θ ). This parti ular resultdepends ompletely on the hoi e of the ompa ti(cid:28) ation latti e.Fixed pointsA di(cid:27)erent way to al ulate the net number of generations for a given model is to (cid:28)nd theEuler number χ of the orbifold under investigation. In ase of the standard embedding, χ gives the number of generations multiplied by 2. We are now interested to he k the71alidity of our previous result by determining χ . The Euler number is provided by theformula χ = 1 | G | X [ θ i ,θ j ]=0 χ θ i ,θ j , i, j = 1 , , , (4.50)where | G | is the order of the orbifold group (in this ase 2) with elements θ i , θ j and χ θ i ,θ j is the number of points whi h are simultaneously (cid:28)xed under the a tion of θ i and θ j . Again we de ide to onsider only the θ twisted se tor where ea h (cid:28)xed torus willprovide ertain (cid:28)xed points under the a tion of θ and θ . The ondition ( T − θ T ) = (2 x , x , , , , is satis(cid:28)ed by the four points (0 , , , , , , (1 , , , , , , (0 , , , , , , (1 , , , , , . (4.51)For the (cid:28)xed torus T ( T − θ T ) = (2 x , x , , , , , whi h is satis(cid:28)ed by the four points (0 , , , , , , (1 , , , , , , (0 , , , , , , (1 , , , , , . (4.52)Finally, ( T − θ T ) = (2 x , x , , , , has no solutions, su h as the torus T . The solutions (4.51(cid:21)4.52) are invariant under θ ,obviously invariant under θ (sin e we are investigating the (cid:28)xed tori under θ se tor).Therefore, invarian e under θ is guaranteed. We have identi(cid:28)ed eight (cid:28)xed points of θ se tor under all the three twisted se tors so χ θ ,θ = 8 . In the same way we (cid:28)nd theother ontributions χ θ ,θ and χ θ ,θ whi h will totally give χ = χ θ ,θ + χ θ ,θ + χ θ ,θ · . (4.53)The number of generations is then N = χ/ . This on(cid:28)rms our previous result onthe net number of generations.4.3.2 Introdu tion of Wilson linesA Wilson line is a va uum expe tation value for an internal gauge (cid:28)eld omponent A i , where the index labels the dire tion along the latti e ve tors (4.21). As we havementioned already, the maximum number of independent Wilson lines depends on the ompa ti(cid:28) ation latti e and for the SO (4) skew ase we get only three possible inde-pendent Wilson lines that an be added. The orbifold a tion on the ve tors generatingthe SO (4) latti e (see table 4.24) provides the onsisten y ondition for these Wilsonlines A i , A + A , A + A , A + A ∈ Λ E × E , i = 1 , . . . , . (4.54)72e note that the (cid:28)rst ondition holds for any Wilson lines, for any latti e.The e(cid:27)e ts of Wilson lines in the orbifold onstru tion are threefold. First, themodular invariant onditions are more restri ted for the hoi e of the embedding V I and new onstraints are introdu ed. Se ondly, in the untwisted se tor they introdu enew proje tions, breaking the gauge group. Finally, in the twisted se tors, the masslessequations hange with respe t to ea h (cid:28)xed point and this provides di(cid:27)erent left statesfrom the ase with no Wilson lines. Moreover, the transformation laws of these states hange, a ordingly to the formula | p + kV + n a A a > L → e πi ( p + kV + n a A a ) · ( lV + m a A a ) | p + kV + n a A a > L , (4.55)where the (cid:28)xed point onsidered here is given by the onstru ting element ( θ k , n a e a ) andthe proje tion is performed under the elements of the entraliser h = ( θ l , m a e a ) . The laststep in the derivation of the spe trum is tensoring left-moving and right-moving statesto obtain invariant obje ts under the full spa e group. The modi(cid:28) ation introdu ed bythe Wilson lines is that now the states have to be invariant under the entraliser, whi his a subset of S . The parti ular hoi e of our ompa ti(cid:28) ation latti e does not allow usto redu e the total number of generations with the introdu tion of Wilson lines, as weexplain in detail at the end of the hapter. The other interesting impli ation due to thepresen e of Wilson lines is the breaking of the symmetry group and we will show howthis is realised in a parti ular ase. In [108, 111℄ the visible gauge group has been brokeninto SO (10) or SU (5) or into the Standard Model gauge group SU (3) × SU (2) × U (1) plus additional U (1) s. The ni e breaking pattern is not enough to get semi-realisti orbifolds, sin e in fa t in the previous examples many phenomenologi al requirements ould not be implemented.In this se tion we show how the breaking of the hidden E ′ → SO (8) ′ × SO (8) ′ isrealised, in order to explain some te hni al details regarding this sort of al ulation.Few remarks on the hoi e of Wilson lines are listed below. • We note that Wilson lines with entries ∈ { , ± / , ± } break the initial gaugesymmetry to SO (2 n ) subgroups, while entries ∼ ± / produ e SU ( n ) algebras. • If we want to break only the hidden (observable) se tor, the Wilson lines have tohave only non-zero entries in the se ond ((cid:28)rst) 8 dimensional ve tor. • Wilson lines ontaining a single entry equal to proje t the spinorial roots (4.61)or (4.63) in the untwisted se tor by the proje tion ondition p I · A I = 0 (mod 1). • The modular invariant onditions in eqs.(4.14) have to hold for any hoi e ofWilson lines.Keeping in mind the previous observations, we pro eed by introdu ing the following73ilson lines A = A = (cid:0) (cid:1) (cid:18) (cid:19) , , , , ! ,A = A = (cid:0) (cid:1) (cid:0) , , (cid:1) ,A = A = (cid:0) (cid:1) , , , , − , (cid:18) (cid:19) ! . (4.56)It is easy to verify their modular invarian e: V , · A α = 0 (mod 1); A α · A β = 0 (mod 1), α = β ; A α = 0 (mod 2), α, β = 1 , .. .Ea h SO (8) fa tor has rank four, thus the total initial rank is not redu ed. We knowhow many roots to expe t for the algebra of ea h SO (8) by using the relation D SO (8) − R SO (8) = T.R. SO (8) → − , where D is the dimension of the group, R its rank and T.R. the number of total rootweights. By applying the proje tions indu ed by the Wilson lines on the initial roots of E ′ , in eqs.(4.62) and (4.63), only the following roots survive p I = (0 )( ± , ± , , , , , , , p I = (0 )(0 , , , , ± , ± , , , (4.57)providing in fa t the algebra of a SO (8) × SO (8) group.4.3.3 Massless Spe trumThe massless spe trum of the model is produ ed by the solutions of the eqs.(4.16) and(4.19) in the untwisted and in the twisted se tors respe tively. An invariant solutionis obtained by tensoring right- and left-moving solutions whi h survive the orbifoldproje tions.Untwisted spe trumThe untwisted massless spe trum is derived by solving equation (4.16). Subsequently,we have to look at the invariant states under the a tion of the orbifold group of Z × Z ,where a generi element is indi ated by G = ( θ i , n a e a ; V Ii , n a A Ia ) . We write expli itlythe de(cid:28)nitions of the os illator number operators N and ˜ N in the Neveu S hwarz (NS)and in the Ramond (R) se tor. We remind that the right se tor is supersymmetri whilethe left one only ontains bosoni os illators. ˜ N NS = ∞ X n =1 ˜ α i − n ˜ α in + ∞ X n =1 / n ˜ b i − n ˜ b in − , ˜ N R = ∞ X n =1 ˜ α i − n ˜ α in + ∞ X n =0 n ˜ d i − n ˜ d in ,N = ∞ X n =1 α i − n α in + α I − n α In − , ˜ d i the fermioni os illators in the Ramond se tor and in luded thevalues of a L,R .The total set of bosoni and fermioni os illators whi h an transform under theorbifold a tion is given below in the light- one gauge. For brevity we drop the tilde onthe right os illators and di(cid:27)erentiate the bosoni left and right os illators with the label
L, R when needed. Moreover, the omplex onjugate os illators are indi ated by a bar. α µn , α in , α in , b µ − n , b i − n , b i − n , d µn , d i − n , d i − n . It is onvenient to use here the omplex notation Z i for the bosoni and Ψ i for thefermioni oordinates in the ompa t dimensions, i = 1 , , . As anti ipated before, thetransformation properties for these os illators in the ompa t dimensions are Z i → e πiv i Z i , Ψ i → e πiv i Ψ i . If we onsider only the massless ontributions, we obtain the terms b µ − / , b i − / , b i − / , d µ , d i , d i , α µ − , α i − , α i − . The right moving solutions are obtained from the massless equation (4.16). The orre-sponden e between SO (8) weight roots and os illators is given in the table below, wherethe transformation laws under θ and θ are also provided.Right Os illator Weight θ θ b µ =1 , − / ( ± , , ,
0) 1 1 b i =1 − / (0 , , , e iπ b i =2 − / (0 , , , e − iπ e iπ b i +3 − / (0 , , ,
1) 1 e − iπ d µ =1 , ± (1 / , / , / , /
2) 1 1 d i =10 (1 / , − / , / , / e − iπ d i =20 (1 / , / , − / , / e iπ e − iπ d i =30 (1 / , / , / , − /
2) 1 e iπ (4.58)The phases of α µ − , α i − , α i − in the right and in the left se tor are analogousto b µ − / , b i − / , b i − / . The os illators of the gauge degrees of freedom α I =1 − are in-variant under the a tion of the twists. The orrespondent omplex os illators transformobviously with opposite phases. In the left se tor the solutions of the massless equation an be os illators and momenta p I , roots of E × E ′ latti e. The orbifold proje tion forthe p I is given by G ( p ) = e iπp I .V I = 1 . (4.59)Its solutions give rise to the gauge bosons whi h des ribe the symmetry of the theory.Solutions p I whi h pi k a phase under the previous proje tion an still survive the75otal proje tion of the orbifold when they are tensored with non invariant right states,transforming with opposite phase w.r.t. the left ontribution. These are the hargedmatter states. We show now how the proje tions produ e the bosons of the unbrokengauge group. The roots of the E × E ′ latti e are of the form • p I = ( ± , ± , , , , , , (4.60) • p I = ( ± / , ± / , ± / , ± / , ± / , ± / , ± / , ± / (4.61) • p I = (0) ( ± , ± , , , , , , (4.62) • p I = (0) ( ± / , ± / , ± / , ± / , ± / , ± / , ± / , ± / (4.63)The roots (4.60(cid:21)4.61) produ e the observable se tor while the ve tors (4.62(cid:21)4.63) givethe hidden se tor. In the standard embedding the shift ve tors V and V are V = (1 / , − / , (0) )(0) , V = (0 , / , − / , (0) )(0) , hen e it is straightforward that no roots are proje ted out for the hidden se tor, whenapplying ondition (4.59). The surviving roots from the observable se tor are instead p I = (0 , , , ± , ± , , , ,p I = ((1 / ( ± / )(0) ( even − ) , p I = (( − / ( ± / )(0) ( odd − ) . Finally, we have obtained invariant roots for the hidden E and invariant roots forthe observable se tor. The last ones represent the weight ve tors of the ex eptional Liegroup E , as it is derived by the analysis of the simple roots [115℄. From the ompleteset of roots at hand, only fourteen are simple roots, orresponding to a rank algebra.However, the rank of the gauge group is not redu ed, meaning that there are twoadditional U (1) symmetries. The (cid:28)nal gauge group is E × U (1) × E ′ . In the table below we list all the invariant momenta and the matter states with theirtransformation laws. P I θ θ (0 , , , ± , ± , , , / , ( ± / )(0) , ( even − ) 1 1(( − / , ( ± / )(0) , ( odd − ) 1 1( a )( ± , , , ± , , , , e ± iπ b )(0 , ± , , ± , , , , e ± iπ e ± iπ ( c )(0 , , ± , ± , , , , e ± iπ ( a )(0 , ± , ± , (0) )(0) e ± iπ b )( ± , , ± , (0) )(0) e ± iπ e ± iπ ( c )( ± , ± , (0) )(0) e ± iπ ( a )( ± (1 / , / , − / , ( ± (1 / )(0) * e ± iπ ( b )( ± (1 / , − / , / , ( ± (1 / )(0) * e ± iπ e ± iπ ( c )( ± ( − / , / , / , ( ± (1 / )(0) * e ± iπ ∗ indi ates that we have to onsider an odd number of (cid:16) − (cid:17) for the last (cid:28)ve entriesif the (cid:28)rst three entries have a (cid:16) + (cid:17) sign in front, in the other ase we take an evennumber of − / entries. The untwisted massless spe trum is summarised below.Right mover Left mover Parti le ( ± , , , ⊗ ( ± , , , G µν , B µν , φ ± (1 / , / , / , / ⊗ ( ± , , ,
0) Ψ αµ + h. . ( ± , , , ⊗ (0 , , , ± , ± , , , ⊗ (( − / , ( ± / )(0) A µ ⊗ ((+1 / , ( ± / )(0) ± (1 / , / , / , / ⊗ (0 , , , ± , ± , , , ⊗ (( − / , ( ± / )(0) λ α ⊗ ((+1 / , ( ± / )(0) b − / , d ⊗ ˜ α − b − / , d ⊗ ˜ α − b − / , d ⊗ ˜ α − b − / , d ⊗ ( a ) b − / , d ⊗ ( b ) b − / , d ⊗ ( c ) In the table above we have used an analogous notation for left ompa t os illators,see table 4.58, where, as usual, the (cid:28)rst entry of the ve tor orresponds to the om-plexi(cid:28)ed transverse spa etime dimension, while the last three are the omplex ompa tdimensions. The (cid:28)rst set of states provides the supergravity multiplet and the superYang Mills multiplet, the se ond gives rise to the moduli and the last provides 4 3 (27 , ∈ E × E ′ as the solutions are given in three ombinations a , b and c .Twisted se torsWe derive the spe trum for one twisted se tor only sin e the analysis is analogous inthe other ases. For instan e, we solve the massless equations for the (cid:28)xed tori of θ N = 1 supersymmetry in fourdimensions. 77e tor T : m L p + V ) + N − , (4.64) T : m L p + V + A + A ) + N − , (4.65) T : m L p + V + A ) + N − , (4.66) T : m L p + V + A ) + N − , (4.67)while the right massless equation does not hange for the di(cid:27)erent tori and it has beenpresented in eq.(4.19). As we said, for the Z × Z orbifold δ c = / , thus the twistedright movers have to be solutions of ( q + v ) = 1 / . These solutions are showed in thetable below with their transformation propertiesRight movers θ θ q ,sh = (0 , , − / , − / e iπ q ,sh = (0 , , / , / e − iπ q ,sh = ( − / , / , , e iπ q ,sh = (1 / , − / , , e − iπ where ¯ q ,sh and ¯ q ,sh orrespond to the Ramond shifted os illators. We only onsiderthe q ,sh solution tensored with the left twisted states sin e the q ,sh is exa tly the right ontribution of the orrespondent antiparti les. In fa t, we will (cid:28)nd a ertain number ofleft states, solutions of the left massless equation, whi h transform with opposite phaseof q ,sh , providing invariant states. At the same time the same number of left movers ispresent in the massless spe trum with opposite transformation phases, giving invariantsif ombined with the q ,sh . Only one set of these solutions has to be onsidered, asanti ipated before. Moreover, we note that the spe trum is supersymmetri sin e anyleft solution tensored with q ,sh , for instan e, and providing an invariant state, is alsoautomati ally invariant when multiplied by the Ramond right ¯ q ,sh .For ompleteness, we provide the right os illators with their transformations in thetwisted se tor θ α in → α n ; α n ; α n − / ; α n +1 / ; α n − / ; α n +1 / ; (4.68) Ψ n + ρ → Ψ n + ρ ; Ψ n − ρ ; Ψ n − / ρ ; Ψ n +1 / − ρ ; Ψ n − / ρ ; Ψ n +1 / − ρ ; (4.69)where ρ = 0 in the Ramond ase ( d i os illators) and ρ = 1 / in the NS ( b i os illators)respe tively. For the left ompa t os illators we have analogous expressions to eqs.(4.68),while the os illators for the gauge degrees of freedom do not transform under the twists.78e solve eq.(4.64) to obtain the ontribution to the massless spe trum from thetrivial torus of θ se tor. We distinguish two ases, when N = 0 and when N = 1 / . Aremark is to be done at this point. When looking for the p Ishift , not only we onsider theroots (4.60), (4.61), (4.62) and (4.63), but also all their linear ombinations, as long asthey still satisfy the massless equation. Keeping this in mind, we obtain the followingresults: for N = 0 , 56 p Ishift are found, half of whi h take a phase e iπ while the otherstransform with opposite phases (as explained before, only one set of these solutions is onsidered); if N = 1 / , only one p Ishift satis(cid:28)es the massless equation. In total thetrivial torus provides the states in the table below.Os illators P Ishift
Right os illator number of solutions N L = 0 ( ± , − / , − / , )(0 ) q ,sh (0 , / , / , ± , )(0 ) q ,sh ( − / , , , ( ± / )(0) even q ,sh N L = 1 / : α L (0 , / , − / , )(0 ) q ,sh N L = 1 / : α L (0 , − / , / , )(0 ) q ,sh E and SO (8) ′ × SO (8) ′ . Ea h multiplet is identi(cid:28)ed by groupingthe states with same U (1) harges. If we indi ate with α i , i = 1 , .. , the simple rootsof E , given in (C.1) in Appendix C, and with α j , j = 9 , .. and α k , k = 13 , .. , thesimple roots of the two SO (8) gauge groups (we are not interested here in lassifyingthe states under the hidden gauge group, sin e the potential standard model parti lesare singlets under it), then for every root we need to al ulate p IDL E = ( α · p I , α · p I , α · p I , α · p I , α · p I , α · p I , ) Q ,Q ,p IDL SO (8) ′ = ( α · p I , ..., α · p I ) , p IDL SO (8) ′ = ( α · p I , ..., α · p I ) , (4.70)where the Q and Q harges are obtained by Q = H − H and Q = H + H − H .This pro edure is shown in Table .1 in Appendix C.We provide below the (cid:28)nal result for the ontribution of the massless states for thetrivial (cid:28)xed torus T , where the notation indi ates the representation of the multipletsunder the gauge group E × SO (8) ′ × SO (8) ′ and the apex gives the Q , harges: ˜ N = 12 (1 , , − , , (1 , , , − , ˜ N = 0 (1 , , , − , ( , , − , − . (4.71)This torus provides a generation under the E gauge group. By performing the same al ulation for the other (cid:28)xed tori of θ we (cid:28)nd out that the T torus provides exa tlythe same ontent of T , and this is due to the property A + A ∈ Λ × . If no Wilsonlines are introdu ed in our model, we expe t eqs.(4.66) and (4.67) to redu e to (4.64),79roviding a generation and an anti-generation, plus a ertain number of singlet states.Then, the total ontribution from T and T to the net number of generations is zero.When we swit h on the Wilson line A we automati ally get a huge hange in botheqs.(4.66) and (4.67), giving obviously the same ontribution. In this ase the hoi eof Wilson lines (4.56) only produ es hidden harged states, proje ting the generationsunder the observable gauge group.To on lude, the parti ular hoi e of our ompa ti(cid:28) ation latti e redu es the numberof (cid:28)xed tori to four par ea h twisted se tor, providing a total number of nine generations(from the (cid:28)xed tori (4.25),(4.26), (4.27), (4.29), (4.30), (4.31), (4.33), (4.34), (4.35))and three anti-generations (from the (cid:28)xed tori (4.28), (4.32), (4.36)). We showed that,independently on the hoi e of Wilson lines, there is no way to proje t out any of thesegenerations. This is in fa t a limitation of the SO (4) latti e.80hapter 5Constru tion of partition fun tionsin heteroti E × E modelsIn this hapter we dis uss some examples of heteroti superstring models ompa ti(cid:28)edon shift orbifolds. In parti ular the ases presented are four dimensional shift orbifoldson whi h a Z or a Z × Z proje tion a ts on the internal tori.As we explained in hapter 4, in the standard orbifold ompa ti(cid:28) ation the string oordinates are identi(cid:28)ed under internal inversion operations, for instan e the Z gen-erators orrespond to π rotations. The shift orbifolds are instead reated by the a tionof dis rete shifts on the basis ve tors of the ompa ti(cid:28) ation latti e. The result of thisoperation an lead to the implementation of the S herk-S hwarz me hanism for thespontaneous supersymmetry breaking. In quantum (cid:28)eld theory the same me hanismis obtained by shifts on the internal Kaluza Klein momenta [10, 116℄, while in stringtheory a more general pro edure is given when introdu ing momentum or winding shiftsalong the ompa t dire tions [117, 118, 119℄, while preserving modular invarian e. Thedi(cid:27)erent hoi e for the two types of the shift will produ e the so- alled S herk-S hwarzbreaking or the M-theory breaking [120, 121℄.In this thesis we will onsider the simple ase of a one-dimensional momentum shift-orbifold with Z or Z × Z a tion.5.1 Shift orbifoldIn this se tion we are interested in looking at a simple example of shift orbifold realisedin heteroti models. Thus, we start from the partition fun tion of the E × E heteroti string in 10 dimensions. Z + E × E = ( V − S )( O + S )( O + S ) . (5.1)The next step is to ompa tify on a fa torisable six torus of the form T × T × T andintrodu e the shift in one ompa t dimension x δ : x → x + πR, δ = 1 . (5.2)81he shift orbifold is generated by the elements (1 , ǫ ( − F ξ δ, ǫ ( − F ξ δ, ǫ ǫ ( − F ξ + F ξ ) = (1 , ǫ a, ǫ b, ǫ ǫ ab ) , where F ξ is an internal fermion number in the se tor des ribing the (cid:28)rst E gaugegroup and F ξ is an internal fermion number in the se tor des ribing the se ond E gauge group. The parameters ǫ , ∈ {± } lead to di(cid:27)erent models. In this se tion we onsider the ase with the group elements (1 , a, b, ab ) , where ǫ , = 1 , and show in detailthe derivation of the resulting partition fun tion.An other interesting ase is when the group elements are given by (1 , − a, b, − ab ) ,obtained when ǫ = − and ǫ = 1 , and show in detail the derivation of the resultingpartition fun tion. This result will be presented brie(cid:29)y in se tion 5.3.Let us note (cid:28)rst that the a tion of the previously introdu ed operators on the latti eand on the SO (2 n ) hara ters is given by δ : Λ m,n → ( − m Λ m,n ( − F ξi : ( O /V ) i → ( O /V ) i ( S /C ) i → ( − S / − C ) i , ( i = 1 , . (5.3)Now let us introdu e the proje tion operator ∓ ( − F ξ δ × − F ξ δ { ∓ ( − F ξ δ + ( − F ξ δ ∓ ( − F ξ + F ξ } , (5.4)where the sign + refers to the (cid:28)rst ase , and the − refers to the se ond ase.The partition fun tion in eq.(5.1) after the ompa ti(cid:28) ation on the six-torus be omes Z + E × E = ( V − S )Λ Λ Λ m ′ ,n ′ Λ m,n ( O + S )( O + S ) . (5.5) Λ and Λ are the two latti es for two-dimensional tori, while the third two-torus hasbeen fa torised into two ir les to fa ilitate the implementation of the shift. The fullpartition fun tion whi h is obtained from eq.(5.5) and is invariant under the orbifoldgroup (5.4) is given by Z tot = Z + X i Z i + X i ( Z i + Z ii ) + c X i = j Z ij , (5.6)where i, j ∈ { a, b, ab } and the onstant c , alled the dis rete torsion, multiplies a mod-ular invariant orbit. The (cid:28)rst two terms in eq.(5.6) orrespond to the total ontributionof the untwisted se tor of the orbifold and are given by Z = Z , + Z ,ab + Z ,a + Z ,b == 14 ( V − S )Λ Λ Λ m ′ ,n ′ Λ m,n [( O + S )( O + S ) + ( O − S )( O − S )+( − m { ( O − S )( O + S ) + ( O + S )( O − S ) } ] . (5.7)82he term Z ,ab is obtained by a ting on the eq.(5.5) with the operator ab , and the thirdand the forth ontributions ( Z ,a , Z ,b ) are respe tively given by a ting with operators a and b on the eq.(5.5). In a similar way the last two terms in (5.6) orrespond to thetwisted se tor whi h ontributions have to be al ulated. In our model we hoose thevalue of c to be +1 . We an rewrite the untwisted se tor (5.7) as Z = 12 ( V − S )Λ Λ Λ m ′ ,n ′ Λ m,n [( O O + S S ) + ( − m ( O O − S S )] , (5.8)that an be rearranged, by using the formula (A.12) in Appendix A, into the form Z = ( V − S )Λ Λ Λ m ′ ,n ′ [Λ m,n ( O O ) + Λ m +1 ,n ( S S )] . (5.9)The derivation of the twisted se tor, negle ting for the moment the torsion ontribution,is given by the a tion of T and S transformations of ea h term in (5.8). We illustrate thepro edure with a s hemati pi ture below. These terms are given in (D.1) in AppendixD. T invariant z }| { Z ,ab S ←→ Z ab, T ←→ S invariant z }| { Z ab,ab T invariant z}|{ Z ,a S ←→ Z a, T ←→ S invariant z}|{ Z a,a T invariant z}|{ Z ,b S ←→ Z b, T ←→ S invariant z}|{ Z b,b Z a,b T ←→ Z a,ab S ←→ Z ab,a l S l T Z b,a T ←→ Z b,ba S ←→ Z ba,b (5.10)We note that the al ulation of the terms whi h ontribute to the torsion is more subtlesin e we have to de(cid:28)ne the way the proje tions a t in a twisted se tor, while preservingmodular invarian e. If, for instan e, we take the element Z a, , its a proje tion wouldprovide a di(cid:27)erent result w.r.t. the element Z a,a , obtained by a T transformation of Z a, . This means that we have to reprodu e the same pattern of a tion when theproje tor b a ts onto Z a, . The b operator ontains the shift δ whi h, in the twistedse tor a produ es a hange on the latti e equal to ( − m Λ m,n +1 / . The (cid:28)rst group ofgauge hara ters, whi h transforms a ordingly to a T transform for the Z a, element,in the b proje tion is untou hed, while b a ts on the se ond set of hara ters in the usualway, as if we are onsidering an untwisted element. The formula below summarises thispro edure Z a,b = (cid:2) ( − F ξ δ (cid:3) a { ( V − S )Λ Λ Λ m ′ ,n ′ Λ m,n +1 / [( V + C )( O + S )] } = ( V − S )( − m Λ Λ Λ m ′ ,n ′ Λ m,n +1 / [( V + C )( O − S )] . (5.11)83t this point the remaining ontributions are simply derived by an S and T transfor-mation hain Z a,b T −→ Z a,ab S −→ Z ab,a T −→ Z ab,b S −→ Z b,ab T −→ Z b,a . (5.12)These expressions omplete the list of terms to get the full twisted se tors.ab twisted se tor Z ab, + Z ab,ab + Z ab,a + Z ab,b = 14 ( V − S )Λ Λ Λ m ′ ,n ′ Λ m,n [( V + C )( V + C ) + ( V − C )( V − C )] + c ( − m [( − V + C )( V + C ) + ( V + C )( − V + C )] . (5.13)a twisted se tor Z a, + Z a,a + Z a,b + Z a,ab = 14 ( V − S )Λ Λ Λ m ′ ,n ′ Λ m,n +1 / [( V + C )( O + S ) + ( − m ( − V + C )( O + S )] + c [( − m ( V + C )( O − S ) + ( − V + C )( O − S )] . (5.14)b twisted se tor Z b, + Z b,b + Z b,a + Z b,ab = 14 ( V − S )Λ Λ Λ m ′ ,n ′ Λ m,n +1 / [( O + S )( V + C ) + ( − m ( O + S )( − V + C )] + c [( − m ( O − S )( V + C ) + ( O − S )( − V + C )] . (5.15)The S and T transformations used to derive the previous terms are given in Appendix(A.9). Putting these results into (5.6) we (cid:28)nally obtain Z − = ( V − S )Λ Λ Λ m ′ ,n ′ (cid:2) Λ m,n ( O O + C C ) + Λ m +1 ,n ( S S + V V )+ Λ m,n + ( O C + C O ) + Λ m +1 ,n + ( V S + S V ) (cid:3) . (5.16)At this level, the model presents N = 4 supersymmetry in four dimensions and a SO (16) × SO (16) gauge group. This model ontains gravity and Yang Mills (cid:28)elds asits massless ex itations. In the next se tion we examine the Z orbifold of (5.16) anddis uss its massless spe trum. 84.2 Partition fun tion of the heteroti E × E shift orbifoldsuperstring with Z a tionIn this se tion we onsider the Z orbifold of the partition fun tion (5.16). The modelobtained by this further a tion has N = 2 supersymmetry in four dimensions and SO (4) × SO (12) × SO (16) gauge symmetry.The Z is generated by the elements (1 , h ) where h a ts on the ( omplex) oordinatesof the internal fa torised torus T = T × T × T as Z → e iπ Z , Z → e iπ Z , Z → Z . We onsider the standard embedding, thus the element h a ts non-trivially on the gaugedegrees of freedom of the heteroti string as well. For this reason it is onvenient tode ompose the SO (2 n ) hara ters in su h a way to keep O , V , S and C fa tors (onwhi h the element h a ts non-trivially) expli it. The new partition fun tion reads like Z T ot = Z + Z h + Z h + Z hh , (5.17)where Z is the untwisted term with no proje tion that orresponds exa tly to (5.16).The following term Z h is obtained by a ting with h onto the previous, while an Stransformation produ es the third term whi h, after a T transformation, provides Z hh .If we de ompose the hara ters by applying formula (A.8), then the (cid:28)rst term in (5.17)be omes Z = 14 (cid:2) V O + O V − S S − C C (cid:3) Λ Λ Λ m ′ ,n ′ × (cid:2) (Λ m,n + Λ m,n + )( O O + V V + C S + S C )( O + C )+(Λ m,n − Λ m,n + )( O O + V V − C S − S C )( O − C )+(Λ m +1 ,n + Λ m +1 ,n + )( V O + O V + S S + C C )( V + S )+(Λ m +1 ,n − Λ m,n + )( V O + O V − C C − S S )( V − S ) (cid:3) . (5.18)The a tion of the twist, imposed by the Z a tion on the hara ters, has to be onsistentwith worldsheet supersymmetry [47, 122, 100, 101℄ and an be shown expli itly byapplying the properties of the θ -fun tions into the de(cid:28)nitions of the hara ters in (A.5).These properties hold for the spa etime degrees of freedom O → O , V → − V ,S → − S , C → C , (5.19)85nd for the gauge degrees of freedom as well, O = O O + V V → O O − V V ,V = V O + O V → − V O + O V ,S = S S + C C → − S S + C C ,C = S C + C S → − S C + C S , (5.20)where we have used the SO (4) × SO (12) de omposition of the SO (16) hara ters. We(cid:28)nally get Z h = 14 (cid:2) − V O + O V + S S − C C (cid:3) Λ m ′ ,n ′ | ηθ | × (cid:8) (Λ m,n + Λ m,n + )( O O − V V + C S − S C )( O + C )+(Λ m,n − Λ m,n + )( O O − V V − C S + S C )( O − C )+(Λ m +1 ,n + Λ m +1 ,n + )( − V O + O V − S S + C C )( V + S )+(Λ m +1 ,n − Λ m,n + )( − V O + O V − C C + S S )( V − S ) (cid:9) . (5.21)The twisted se tor is obtained by performing S and T transformations on ea h termof the previous expression. In parti ular the S transformation of Z h gives Z h whilethe T transform of the last one provides the Z hh . It is indi ative at this point to showexpli itly the pro edure for at least the (cid:28)rst ontribution of (5.21).From se tion A.1.2 in Appendix A we get the following S transformation laws: ( O + V ) , , → ( O + V ) , , , ( O − V ) , , → ( S + C ) , , , ( S − C ) , → ( − S + C ) , , ( S − C ) → ( S − C ) , ( O + C ) → ( O + C ) , (5.22)where the indi es refer to the hara ters of SO (4) , SO (12) and SO (16) respe tively,and provides ( − V O + O V + S S − C C ) = 12 [ − ( O + V ) , ( O − V ) , + ( O − V ) , ( O + V ) , +( S + C ) , ( S − C ) , + ( S − C ) , ( S + C ) , ] S −→
12 [ − ( O + V )( S + C ) + ( S + C )( O + V ) − ( O − V )( S − C ) − ( S − C )( O − V )] , = ( − OS − V C + SV + CO ) , . (5.23)In se tion (5.2.1) we will show the transformation laws of the bosoni ontributionsand in Appendix A their modular transformations are presented. By applying thoseexpressions we an write Λ m ′ ,n ′ | ηθ | × (Λ m,n + Λ m,n + ) S −→ Λ m ′ ,n ′ | ηθ | × (cid:0) Λ m,n + Λ m,n + (cid:1) . ( O O − V V + C S − S C ) = 12 [( O + V )( O − V ) + ( O − V )( O + V ) − ( C + S )( C − S ) + ( C − S )( C + S )] , S −→
12 [( O + V )( S + C ) + ( S + C )( O + V )+( O − V )( C − S ) − ( C − S )( O − V )] , = ( OC + V S + CV + SO ) , , (5.24)where the notation has been explained before.We now apply the same pro edure to all the other terms ontained in Z h , obtainingthe expression Z h = 14 (cid:2) − V C + S V − O S + C O (cid:3) Λ m ′ ,n ′ × | ηθ | × (cid:8) (Λ m,n + Λ m,n + )( V S + S O + O C + C V )( O + C )+(Λ m +1 ,n + Λ m +1 ,n + )( O S + S V + V C + C O )( V + S )+(Λ m,n − Λ m,n + )( − V S + S O − O C + C V )( O − C )+(Λ m +1 ,n − Λ m +1 ,n + )( − O S + S V − V C + C O )( V − S ) (cid:9) . (5.25)The al ulation whi h provides Z hh onsists in applying the T transformation for ea hterm in the above result. As the pro edure is analogous for every ontribution, we showonly the T a tion on the (cid:28)rst term, whi h we reprodu e here again ( − V C + S V − O S + C O )Λ m ′ ,n ′ × | ηθ | × (Λ m,n + Λ m,n + )( V S + S O + O C + C V )( O + C ) . (5.26)In the formulae below we fa torise a global sign obtained from the phase-prefa tor ofthe T transformation, given in (A.9) T SO (4) = e − iπ/ diag (1 , − , i, i ) ,T SO (12) = e − iπ/ diag (1 , − , − i, − i ) ,T SO (16) = e − iπ/ diag (1 , − , , ,X X X T −→ e iπ/ e − πi/ X X X = − X X X . (5.27)The spa etime fa tors transform as − V C + S V − O S + C O → i ( − V C + S V + O S − C O ) , while the bosoni ontribution is given by the expression below (here we are omittingthe transverse bosons) | ηθ | × (Λ m,n + Λ m,n + ) → | ηθ | × (Λ m,n + Λ m,n + ) . V S + S O + O C + C V → − i ( − V S + O C − S O + C V ) . (5.28)We observe that the term ( O + C ) → ( O + C ) remains invariant also under T. The ombination of these results leads to the (cid:28)nal expression Z hh = 14 (cid:2) − V C + S V + O S − C O (cid:3) Λ m ′ ,n ′ × | ηθ | × (cid:8) (Λ m,n + Λ m,n + )( V S + S O − O C − C V )( O + C )+(Λ m +1 ,n − Λ m +1 ,n + )( − O S − S V + V C + C O )( − V + S )+(Λ m,n − Λ m,n + )( − V S + S O + O C − C V )( O − C )+(Λ m +1 ,n + Λ m +1 ,n + )( O S − S V − V C + C O )( − V − S ) (cid:9) . (5.29)5.2.1 The bosoni ontributionIn this se tion we present some useful details, used already in the onstru tion of the un-twisted and twisted heteroti Z partition fun tion, on erning the bosoni ontributionto ea h of these se tors.Untwisted se tor: | η | × | η | | {z } µ =2 , spa etime × | η | × | η | × | η | × | η | × | η | × | η | | {z } ompa ti(cid:28)ed dimensions , where | η | = 1 η × η . Now, sin e the Z a tion gives η → ηθ when a e iπ a ts on ea h omplex dimension, weobtain Z → η η ; Z h → η θ η θ . Twisted se tor: It is su(cid:30) ient to apply S and T transformations on the previousresult ( 4 η θ ) S −→ ( 4 η θ ) T −→ ( 4 η θ ) , for the right ontribution, reminding that the analogous result holds for the left bosoni part. Combining both se tors, the bosoni ontribution resulting in the twisted ontri-butions is given respe tively by Z h → η θ η θ ; Z hh → η θ η θ .
88t is useful for later purposes to expand in powers of q the right ( and in ¯ q for the left)bosoni ontribution. In (cid:28)rst approximation Z → η ∼ q − / (1 + 8 q + .. ) , Z h → η θ ∼ q − / (1 − q / + .. ) , Z h → η θ ∼ q − / (1 + .. ) , Z hh → η θ ∼ q − / (1 + 4 q / + .. ) . (5.30)5.2.2 Spe trumWe have now all the ingredients to provide the untwisted spe trum of our model, whi his obtained by the following sum Z + Z h ∼ Λ m ′ ,n ′ × h Λ m,n (cid:8)(cid:0) O V − C C (cid:1)(cid:2) ( O O O + C S C ) (cid:3) +( V O − S S (cid:1)(cid:2) ( V V O + S C C ) (cid:3)(cid:9) +Λ m,n + (cid:8)(cid:0) O V − C C (cid:1)(cid:2) ( O O C + C S O ) (cid:3) +( V O − S S (cid:1)(cid:2) ( V V C + S C O ) (cid:3)(cid:9) +Λ m +1 ,n (cid:8)(cid:0) O V − C C (cid:1)(cid:2) ( O V V + C C S ) (cid:3) +( V O − S S (cid:1)(cid:2) ( V O V + S S S ) (cid:3)(cid:9) +Λ m +1 ,n + (cid:8)(cid:0) O V − C C (cid:1)(cid:2) ( O V S + C C V ) (cid:3) +( V O − S S (cid:1)(cid:2) ( V O S + S S V ) (cid:3)(cid:9)i . (5.31)As announ ed previously, we are interested in the massless states, whose expansion isprovided below. The (cid:28)rst two terms give the untwisted right ontributions, the last fourprovide the untwisted left massless terms. O V /C C η ∼ q / q / (1 + 6 q − + .. ) q / (4 q − / + .. ) ∼ q + ..V O /S S η ∼ q + ..O O O η ∼ q − / (1 + 8 q + .. ) q − / (1 + 6 q + .. ) q − / (1 + 66 q + .. ) q − / (1 + 120 q + .. ) ∼ q + 6 q + 66 q + 120 q + ..V V O η ∼ q − / q − / (4 q / + .. ) q − / (4 q / + .. ) q − / (1 + 120 q + .. ) ∼ q + .. (5.32)Summing up, in the untwisted se tor one has N = (1 , , D = 6 SUGRA multiplet andthe Yang(cid:21)Mills multiplet, partially proje ted by the Z a tion, provided by the terms O O O η × O V − C C η , V V O η × V O − S S η . (5.33)89 similar al ulation is performed for the twisted se tor, where we present the fullspe trum Z h + Z hh ∼
162 Λ m ′ ,n ′ × h Λ m,n (cid:8)(cid:2)(cid:0) − V C + S V (cid:1) ( V S C + S O O )+( − O S + C O (cid:1) ( O C C + C V O ) (cid:3) × ( 1 | η | | θ | + 1 | η | | θ | )+ (cid:2)(cid:0) − V C + S V (cid:1) ( O C C + C V O )+( − O S + C O (cid:1) ( V S C + S O O ) (cid:3) × ( 1 | η | | θ | − | η | | θ | ) (cid:9) +Λ m,n + (cid:8)(cid:2)(cid:0) − V C + S V (cid:1) ( V S O + S O C )+( − O S + C O (cid:1) ( O C O + C V C ) (cid:3) × ( 1 | η | | θ | + 1 | η | | θ | )+ (cid:2)(cid:0) − V C + S V (cid:1) ( O C O + C V C )+( − O S + C O (cid:1) ( V S O + S O C ) (cid:3) × ( 1 | η | | θ | − | η | | θ | ) (cid:9) +Λ m +1 ,n (cid:8)(cid:2)(cid:0) − V C + S V (cid:1) ( S V V + V C S )+( − O S + C O (cid:1) ( O S S + C O V ) (cid:3) × ( 1 | η | | θ | + 1 | η | | θ | )+ (cid:2)(cid:0) − V C + S V (cid:1) ( O S S + C O V )+( − O S + C O (cid:1) ( S V V + V C S ) (cid:3) × ( 1 | η | | θ | − | η | | θ | ) (cid:9) +Λ m +1 ,n + (cid:8)(cid:2)(cid:0) − V C + S V (cid:1) ( S V S + V C V )+( − O S + C O (cid:1) ( O S V + C O S ) (cid:3) × ( 1 | η | | θ | + 1 | η | | θ | )+ (cid:2)(cid:0) − V C + S V (cid:1) ( O S V + C O S )+( − O S + C O (cid:1) ( S V S + V C V ) (cid:3) × ( 1 | η | | θ | − | η | | θ | ) (cid:9)i . (5.34)The twisted massless states form a N = 1 , D = 6 half hypermultiplet are provided by C V O η θ × O S − C O η ¯ θ , S O O η θ × O S − C O η ¯ θ . (5.35)90he relevant expansions for the massless ontributions in eq.(5.35) are presented below,for the right and left ontributions respe tively O S /C O η θ ∼ q / q / (1 + 6 q − + .. )2 q − / (1 + 4 q − + .. ) ∼ q + ..V C /S V η θ → massiveC V O η θ ∼ q − / q / (1 + 4 q + .. ) q − / q / q − / (1 + 120 q + .. ) ∼ q + ..S O O η θ ∼ q − / q / q / (1 + 4 q + .. ) q − / (1 + 66 q + .. ) q − / (1 + 120 q + .. ) ∼ q + ... (5.36)5.3 A string model with no gravityIt is interesting to onsider a variation of the previous model, obtained by a the shift orb-ifold with group elements (1 , − a, b, − ab ) , with the hoi e of the torsion onstant c = 1 .The modular invariant string theory derived from this orbifold a tion is hara terisedby the absen e of the graviton in its full spe trum. This result leads us to the possibleinterpretation of a little heteroti string, in onne tion with [123, 124, 125, 126℄. Manyinteresting properties of this kind of model an be investigated by the string thermo-dynami s at nonzero temperature [127, 128, 129, 130℄. The basi idea is to generalisethe partition fun tion by adding the temperature dependen e and obtaining (5.16) and(5.37) as parti ular ases [131℄.We quote the expression of the partition fun tion after the orbifold a tion, whi h isindi ated by Z ′− to distinguish from the partition (5.16). Z ′− = ( V − S )Λ Λ Λ m ′ ,n ′ (cid:2) Λ m,n ( S O − C C ) + Λ m +1 ,n ( O S − V V )+ Λ m,n + ( S C − C O ) + Λ m +1 ,n + ( O V − V S ) (cid:3) . (5.37)We observe that, although (5.37) is a modular invariant string va uum, it does notrespe t the spin-statisti s, sin e the bosoni ontributions to the partition fun tionshould arise with positive terms while the fermioni ontribute with negative terms.This prin iple obviously does not hold for the partition fun tion (5.37).From the expression above one an see that the zero mass spe trum of the model ontains no gravity be ause of the absen e of the term O O in the left se tor. More-over the adjoint representation is missing as well so the gauge group annot be de(cid:28)nedeither. We onsider a Z a tion of the orbifold Z ′− and provide its spe trum. The de-tails of the te hniques used for the Z proje tions have been widely explained in se tion5.2. 91ntwisted se tor Z ′ + Z ′ h ∼ Λ m,n × h Λ m,n (cid:8)(cid:0) O V − C C (cid:1)(cid:2) ( C C O − C S C ) (cid:3) +( V O − S S (cid:1)(cid:2) ( S S O − S C C ) (cid:3)(cid:9) +Λ m,n + (cid:8)(cid:0) O V − C C (cid:1)(cid:2) ( C C C − C S O ) (cid:3) +( V O − S S (cid:1)(cid:2) ( S S C − S C O ) (cid:3)(cid:9) +Λ m +1 ,n (cid:8)(cid:0) O V − C C (cid:1)(cid:2) ( O O V − O V S ) (cid:3) +( V O − S S (cid:1)(cid:2) ( V V V − V O S ) (cid:3)(cid:9) +Λ m +1 ,n + (cid:8)(cid:0) O V − C C (cid:1)(cid:2) ( O O S − O V V ) (cid:3) +( V O − S S (cid:1)(cid:2) ( V V S − V O V ) (cid:3)(cid:9)i . (5.38)The massless untwisted ontributions are given by C C O η × O V − C C η , S S O η × V O − S S η , (5.39)sin e right and left ontributions give C C O η ∼ q , S S O η ∼ q ,O V η ∼ q , V O η ∼ q ,S S η ∼ q , C C η ∼ q . From the expressions above one an read the ontent of the massless spe trum interms of the six-dimensional N = (1 , (whi h gives in D = 4 N = 4 supersymmetryupon the dimensional redu tion to four dimensions) SUSY multiplets. In parti ular onehas massless (1 , multiplets, whose bosoni part ontains one ve tor and four s alar(cid:28)elds.We remind that this string solution is not physi al, sin e the graviton does notappear in the untwisted spe trum. However, it represents a onsistent solution for itsmodular invarian e. Thus, the question arises: what role this solution plays in thestring theory, if any?For ompleteness, we pro eed the al ulation by presenting the twisted se tor.92wisted se tor Z ′ h + Z ′ hh ∼
162 Λ m ′ ,n ′ × h Λ m,n (cid:8)(cid:2)(cid:0) − V C + S V (cid:1) ( V C O − V S C )+( − O S + C O (cid:1) ( O S O − O C C ) (cid:3) × ( 1 | η | | θ | + 1 | η | | θ | )+ (cid:2)(cid:0) − V C + S V (cid:1) ( O S O − O C C )+( − O S + C O (cid:1) ( V C O − V S C ) (cid:3) × ( 1 | η | | θ | − | η | | θ | ) (cid:9) +Λ m,n + (cid:8)(cid:2)(cid:0) − V C + S V (cid:1) ( V C C − V S O )+( − O S + C O (cid:1) ( O S C − O C O ) (cid:3) × ( 1 | η | | θ | + 1 | η | | θ | )+ (cid:2)(cid:0) − V C + S V (cid:1) ( O S C − O C O )+( − O S + C O (cid:1) ( V C C − V S O ) (cid:3) × ( 1 | η | | θ | − | η | | θ | ) (cid:9) +Λ m +1 ,n (cid:8)(cid:2)(cid:0) − V C + S V (cid:1) ( C V V − C O S )+( − O S + C O (cid:1) ( S O V − S V S ) (cid:3) × ( 1 | η | | θ | + 1 | η | | θ | )+ (cid:2)(cid:0) − V C + S V (cid:1) ( S O V − S V S )+( − O S + C O (cid:1) ( C V V − C O S ) (cid:3) × ( 1 | η | | θ | − | η | | θ | ) (cid:9) +Λ m +1 ,n + (cid:8)(cid:2)(cid:0) − V C + S V (cid:1) ( C V S − C O V )+( − O S + C O (cid:1) ( S O S − S V V ) (cid:3) × ( 1 | η | | θ | + 1 | η | | θ | )+ (cid:2)(cid:0) − V C + S V (cid:1) ( S O S − S V V )+( − O S + C O (cid:1) ( C V S − C O V ) (cid:3) × ( 1 | η | | θ | − | η | | θ | ) (cid:9)i . (5.40)The massless twisted ontributions are given by O S O η θ × O S − C O η ¯ θ , (5.41)sin e right and left ontributions give C C O η θ ∼ q , O S η ¯ θ ∼ q , C O η ¯ θ ∼ q . As it was for the ase of the untwisted se tor, one an group the massless spe trumin terms of six dimensional supersymmetry multiplets. In the twisted se tor one has93 = 6 N = 1 supersymmetry (whi h gives in D = 4 N = 2 upon the redu tion to fourdimensions) and the massless spe trum forms half-hypermultiplets.5.4 Supersymmetri Z × Z shift orbifold modelIn this se tion we present the partition fun tion for the shift orbifold (5.16) with thea tion of Z × Z orbifold. The main di(cid:27)eren e w.r.t. the ase treated in se tion 5.2is that the spe trum is not anymore ompletely determined by the modular invarian eof one-loop torus amplitude and an ambiguity is present when proje ting the twistedse tors.The fa t that many di(cid:27)erent hoi es ( onsistent with modular invarian e) an bemade is des ribed by a phase ǫ , alled the dis rete torsion, whi h dis onne ts the modu-lar orbits [132, 133℄. An analogous situation was presented in the derivation of eq.(5.11).In the ase of Z × Z orbifold, the elements a ting on the torus T are given by , g = (+ − − ) , f = ( − + − ) , h = ( − − +) , where the notation means that ea h " + " or " − " a ts on the omplex oordinates ofea h two-torus. The elements g , f and h generate three independent twisted se tors.The a tion of the orbifold group elements on the SO (2) hara ters orresponding to thethree two-tori is given by the table below. T × T × T O V S C O V S C O V S C g: + + + + + − i − i + − − i i h: + − i − i + − − i i + + + +f: + − i − i + + + + + − − i i In hapter 2 we presented the spin stru tures. They represent the building blo ksfor the partition fun tion in orbifolds models. Their modular transformation properties an be presented with the s hemati pi ture below T : S : ab aba ab ba -1 Figure 5.1: Modular transformations for a generi amplitude in orbifold models,where a , b ∈ { , g, h, f } for a Z × Z orbifold.whi h shows that the Z × Z orbifold needs at least two independent modular or-bits. For example, the element ( g, h ) annot be derived from any untwisted amplitude.Therefore, to obtain the full partition fun tion, we have to al ulate ea h of the on-tributions shown in (cid:28)g.5.2, where the empty and the oloured boxes are asso iated to94wo independent orbits. We remind that the full partition has to be modular invariant.At the end of this se tion we provide the result of the partition fun tion in a ompa t o gfho g f ho g f ho g f ho g f hog fh ogf h ogfh Figure 5.2: Modular orbits in Z × Z orbifolds.form in the ase without dis rete torsion ( ǫ = 1 ). In general, the value of the phase forthe Z × Z ase an be ǫ = ± sin e it has to be of the same order as the generatorsof the orbifold. The explanations on erning the (cid:28)nal form for the partition fun tion(5.43) are presented in the next se tions and the de(cid:28)nitions on erning the terms T ij and G ij are given in Appendix D.The generi expression for a Z × Z orbifold partition fun tion an be indi ated as Z Total = T r ( un. + tw. ) g + f + h Z , (5.42)where Z in this ase is given by eq.(5.16). The expli it al ulation of (5.42) gives Z Total = 14 (cid:26) T oo Λ Λ Λ m ′ ,n ′ [Λ m,n ( O O + C C ) + Λ m +1 ,n ( S S + V V )+ Λ m,n + ( O C + C O ) + Λ m +1 ,n + ( V S + S V )]+ T og Λ | ηθ | G g + T of Λ | ηθ | G of + T oh Λ m ′ ,n ′ | ηθ | (cid:8) Λ m,n G oh + Λ m +1 ,n G ′ h + Λ m,n +1 / G ′′ h + Λ m +1 ,n +1 / G ′′′ h (cid:9) + T go G g Λ | ηθ | + T gg G gg Λ | ηθ | + T fo G fo Λ | ηθ | + T ff G ff Λ | ηθ | + T ho Λ mn | ηθ | (cid:8) Λ mn G h + Λ m +1 ,n G ′ h + Λ m,n +1 / G ′′ h + Λ m +1 ,n +1 / G ′′′ h (cid:9) + T hh Λ mn | ηθ | (cid:8) Λ mn G hh + Λ m +1 ,n G ′ hh + Λ m,n +1 / G ′′ hh + Λ m +1 ,n +1 / G ′′′ hh (cid:9) + ( T gh G gh + T gf G gf + T fg G fg + T fh G fh + T hg G hg + T hf G hf ) | η θ θ θ | (cid:27) , (5.43)where Λ , Λ and Λ m ′ n ′ Λ mn denote the three latti e sums asso iated to the three internaltori, as usual. The ontributions of the transverse bosons is impli it here.95he details for the derivation of eq.(5.43) are presented in the following se tions,where we provide the main steps. In fa t all the ingredients and the general methodshave been presented extensively in the previous part of this hapter. The main di(cid:30) ultyfor the Z × Z orbifold onsists in handling orre tly the numerous produ ts of hara -ters whi h have to be transformed under S transformation. In fa t they generate hugesums of terms whi h need some rearrangement to obtain a ompa t readable result. Forthis purpose a simple mathemati a program has been used.5.4.1 Untwisted spe trumAs in the al ulation of eq.(5.7), the untwisted ontribution is given by the sum of theproje tions w.r.t. the elements of the orbifold group of the initial partition fun tion.The (cid:28)rst four rows of eq.(5.43) indi ate the total untwisted se tor.The twisted se tor is given by the sum of two pie es, the (cid:28)rst is the S and T trans-formation of the untwisted ontribution, the se ond is the new independent modularorbit with its S and T transforms. We provide some further information on erning thederivation of the twisted se tor in the next se tion.5.4.2 Twisted se tor hWe present in the following the details for one twisted se tor only, in parti ular the h se tor, where the element G hg (cid:28)xes the hoi e of the independent orbit for our modelby a onsistent proje tion. In the other two twisted se tors there is no need for su h a hoi e sin e all elements are determined by modular transformations from the previousones. For the determination of a twisted se tor h we pro eed, as usual, by taking the S transform of the untwisted element proje ted by h . An S transformation of G h gives G h = (cid:8) S C O O + C S V V + S S V O + C C O V + C S O O + S C V V + C C V O + S S O V (cid:9) O + (cid:8) V O S C + O V C S + O O S S + V V C C + V O C S + O V S C + O O C C + V V S S (cid:9) C . (5.44)The T transformation of (5.44) provides the ontribution G hh and in its expression wein lude the total phase arising from the following overall phases G h T −→ iG hh , T h T −→ iT hh ⇒ i × i × − , where the last − in the formula is the global prefa tor obtained in the T transformations(see eq.(5.27)). 96 hh = (cid:8) S C O O + C S V V − S S V O − C C O V + C S O O + S C V V − C C V O − S S O V (cid:9) O + (cid:8) V O S C + O V C S − O O S S − V V C C + V O C S + O V S C − O O C C − V V S S (cid:9) C . (5.45)The independent orbit G hg is obtained by the a tion of the g element onto G h . Theoverall phase is in luded in the (cid:28)nal expression (5.46) and results from G h g −→ iG hg , T h g −→ iT hg ⇒ i × i = − .G hg = (cid:8) S C O O + C S V V + S S V O + C C O V − C S O O − S C V V − C C V O − S S O V (cid:9) O +( − (cid:8) − V O S C − O V C S − O O S S − V V C C + V O C S + O V S C + O O C C + V V S S (cid:9) C . (5.46)We observe that the hoi e for our proje tion is not the onventional one sin e afterperforming the g a tion onto the gauge degrees of freedom we also added a minus signin front of all the terms multiplying C . This operation provides a natural result for G hg , meaning that the omposition of the hara ters is analogous to G h and G hh andassures the modular invarian e of the partition fun tion.The T transformation of eq.(5.46) provides G hf whi h, as usual, in ludes the totalphase from G hg T −→ iG hf , T hg T −→ iT hf ⇒ i × i × − .G hf = (cid:8) S C O O + C S V V − S S V O − C C O V − C S O O − S C V V + C C V O + S S O V (cid:9) O + (cid:8) V O S C + O V C S − O O S S − V V C C − V O C S − O V S C + O O C C + V V S S (cid:9) C . (5.47)We have an important omment to make before dis ussing the relevant parts of thespe trum. In the untwisted se tor generated by the h element there are gauge ontri-butions ( G ′ h , G ′′ h , G ′′′ h ) whi h are multiplied by massive latti es, not providing anylow energy states. In the h twisted part these terms an still give a ontribution tothe massless spe trum (sin e we rearrange the latti es with the transformations (A.13).The presen e of the terms G ′ h , G ′′ h , G ′′′ h and their T transformations will not providemassless states. Thus, we an negle t these ontributions when we dis uss the relevantpart of the spe trum. 97.4.3 Torus amplitudes for the right and for the left se torIn the result (5.43) we have used the torus amplitudes de(cid:28)ned in terms of the quantities T ij , i = 0 , g, h, f , providing a simple and ompa t form for the partition fun tion. T i = τ i + τ ig + τ ih + τ if , T ig = τ i + τ ig − τ ih − τ if ,T ih = τ i − τ ig + τ ih − τ if , T if = τ i − τ ig − τ ih + τ if , (5.48)where the Z × Z hara ters τ ij are produ ts of the four level-one hara ters, de(cid:28)nedexpli itly in (D.3). The ordering of the four fa tors refers to the eight transverse di-mensions of spa etime. The (cid:28)rst fa tor is asso iated to the two transverse spa e timedire tions. For the left se tor we have G i = g i + g ig + g ih + g if , G ig = g i + g ig − g ih − g if ,G ih = g i − g ig + g ih − g if , G if = g i − g ig − g ih + g if . (5.49)The ontent of the above de(cid:28)nitions is given in (D.2). There we also provide the expli itexpressions for the gauge ontributions G ′ h , G ′′ h , G ′′′ h for the untwisted se tor and G ′ h , G ′′ h and G ′′′ h for the twisted se tor.5.4.4 Massless spe trumThe formula (5.43) presents the full modular invariant partition fun tion for the shiftorbifold (5.16) with Z × Z a tion. We noti e that the only ontributions from theuntwisted spe trum, where we have negle ted the a ented expressions, ome fromthe ombinations τ g + τ g g g + τ h g h + τ f g f . (5.50)Our main interest is as usual the low energy physi s of the model hen e we will presenthere the massless terms, whi h an be expanded in powers of q by applying the relationsof se tion A.3 in Appendix A. [ V O O O − S S S S − C C C C ] × [ O O O O O ] , [ O V O O − C C S S − S S C C ] × [( O V V O + V O O V ) O ] , [ O O O V − C S S C − S C C S ] × [( V V O O + O O V V ) O ] , [ O O V O − C S C S − S C S C ] × [( V O V O + O V O V ) O ] . (5.51)The gauge group of this model is given by G = SO (2) × SO (2) × SO (2) × SO (10) × SO (16) and the representations of the untwisted matter is provided in the following.98 Ve torial supermultiplet: [ V O O O − S S S S − C C C C ] × [ O O O O O ] → [(2 , , ,
1) + (1 + , + , + , + ) + (1 − , − , − , − )] × (1 , , , , • Two hiral supermultiplets: [ O V O O − C C S S − S S C C ] × [( O V V O + V O O V ) O ] → [(1 , , ,
1) + (1 − , − , + , + ) + (1 + , + , − , − )] × [(1 , , , ,
1) + (2 , , , , • Two hiral supermultiplets: [ O O O V − C S S C − S C C S ] × [( V V O O + O O V V ) O ] → [(1 , , ,
2) + (1 − + + − ) + (1 + − − + )] × [(2 , , , ,
1) + (1 , , , , • Two hiral supermultiplets: [ O O V O − C S C S − S C S C ] × [( V O V O + O V O V ) O ] → [(1 , , ,
1) + (1 − + − + )(1 + − + − )] × [(2 , , , ,
1) + (1 , , , , . In our notations we indi ate with ± the two di(cid:27)erent hiralities of a spinor in the SO (2) representation. This model has N = 1 in four dimensions.The twisted se tor gives rise to the only non-vanishing terms τ g g g + τ gg g gg + τ gh g gh + τ gf g gf + τ h g h + τ hg g hg + τ hh g hh + τ hf g hf + τ f g f + τ fg g fg + τ fh g fh + τ ff g ff , (5.52)whose massless ontributions have been indi ated in the following [ O O C C − C S O O ] × [( C O O C + V C C O + O S S V ) O ] , [ O O S S − S C O O ] × [( S O O S + V S S O + O C C V ) O ] , [ O C C O − C O O S ] × [( C C V O + S S O V ) O ] , [ O S S O − S O O C ] × [( S S V O + C C O V ) O ] , [ O S O S − S O C O ] × [( O S O S + S V S O + C O C V ) O ] , [ O C O C − C O S O ] × [( O C O C + C V C O + S O S V ) O ] . (5.53)These hiral supermultiplets fall into the representations presented below.99 Three hiral supermultiplets: [ O O C C − C S O O ] × [( C O O C + V C C O + O S S V ) O ]+[ O O S S − S C O O ] × [( S O O S + V S S O + O C C V ) O ] → [(1 , , ± , ± ) + (1 ± , ± , , × [(1 ± , , , ,
1) + (2 , ± , ± , ,
1) + (1 , ± , ± , , • Two hiral supermultiplets: [ O C C O − C O O S ] × [( C C V O + S S O V ) O ]+[ O S S O − S O O C ] × [( S S V O + C C O V ) O ] → [(1 , ± , ± ,
1) + (1 ± , , , ± )] × [(1 ± , ± , , ,
1) + (1 ± , ± , , , • Three hiral supermultiplets: [ O S O S − S O C O ] × [( O S O S + S V S O + C O C V ) O ]+[ O C O C − C O S O ] × [( O C O C + C V C O + S O S V ) O ] → [(1 , ± , , ± ) + (1 ± , , ± , × [(1 , ± , , ,
1) + (1 ± , , ± , ,
1) + (1 ± , , ± , , . We noti e that the twisted massless spe trum ontains two hiral supermultipletsin the spinorial representation of SO (10) , plus few supermultiplets in the fundamentalof SO (10) in four dimensions. This result on ludes our hapter.100hapter 6Con lusionsIn this thesis we fo us our study on heteroti superstring theories and their appli a-tions to parti le physi s. In parti ular, we are interested in the sear h of semi-realisti four-dimensional superstring va ua whi h an reprodu e, at low energy, the StandardModel physi s. Motivated by the SO (10) embedding of matter in heteroti models, weinvestigate di(cid:27)erent s hemes of ompa ti(cid:28) ation of the E × E heteroti string from tento four dimensions. A very su essful approa h is given by free fermioni models. Theygive rise to the most realisti three generation string models to date. Their phenomenol-ogy is studied in the e(cid:27)e tive low energy (cid:28)eld theory by the analysis of supersymmetri (cid:29)at dire tions. In the (cid:28)rst example illustrated in hapter 3, the model ontent onsistsof MSSM states in the observable Standard Model se tor. In that model, for the (cid:28)rsttime, we apply a new general me hanism that allows the redu tion of Higgs ontentat the string s ale by an opportune hoi e of asymmetri boundary onditions for theinternal fermions of the theory. An additional result for minimal Higgs spe trum modelsis the fa t that the supersymmetri moduli spa e is redu ed as well, and this in reasesthe predi tive power of the theory.A ommon feature of free fermioni models is the presen e of an anomalous U (1) whi h gives rise to a Fayet-Iliopoulos D-term that breaks supersymmetry at one-looplevel in string perturbation theory. Supersymmetry is restored by imposing D and F(cid:29)atness on the va uum. Generally, it has been assumed that in a given string modelthere should exist a supersymmetri solution to D and F (cid:29)atness onstraints. Never-theless, in the se ond model presented in hapter 3, su h as in the previous example, no(cid:29)at solutions are found after employing the standard analysis for (cid:29)at dire tions. TheBose-Fermi degenera y of the spe trum implies that the osmologi al onstant vanisheswhile supersymmetry remains broken at the perturbative level. This unexpe ted resultmay open new possibilities for the supersymmetry breaking me hanism in string theory.By looking at a very di(cid:27)erent ba kground, the one given by the orbifold onstru tion, itis possible to obtain omplementary advantages in the understanding of semi-realisti models, su h as a more geometri pi ture of those. Moreover, in the ase of Z × Z Z × Z orbifold with a non-fa torisable skewed om-pa ti(cid:28) ation latti e has been analysed, where the redu tion of the number of familiesis realised and suggests new way of investigating orbifold ompa ti(cid:28) ations. No semi-realisti models are presented in this set up yet, nevertheless the possible ombinationsof a proper hoi e for the ompa ti(cid:28) ation latti e plus the presen e of suitable Wilsonlines provides new han es in the onstru tion of semi-realisti models. A hallengingoutlook in this set up is the introdu tion of asymmetri shifts and twists. Indeed, theseelements seem to be related with free fermioni models where asymmetri boundary onditions are imposed on the ompa t dimensions and are responsible for the mostsu essful phenomenologi al features of these models.In the last hapter we present the formalism for the onstru tion of modular invariantpartition fun tions in heteroti orbifold models and, among a few examples, the ase ofa Z × Z shift orbifold model. The study of orbifolds with di(cid:27)erent proje tions shouldlighten the properties of the low energy spe trum and possibly provide some sele tionme hanism for semi-realisti va ua. For instan e, a hallenging proje t would be therealisation of the Higgs-matter splitting. This me hanism is viable with an orbifoldproje tion that will allow to obtain string states uniquely from the untwisted se tor andthe matter states from the twisted se tors. This me hanism is already well-known inthe free fermioni ase. 102ppendix AA.1 η and θ -fun tions and modular transformationsThe Dedekind η fun tion is de(cid:28)ned as η ( τ ) = q ∞ Y p =1 (1 − q p ) . (A.1)We provide the modular transformations of η and of the Tei hmuller parameter τ interms of omplex fun tion and real omponents. T : η (1 + τ ) = e iπ/ η ( τ ) , S : η ( − τ ) = ( − iτ ) η ( τ ) . (A.2) T : τ → τ + 1 ; τ → τ ; dτ dτ → dτ dτ .S : τ + iτ → − τ + iτ = − ¯ ττ ¯ τ ; dτ d ¯ τ → dτ d ¯ τ | τ ¯ τ | . The de(cid:28)nition of the θ fun tion is given in both notations, as sum and as produ tformulae θ (cid:20) αβ (cid:21) (0 | θ ) = ∞ X n = ∞ q ( n + α ) e πi ( n + α ) β = e πiαβ q α ∞ Y n (1 − q n )(1 + q n + α − e πiβ )(1 + q n − α − e − πiβ ) (A.3)and their modular transformations T : θ (cid:20) αβ (cid:21) (0 | τ + 1) = e − πiα ( α − θ (cid:20) αβ + α − (cid:21) (0 | τ ) ,S : θ (cid:20) αβ (cid:21) (0 | − τ ) = ( − iτ ) e πiαβ θ (cid:20) β − α (cid:21) (0 | τ ) . (A.4)103.1.1 SO (2 n ) hara ters in terms of θ -fun tions O n = θ n + θ n η n ; V n = θ n − θ n η n ; S n = θ n + i − n θ n η n ; C n = θ n − i − n θ n η n . (A.5)It is useful to present the expli it expansions of the previous fun tions and the η fun tionin terms of powers of q , where q = e iπτ O n = Π ∞ p =1 (1 − q p ) n (1 + q p − ) n + Π ∞ p =1 (1 − q p ) n (1 − q p − ) n q n Π ∞ p =1 (1 − q p ) n = q − n (1 + n (2 n − q + ... ) ,V n = Π ∞ p =1 (1 − q p ) n (1 + q p − ) n − Π ∞ p =1 (1 − q p ) n (1 − q p − ) n q n Π ∞ p =1 (1 − q p ) n = q − n (2 nq + ... ) ,S n /C n = q n Π ∞ p =1 (1 − q p ) n (1 + q p ) n (1 + q p − ) n q n Π ∞ p =1 (1 − q p ) n = 2 n − q n (1 + 2 nq + ... ) , η n = q − n (1 + nq + ... ) , (A.6)where the de(cid:28)nition of θ -fun tions and the binomial expansion below have been applied, ( a + b ) n = n X i =0 C (cid:18) ni (cid:19) a n − i b i = a n + C (cid:18) ni (cid:19) a n − b + ... . The de omposition of an SO ( x + y ) hara ter into the produ t of an SO ( x ) with an SO ( y ) hara ter is given by the expressions below: O n = O x O y + V x V y , V n = V x O y + O x V y ,C n = S x C y + C x S y , S n = S x S y + C x C y , (A.7)where n = x + y and x, y are even.In the study of the S transformations of the previous expansions it an be useful torearrange (A.7) with the relations aa + bb = 12 [( a + b )( a + b ) + ( a − b )( a − b )] aa − bb = 12 [( a − b )( a + b ) + ( a + b )( a − b )] ab + ba = 12 [( a + b )( a + b ) − ( a − b )( a − b )] ab − ba = 12 [( a − b )( a + b ) − ( a + b )( a − b )] (A.8)where a and b an be any of O n , V n , S n , C n .104.1.2 Modular transformations for SO (2 n ) hara tersThe modular S and T transformations a t on the hara ters as O n V n S n C n S −→ − − − i − n − i − n − − i − n i − n O n V n S n C n ; O n V n S n C n T −→ e − inπ/ − e inπ/
00 0 0 e inπ/ O n V n S n C n . (A.9)A.2 De(cid:28)nition of latti eThe partition fun tion of a ompa t s alar on a ir le of radius R is Λ m,n = 1 ηη X m,n q α ′ p L / q α ′ p R / , (A.10)where the hiral momenta are de(cid:28)ned as p L,R = mR ± nRα ′ . Therefore, if one of the non- ompa t oordinates of a riti al string is repla ed witha ompa t one, the ontinuous integration over internal momenta is repla ed by thelatti e sum √ τ η ( τ ) η ( τ ) → Λ m,n .For the ase of a d-dimensional torus the eq.(A.10) is generalised to ~ Λ m,n = 1 η d η d X m,n q α ′ p TL g − p L / q α ′ p TR g − p R / (A.11)where p L,a = m a + α ′ ( g ab − B ab ) n b , p R,a = m a + α ′ ( g ab + B ab ) n b , g ab is the metri onthe torus and B ab is an antisymmetri NS-NS (cid:28)eld.A.2.1 De(cid:28)nition of shifted latti esIn this se tion we present ombinations obtained with the standard latti e Λ mn whenthe shift δ : Λ m,n → ( − m Λ m,n a ts on it. Moreover we show their main propertieswhose demonstration is given in se tion A.2.2. Λ m,n = 1 + ( − m m,n , Λ m +1 ,n = 1 − ( − m m,n , Λ m,n + = 1 + ( − m m,n + , Λ m +1 ,n + = 1 − ( − m m,n + . (A.12)105ransformation properties S , T invariant z }| { Λ m,n ; S , T invariant z }| { Λ m,n + Λ m,n + T invariant z }| { ( − m Λ m,n S −→ Λ m,n + T −→ S invariant z }| { ( − m Λ m,n + T invariant z }| { Λ m,n − Λ m,n + S −→ Λ m +1 ,n + Λ m +1 ,n + T −→ S invariant z }| { Λ m +1 ,n − Λ m +1 ,n + . (A.13)A.2.2 Proof for the transformation properties (A.13)In this se tion we show how to derive some of the properties presented in the previousse tion.1) S invarian e for Λ m,n .2) T invarian e for Λ m,n and ( − m Λ m,n .3) ( − m Λ m,n S −→ Λ m,n +1 / .The other relations shown in (A.13) an be derived with the same te hniques below.It is useful to keep in mind the de(cid:28)nitions of the general latti e (A.10) and the hiralmomenta p L,R . Moreover we an rewrite q and ¯ q in the onvenient way q = e πiτ = e πi ( τ + iτ ) = e π ( iτ − τ ) , ¯ q = e − πi ¯ τ = e − πi ( τ − iτ ) = e − π ( iτ + τ ) . The Poisson resummation formula will be applied onstantly in the demonstration ofthe previous statements, thus we provide its general expression below X m i ∈ Z e − πm i · m j A ij + πB i m i = 1 √ det A X m k ∈ Z e − π ( m k + iBk )( A − ) kl ( m l + iBl ) (A.14)We start by demonstrating point 1).The best way to pro eed is to rewrite the latti e sum in the more onvenient form Λ m,n = X m,n e π ( iτ − τ ) α ′ ( mR + nRα ′ ) e π ( − iτ − τ ) α ′ ( mR − nRα ′ ) . (A.15)We noti e that the η ¯ η fa tor has been dropped for onvenien e.Let us simplify the two exponentials and rewrite Λ m,n = X m,n e πiτ mn e − πτ α ′ ( m R + n R α ′ ) . (A.16)If we perform a Poisson resummation w.r.t. m we have a = α ′ τ R → √ detA = R √ α ′ τ , b = 2 iπn R √ α ′ τ X m,n e − π ( m ′ − τ n ) R α ′ τ e − πτ n R α ′ . (A.17)We expand the square and we obtain an exponential with four terms. Two of them anbe rewritten as − πR α ′ ( τ τ + τ ) n = − πR α ′ | τ | τ n . We apply now the resummation w.r.t. na = R | τ | α ′ τ → √ detA = √ α ′ τ R | τ | , b = 2 R τ m ′ τ α ′ whi h transforms (A.17) into R √ α ′ τ √ α ′ τ Rτ X m,n e − π ( m ′ πR α ′ τ ) e − πτ α ′ R | τ | ( n ′ + im ′ R τ τ α ′ ) . (A.18)We expand the exponents and use the equivalen e πR m ′ α ′ τ ( − τ | τ | ) = − τ πR m ′ | τ | α ′ to get (cid:28)nally ⇒ | τ | X m ′ ,n ′ e − πi τ | τ | m ′ n ′ e − πτ | τ | ( m ′ R α ′ + n ′ α ′ R ) . (A.19)The expression above is equivalent to Λ m,n if we rede(cid:28)ne − τ | τ | = τ ′ , τ | τ | = τ ′ , (A.20)whi h is in fa t the S transformation of τ → − /τ . The prefa tor / | τ | in (A.19)belongs to the transformation of ηη (whi h we dropped at the beginning), showing that(A.19) is the S transformation of (A.10).The explanation for point 2) is very simple sin e the invarian e under T is trivial τ T −→ τ + 1 = ( τ + 1) + iτ → Λ m,n = X m,n e π ( iτ mn ) e πimn | {z } e − πτ α ′ ( m R + n R α ′ ) . The quantity ( − m Λ m,n is obviously invariant under T transformation as well.More algebra is involved for the proof of point 3).The main idea here is to show that ( − m Λ m,n ( τ ) an be rewritten as Λ m,n +1 / ( τ ′ ) ,where τ ′ is given by (A.20). Let us start with the de(cid:28)nition ( − m Λ m,n = X m,n e πim ( τ n +1 / e − πτ α ′ m R e − πτ n R α ′ . (A.21)By applying the Poisson resummation w.r.t. ma = τ α ′ R , b = 2 i ( τ n + 1 / , ⇒ X m,n e − π ( m ′ − ( τ n +1 / R τ α ′ e − πτ n R α ′ . (A.22)Rearranging the exponential and using the relation − πR n α ′ ( τ τ ) = − | τ | πR n τ α ′ we get R √ τ α ′ X m ′ ,n e − πR n | τ | τ α ′ e π ( m ′ − / n τ R τ α ′ e − π ( m ′ − / R τ α ′ . (A.23)A Poisson resummation of (A.23) w.r.t. n , where a = R | τ | τ α ′ , b = 2( m ′ − / τ R τ α ′ , will provide ⇒ | τ | X m ′ ,n ′ e − π τ α ′ R | τ | ( n ′ + i ( m ′ − / τ R τ α ′ ) e − π ( m ′ − / R τ α ′ = 1 | τ | X m ′ ,n ′ e − π τ α ′ R | τ | n ′ e − π ( m ′ − / R τ | τ | α ′ e − iπn ′ ( m ′ − / τ | τ | = Λ n ′ ,m ′ +1 / . (A.24)As we said, on e rede(cid:28)ning n ′ → n , m ′ → m and identifying the transformed τ ′ param-eter, we have obtained exa tly the S transformation of the initial (A.21).A.3 Expansion of SO (2 n ) hara ters in powers of qThis se tion presents the expli it expansions of the hara ters used in se tions 5.2-5.4for the sear hing of the massless spe trum. V = q − (2 q + ... ) , O = q − (1 + q + ... ) , S /C = q (1 + 2 q + ... ) ,V = q − (4 q + ... ) , O = q − (1 + 6 q + ... ) , S /C = 2 q (1 + 4 q + ... ) ,V = q − (10 q + ... ) , O = q − (1 + 45 q + ... ) , S /C = 2 q (1 + 10 q + ... ) ,V = q − (12 q + ... ) , O = q − (1 + 66 q + ... ) , S /C = 2 q (1 + 12 q + ... ) ,V = q − (16 q + ... ) , O = q − (1 + 120 q + ... ) , S /C = 2 q (1 + 16 q + ... ) , The latti e sum ontributes to the spe trum as Λ m,n → q q + ..., Λ m +1 ,n → no massless solutions , Λ m,n + → no massless solutions , Λ m +1 ,n + → no massless solutions . (A.25)10809ppendix BB.1 Tables for two models with redu ed Higgs spe trum F SEC SU (3) × Q C Q L Q Q Q Q Q Q SU (2) ,.., Q Q SU (2) L b (1 , − − − , , , , ,
1) 0 0 Q (3 , − , , , , ,
1) 0 0 d cL (¯3 , − − − , , , , ,
1) 0 0 N cL (1 , − − − , , , , ,
1) 0 0 u cL (¯3 , − − − , , , , ,
1) 0 0 e cL (1 , − , , , , ,
1) 0 0 L b (1 , − − , , , , ,
1) 0 0 Q (3 , − − , , , , ,
1) 0 0 d cL (¯3 , − − − , , , , ,
1) 0 0 N cL (1 , − − − , , , , ,
1) 0 0 u cL (¯3 , − − − , , , , ,
1) 0 0 e cL (1 , − , , , , ,
1) 0 0 L b (1 , − (1 , , , , ,
1) 0 0 Q (3 , − (1 , , , , ,
1) 0 0 d cL (¯3 , − − (1 , , , , ,
1) 0 0 N cL (1 , − − (1 , , , , ,
1) 0 0 u cL (¯3 , − − (1 , , , , ,
1) 0 0 e cL (1 , (1 , , , , ,
1) 0 0 h NS (1 ,
2) 0 − , , , , ,
1) 0 0¯ h (1 ,
2) 0 1 0 0 − , , , , ,
1) 0 0 φ (1 ,
1) 0 0 0 0 0 0 1 0 (1 , , , , ,
1) 0 0 φ ′ (1 ,
1) 0 0 0 0 0 0 − , , , , ,
1) 0 0˜ φ (1 ,
1) 0 0 0 0 0 0 0 0 (1 , , , , ,
1) 0 0 φ (1 ,
1) 0 0 0 0 0 0 0 0 (1 , , , , ,
1) 0 0 φ (1 ,
1) 0 0 0 0 0 0 1 0 (1 , , , , ,
1) 0 0 φ ′ (1 ,
1) 0 0 0 0 0 0 − , , , , ,
1) 0 0˜ φ (1 ,
1) 0 0 0 0 0 0 0 0 (1 , , , , ,
1) 0 0 C − ++ b (1 ,
1) 0 − − , , , , ,
1) 1 0 C − + − + β + 2 γ (1 ,
1) 0 1 − , , , , , − D + (1 ,
2) 0 0 − − − , , , , ,
1) 1 0 D − (1 ,
2) 0 0 − , , , , , − C + − + (1 ,
1) 0 − − , , , , ,
1) 1 0 C + −− (1 ,
1) 0 1 − , , , , , − Table 3.a. 110
SEC SU (3) × Q C Q L Q Q Q Q Q Q SU (2) ,.., Q Q SU (2) T + b (¯3 , − − − , , , , ,
1) 0 1 C − + β (1 , − − , , , , ,
1) 0 − C + (1 , − − , , , , ,
1) 0 1 T − (3 , − , , , , ,
1) 0 − D b + 2 γ (1 ,
1) 0 0 0 −
12 12 − , , , , ,
2) 0 0 S (1 ,
1) 0 0 0 −
12 12 − , , , , , − − S ′ (1 ,
1) 0 0 0 −
12 12 − , , , , ,
1) 1 1˜ S (1 ,
1) 0 0 0 −
12 12 12 , , , , , − S ′ (1 ,
1) 0 0 0 −
12 12 12 , , , , ,
1) 1 − S b + 2 γ (1 ,
1) 0 0 − , , , , , − S ′ (1 ,
1) 0 0 − , , , , ,
1) 1 − D (1 ,
1) 0 0 − , , , , ,
2) 0 0˜ S (1 ,
1) 0 0 − − , , , , , − − S ′ (1 ,
1) 0 0 − − , , , , ,
1) 1 1 S b + 2 γ (1 ,
1) 0 0 − − (1 , , , , , − S ′ (1 ,
1) 0 0 − − (1 , , , , ,
1) 1 − S (1 ,
1) 0 0 − − − (1 , , , , , − − S ′ (1 ,
1) 0 0 − − − (1 , , , , ,
1) 1 1˜ D (1 ,
1) 0 0 − − − (1 , , , , ,
2) 0 0 A + b + 2 γ (1 ,
1) 0 0 − − (2,1,1,1,1,1) A − (1 ,
1) 0 0 (2,1,1,1,1,1) − D b + (1,1) −
12 12 12 (1,2,1,2,1,1) D ′ b + 2 γ (1,1) −
12 12 12 (2,1,1,1,2,1) D b + (1,1) − − (2,1,1,1,2,1) D ′ b + 2 γ (1,1) − − (1,2,1,2,1,1) b + b (1,1)
34 12 14 − −
14 12 − (1,1,1,2,1,1) − b ± γ (1,1) − − −
14 14 14 − − (1,1,1,2,1,1) −
12 12 ˜ D ′ b + (1,1) − − − (2,1,1,1,2,1) D b + 2 γ (1,1) − − (1,2,1,2,1,1) b + (1,1) − − −
12 12 (2,1,1,1,1,1) b + b + (1,1) −
12 12 (2,1,1,1,1,1) β + 2 γ (1,1) − − − (2,1,1,1,1,1) (1,1) − − − (2,1,1,1,1,1) D (3 , − + b + (1,1) − − (1,1,2,2,1,1) D (5)+ b + b (1,1) − − − (1,1,1,1,2,1) D (5) − α + 2 γ (1,1) − (1,1,1,1,2,1) − D (3 , − (1,1) − − (1,1,2,2,1,1) D (6)+ − ± γ (1,1)
34 12 14 14 14 12 − (1,1,1,1,1,2) − D (6) −− (1,1)
34 12 14 14 14 − − (1,1,1,1,1,2) − D (6)++ (1,1) − − − − −
14 12 − (1,1,1,1,1,2) −
12 12 D (6) − + (1,1) − − − − − − − (1,1,1,1,1,2) −
12 12
Table 3.a ontinued. 111
SEC SU (3) × Q C Q L Q Q Q Q Q Q SU (2) ,.., Q Q SU (2) D (3) −− b + b (1,1)
34 12 −
14 14 − − − (1,1,2,1,1,1) −
12 12 D (3)+ − ± γ (1,1)
34 12 −
14 14 − −
12 12 (1,1,2,1,1,1) −
12 12 D (3) − + (1,1) − −
12 14 −
14 14 − − (1,1,2,1,1,1) − D (3)++ (1,1) − −
12 14 −
14 14 −
12 12 (1,1,2,1,1,1) − F b + α (1,1) −
12 14 − −
14 12 (1,1,2,1,1,1) − − F ′ β ± γ (1,1) −
12 14 − −
14 12 (1,1,1,1,1,2)
12 12 ˜ F (1,1) −
34 12 −
14 14 14 12 − (1,1,2,1,1,1)
12 12 ˜ F ′ (1,1) −
34 12 −
14 14 14 12 − (1,1,1,1,1,2) − − F b + b (1,1) −
12 14 14 14 12 12 (1,1,2,1,1,1) − − F β ± γ (1,1) −
12 14 14 14 12 − (1,1,1,1,1,2)
12 12 F (1,1) −
34 12 − − −
14 12 12 (1,1,2,1,1,1)
12 12 F (1,1) −
34 12 − − −
14 12 − (1,1,1,1,1,2) − − b (1,1)
34 12 −
14 14 − − (1,1,1,2,1,1) − ± γ (1,1) − −
12 14 −
14 14 − (1,1,1,2,1,1) −
12 12 b + b (1,1)
34 12 14 − −
14 12 − (1,2,1,1,1,1) −
12 12 ± γ (1,1) − − −
14 14 14 − − (1,2,1,1,1,1) − b + b + b (1,1)
34 12 −
14 14 − − − (1,2,1,1,1,1) −
12 12 + b ± γ (1,1) − −
12 14 −
14 14 − − (1,2,1,1,1,1) − Table 3.a ontinued. 112
SEC SU (3) × Q C Q L Q Q Q Q Q Q SU (2) ,.., Q H SU (2) × SU (4) H L b (1 , − − , , , ,
1) 0 Q (3 , − − , , , ,
1) 0 d c (¯3 , − − − , , , ,
1) 0 N c (1 , − − − , , , ,
1) 0 u c (¯3 , − − − , , , ,
1) 0 e c (1 , − , , , ,
1) 0 L b (1 , − − − , , , ,
1) 0 Q (3 , − , , , ,
1) 0 d c (¯3 , − − , , , ,
1) 0 N c (1 , − − , , , ,
1) 0 u c (¯3 , − − − − , , , ,
1) 0 e c (1 , − − , , , ,
1) 0 L b (1 , − (1 , , , ,
1) 0 Q (3 , − (1 , , , ,
1) 0 d c (¯3 , − − (1 , , , ,
1) 0 N c (1 , − − (1 , , , ,
1) 0 u c (¯3 , − − (1 , , , ,
1) 0 e c (1 , (1 , , , ,
1) 0 h NS (1 ,
2) 0 − , , , ,
1) 0¯ h (1 ,
2) 0 1 0 0 − , , , ,
1) 0Φ (1 ,
1) 0 0 0 0 0 0 1 1 (1 , , , ,
1) 0¯Φ (1 ,
1) 0 0 0 0 0 0 − − , , , ,
1) 0Φ ′ (1 ,
1) 0 0 0 0 0 0 1 − , , , ,
1) 0¯Φ ′ (1 ,
1) 0 0 0 0 0 0 − , , , ,
1) 0¯Φ (1 ,
1) 0 0 0 0 0 − − , , , ,
1) 0Φ ′ (1 ,
1) 0 0 0 0 0 1 0 − , , , ,
1) 0¯Φ ′ (1 ,
1) 0 0 0 0 0 − , , , ,
1) 0Φ (1 ,
1) 0 0 0 0 0 1 0 1 (1 , , , ,
1) 0 ξ , , (1 ,
1) 0 0 0 0 0 0 0 0 (1 , , , ,
1) 0Φ NS (1 ,
1) 0 0 0 0 0 1 1 0 (1 , , , ,
1) 0¯Φ (1 ,
1) 0 0 0 0 0 − − , , , ,
1) 0Φ ′ (1 ,
1) 0 0 0 0 0 1 − , , , ,
1) 0¯Φ ′ (1 ,
1) 0 0 0 0 0 − , , , ,
1) 0Φ αβ b + b (1 ,
1) 0 0 0 0 0 −
12 12 , , , ,
1) 0¯Φ αβ α + β (1 ,
1) 0 0 0 0 0 − , , , ,
1) 0Φ αβ (1 ,
1) 0 0 0 0 0 −
12 12 , , , ,
1) 0¯Φ αβ (1 ,
1) 0 0 0 0 0 − , , , ,
1) 0 V b + 2 γ (1 ,
1) 0 0 0 −
12 12 12 , , , ,
6) 0 V (1 ,
1) 0 0 0 −
12 12 − , , , , − V (1 ,
1) 0 0 0 −
12 12 − , , , ,
1) 2 V b + 2 γ (1 ,
1) 0 0 − − , , , ,
6) 0 V (1 ,
1) 0 0 − , , , , − V (1 ,
1) 0 0 − , , , ,
1) 2
Table 3.b. 113
SEC SU (3) × Q C Q L Q Q Q Q Q Q SU (2) ,.., Q H SU (2) × SU (4) H V b + 2 γ (1 ,
1) 0 0 − − (1 , , , ,
6) 0 V (1 ,
1) 0 0 − − − (1 , , , , − V (1 ,
1) 0 0 − − − (1 , , , ,
1) 2 V b + (1 ,
1) 0 0 0 −
12 12 − , , , ,
1) 0 V b + 2 γ (1 ,
1) 0 0 0 −
12 12 − , , , ,
1) 0 V b + (1 ,
1) 0 0 − , , , ,
1) 0 V b + 2 γ (1 ,
1) 0 0 − , , , ,
1) 0 V b + (1 ,
1) 0 0 − − − (2 , , , ,
1) 0 V b + 2 γ (1 ,
1) 0 0 − − − (1 , , , ,
1) 0 H b + α (1 ,
2) 0 0 0 0 0 0 −
12 12 (2 , , , ,
1) 0¯ H (1 ,
2) 0 0 0 0 0 0 − (2 , , , ,
1) 0 H b + β (1 ,
2) 0 0 0 0 0 − (1 , , , ,
1) 0¯ H (1 ,
2) 0 0 0 0 0 − (1 , , , ,
1) 0 H b ± γ (¯3 , −
12 14 14 − − (1 , , , ,
1) 1 H (1 , −
34 12 14 14 − − (1 , , , ,
1) 1 H (1 , − − −
34 14 − − (1 , , , ,
1) 1 H (1 , − −
12 14 − − − (1 , , , ,
1) 1 H (1 , − −
12 14 14 34 − (1 , , , ,
1) 1¯ H (3 , −
14 12 − −
14 14 (1 , , , , − H (1 , − − −
14 14 (1 , , , , − H (1 ,
34 12 34 −
14 14 (1 , , , , − H (1 ,
34 12 −
14 34 14 (1 , , , , − H (1 ,
34 12 − − − (1 , , , , − H b + b (1 , − − −
14 14 12 , , , , − H β ± γ (1 , −
34 12 14 14 − − , , , ,
1) 1 H b (1 , − − −
14 14 12 , , , ,
1) 1¯ H + β ± γ (1 , −
34 12 14 14 − − , , , , − H b + b (1 , −
34 12 14 14 − − , , , ,
1) 1¯ H α ± γ (1 , − − −
14 14 , , , , − H b + (1 , −
34 12 14 14 − − , , , , − H + α ± γ (1 , − − −
14 14 , , , ,
1) 1 H b + α (1 ,
34 12 14 14 14 −
12 12 , , , ,
4) 0 H + β ± γ (1 , − − − − − −
12 12 , , , , ¯4) 0 H b + α (1 ,
34 12 14 − − − − (1 , , , ,
4) 0 H + β ± γ (1 , − − −
14 14 14 − − (1 , , , , ¯4) 0 H b + α (1 ,
34 12 −
14 14 − − (1 , , , ,
4) 0 H + β ± γ (1 , − −
12 14 −
14 14 − (1 , , , , ¯4) 0 Table 3.b ontinued. 114D Q ( A ) Φ Φ ′ ¯Φ ′ V V V V V V N c N c N c Φ ′ Φ ¯Φ e c e c e c H H H D ′ D ′ D ′ D ′ D ′ D ′ D ′ D ′ D ′ D ′ D ′ D ′ D ′ D -Flat dire tion basis of non-abelian singlet (cid:28)elds. Column 2 spe i(cid:28)es theanomalous harge and olumns 3 through 16 spe ify the norm-square VEV omponentsof ea h basis dire tion. The six (cid:28)elds e ci and H , , arry hyper harge, the remainingdo not. A negative omponent indi ates the ve tor partner of a (cid:28)eld (if it exists) musttake on VEV rather than the (cid:28)eld. 115D VEV D ′ V D ′ V D ′ V D ′ e c D ′ N c D ′ e c D ′ Φ D ′ ¯Φ D ′ Φ ′ D ′ V D ′ N c D ′ V D ′ V Table 3.d. Unique VEV asso iated with ea h non-abelian singlet (cid:28)eld D -Flat basisdire tion. 116D Q ( A ) Φ Φ ′ ¯Φ ′ V V V V V V N c N c N c Φ ′ Φ ¯Φ e c e c e c H H H ¯Φ αβ , H H ¯ H ¯ H H H H V V V H H H V V V V V V Q Q Q d c d c d c u c u c u c H h L L L H ¯ H ¯ H D -2 4 1 1 0 0 0 0 0 0 6 0 0 0 00 0 0 -6 -1 -4 -70 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 012 0 0 D -2 1 1 4 0 0 0 0 0 0 0 6 0 0 00 0 0 -6 -1 -7 -40 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 012 0 0 D -1 2 -1 2 0 0 0 0 0 0 0 0 3 0 00 0 0 -3 -2 -2 -20 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 06 0 0 D -1 2 -1 8 0 0 0 0 0 0 0 0 0 0 00 0 0 -6 4 1 -50 0 0 0 0 12 0 0 0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 D -1 8 -1 2 0 0 0 0 0 0 0 0 0 0 00 0 0 -6 4 -5 10 0 0 0 0 0 12 0 0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 D -1 2 5 2 0 0 0 0 0 0 0 0 0 0 00 0 0 -6 -2 1 10 0 0 0 0 0 0 12 0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0Table 3.e D -Flat dire tion basis of all (cid:28)elds.117D Q ( A ) Φ Φ ′ ¯Φ ′ V V V V V V N c N c N c Φ ′ Φ ¯Φ e c e c e c H H H ¯Φ αβ , H H ¯ H ¯ H H H H V V V H H H V V V V V V Q Q Q d c d c d c u c u c u c H h L L L H ¯ H ¯ H D -1 1 0 1 0 0 0 0 0 0 0 0 0 0 00 0 0 -2 -1 -2 -20 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0 0 0 0 3 0 0 0 02 0 0 D -1 -1 -1 2 0 0 0 0 0 0 0 0 0 0 00 0 0 -6 -2 -5 -50 0 0 -6 0 0 0 0 0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 06 0 0 D -1 -1 2 2 0 0 0 0 0 0 0 0 0 0 00 0 0 -6 -2 -5 -50 -6 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 06 0 0 D D D Q ( A ) Φ Φ ′ ¯Φ ′ V V V V V V N c N c N c Φ ′ Φ ¯Φ e c e c e c H H H ¯Φ αβ , H H ¯ H ¯ H H H H V V V H H H V V V V V V Q Q Q d c d c d c u c u c u c H h L L L H ¯ H ¯ H D D D D D Q ( A ) Φ Φ ′ ¯Φ ′ V V V V V V N c N c N c Φ ′ Φ ¯Φ e c e c e c H H H ¯Φ αβ , H H ¯ H ¯ H H H H V V V H H H V V V V V V Q Q Q d c d c d c u c u c u c H h L L L H ¯ H ¯ H D D D D D D Q ( A ) Φ Φ ′ ¯Φ ′ V V V V V V N c N c N c Φ ′ Φ ¯Φ e c e c e c H H H ¯Φ αβ , H H ¯ H ¯ H H H H V V V H H H V V V V V V Q Q Q d c d c d c u c u c u c H h L L L H ¯ H ¯ H D D D D D D Q ( A ) Φ Φ ′ ¯Φ ′ V V V V V V N c N c N c Φ ′ Φ ¯Φ e c e c e c H H H ¯Φ αβ , H H ¯ H ¯ H H H H V V V H H H V V V V V V Q Q Q d c d c d c u c u c u c H h L L L H ¯ H ¯ H D D D D D D Q ( A ) Φ Φ ′ ¯Φ ′ V V V V V V N c N c N c Φ ′ Φ ¯Φ e c e c e c H H H ¯Φ αβ , H H ¯ H ¯ H H H H V V V H H H V V V V V V Q Q Q d c d c d c u c u c u c H h L L L H ¯ H ¯ H D D D D D D Q ( A ) Φ Φ ′ ¯Φ ′ V V V V V V N c N c N c Φ ′ Φ ¯Φ e c e c e c H H H ¯Φ αβ , H H ¯ H ¯ H H H H V V V H H H V V V V V V Q Q Q d c d c d c u c u c u c H h L L L H ¯ H ¯ H D D D D D D Q ( A ) Φ Φ ′ ¯Φ ′ V V V V V V N c N c N c Φ ′ Φ ¯Φ e c e c e c H H H ¯Φ αβ , H H ¯ H ¯ H H H H V V V H H H V V V V V V Q Q Q d c d c d c u c u c u c H h L L L H ¯ H ¯ H D D D D N c D ¯ H D u c D V D V D N c D ¯ H D h D H D H D N c D H D e c D V D V D H D Φ ′ D e c D V D V D H D ¯Φ D V D V D L D H D ¯Φ αβ , D V D V D V D H D ¯ H D u c D V D V D ¯ H D V D Q D d c D d c D H D Q D u c D L D d c D Φ D Q D H D V D L Table 3.f. Unique VEV asso iated with ea h D -Flat basis dire tion.126uinti superpotential: W = Q H L ¯ H ξ + Q H L ¯ H ξ + Q u c ¯ H ¯ H H + Q u c H ¯ H ¯ H + d c u c H ¯ H ξ + d c H H Φ V + d c u c H ¯ H ξ + d c H H ¯Φ ′ V + H ¯ H ¯ H ¯ H H + H ¯ H H ¯ H ¯ H + H ¯ H ¯ H ¯ H ¯Φ αβ + H ¯ H Φ αβ H H + H ¯ H ¯Φ αβ ¯ H ¯ H + L ¯ H N c ¯ H H + L H N c ¯ H ¯ H + H H ¯Φ ′ H H + H H Φ N c V + H H ¯Φ ′ N c V + H H ¯Φ ′ ¯ H ¯ H + H ¯ H ¯ H ¯ H H + H ¯ H ¯ H H ¯ H + H ¯ H ¯ H ¯ H ¯Φ αβ + H ¯ H Φ αβ H H + H ¯ H ¯Φ αβ ¯ H ¯ H + H H ξ H H + H H ¯Φ ′ Φ αβ ¯ H + ¯ H ¯ H Φ ′ ¯ H ¯ H + ¯ H ¯ H Φ ′ H H + ¯ H ¯ H ¯ H H H + ¯ H ¯ H H H ¯ H + ¯ H ¯ H ¯ H H H + ¯ H ¯ H H ¯ H H + ¯ H ¯ H H ¯ H H + ¯ H ¯ H H ¯ H H + ¯ H ¯ H ξ ¯Φ αβ ¯ H + ¯ H ¯ H Φ αβ ¯Φ αβ H + ¯ H ¯ H Φ αβ ¯Φ αβ H + ¯ H ¯ H H H ¯ H + ¯ H ¯ H H H ¯ H + ¯ H ¯ H H ¯ H H + ¯ H ¯ H H ¯ H H + ¯ H H H H H + ¯ H ¯ H H H ¯ H + ¯ H ¯ H Φ ′ ¯Φ αβ H + ¯ H H H ¯ H ¯ H + ¯ H H H ¯ H ¯ H + ¯ H H H ¯ H ¯ H + ¯ H H Φ αβ ¯Φ αβ ¯ H + ¯ H H Φ αβ ¯Φ αβ ¯ H + ¯ H H H ¯ H ¯ H + ¯ H H H ¯ H ¯ H + ¯ H H ¯ H ¯ H H + ¯ H H ¯ H ¯ H H + ¯ H ¯ H H ¯ H ¯Φ αβ + ¯ H ¯ H ¯ H ¯ H Φ ′ + ¯ H ¯ H ¯Φ ′ ¯Φ αβ ¯Φ αβ + ¯ H ¯ H ¯Φ ′ ¯Φ αβ ¯Φ αβ + ¯ H H Φ αβ H H + ¯ H H ¯Φ αβ ¯ H ¯ H + ¯ H ¯ H ¯ H H ¯Φ αβ + ¯ H ¯ H Φ ′ ¯ H ¯ H + ¯ H ¯ H H ¯ H ¯Φ αβ + ¯ H ¯ H H ¯ H ¯Φ αβ + ¯ H ¯ H H ¯ H ¯Φ αβ + ¯ H ¯ H Φ αβ ¯Φ αβ ¯Φ αβ + ¯ H ¯ H Φ αβ ¯Φ αβ ¯Φ αβ + ¯ H ¯ H Φ αβ ¯Φ αβ ¯Φ αβ + ¯ H ¯ H ¯Φ αβ H ¯ H + ¯ H ¯ H ¯Φ αβ H ¯ H + ¯ H ¯ H ¯Φ αβ ¯ H H + ¯ H ¯ H ¯Φ αβ ¯ H H + H H H H ¯Φ ′ + H H Φ ′ Φ αβ Φ αβ + H H Φ ′ Φ αβ Φ αβ + H H Φ ′ H H + H H Φ H H + H ¯ H ¯Φ αβ H H + H H ¯Φ ′ H H + ¯ H ¯ H Φ ′ H H + ¯ H ¯ H Φ ′ ¯Φ αβ ¯Φ αβ + ¯ H ¯ H Φ ′ ¯Φ αβ ¯Φ αβ + ¯ H H Φ αβ H H + ¯ H H ¯Φ αβ ¯ H ¯ H + H H ¯Φ ′ Φ αβ Φ αβ + H H ¯Φ ′ Φ αβ Φ αβ + H H ¯Φ ′ H H + ¯Φ ′ ¯Φ αβ ¯Φ αβ H H + ¯Φ ′ ¯Φ αβ ¯Φ αβ H H + Φ N c V ¯ H ¯ H + Φ N c V ¯ H ¯ H + Φ ′ N c V ¯ H ¯ H + Φ ′ N c V ¯ H ¯ H + Φ ′ ¯ H ¯ H ¯ H ¯ H + Φ ′ ¯ H ¯ H ¯ H ¯ H + ¯Φ ′ H H H H + ¯Φ ′ H H H H + Φ N c V H H + Φ N c V H H + Φ N c V H H + Φ N c V H H + Φ ¯Φ αβ ¯Φ αβ H H + Φ ¯Φ αβ ¯Φ αβ H H + N c V ¯Φ αβ H ¯ H + N c ¯Φ αβ H ¯ H V + H ¯ H Φ αβ H H + H ¯ H ¯Φ αβ ¯ H ¯ H + H ¯ H Φ αβ H H + H ¯ H ¯Φ αβ ¯ H ¯ H + H ¯ H Φ αβ H H + H ¯ H ¯Φ αβ ¯ H ¯ H + Φ αβ Φ αβ ¯Φ αβ H H + Φ αβ Φ αβ ¯Φ αβ H H + Φ αβ ¯Φ αβ ¯Φ αβ ¯ H ¯ H + Φ αβ H H ¯ H H + Φ αβ H H ¯ H H + Φ αβ H ¯ H H H + Φ αβ H ¯ H H H + Φ αβ Φ αβ ¯Φ αβ H H + Φ αβ ¯Φ αβ ¯Φ αβ ¯ H ¯ H + Φ αβ ¯Φ αβ ¯Φ αβ ¯ H ¯ H + ¯Φ αβ H H H H + ¯Φ αβ H ¯ H ¯ H ¯ H + ¯Φ αβ H ¯ H ¯ H ¯ H + ¯Φ αβ ¯ H ¯ H ¯ H H + ¯Φ αβ ¯ H ¯ H ¯ H H . (B.1)127ppendix CC.1 Weight roots of E representations in the twisted se tor θ of the SO (4) model p sh = p − V p sh,DL ( E ) (1 , − / , − / , ) (0 , , , , , − , − / , − / , ) (1 , , − , , , , / , / , , ) (0 , − , , , , − , / , / , − , ) ( − , , , , , , / , / , , , ) (0 , , , , , − , / , / , , − , ) (0 , − , , , , , / , / , , , ) ( − , , , , − , , / , / , , − , ) (0 , , , − , , , / , / , , ,
0) ( − , , , − , , , / , / , , − ,
0) (0 , , , , , − , / , / , ,
1) (0 , , , , − , − , / , / , , −
1) ( − , , , , − , − / , , , / , / , / , / , /
2) (0 , , , , − , − / , , , − / , − / , / , / , /
2) (0 , , , , − , − / , , , − / , / , − / , / , /
2) (0 , , , − , , − / , , , − / , / , / , − / , /
2) (0 , , − , , − , − / , , , − / , / , / , / , − /
2) ( − , , − , , , − / , , , / , − / , − / , / , /
2) (1 , − , , − , , − / , , , / , − / , / , − / , /
2) (1 , − , , , , − / , , , / , − / , / , / , − /
2) (0 , − , , , , − / , , , / , / , − / , − / , /
2) (1 , , , , − , − − / , , , / , / , − / , / , − /
2) (0 , , , − , , − / , , , / , / , / , − / , − /
2) (0 , , − , , , − / , , , − / , − / , − / , − / , /
2) ( , , , , , )( − / , , , − / , − / , − / , / , − /
2) (0 , , , − , , − / , , , − / , − / , / , − / , − /
2) (0 , , − , , , − / , , , − / , / , − / , − / , − /
2) (0 , , − , , , − / , , , / , − / , − / , − / , − /
2) (1 , − , , , , Table .1 ontains the 28 roots whi h ful(cid:28)l the massless equation for the twistedse tor θ for the (cid:28)xed torus T . The solutions p sh , shifted by V , are showed in the (cid:28)rst olumn. In the se ond olumn the roots are rewritten in Dynkin labels with respe t to E . The (cid:28)rst root is a singlet of E . The other 27 belong to the same multiplet and form128n fa t the of E . The highest weight of the representation is ( , , , , , ) .These states are singlets under the hidden E ′ gauge group.The simple roots of E are given below : α = ( − / , − / , − / , / , − / , − / , − / , / α = (0 , , , − , , , , α = (1 / , / , / , / , − / , − / , / , / α = (0 , , , , , , − , α = (0 , , , , , − , , − α = ( − / , − / , − / , − / , − / , / , / , − / . (C.1)129ppendix DD.1 Total amplitude ontributions of the shift orbifold ineq.(5.16) Z o,ab = ( V − S )Λ Λ Λ m ′ ,n ′ Λ m,n [( O − S )( O − S )] S −→Z ab,o = ( V − S )Λ Λ Λ m ′ ,n ′ Λ m,n [( V + C )( V + C )] T −→Z ab,ab = ( V − S )Λ Λ Λ m ′ ,n ′ Λ m,n [( V − C )( V − C )] → S invariant Z o,a = ( V − S )Λ Λ Λ m ′ ,n ′ ( − m Λ m,n [( O − S )( O + S )] S −→Z a,o = ( V − S )Λ Λ Λ m ′ ,n ′ Λ m,n +1 / [( V + C )( O + S )] T −→Z a,a = ( V − S )Λ Λ Λ m ′ ,n ′ ( − m Λ m,n +1 / [( − V + C )( O + S )] → S invariant Z o,b = ( V − S )Λ Λ Λ m ′ ,n ′ ( − m Λ m,n [( O + S )( O − S )] S −→Z b,o = ( V − S )Λ Λ Λ m ′ ,n ′ Λ m,n +1 / [( O + S )( V + C )] T −→Z b,b = ( V − S )Λ Λ Λ m ′ ,n ′ ( − m Λ m,n +1 / [( O + S )( − V + C )] → S invariant Z a,b = ( V − S )Λ Λ Λ m ′ ,n ′ ( − m Λ m,n +1 / [( V + C )( O − S )] T −→Z a,ab = ( V − S )Λ Λ Λ m ′ ,n ′ Λ m,n [( C − V )( O − S )] S −→Z ab,a = ( V − S )Λ Λ Λ m ′ ,n ′ ( − m Λ m,n +1 / [( − V + C )( V + C )] T −→Z ab,b = ( V − S )Λ Λ Λ m ′ ,n ′ ( − m Λ m,n +1 / [( V + C )( − V + C )] → S Z b,ab = ( V − S )Λ Λ Λ m ′ ,n ′ Λ m,n +1 / [( O − S )( C − V )] T −→Z b,a = ( V − S )Λ Λ Λ m ′ ,n ′ ( − m Λ m,n [( O − S )( V + C )] . (D.1)D.2 Left amplitudes of Z × Z orbifold model in eq.(5.43)In this se tor we assume that the (cid:28)rst three elements of ea h produ t orrespond to the ompa t spa e, hen e they feel the a tion of the Z × Z orbifold.130ntwisted g = ( O O O O + V V V V ) O + ( S S S C + C C C S ) C ; g g = ( O V V O + V O O V ) O + ( S C C C + C S S S ) C ; g h = ( V V O O + O O V V ) O + ( C C S C + S S C S ) C ; g f = ( V O V O + O V O V ) O + ( C S C C + S C S S ) C . (D.2) G ′ oh = ( S S S S + C C S S − C S C S − S C C S − C S S C − S C S C + S S C C + C C C C ) S +( − V O O O − O V O O + O O V O + V V V O + O O O V + V V O V − V O V V − O V V V ) V ; G ′′ oh = ( S S S C + C C S C − C S C C − S C C C − C S S S + S S C S + C C C S − S C S S ) O +( O O O O + V V O O − V O V O − O V V O − V O O V − O V O V + O O V V + V V V V ) C ; G ′′′ oh = ( S S S S + C C S S − C S C S − S C C S − C S S C − S C S C + S S C C + C C C C ) V +( − V O O O − O V O O + O O V O + V V V O + O O O V + V V O V − V O V V − O V V V ) S ; (D.3)where ea h of these (D.3) ontributions do not play a role in the massless untwistedspe trum, beside they an ontribute in the twisted massless se tor.Twisted se tor h g h = ( S C O O + C S V V ) O + ( V O S C + O V C S ) C ; g hg = ( S S V O + C C O V ) O + ( O O S S + V V C C ) C ; g hh = ( C S O O + S C V V ) O + ( V O C S + O V S C ) C ; g hf = ( C C V O + S S O V ) O + ( O O C C + V V S S ) C . (D.4)131wisted se tor g g g = ( O C S O + C V O S + S O V C + V S C V )( O + C )+ ( O C C O + C V O C + S O V S + V S S V ) S + ( S O O C + V S C O + C V V S + O C S V ) V ; g gg = ( O S C O + S V O C + C O V S + V C S V )( O + C )+ ( O S S O + S V O S + C O V C + V C C V ) S + ( C O O S + V C S O + S V V C + O S C V ) V ; g gh = ( S O O S + V S S O + C V V C + O C C V )( O + C )+ ( S O O C + V S C O + C V V S + O C S V ) S + ( O C C O + C V O C + S O V S + V S S V ) V ; g gf = ( C O O C + V C C O + S V V S + O S S V )( O + C )+ ( C O O S + V C S O + S V V C + O S C V ) S + ( O S S O + S V O S + C O V C + V C C V ) V . (D.5)Twisted se tor f g f = ( C O S O + V C O S + O S V C + S V C V )( O + C )+ ( C O C O + V C O C + O S V S + S V S V ) S + ( O S O C + S V C O + V C V S + C O S V ) V ; g fg = ( O C O C + C V C O + V S V S + S O S V )( O + C )+ ( O C O S + C V S O + V S V C + S O C V ) S + ( S O S O + V S O S + O C V C + C V C V ) V ; g fh = ( O S O S + S V S O + V C V C + C O C V )( O + C )+ ( O S O C + S V C O + V C V S + C O S V ) S + ( C O C O + V C O C + O S V S + S V S V ) V ; g ff = ( S O C O + V S O C + O C V S + C V S V )( O + C )+ ( S O S O + V S O S + O C V C + C V C V ) S + ( O C O S + C V S O + V S V C + S O C V ) V . (D.6)132or ompleteness we present also the twisted amplitudes whi h do not ontribute tothe low energy spe trum G ′ h = ( C C O O + S S O O + S C V O + C S V O + S C O V + C S O V + C C V V + S S V V ) V +( O O S C + O O C S + V O C C + V O S S + O V C C + O V S S + V V S C + V V C S ) S ; G ′′ h = ( O O C C + O O S S + V O S C + V O C S + O V S C + O V C S + V V C C + V V S S ) O +( S C O O + C S O O + C C V O + S S V O + C C O V + S S O V + S C V V + C S V V ) C ; G ′′′ h = ( O O S C + O O C S + V O C C + V O S S + O V C C + O V S S + V V S C + V V C S ) V +( C C O O + S S O O + S C V O + C S V O + S C O V + C S O V + C C V V + S S V V ) S . (D.7)One obtains analogous expressions by applying T transformations on ea h of the previ-ous amplitudes, giving rise to G ′ hh , G ′′ hh and G ′′′ hh respe tively.133.3 Right amplitudes of Z × Z orbifold model in eq.(5.43)We assume that the (cid:28)rst element of the following produ ts orresponds to spa etimedegrees of freedom, hen e the a tion of the Z × Z orbifold applies on the last threeelements. τ = V O O O + O V V V − S S S S − C C C C ,τ g = O V O O + V O V V − C C S S − S S C C ,τ h = O O O V + V V V O − C S S C − S C C S ,τ f = O O V O + V V O V − C S C S − S C S C ,τ g = V O S C + O V C S − S S V O − C C O V ,τ gg = O V S C + V O C S − S S O V − C C V O ,τ gh = O O S S + V V C C − C S V V − S C O O ,τ gf = O O C C + V V S S − S C V V − C S O O ,τ h = V S C O + O C S V − C O V C − S V O S ,τ hg = O C C O + V S S V − C O O S − S V V C ,τ hh = O S C V + V C S O − S O V S − C V O C ,τ hf = O S S O + V C C V − C V V S − S O O C ,τ f = V S O C + O C V S − S V S O − C O C V ,τ fg = O C O C + V S V S − C O S O − S V C V ,τ fh = O S O S + V C V C − C V S V − S O C O ,τ ff = O S V C + V C O S − C V C O − S O S V ,,