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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. hrce al of table Character G group gauge the of Higgsing F bootstrap Conformal E geometry toric and manifolds Calabi-Yau D characters and algebra Superconformal C lattices on notes Some B conventions - objects modular some of Definition A Outlook and Conclusion 6 . acltn h rmvWte/oaua-aainvaria Gromov-Witten/Gopakumar-Vafa the Calculating 5.5 3-folds Calabi-Yau on compactifications prepoten II moduli Type vector the and 5.4 couplings Gravitational 5.3 orbifolds CHL 5.2 . aaiYumnflsa yesrae ntrcambient toric in hypersurfaces as manifolds Calabi-Yau D.3 proj weighted in hypersurfaces as manifolds Calabi-Yau properties and D.2 definitions Basic D.1 C.2 (Extended) C.1 forms Jacobi Weak functions A.3 theta Jacobi and A.2 series Eisenstein A.1 CY dual the finding - duality II Type - Heterotic 5.6 spaces prepotentia bootstrap the Conformal - invariants Gromov-Witten zero 5.5.2 Genus 5.5.1 prepotential the and couplings Gravitational 5.4.2 Spectrum 5.4.1 and corrections couplings/threshold Gravitational spectrum 5.2.2 The 5.2.1 index supersymmetric new The 5.1.2 N characters 4 = esmercindex persymmetric N characters 2 = M 24 η -function 5 cieambient ective tial spaces nts h e su- new the l 85 90 83 76 96 81 93 79 79 83 71 64 79 70 61 84 81 59 86 66 57 73 87 85 66 54 70 68 Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. 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Skarke, H. Schimpf, M. Kidambi, A. Chowdhury, A. Banlaki, A. ] .Bnai .Coduy .Kdmi .Schimpf, M. Kidambi, A. Chowdhury, A. Banlaki, A. ] .Bnai .Coduy .Rue,T Wrase, T. Roupec, C. Chowdhury, A. Banlaki, A. ] .Bnai .Catpdyy,A iab,T ciank .S M. Schimannek, T. Kidambi, A. Chattopadhyaya, A. Banlaki, A. ] n rmvWte invariants Gromov-Witten and 22)23 [1911.09697]. 203, (2020) h swampland the rtcsrnson strings erotic aaiYumnflsadsoai groups sporadic and manifolds Calabi-Yau 35 .TeMntrmdl a lobe eae o3dmninlg dimensional 3 to related been also has module Monster The ]. 36 ]. HP10 21)05 [1811.07880] 065, (2019) 1903 JHEP , 37 ( .I section In ]. K 3 × K T 3 K × 2 3 ) /Z × T 2 T 3 5 HP0 22)082,[1811.11619] (2020) 02 JHEP , ssuid hog ult ihtp Istring II type with duality Through studied. is 2 n hi ulClb-a threefolds Calabi-Yau dual their and h onigfnto fdos(lcrcand (electric dyons of function counting The . ahe onhn nCLobflso het- of CHL-orbifolds in moonshine Mathieu 8 HP10 21)19 [1711.09698] 129, (2018) 1802 JHEP , 31 , cln iiso Svcaand vacua dS of limits Scaling 38 a n enrmodels.This Gepner and dal ]. tcgnsof genus ptic section n nMtiuMoonshine Mathieu On oncin.W start We connections. e oa aaiYuman- Calabi-Yau ional sas enconnected been also as sas enexplored been also as tosptnilrela- potential ations ulC.W explain We CY. dual tr n calculating and cters .Te nsection in Then h. papers ftemicrostates the of t Yumnflsfor manifolds -Yau nb bandby obtained be an 2 sdrtp II type nsider nsection In . chimpf, K ,se e.g, see, 3, HP04 JHEP , aiyby ravity rase, Het- 4 3 Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. 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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. h pcso ekJcb om fee egtaditgrindex integer and weight even integer of half forms and Jacobi weight weak even of of spaces forms the Jacobi weak of spaces The h pc fwa aoifrso vnweight even of forms Jacobi weak of space The h iesenseries Eisenstein the om fwih n index and 0 weight of forms oepeieyoefinds one precisely More nAppendix in .Fo h bv rprisi olw that follows it properties above the From 3. nt dimensional. finite n ti alda called is it and ial faJcb omstse h e ekrcondition weaker yet the satisfies form Jacobi a if Finally n h pc falhlmrhcrs.csia,wa,wal ho weakly weak, weight cuspidal, of resp. forms holomorphic all of space The ti called is it called uniy∆=4 = ∆ quantity c hc losfraFuirexpansion Fourier a for allows which where ( ,r n, ≥ = ) n J J J J J ˜ ˜ ˜ ˜ ˜ holomorphic 0 q 0 0 0 0 0 A , , , , , 5 4 1 2 3 o oefixed some for := n a aiycnic nsl httespace the that oneself convince easily can One . = = = = = C aoics form cusp Jacobi a (4 { { { { { e φ φ φ φ φ 2 nm πiτ E 0 5 0 4 0 3 0 0 2 mn , , , , , 1 1 1 1 1 k 4 y , E , E , E , } E , ekJcb form Jacobi weak ( − n index and τ − if ) 4 4 4 4 r := E , φ φ φ φ 2 r C − 2 − 2 − 2 − 2 r , 2 m c c (∆ 6 2 2 2 2 e n scle the called is ( ( ( , , , , where ) 1 1 1 1 2 ,r n, ,r n, τ c sgnrtdby generated is 0 φ φ φ φ } πiz r , ( n h functions the and ) k,m 0 3 0 2 0 ,r n, ∈ , , , aihsfr∆ for vanishes ) 1 1 1 npriua h rpry( property the particular In . nes4 unless 0 = ) nes4 unless 0 = ) m E , E , E , ( Z unless 0 = ) ,z τ, if 6 6 6 sdntdby denoted is ti called is it φ φ φ C C = ) − 3 − 3 − 3 (∆ (∆ 16 if 2 2 2 , , , 1 1 1 discriminant r , r , φ φ } X n,r 0 2 0 aihsfr∆ for vanishes ) nydpnson depends only ) , , φ 1 1 k m E , E , k,m c n nee index integer and ( ,r n, ai lmnsfor elements basis 4 4 2 2 ( ekyhlmrhcJcb form. 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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. to subgroup A ie w rus( groups two Given 1 Identity: (1) fulfilled: are properties following the that such enlof kernel and itself oehrwt h etito ftebnr prto to operation binary the of restriction the with together eainhp[ relationship ntefloigw ilmsl utwrite just mostly will we following the In 2 Associativity: (2) hwta h ubro ieridpnetcaatr sequ is characters group independent the linear of representation of in irreducible appropriate number an the defining by that and show space vector a form functions homomorphism representation A to casfntos,ie,te r osato equivalence on constant are they i.e., ‘class-functions’, have of 3 nes element: Inverse (3) called necagal n ipywrite simply and interchangeably ru sapi ( th in pair a properties rol is important important group most an A their play ab and them groups, subsection introduce sporadic the will called in so discussed groups, objects finite modular the from Apart groups Sporadic 2.2 ewl fe s h hrhn notation shorthand the use often will we g subgroup A ch ∈ R 7 8 H H G/kerφ edefine We ycmo bs fntto,frarepresentation a for notation, of abuse common By G : sanra ugopof subgroup normal a is , aN epcigtebnr prto,i.e., operation, binary the respecting a Sc neeeti unique.) is element an (Such G utetgroup quotient 7 − .. if i.e., , 1 → · φ ∈ { bN . eoe by denoted , e C G N } N aN 45 g , . = s.t. , fagroup a of hr xs nelement an exist There fagroup a of ] N := ρ abN 7→ R : { sivratudrcnuain o omlsubgroup normal a For conjugation. under invariant is G, G, G a ab ( = Tr ec h e flf cosets left of set the Hence . group A . o all For · | → · b V · a ossigo set a of consisting ) ,( ), ,ρ V, ∈ ( − o each For ρ e φ ker Hom( 1 N ( G ,⋆ H, J = g G ˜ fagroup a of ) } 2 )) , k,m scalled is eoe by denoted , ,b c b, a, a Na g .. h e feeet in elements of set the i.e., , ru homomorphism group a ) − 8 V nta of instead + G ytecci rpryo h rc,caatr are characters trace, the of property cyclic the By . 1 G a .Gvnarepresentation a Given ). := 2 1 h mg of image The . a scalled is = = ∈ { ∈ ba G e φ normal Teivreeeeti unique.) is element inverse (The . G | 0 ( , b G , 3 2 G e ρ ∈ hr xssa nes lmn,dntdby denoted element, inverse an exists there a J ( 17 stpecnitn favco space vector a of consisting tuple is ˜ hrceso nt rusmyb listed be may groups finite of Characters . ∈ ab g N N 2 · ) simple k,m φ G G b } eoe by denoted , G ( ≤ ) . for a − n iayoperation binary a and hnrfrigt ru ( group a to referring when · s.t. , · 1 G c m , b a = sa subset as is , = ) G G/N · fisol omlsbrusare subgroups normal only its if R b a saasbru of subgroup a a is . ( = e φ · ∈ ( ( · lse ftegroup the of classes b a = ,ρ V, a N Z φ ) · ⊳G N . φ ⋆ c { = R G : × holds. ) ewl fe use often will we ) aN G edfiethe define we , a ( N apdt h identity the to mapped b → | · o all for ) N if , ssubsection. is a nMosie We Moonshine. in e e rdc n may one product ner gi om group. a forms again e lt h ubrof number the to al ∈ v,cranspecial certain ove, H ⊆ = gNg G G sampfrom map a is a } · o all for − ..teset the s.t. H omagroup, a form ,b a, : 1 G = isomorphic G, × ∈ G ⊳ N G character N R V · These . G G .Also ). a (2.34) o all for and The . n a and ∈ → we G N G G G V . Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. hs ru fteLehltie(e Appendix (see lattice Leech the of group phism eas fissz hc is which size its of because Co 0o h 6soai rusaesbrusof subgroups are groups sporadic 26 the of 20 ncaatrtbe ihclmslble yeuvlnecla equivalence by labelled columns with tables character in n 26 and pi pit rusblnigt n‘nnt aiy,which family’, ‘infinite an to [ belonging Groups’ groups Finite into of up ‘ATLAS split the in collected been have th from mainly follows This fi groups. all simple classify finite to all order classify In representations. irreducible the h ags ahe group Mathieu largest The fr apart are us to groups importance Mathieu special the of ar be will groups that sporadic groups The 6 al remaining are group the Monster pariah. whereas the th family’ of for ‘happy subgroups stands as the groups appear two which between groups line The a where groups sporadic tutdb resi 90[ 1980 in Griess by structed h ags prdcgopi h ocle (Fischer-Griess) called so the is group sporadic largest interest. of The be will groups sporadic ,B... A, hc ttsta o xdgroup fixed a for that states which uhthat such osbepermutation. possible lastesm n h utetgroups quotient the and same the always olcieeoto ag ubro ahmtcas(e,e (see, mathematicians of number large a of effort collective | M 9 1 • • • hs r omlylble yteodro neeeti h ls follo class the in element an of order the by labelled normally are These lsicto falfiiesml ruswsahee nthe in achieved was groups simple finite all of Classification | a eotie rmteCna group Conway the from obtained be can fteei oete n qiaec ls iheeet ftesam the of elements with class equivalence one then more is there if , 2 = h ylcgroups cyclic the 6fmle fLetype Lie of families 16 h lentn groups alternating the prdcgroups sporadic 46 H · i 3 snra in normal is 20 · 5 9 G · M 7 = 6 24 Z hc ontbln oayifiiefml.Fru otythe mostly us For family. infinite any to belong not do which · H M , 11 p 0 ,( H 2 ⊃ | p 23 M M i · − A prime) a 47 13 H 24 1 24 swl steCna group Conway the as well as n for , ) omlydntdby denoted normally ]), 1 and 3 | a size has ⊃ 2 = · 17 G H H n 10 · 2 i n n etdsequence nested any and 19 ≥ − · ⊃ · · · ⊃ 1 3 /H 18 5 · 3 23 H · i 5 i − · ssml,telnt ftesqec is sequence the of length the simple, is Co · M 29 1 7 /H H Figure . B 0 · · k 11 ). yiscentre. its by 31 i ⊃ pern r h aeu oa to up same the are appearing · · H 23 41 46 k iegop ti ucetto sufficient is it groups nite . +1 .Tecasfiaincnbe can classification The ]. · 1 M 47 sses hw irmo l the all of digram a shows = e Co totie t name its obtained It . are · Jordan-H¨older theorem g [ .g, { mteMntrgroup Monster the om 59 1 9 e ose group Monster . ugoprelation. subgroup e } n oslble by labelled rows and Co e yacptlletter, capital a by wed ≈ · order. e oeie called sometimes e 1 , 0 al 90 ya by 1980s early orfre oas to referred so 2 8 steautomor- the is ) h results The ]). × 10 53 . (2.35) (2.36) (con- Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. pern nuba onhn.W alagroup a call We G moonshine. umbral in appearing hs ru fteetne iayGlycode Golay binary extended the of group phism n h translation the and tcnb huh fa ugopo h emtto group group permutation The the of subgroup as of thought be can It { fdt uhta n -i ro a ecretd tcnit fa12-dimen a of consists It corrected. be can error 3-bit any that such data of eoe h ffielna ru endtruhtentrlacti natural the through defined group linear affine the denotes points, For sdfie ob h ru finvertible of group the be to defined is ti endt etesbru of subgroup the be to defined is It for 0 10 11 , ≃ 1 i h xeddGlycd sa ro-orcigcd htecds12 encodes that code error-correcting an is code Golay extended The emtto group permutation A } q eedwt oentto htwl enee npriua fo particular in needed be will that notation some with end We 1 = 24 2 rm oe,w define we power, prime a .H a f2-i od uhta n w itnteeet of elements distinct two any that such words 24-bit of k , . . . , 1 a , . . . , if , M G . iue1 Figure 23 k a omlsubgroup normal a has and so order of is b x 1 b . . . 7→ h prdcsml rus iuefo [ from Figure groups. simple sporadic The : G k x cigon acting , + a | i M v 6= 23 F for a q | j 2 = ob h nt edwith field finite the be to and n v M ∈ onsi called is points 7 24 Z n b F i · × fodr2(hence 2 order of htkesoepitfixed. point one keeps that 3 6= q n 19 2 . n b · j arcswt offiinsin coefficients with matrices 5 for · 7 10 i · 6= nparticular In . 11 G k j -transitive hr xssa exists there , · h obecvro group a of cover double the 23 W . ie na es coordinates. 8 least at in differ Z q scnrl s.t. central) is S lmns Then elements. ffraytogv esof sets give two any for if 24 M hti h automor- the is that nof on i fdt no24-bit into data of bit g inlsubspace sional 24 ∈ 48 s5-transitive is G ]. GL F with h groups the r q . n G/Z ( AGL q g on ) ( GL a (2.37) i ≃ = ) W n n ( ( H H F 11 q q of b q n ) ) i , . . Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. ahmtc.Frhrlgtwsse nti ujc yJh Th John by subject this on shed was light Further co lar previously between mathematics. between coincidence connection some surprising just a was this constituted if clear not was it endi ( in defined h xaso offiinso h li--ucina n( in as Klein-J-function the John of by observation coefficients the expansion with the started moonshine of subject The Moonshine Monstrous 3.1 studied well moonshine. and known Monstrous examples discuss with briefly known starting will and we section encounters this first In - Moonshine 3 uso iesoso reuil ersnain fteMo the of representations write irreducible may of dimensions of sums n otncnetrdta o each for that conjectured Norton and repr irreducible of dimensions the are r.h.s. the on numbers rpsdteeitneo rddrepresentation graded a of existence the proposed orsodt apmdl fvrosgns0mdlrgroups. ( modular observations 0 the genus that various clear of Hauptmoduls to correspond hr h ubr ntelhs r h xaso coefficients expansion the are l.h.s. the on numbers the where with ersnainof representation hswsdn yCna n otn[ Norton and Conway by done was This eea eis o called now series, general twste h ugsinb hmsnt osdrfrevery for consider to Thompson by suggestion the then was It V − 1 = 3.4 ρ 0 V , 6297 4692 1986+2 + 21296876 + 842609326 = 864299970 )cicdswt h nqeHutou ihexpansion with Hauptmodul unique the with coincides )) 1970=2267 983+1 + 196883 + 21296876 = 21493760 1 M J 984=128 1 + 192883 = 196884 = T ( g z ( ρ ree ydmninsc that such dimension by ordered dim = ) z 1 =ch := ) ⊕ . . . ρ ca-hmsnseries McKay-Thompson 0 V , 3.1 q V,q V ( 2 V eentmr oniecs oepeieyConway precisely More coincidences. mere not were ) = ( = =dim( := ) g ch = ) ρ V 2 − ⊕ 1 g ⊕ ρ ∈ V 1 20 V − V ⊕ M 1 1 3 − ( h on htvarious that found who ] ⊕ g ρ 1 ) ) h ca-hmsnseries McKay-Thompson the 0 V q q V , − − 2 1 1 3 ⊕ + + V = , . . . X X i i fteMntrgroup Monster the of ... ∞ =1 ∞ =1 · 983+2 + 196883 and ch dim( pee neae at of parts unrelated mpleted 2.16 snain of esentations V i enmeso fi really it if or numbers ge se ru.Ta swe is That group. nster ( ρ xmlso Moonshine of examples V g i a eepesdas expressed be can ) ) i ) q en h irreducible the being q ca n17 that 1978 in McKay i . i hrb tbecame it Thereby . n( in g msn[ ompson · 1 ∈ 2.16 T M M g q ( n the and ) − z Initially . h more the T indeed ) 1 49 g + ( z who ] (3.2) (3.4) (3.3) (3.1) O (as ) ( q ) Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. coincidence. enrivltos ti h omlsro Γ of normaliser the is It involutions. Lehner o hc Γ which for ftesrs nrytno.Te aif h elknown well the satisfy They relations tensor. energy stress the of near h owyNro ojcuewsvrfidb ti,Fn n S and Fong Atkin, by verified was conjecture. conjecture Conway-Norton Conway-Norton the The by explained somewhat is this a edter CT.Teifiieia ofra transforma conformal infinitesimal The (CFT). theory field mal vect Leec no [ the with follow lattice on 24-dimensional bosons even 24 self-dual, unique considering by starts construction This [ mo (FLM) module graded Monster Meurman the envisioned and the Lepowsky of Frenkel, construction by explicit An ically. ftecci rpryo h rc.TeMntrgophs194 has group Monster at The evaluated at trace. evaluated character the a of Moreover property cyclic the of rvosyi a led enntdb nrwOgi 95[ 1975 in Ogg Andrew by noted been already had it Previously 171. is series Thompson ca-hmsnseries McKay-Thompson a ed.W ilntg nodphhr ahrw ilexplai will we rather here depth [ into in go given not approach formul will precise We mathematically field). a mal is (which algebra operator sanra ugopwhere subgroup normal a as ftecmlxpaeaegnrtdb h momenta the by generated are plane complex the of where n r xcl h rm iioso h re fteMonster the of order the of divisors prime the exactly are and ler a egvnby given be may algebra 13 12 aosyOgoee oteo akDne’ hse otefis person first the to whiskey Daniel’s Jack of bottle a offered Ogg Famously Γ 0 z ( p c 7 +i h ru eeae yΓ by generated group the is )+ → n ei h icsinwt oebsc oin f2dimens 2 of notions basics some with discussion the begin and ] ∈ i R 0 ∞ ( p g stecnrlcag fteter.Rpeettoso h Vi the of Representations theory. the of charge central the is )+ − o oegnszr ugopΓ subgroup zero genus some for 1 12 h oa ubro apmdl rsn rmteMacKay- the from arising Hauptmoduls of number total The . [ 7 L a eu are 0 genus has { ]. 2 V m , L , ♮ 3 h L osrcinmksueo h ocp fvertex of concept the of use makes construction FLM The . , primary 5 n T ( = ] , g 11 ( z N , nydpnso h ojgc ls of class conjugacy the on depends only ) 13 m L ∈ n , − fields 17 0 N = g ( n p , , oehrwith together ) I ilb h ope ojgt ftecharacter the of conjugate complex the be will ) 19 N 0 L ( p | , m φ 2 in )  dz 23 21 ( πi + ord z n , facrancnomlweight conformal certain a of ) z SL 29 + n ( +1 g , (2 12 52 c 31 ) g T , · ( R .Ti ouei yial called typically is module This ]. ( , ≤ m gcd z ). 41 √ 1 (3.6) ) 3 p SL , (24 − 47 − iaoocmuao algebra commutator Virasoro 1 0 2 m p , ( ro eghls hn2 We 2. than less length of or ord , 59 0 R )
δ ,wihcnan Γ contains which ), hc r nw sAtkin- as known are which , , m to facia confor- chiral a of ation 71 ( ru ( group + uewsfial given finally was dule g atc,wihi the is which lattice, h n, 50 } )) 0 h oephysical more the n tions  htteprimes the that ] ojgc classes. conjugacy oieta the that Notice . ih[ mith 2.35 z oepanthis explain to oa confor- ional → ) 51 13 g z h Clearly . numer- ] because + defined rasoro ǫz 0 (3.7) (3.5) ( n N +1 p ) Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. nodrt banirdcberepresentations. irreducible obtain to order in ofra edter ossso et n right-moving a ( and weight left- conformal by a given of part consists holomorphic) theory field conformal A nsc ihs egtsaeoeotisarpeetto o representation a obtains one state for weight which highest a such on fulfils rmr edo h aumoeotisa t obtains singular one the vacuum all the state explicitly on only field we primary r.h.s. the on where yteroeao rdc xaso OE ihteenergy-m the with (OPE) expansion product operator their by h hoyi endas defined is theory the nsc a rmapiayfil fcnomlweight conformal of field primary a from way a such in enwcnie h oceeeapeo 4fe bosons free 24 w of we example which concrete following the the in consider in now pa We interested holomorphic be the will consider we can situation one and parts right-moving and em module Verma hr h rc stknoe h ibr pc Vramodule) (Verma space Hilbert the over taken q is trace the where nti context this In oe ihantrlitgrgrading integer natural a with comes where endby defined theory. ics ndti ee For here. detail in discuss nnt ubro rmr ed.I pcfi iutos( situations specific In fields. primary of number infinite u togrqieet ntepsil omof form possible the on requirements strong put =exp(2 := 14 ngnrluiaiyrsrcstevle of values the restricts unitarity general In L N 0 Z h, | c h > c h ¯ ( πiτ i q, onstenme fpiayfilso ofra egt( weight conformal of fields primary of number the counts = q ¯ ), and 1 hudb nain ne oua rnfrain,( transformations, modular under invariant be should ) h T n o h ntr,irdcbecs eoe by denoted case irreducible unitary, the for and τ n | ch ( h z en h oua aaee ftetrs twl eo h for the of be will It torus. the of parameter modular the being i sas aldthe called also is L 1 ) L , > h h,c φ ( n ( z q | 2 h ilb ntr n irreducible and unitary be will 0 := ) L ) i h,c n ∼ Z Z o all for 0 = c c > c ( ( q ( := q, z q, h 1 − q ¯ q { ¯ − 24 = ) c h Tr = ) v npriua tlast h eurmn fan of requirement the to leads it particular in 1 z X n ∈ ∞ 2 =0 ) X level L h, 2 L > n (dim φ h ¯ h,c h h,c H ( 22 N and z | h hrce fteVramdl is module Verma the of character The .  L 2 = .B cigwt h generators the with acting By 0. h, q L + ) 0 L h ¯ ihs egtstate weight highest v q,c n L c ch 0 − n n utfco u oeta ulstates null potential out factor must one and ( = ) h, ( n L q 24 c z n L h,c 1 h ¯ q ¯ h h,c n = L .Tetrspriinfnto of function partition torus The ). − N ¯ 1 ch + 0 − h, z Q where L h n ¯ 24 2 c ¯ ) hc ehwvrwl not will however we which , h,c ¯ ) n ∞  3.12 h ∂ v =1 14 q 2 } si eea eerdt as to referred general in is X φ h ersnainarising representation A . . (1 − ( a atrz noleft- into factorize may ) i z rs yatn iha with acting By erms. 24 ( − c 2 l eciebelow. describe ill z, h iaooalgebra Virasoro the f + ) mnu tensor omentum q z ¯ (holomorphic-/anti- ) tol.I ssc a such is It only. rt , ) L n i , . . . fteter and theory the of | . h,c h i 1 = 2.1 hsmodule This . = ,of ), h, . . . φ (0) L h ¯ 24 fthe of ) n τ < n , | This . , (3.10) (3.11) (3.12) 0 (3.8) (3.9) i na on It . m 0 Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. yatn with acting By o every For atto ucino h hoyteeoebecomes therefore theory the of function partition ihcmuainrltosgvnby given relations commutation with crepnigt h factorization the to (corresponding hi oeepninmyb rte as written be may expansion mode Their where vn efda nee atc Λ lattice integer self-dual even, way where e.g., o h oceechoice concrete the For h rmr ed fteter r ie ypriua produ particular by given are theory the one of fields primary The opi n nihlmrhcpr.Ta sw a split can we is That part. anti-holomorphic and morphic X r h opnnso h cntn)antisymmetric (constant) the of components the are sgvnby given is a 2 + U b S 1 currents (1) ab x π X = i ( ( = n i z 2 β a r rehlmrhcfilswihfulfil which fields holomorphic free are ) 1 ∼ π ± / ∈ 2), g Z X ab ihs egtsaeo hster sdfie by defined is theory this of state weight highest a Λ ∂X n a , i α a 2 + − j ∈ ≶ n i i ∂X ¯ ( Z π z nsc tt h hl okspace Fock whole the state a such on b = ) ( h erci endby defined is metric The . [ i λ q n a hwta h ibr pc atrzsit holo- a into factorizes space Hilbert the that show can one p + i i i x p , / | i∂x i 2) β B ( j i z = ] ij , Z X = ) = i V ∂X o all for ( e n h exponentials the and β ( q, / a β z, − ( = 2 B z q q i i ¯ iδ ∂X z i =: ) ¯ ¯ | = ) e β = ) − N ij b < i = h, i λ , ip α ,

e = h ¯ e [ η 1 2 e i α 1 iβ ∈ n = ( 23 ln , . . . , a ( n i 2 · q 1 | x x 1 · α , β π ) ( N and Λ z ( e 24 z i z ) Z b + h m j + ) 0 = : N X e β o 2 mod β , = ] ∈ i 24 g h ¯ Λ X n ab x , g 6=0 n( in / > q ∂X ∈ nδ ab (¯ x X for β z i 2 B α )) ( (3.20) Λ = i , / ij n 3.12 z fdmnin2.Teaction The 24. dimension of 2 2 a n i > n − δ .

( = ) ∂X ¯ e z n edwrt h atc frame. lattice the w.r.t. field ∼ + a − )). m, n · b 0 e x e + 0 . a X i b . ( ) and z i F b ( X +2 ) ab z, β t ftedimension the of cts a ∂X z ¯ sgnrtd The generated. is b ntefollowing the in ) n hence and ab πλ a ∂X ¯ = o all for , B b ij ( e a λ ) X (3.18) (3.17) (3.16) (3.19) (3.13) (3.15) (3.14) i ( ∈ a e b Λ. ∼ ) j Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. F h ibr space Hilbert The h atc Λ group. orbif lattice expli Monster the more The the theory of exactly orbifolded is group the group symmetry symmetry study a will is we following hence symmetr the which two group These Monster spe states. the roughly untwisted which and symmetry twisted s additional untwisted exchanging and an ac twisted has exchange group however not symmetry theory does This that way a factors. in theory cocyle the through enlarged onsaecasfidb Λ by classified are points osnthv onso (length) of points have not does eas ftecccefcosta rs ntedfiiinof definition the in arise that factors cocycle the of Because .. h oua invariant modular the i.e., hr Θ where fdimension of h ymtyo h hoybfr h riodi culyan actually is orbifold the before theory the of symmetry the atc .. vn efda fdmnin2 - 24 dimension of self-dual even, i.e., - lattice nadtoa diiecntn 4 h osataie fro fields arises constant primary The 24 the 24. by constant additive additional an h atto ucino h ooopi ato h hoyis theory the of part holomorphic the of function partition The h riodteatmrhs ru fteLehltiei th fo is lattice the Leech in the theory of the group of automorphism group the symmetry orbifold the the enlarges also it but nadditional an rtCna group Conway first symmetry discrete the by orbifolds h bla ru ( group abelian the o h oceeltiew r neetdi,nml h Lee the namely in, interested are we lattice concrete the For soitv.Ascaiiycnb etrdb nldn fa a including by restored be can Associativity associative. oylsntrlyapa ncnrletnin of extensions central in appear naturally Cocycles htsatisfies, that β 15 dfie rud( around (defined o naeingroup abelian an For Λ ( q = ) Z Leech β c Z Leech ( 2 2 α P / ) .TeOEo h etxoperators vertex the of OPE The 2. riod soiial osdrdb L.Cnrtl n si one Concretely FLM. by considered originally as orbifold, c β a oivratsbatcsudrti ymtyadtefixed the and symmetry this under sublattices invariant no has ( H Z ( ∈ q Co β 3.18 Λ 2 = ) (+) = ) ) A q 24 1 ǫ β hc steqoin of quotient the is which , ( , h ymtygopo h rioddLehltiei the is lattice Leech orbifolded the of group symmetry The . 2 ) h ibr pc ftetitdter ilte consist then will theory twisted the of space Hilbert The )). eoeobfligi h ietpouto h okspaces Fock the of product direct the is orbifolding before ,β α, / Θ / s ǫ i∂x 2 2Λ. J ( η ∈ Leech ,β α, steteafnto fteltieΛ o u hieof choice our For Λ. lattice the of function theta the is Z fnto hc hwdu nmntosmosiewith moonshine monstrous in up showed which -function 24 + ) N i ( map a , q fwih .Teemyb eoe yperforming by removed be may These 1. weight of ) ǫ = ) = 2 ( c α ( ,w find we 2, = α + 1 q q + ,γ β, 4+196884 + 24 + − x ǫ c/ β : 24 − → A = ) ,where ), 24 Tr × ǫ A A q ( ,γ β, L . x → 0 Z hsrmvsteuwne states unwanted the removes This . = ( + ) Z q ǫ /s smdlrinvariant. modular is ) ( Co η ,β α, q ǫ Z ( Θ ( q ,β α, + 0 scle -oyei [ if 2-cocyle called is Λ ) 24 yiscnr.Ti gi gets again This centre. its by sa2-cocycle a is ) · · · + V β = γ ) sdfie bv snot is above defined as . il n hwta the that show and citly h etxoperators vertex the h ttsgenerated states the m J hltieΛ lattice ch ctor ( owygroup Conway e ae.Teorbifolded The tates. e oehrgenerate together ies q so h orbifolded the on ts xeso of extension kn ol onto down boils aking o ie by given now lwn a.Before way. llowing 24 + ) c le hoy In theory. olded ( β 15 nter.h.s. the on ) , 4 . ] Leech which , Co (3.23) (3.22) mply (3.21) Co 0 V by 0 β . Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. with wse theory twisted h opeepriinfnto sthen is function partition complete The eeatyoti the obtain exactly we e hti sntlre eflo L n en pca produ special a define and FLM follow the we of larger part not be is will it group t that Monster exactly see is the theory that the argued of already symmetry have the that show to remains It fteivratsae fteutitdtheory, untwisted the of states invariant the of hr ntels iew aeitoue h oee number Coxeter is the introduced cases have our we for line last the in where atto ucino h nain ttsi hssco on sector this in states invariant the of function partition ibr pc smr utei atclrdet h two-cocyc ( the operators to the due of particular construction in the subtle in more is space Hilbert h wse ibr pc rssfo ed satisfying fields from arises space Hilbert twisted The hc aetemd expansion mode the have which hs ttsaeepiil ie by given explicitly are states These n h atto ucino h nwse etrcnb cal be can sector untwisted the of function partition the and ,g g, Z tw Z 2 H ( ( + ,oedefines one 1, = α − q (+) ) n i + = ) ( q byn omtto eain si ( in as relations commutation obeying = ) = Z (+) { + Z q α − (+) + − i ( H 1 1 q | n Λ 24 Tr + ) 1 ( ( 2 + − q | α . . . H = ) ) n hc aihsfrteLehltie i.e., Leech-lattice, the for vanishes which and oecnrt fw eoeteato h riodato by action orbifold the action the denote we if concrete More . ( + Z − ) ( J + q − − i q 2 L ) − n k ucina u atto function. partition our as function ( 0 2 1 q k Tr = = ) ( x H | = = β H i q ( (+) + i 1 (+) + z / 1 1 1 2 2 2 + ˜ = ) 2 q ( 2 1 Θ Θ = J L |−  Λ Λ 0 ( { q Leech Leech q β = η η v Q i 24( + ) i 24 24 + 3.20 ∈ ) n ∞ 2 1 { ∪ } =1 ( ( i H 25  q q m .Frtedtisw ee o[ to refer we details the For ). ) ) (1 (+) ∈ Θ X + + Z h α Λ − η +1 | 1 Leech − ) + 1)) + i gv 24 3.17 1 1 2 2 1 q / n 2 η H 2 n 1 12 1 + = 24 − α α . . . (+) n + .Tecntuto ftetwisted the of construction The ). + 1 n i " /  2 z  ( Q ) v θ n h nain ttso the of states invariant the and − 24 − i 2 1 θ x 3 } η 2 n n ∞ 2 θ n k . ( i , =1 +1 ( 2 4 −  J k e ) finds e +1 ( 12 12 2 1+ (1 q q Q πi ( − ) + ( + uae obe to culated | z 1 n ∞ − β eMntrgop We group. Monster he = ) h =1  |− − i q ymtygop To group. symmetry θ 4 = 24) fteltie which lattice, the of efcosta arise that factors le n 1+ (1 θ 2 η t(‘cross-bracket’) ct ) θ h 4 − 24 3  Leech ) x 1  12 12 ( β q z J . n i − − )(modΛ − ) ,s that so 0, = 12 + } 1 7 (  / .Frthe For ]. θ 2 1 θ ) η θ 24 3 h, 4  ) (3.27) (3.28) (3.26) (3.24) (3.25)  Leech 12 12 .  # ) Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. oced n tes sterelation the is others) and Borcherds where r nedtegnszr apmdl ojcue.Ti a be can This the conjectured. by Hauptmoduls fulfilled zero tities genus the indeed are a P nteFuircmoet ffilso egt2by 2 weight of fields of components Fourier the on (2) 3 hr xssalift a exists There (3) where eeaiaino osru onhn ntefloigman following the in g moonshine Monstrous of generalization eesr dniisgnrlztoso ( of identities/generalizations Kac-Moody generalized necessary this for identities the denominator for The identities to algebr identity of Kac-Moody this and generalize Lie to able from known identities denominator the on s bosons have 24 we autom so of the group group that Monster symmetry known the the is precisely that It is algebra Griess subalgebra. 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Lie Monster the particular in and ] X n a , 3.30 . n ∈ ) φ , ) 26 T Q b a d c 4 a endo h pe afplane half upper the on defined g a , m Tr n j h otbscoe(on yZagiers, by (found one basic most The . [ ) · ) − n 5 V 1 h tutr f( of structure The . ∈ 6 = [ + ] ( ]. g Z SL ) J n 2 hq ˜ ( φ ρ (2 riodo h ec atc has lattice Leech the of orbifold n j ) n +1 T , − q − Z g epninof -expansion 1 φ , sb nrdcn h notion the introducing by ’s . and ) J V ( m i ( τ V − g (3.30) ) ler hnlast the to leads then algebra ( = ) 1 T h P fsc fields such of OPE the f ,h g, g )(3.29) ]) e.Frec element each For ner. γ g s.t. ) ler tefappears itself algebra l h requirements the all s ceddi showing in ucceeded Tr = ∈ feeet in elements of ) L s oced was Borcherds as. C 3.30 a rhs ru of group orphism oeuigiden- using done n γ , i ∈ H J hc xsall fixes which Q 24 (+) ssmlrto similar is ) + yti cross- this by i.e., , V ross-bracket 1. = ugse a suggested + ( H g H ) ( + − n J ..the s.t. ) gq ( fthe of (3.31) q L = ) 0 M − 1 Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. n tde n[ in studied and (4) aiirlnug fCT n[ In CFTs of language familiar of moonshine subgroups Monstrous zero genus that discrete seen certain have between we nection section previous the In moonshine Conway 3.2 hrlSF ossigo bosons 8 of consisting SCFT chiral a s.t. module’ ‘Conway dimensional infinite an find can one case Γ atc rioddb the by orbifolded lattice ie h nweg fteMntrmdl,w soit t M its associate we module, Monster ries the of knowledge the given ru of group ru.Mr rcsl oa element an to precisely More group. h orsodn ca-hmsnsre h offiinsof coefficients the - series McKay-Thompson corresponding the values (5) function olw.Fra element an For follows. g on of nue nebdigΓ embedding an induces nesodfo h osrcino h ose ouea ch a as module context Monster less this or the In more of above. can sented construction (4) the of from exception the understood with properties these All aual ufi h rpris(-)ad() h generalis The [ in (5). proven and been (1-3) have properties the fulfil naturally ucini pt osattefnto soitdt h 2 the to associated function the constant a to up is function group, with omlzdpicpemdlsatce otegnszr grou zero genus the to attached modulus principle normalized g nti oue uhamdl a kthdi [ in sketched was module a Such module. this in T SL < H T T m T safis tpoemyosrethat observe may one step first a As . 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g SL T dmosieconjectures moonshine ed hc hni h unique the is then which m B h ( m (2 steivrac group invariance the is lmn fteMonster the of element τ Leech ). Γ p , ca-hmsnse- acKay-Thompson SL < T tagtowrl be straightforwardly h functions the R g s n h Monster the and ) salse con- a establishes g ψ r hntae of traces then are ⊗ rlCTa pre- as CFT iral i [ 9 Z nthe on (2 .As nthis in Also ]. C , R iheigen- with s.t. ) nesub- ence E groups T (3.34) (3.32) (3.33) 8 g s root ore are T m Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. Z rssi o biu.Oemyueadffrn osrcinof construction fermions different chiral a free 24 use of may out One it constructing obvious. not is arises rmti tsesta h CThsa has SCFT the that seems it this From h rtfwcecet a esltit reuil repres irreducible into split be may coefficients few first The uesmer.Oecncntuta construct can One supersymmetry. a aiyb computed be easily can ya element an By a is weight orbifolding after theory The N coefficients exists there h 4fe emoshv manifest a have fermions free 24 The sdsusdi [ in supe discussed a As be to tensor momentum energy the of with choice OPE proper the has N hc seult ( to equal is which hc mlmnsteflwfo the from flow the implements which needn rudsae nteRmn etr rae from created sector, modes Ramond zero the fermion in states ground independent 2 urn ilbekSi(4 xcl to exactly Spin(24) break will current 1 = 2 = action , h n orsodn ugop of subgroups corresponding and 3 = W λ i Z 2 1 − → ilbekSi(4 n twssoni [ in shown was it and Spin(24) break will NS,E and s 58 λ 8 ∈ ( i N h atto ucino hster nthe in theory this of function partition The . τ hsmto a culyb eeaie ofidmdl with models find to generalized be actually may method this ] 3.35 F λ )= )) 2 12 i uesmer.Tepriinfnto nteN sector NS the in function partition The supersymmetry. 1 = 0.Te let Then (0). 10 7 9 71+8855 + 1771 + 299 + 276 + 1 = 11202 = , c 08=1+26+1771 + 276 + 1 = 2048 ynntiiliette ufildb h ht functions. theta the by fulfilled identities non-trivial by ) s 7 276 = 276 F q 2 1 ∈ Z − . . .  = . 1 C NS,ferm. / E 2 {− 4 s.t. 276 + 0 + ( τ c 1 ) W η / θ , 2ter ihn rmr ed fconformal of fields primary no with theory 12 = 12 2 3 ( ( τ , W = NS τ, 1 = ) / s Spin 0) s N 2 X ∈ Co 28 etecrepnigwih 3 weight corresponding the be q } F 4 othe to Co 1 1 2 2 12 / uecreti h olwn way. following the in supercurrent 1 = ednt n fte2 the of one denote we 16 + 1 2 2)smer,btn explicit no but symmetry, (24) X i c 0 =2 λ 2048 + ymtyhwvrhwti symmetry this how however symmetry 4 s Co W 1 , ssmer groups. symmetry as λ , . . . , θ θ θ 0 s R i η 4 2 ( ( ( ( τ, τ, τ, τ etr twssoni [ in shown was It sector. q ) 0) 0) 0) 12 11202 + 24 12 4 4 hc r rioddb the by orbifolded are which 8 16 + httesial chosen suitable the that ] , nain of entations q θ θ 3 4 2 / ( ( NS h aeSF by SCFT same the 2 h aumb the by vacuum the τ, τ, + 0) 0) 12 . . . . etri simply is sector 4 4 cret Any rcurrent. 06linear 4096 =  / pnfield spin 2 Co 1 N 8 (3.38) (3.36) (3.37) (3.35) (3.40) (3.39) that ] 1 = Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. rtce hnoemvsaon ntemdl pc ftetheo the of space two moduli the i.e., in unchanged, around moves one when protected hrces h xaso offiinscnb eae odime to related group be Mathieu the can of coefficients representations expansion the characters, of eeteelpi eu onstenme of number the counts genus elliptic the Here where esatb enn the work. this defining of by rest start the of We concern which main Moonshine, Mathieu the discuss be will we subsection this In moonshine Mathieu 3.3 the rpriso h F oehrwt h pcrlflwadunita and flow index th spectral of the the Jacobi-form of with weak moduli together a the CFT of the independent of is properties and theory the of states n[ In ¯ on depend not fo will hence states, genus ground moving right the from contributions olne one yteelpi genus. [ elliptic In the by counted longer no hoy(CT ihcnrlcags( charges central with (SCFT) theory omc-oua om [ forms mock-modular to eta hre( charge central F utteWte ne n ofrater ihdsrt spectru discrete with theory righ a The for theory. so and the index of Witten sector the Ramond-Ramond just the over taken is the L 16 N F , 62 o o-iceesetu h litcgnswl ngnrlb n be general in will genus elliptic the spectrum non-discrete a For 10 2 1 SU BSsae fteter,temlilct fthe of multiplicity the theory, the of states -BPS R uecnomlcaatr ie nAppendix in given characters superconformal 4 = h litcgnso a of genus elliptic the ] twsntdta hnoeepnsthe expands one when that noted was it ] y r h epcielf n ih oigfrinnme opera number fermion moving right and left respective the are Z 2 -ymtyalgebra, R-symmetry (2) K = ell 3 ( e ,y q, 2 πiz Z c, 0ch 20 = ) q , K ell c ¯ 3 Z (6 = ) 8 = = ell 1 4 34 ( BSsae a art omannBSsaewihi then is which state non-BPS a form to pair may states -BPS e ,y q, , " 2 60 πiτ , 2 N  , ) a acltdt be to calculated was 6)) 0 ]. =4 )=Tr = )) litcgenus elliptic q , θ θ Here . 0 h litcgnsi n ne htcut h BPS the counts that index and is genus elliptic The . 2 2 ( N m ( ( ,z τ, ,y q, q, (4 = = 1) ) ) RR L − 6  c J 0 , 2 n egt0[ 0 weight and ch 2  0 )nniersgamdl(LM on (NLSM) model sigma nonlinear 4) stezr oeo h iaooagbaand algebra Virasoro the of mode zero the is + c, ( M stezr oeo h hr opnn of component third the of mode zero the is − c ¯  2 N 24 y[ by ) 1) 29 Z , 0 θ θ =4 oecnrtl fw s h definition the use we if concretely More . , F 3 3 ell 1 2 L ( ( ( 1 4 ,y q, q, + ,z τ, BSsae fteter.Cnrr to Contrary theory. the of states -BPS fa of 59 F 1) R ) ] + ) y K  J 61 0 -litcgnsi emof term in genus 3-elliptic 2 N q X + n L ]. ∞ =1 4 1 0 (2 = −  BSsae sntcompletely not is states -BPS A C.2 24 c θ θ 4 4 n q ¯ ( uhtere h elliptic the theories such r ( L ¯ , ch nhlmrhcadgv rise give and on-holomorphic ,y q, q, 0 n nsthat finds one )sprofra field superconformal 2) − oalrepr ilalso will part large a to 2 N 1) ,n, 24 oy h modularity The eory. c ) ¯ =4 soso irreducible of nsions m   iyipyta tis it that imply rity 2 1 ybtteidxis index the but ry ( 2 ,z τ, 16 # oigpr is part moving t . os h trace The tors. twl nyget only will it (3.43) ) K (with 3 N (3.42) (3.41) 4 = Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. oma nnt iesoa rddmodule graded dimensional infinite an form n ih sueta the that assume might one orsodn,ntncsaiyirreducible, dim necessarily not corresponding, hr h offiinsadtertedcmoiinit dimen into decomposition the their of and representations coefficients the where litcgenera elliptic ua n od [ Kondo and Mukai n ne ne h ugopΓ subgroup a the transform under to 1 expected index are and they itself 0 genus elliptic the to M uoopim htlaete2fr of 2-form the leave that automorphisms A hyfr h nlgeo h ca-hmsnsre nMonstr ( to in similar series expansion McKay-Thompson an admit the should of hence and analogue the form They h esntegroup the reason The hr h u usoe all over runs sum the where osru onhn longtv ausapa ee ic ecutst count we (bosonic/fermionic). since statistics here, their appear on values negative also moonshine Monstrous nelement an hog srnyeet’ oepeieysneteelliptic the since precisely More effects’. ‘stringy through eea e below. see genera, n 17 18 19 23 ieetrpeettoswt qa iesoshv enindic been have dimensions equal with representations Different h eopsto f‘agr offiinsi o nqebtcnb fixed be can but unique not is coefficients ‘larger’ of decomposition The h xsec fsc ouehsbe hw n[ in shown been has module a such of existence The elcdb rcsof traces by replaced tmgtse aua oasm htti ru ipegt e gets simple group this that assume to natural seem might It . H n = | A g n ∈ | Z nodrt ute teghntecneto to connection the strengthen further to order In . M K Z ell 24 3 K ell M ,g 63 3 nteelpi eu,i.e., genus, elliptic the in ,g 24 of , ( M 64 ,z τ, r ie by given are K 24 ti nw httesmlci uoopim of automorphisms symplectic the that known is it ] g a ecniee,wihaedfie ya neto of insertion an by defined are which considered, be may 3 =Tr := ) rsshr eentcmltl la.Fo h eutof result the From clear. completely not were here arises ntersetv representations, respective the in M A H 24 N A A A A − PS BP . . . cso h eso hs P-tts .. htthey that i.e., BPS-states, these of sets the on acts 2 3 1 0 1 . RR ersnain htapa and appear that representations 4 = 6 3 + 431 = 462 = 50=70+ 770 = 1540 = 0=4 + 45 = 90 = = 0=23 = 20 = =  718 17 − 0 g ( M = 2 ( n N ∞ − =0 K 30 of ) 1) nain,i eea omasbru of subgroup a form general in invariant, 3 H − F n M L 19 2 − SL + H ⊗ · 24 F 45 3 1 15 R rpeettosta per with appear, that -representations (2 · y 3.43 431 ]. 1 n N J , 770 0 Z =4 q ,where ), L u o ihtecoefficients the with now but ) 0 − 24 c eu counts genus q ¯ tdb a.Cnrr to Contrary bar. a by ated aoifrso weight of forms Jacobi s L ¯ 0 A tswt independing sign a with ates − ˜ N n 24 c ¯ sn h wndelliptic twined the using in firreducible of sions Tr =  steodro the of order the is . M lre to nlarged u moonshine ous H 24 n 4 1 h twined the , ( -BPS-states H g .Similar ). n K r the are ,i.e., 3, (3.46) (3.44) (3.45) M 24 Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. hr h rc stknoe the over taken is trace the where litcgenera elliptic fahlmrhcobfl.I atclrte r xetdt expected (A) are they particular twis In were orbifold. they properties: holomorphic as a transform of genera elliptic twining twisted hs r endby defined are these hswl edt o-rva hs ntecs when case the in phase non-trivial a to lead will This element l xaso coefficients expansion all that n[ crossings. In wall ‘unphysical’ of absence the on [ NLSM in of done genera twining possible all and o automorphisms of group the that shown was it Kondo and Mukai on representations irreducible or [ in found coeffi were expansion genera few elliptic first twined the all for of pression knowledge the and assumptions acltdi [ in calculated si eea ugopo h owygroup Conway the of subgroup a general in is o ugopof subgroup a not ypetcatmrhs of automorphism symplectic 20 itof list A 69 M litcadmdlrproperties: modular and Elliptic Z Γ ekJcb omo egt0adidx1wt multiplier with 1 index and 0 weight of form Jacobi weak o eti multiplier certain a for eeaie ahe onhn a osdrdadaltet the all and considered was Moonshine Mathieu generalized ] g,h 24 K ell Z g Z 24 3 osbyu oamlile ytm .. ( i.e., system, multiplier a to up possibly , K ,g,h ell ymtyi rsn.Frcsswhere cases For present. is symmetry K of ell , h 3 3 ,g,h g 67 ,g ( SL aτ cτ o ahe onhn sgvni [ in given is moonshine Mathieu for ( .I [ In ]. 11 ( Z aτ cτ ,z τ, (2 + K + ell , Z + + , 3 65 M d K ,g,h b Z ell + d , 68 b 3 ). nareetwt hs eut.I [ In results. those with agreement in ] 24 ,g,h cτ , lτ hscnetr a rvni pyia’wyb demanding by way ‘physical’ a in proven was conjecture this ] of n[ In . cτ z + + Tr = K z + A d l eecluae.Freeycmuigpair commuting every For calculated. were 3 ′ n d = ) = ) 66 = ) K χ RR,g n( in hswsepne oicuelc fsnua NLSM singular of loci include to expanded was this ] g,h h orsodn wndelpi eeawr also were genera elliptic twined corresponding the 3 e χ 3.43 − e g g,h M  : titdRmn etr ti xetdta these that expected is It sector. Ramond -twisted 2 2 Nh h SL π πim 24 i ( ( cd g a edcmoe nosm fdimensions of sums into decomposed be can ) − n[ In . b a d c e (2 ( l 1) 2 2 π τ , ) cτ 31 i F +2 Z mcz e + L 2 ) d 16 lz y π 15 2 K → i 0 ) cτ Z Co Z ,b eeaiaino h eut of results the of generalization a by ], q cz ]. eecnetrdbsdo h work the on based conjectured were 3 L + K K 2 ell ell U 0 d 1 − 3 3 g Z u never but , 1.I atclreach particular In (1). ,g ,g,h 24 c K ell ( ∈ ,z τ, 2.28 ( 3 ( − h, ,z τ, M 11 c 1) ) g 23 scagdto changed is ) a , – ) F ,h h 14 , ∀ R n a eraie san as realized be can and g d q g ¯ ( teghnn h idea the strengthening ] d L 1 6= ¯ b a d c M ∀ ( 15 e wnn characters twining ted 0 ,z τ, − ,l l, χ 24 ufi h following the fulfil o twspoe that proven was it ] ) 24 g,h c ¯ 20 ′ ∈ )  n ngnrlalso general in and ∈ insepii ex- explicit cients ae nthese on Based . , ne subgroup a under Γ LMon NLSM a f Z ∀ 0 itdtwining wisted ( ( N b a d c ) Z . ,h g, ) K ell ∈ 3 ,g,h SL ∈ (3.48) (3.47) M sa is K (2 (3.49) 24 , 3 Z ) , Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. yto(omtn)elements (commuting) two by 2crepn occi ugop.Tecci ugop are subgroups ( cyclic form The the subgroups. of cyclic to correspond 22 wnn eeawr acltdi [ in Th calculated were transformations). genera modular twining by those from obtained genera ooecrepnec ihcnuaycasso bla sub abelian of tr classes modular conjugacy modulo with genera correspondence elliptic one twining to twisted of set the (C) g hr h ugop eeae by generated subgroups the where Z (B) h ae htwl eipratt saei atclrwhere particular in are us to important be will that identically. cases vanish The which of two last the where Γ aihn ouin.Teol nosrce wse twinin twisted un-obstructed shorthand only the The properties the solutions. where vanishing situations i.e., obstructions, to due (D) supin l h wse wnn litcgnr eeca were genera elliptic twining twisted the all assumptions l otoldb 3-cocycle a by controlled all ti ute otltdta hs rpris(nparticul (in properties these that postulated further is It ′ 0 K ell ( ∈ 3 N ,g,h M naineudrcnuaino h pair the of conjugation under Invariance Z For .Frtecases the For ). ofra hrceswihdpnsaogohrtig ncer on things other among c depends which characters conformal where osdrdabove. considered α g Z K ell 24 r nw.W define We known. are g ( 3 [ r,s ell r,s ′ 69 ,g,h g hr ( where ( ) ]. ,y q, = r ueia osat and constants numerical are ξ sspoe ohv eldfie xaso ntrson terms in expansion defined well a have to supposed is ,g e, g,h = ) φ e ( g,h h dniyof identity the , n ec orsodt wnn eea(n h wse twi twisted the (and genera twining to correspond hence and ) k ,s r, saphase. a is ) Z ( φ := K 2 ell ) B, N N 8 8 3 Z { ∈ ,g,h 4 α α N A K ell g g ( (0 ˜ ˜ 2 r,s 3 ( , φ , 0 ,g,h ,z τ, 0) 2 = ) N , . . . , Z Z 4 are , B,A α ell = ) ell Z , ,h g, 3 ( ( ersnigacasin class a representing r,s ell 3 ,y q, , ,y q, φ , ξ 5 M := g,h , 69 − ∈ g 4 + ) egv h xlctepeso n( in expression explicit the give we 7 ) 24 B,A , ( and .Mn fteetitdtiiggnr vanish genera twining twisted these of Many ]. M 1 Z k , } ) 4 K Nβ ell 24 Z Z φ , and 32 3 hr xss5 uhsbruso which of subgroups such 55 exists There . β h K ,g ell K ell 2 g g ˜ ( ( ′ 3 3 r B, r,s r ylc eecmatepesosfor expressions compact Here cyclic. are r,s ′ ,e,h ,k , g ˜ N 8 − ) ) ′ s A ( ,h g, ( 1 then , q q 1 gk,k r uttetiigelpi genera elliptic twining the just are , = sawih oua omunder form modular 2 weight a is ) ) 2 φ , θ η 1 2 ∈ − ord 6 ( 1 q,y ( 3 q hk A, M ) ) ( 3 ( , 24 g A H ,z τ, ′ 3 ( ( : ,wihicue h cases the includes which ), AD bv nyalwfor allow only above (A-D) 3 φ , ,s r, ,s r, cltdi [ in lculated ( ) M , ar 3 rusof groups A,B ) (0 = ) nfrain si one in is ansformations litcgnr,using genera, elliptic g ∀ 24 eann 4twisted 34 remaining e eeae yelements by generated (0 6= k χ U , 1 ( g,h ∈ , ,h g, 1) ne these Under (1)). M ξ , , , 0) 0) ( = ) g,h 24 an2-cocycles tain M 69 N . , 24 .Nt that Note ]. and super- 4 = g 5.32 generated ′ r g , c g (3.50) (3.51) (3.52) ′ and ) s are ) for ) ning Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. oua omwt oeo order of pole with form modular firdcberpeettoso h hmsngop This group. Thompson expressed the be of can representations function irreducible basis of particular one of coefficients esatb enn h Khe lssae stesto holo of set the as plus-space’ ‘Kohnen like transform the that defining by start We Moonshine Thompson A-D-E 3.5 the relating by a singularities done Val is du this A-D-E particular (in models sigma n at ftemdl pc n twssonhwti cinaris 45+ action = this 90 how for shown i.e., was states, it BPS and trivial space moduli the of parts 20 [ of E Appendix the in found be can expressions further o la o xcl hsato rss tgnrcpit o points generic At arises. action this exactly how clear not h aefra cinof action mo an Mathieu for generalized case of the existence above/the discussion The hr xssa exists There [ conjecture following the to [ in strengthened and [ pursued further was and w it space, moduli in around trivial moves one typically as is change NLSM not does the genus of group symmetry automorphism fteemdl a hw.I [ In shown. was module these of table in (given h pc ffntosdfie ntesm a u loe obe to allowed but way [ same In the in cusps. defined functions of space the imirltiewt oee number Coxeter with lattice Niemeier n[ In Moonshine Umbral completel not still moonshi 3.4 is Mathieu moonshine of Mathieu appearance that the say into to insight correct new gives so and M h offiinso h litcgnsaeepce oarewi agree to expected are genus elliptic the of coefficients the up.Teeitneo prpit ouest h etrva vector [ the in conjectured was s.t. functions trace module specific appropriate as arise of forms existence The cusps. 4 1 25 BSsae [ states -BPS 0 6= 24 htthe that ] hr h uhr td yeIAsrn hoyon theory string IIA type study authors the where ] 21 n nedaluba ymtygop tdffrn onsi i in points different at groups symmetry umbral all indeed and , , md4 o hi ore xaso tinfinity at expansion Fourier their for 4) (mod 1 22 oncinrltn h imirltie n hi symmet their and lattices Niemeier the relating connection a ] 73 M ai o hssae a ie.I a ae oe htthe that noted later was It given. was spaces this for basis a ] 20 Z 24 12 -graded , 71 cinmgtaieb obnn ymtygop rmdiffere from groups symmetry combining by arise might action ovco-audmc oua om a ie.Teeya Thereby given. was forms modular mock vector-valued to ) θ , ( 72 τ = ) ,btn xlctcntuto fteNS skonadthe and known is NLSM the of construction explicit no but ], h T M θ -supermodule 3 24 27 (2 K ]: nthe on τ, a develop). can 3 q )udrΓ under 0) W 24 − 4 5 hsie a encie ‘symmetry-surfing’ coined been has idea This 45. h mrlmosiewsrltdt the to related was moonshine Umbral the ] 1 m = 4 1 at m m BSsae fteNS on NLSM the of states -BPS ≡ m 33 M τ ilb ace oa2 a to matched be will 0 ∞ = , 1mod4 → − 0 3 4 n nadto satisfy addition in and (4) i ∞ 20 W .A te prahwstknin taken was approach other An ]. m n eua eaiu tteother the at behaviour regular and K 70 P 22 3 ]. .I [ In ]. n h ouisaeof space moduli the f × hgae iesosof dimensions graded th c resolved. y n e h uhr f[ of authors the led ssm fdimensions of sums as T q sfrtelws non- lowest the for es n m 2 nhn strengthens onshine 15 udmc modular mock lued eoopi tthe at meromorphic otssest the to root-systems ssgetdi [ in suggested as hnlet Then . h eutn has resulting The . ic h elliptic the Since . opi functions morphic opnn mock- component , e tl tseems it Still ne. 23 ouispace moduli t K h existence the ] c .Sili is it Still 3. n o all for 0 = ygroup ry M Fourier (3.53) 1 ! / 2 17 K K 26 be nt 3 3 – ] Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. eta hre ( charges central uecnomlter hc a CY a has which theory superconformal P hr for where h litcgnsi ekJcb omo egt0adindex and 0 weight of form Jacobi weak a is genus elliptic The emo h litcgnsi a in genus elliptic the of term o based be elliptic will this of of properties Most basic dimensions. the various of in some manifolds recalling manifolds Calabi-Yau by of start genera We elliptic The 5-folds. CY study to 4.1 natural be particular in will it subsection see in Calabi-Y moons studied finding of moonshine in genera Mathieu interested to elliptic are we the particular study In will Calabi- dimensions. we section dimensional this higher In in Moonshine for Searching 4 h itnidxads h ihrodrtrsvns nti ca this in vanish terms order higher the so and index Witten the ( in defined as genus elliptic the in interested be will we where e fthe of few a ilfi h rfco ftegenerator the of prefactor the fix will eoe xcl h ue ubro h aaiYu o smal For Calabi-Yau. the of number Euler the exactly becomes saseiclygvnwal ooopi oua omo we of form modular holomorphic weakly given specifically a is hmsnseries Thompson )advnsigee atif part even vanishing and 4) lssae hscnetr a rvni [ in proven was conjecture This space. plus 21 j i =0 oeta o CY a for that Note a manifolds Yau χ j χ ( Y p ( d 1 Y m ) d × χ = ) ≥ p χ epoedb nlsn h litcgnr o ieetdi different for genera elliptic the analysing by proceed We . i h rddcomponent graded the 0 − c, P j ( Y c ¯ r d (3 = ) d =0 Z 2 holds. ) CY ( d Z − fl hti h rdc ftoCYs, two of product the is that -fold d CY 1) T ( d, ,z τ, [ d g r ] 3 h ( ( τ, d p,r τ = ) m ). := ) q )= 0) epnini ie by given is -expansion 21 ≡ X p For . =0 d m md4,sc htfrall for that such 4), (mod 1 χ ≡ J ( m CY − 0 X 0 , ∞ = , 1mode4 d 1) 2 d − d z W 34 3 fl sistre pc.Sc hoyhas theory a Such space. target its as -fold ntrso h ue ubro h Yand CY the of number Euler the of terms in = p ( 0 = χ m 27 X p p strace =0 ( 3.3 d a aihn d atif part odd vanishing has ]. CY ( y nC’ te than other CY’s in , − d 1) ) )teelpi eu eue to reduces genus elliptic the 1) = W y m p d 2 χ − ( g p p ( ) + CY q m O d Y 3.41 iepeoea similar phenomena, hine ( (4.2) ) umnflsi various in manifolds au d q gt1 ight [ n ) = eeao Calabi-Yau of genera , o an for ) g l 45 e Then se. d/ Y ∈ d d / , 1 [ 2 hs properties these nteKohnen the in 2 h T 74 × K 61 .Concretely ]. .A ewill we As 3. Y m h McKay- the d .Tefirst The ]. N 2 , ≡ mensions. χ (2 = (mod 0 p ( (3.54) Y (4.1) d = ) , 2) Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. sbfr nepnini em of terms in expansion an before As n vndmninl2 dimensional even any For 3-folds Calabi-Yau 4.1.3 sign. minus overall an with expansion same the ntergtmvn aodsco Tr sector Ramond moving right the in h xaso coefficients expansion the sete ypetco anti-symplectic. or symplectic either is a edcmoe ntr fdmnin firdcberepre irreducible of dimensions of term in decomposed be may eu of genus For 2-folds Calabi-Yau 4.1.2 For 1-folds Calabi-Yau 4.1.1 ec a nepnini emof term in expansion an has hence into fre embedded to fix-point connected a way by geometric obtained be can which surface, Enriques The torus hr the where se .,[ e.g, (see, h olwn way following the 22 eismlci uoopimo h niussraei nautomorp an is surface Enriques the of automorphism semi-symplectic A d d d n nstefloigepnini em of terms in expansion following the finds one 3 = hr r w o-rva cases non-trivial two are there 2 = T hr xssol n aaiYu(pt smrhss,na isomorphisms), to (up Calabi-Yau one only exists there 1 = 2 t litcgnshwvrvnse.Mr eeal h el the generally More vanishes. however genus elliptic Its . K Z A 75 hw ahe onhn hc led a endsusdin discussed been has already which moonshine Mathieu shows 3 CY n Z ) hw onhn hnmnncnetdto connected phenomenon moonshine a shows ]), r si ( in as are 3 Enr ( ,z τ, M ( ,z τ, 11 = ) = ) M < χ 3.43 n CY -torus 2 φ − A 12 A A A 0 0=11 = 10 3 ,( ), , = 1 1 n 2 1 3 φ . . . ( [ / 0 77 ,z τ, 5+176 + 55 = 45 = 6+2 + 66 = M sdmnino oevco pc.I atclrthey particular In space. vector some of dimension as 2 , 3.44 3 2 T .Iselpi eu sjs afta fthe of that half just is genus elliptic Its ]. ( 11 − ,z τ, 2 10ch = ) n 1 .Snealthe all Since ). , ic t eismlci automorphism semi-symplectic its since − , N aihs hsi u otefrinczr modes zero fermionic the to due is This vanishes. = ) N 1 hrceso h form the of characters 4 = , · hrcesi lopsil hc ed to leads which possible also is characters 2 =  2 2 + 120 χ ( , 35 2 N − CY K 2 , 0 =4 1) , 0 3 n h niussrae h elliptic The surface. Enriques the and 3 ( F  ,z τ, R ch q ¯ · L ¯ 3 N A 4 76 + 144 + ) 0 , 0 − =2 n , 2 1 24 r vnoemyaaninterpret again may one even are c ¯ ( X n ,z τ,  ∞ =1 ∝ N A ch + ) , 2 θ n characters 2 = 2 ch (¯ q, 2 N 2 N ,n, ettosof sentations − , =4 M 0 =2 nouinof involution e )=0. = 1) , 2 1 − imwoelf to lift whose hism ( 12 2 1 ,z τ, ( ,z τ, [ 76 itcgnsof genus liptic eytetwo- the mely ) . .I si a in is It ]. )  22 K M 3.3 a be can and 3 (4.5) (4.3) (4.4) 12 . K K in 3 3 Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. hnepne ntrsof terms in expanded When N index n ne .Ti etrsaei eeae ytetoelemen two the by generated is space vector This ( 2. index and 0 hrces n[ 2 In exhibits it characters. that shown was It nti iuto ait fcnetost prdcgroup sporadic [ to In connections lished. of variety a situation this In with ( equations the by fixed is normalization the where For 5-folds Calabi-Yau 4.1.5 o gnie Y4-folds CY ‘genuine’ For nee ne ekJcb om ihtefunction the with forms Jacobi weak index integer form Jacobi weak the find number Euler the by For 4-folds Calabi-Yau 4.1.4 moonshine. Mathieu to invariants Gromov-Witten fro interesting very not section is In it characters, two includes only exhibits exhibit to shown and h litcgnr curn for occurring genera elliptic the suiul xdb h ue ubro h Y5fl eaecon are we 5-fold CY between the relations of the number of Euler the use by fixed uniquely is o ugopthereof. subgroup a not rm edo neesmdl 11. modulo integers of field prime 2.34 23 24 hrcesadshows and characters 4 = nti otx edfie‘eun’t enCY mean to ‘genuine’ define we context this In Here Z d d bv.Truh( Through above. ) φ CY h litcgnsi neeeto h space the of element an is genus elliptic the 4 = h litcgnr curn ilb ekJcb om of forms Jacobi weak a be will occurring genera elliptic the 5 = 5 2 0 ssae n( in stated As . , 4 L 2 3 M ( 2 ,z τ, and 5 1)sad o h nt ipegroup simple finite the for stands (11) 21 24 ewl td Y3flsfo ieetprpcieb conne by perspective different a from 3-folds CY study will we h ekJcb form Jacobi weak the ] = ) moonshine. N 78 χ 144 hrcesfrcnrlcharge central for characters 2 = h aefnto a xaddi em of terms in expanded was function same the ] CY 4 χ ( 6 1 L φ 2.34 CY ( 4.1 0 φ 2 , 23 4 (11) 1 0 2 ( , and n ( and ) 1 hycnb bandb utpyn h corresponding the multiplying by obtained be can they ) ,z τ, eawy have always we N 5 + N 24 M ) hrcesteatoso [ of authors the characters 1 = χ .M 2 E 23 d hrcesfrcnrlcharge central for characters 4 = onhn.I diini a hw n[ in shown was it addition In moonshine. 0 − 4 4.2 r l rprinlto proportional all are 5 = = 12 φ onhn hnepne in expanded when moonshine E 24 − 2 1 efidta h litcgnsi nqeyfixed uniquely is genus elliptic the that find we ) P onhn hnepne ntrsof terms in expanded when moonshine 2 4 ( , ( 1 φ τ r d exhibits ) 0 2 =0 ) , 36 φ 1 − ( − − χ 2 SL P E , 1) 1 0 d 4 ( flswoehlnm ru is group holonomy whose -folds φ = ,z τ, r h − 2 (2 h 0 2 M , 4.2 ,r , ) 0 1 1 = 11) 2 , sn h omlso ( of formulas the Using . perdi mrlmoonshine. umbral in appeared ) 22 0 + ) c φ n ( and ) + 0 3( = onhn pnepninin expansion upon moonshine J , ˜ 3 2 h χ 0 SL onhn perspective. moonshine a m rmti tflosthat follows it this From . , 0 2 0 , aearaybe estab- been already have s E 4 (2 d fJcb om fweight of forms Jacobi of A.15 2. = 4 , )gvni ( in given 3) + ( φ F 79 τ 0 11 ) , φ hwdta talso it that showed ] 3 2 .A hsexpansion this As ). ) / φ − N sa scerfrom clear is as ts N F c 2 0 ieig Making sidering. 11 ∗ , , 1 1 3 = characters. 2 = ( h prefactor The . where , characters 2 = ,z τ, egt0and 0 weight d ) tn their cting 2 multiplied SU . C.12 F 58 A.3 11 N ( that ] d sthe is (4.6) and ) we ) 4 = in ) Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. odi ipetepouto 3(nius ufc ihaCY a with surface c (Enriques) understood K3 better the a examp through arise of trivial) just would product rather connection the (and or obvious simple 5-folds of is CY think fold for course group of may (Enriques) One Mathieu the to connection Z el the of of expansion terms following in the directly follows this From codn ofrua ( formulas to according hsepninmksi la htfrC -od with 5-folds CY for that clear it makes expansion This Appendix and fsg fthe of sign of refrC -od with 5-folds CY for true For 6-folds Calabi-Yau 4.1.6 faC -odmyhneb nqeyfie ntrso h ue n Euler the of terms in fixed uniquely be hence may 6-fold CY a of offiinsaetesm sfrMtiumosie hra fo whereas moonshine, Mathieu for as χ same the are coefficients a ese rm( from seen be can Erqe)gopw ilsuytetie litcgnr fo genera elliptic twined section the 5 in study CY 5-folds general will between we connection group deeper (Enriques) a actually is there if CY CY 5 5 rmteeosrain ti etil o la fteere there if clear not certainly is it observations these From d Z χ ( = ,z τ, p CY h litcgnsi ekJcb omo egt0adindex and 0 weight of form Jacobi weak a is genus elliptic the 6 = − = 6 = ) 4w n h offiinso niusmosie ic h ove the Since moonshine. Enriques of coefficients the find we 24 = P C = N esethat see we , − r 6 χ 1728 − + χ =0 N CY characters: 2 = 864 24 CY χ X n ( 1 φ φ φ ∞ 48 hrcesi eecneto,teaoesaeetalso statement above the convention, mere a is characters 2 = 6 =1 − CY 5 0 0 0 4.2 φ , ( φ , , 1) 2.34 2 3 5 0 3 2 3 2 3 A χ 0 , ch ch ch h 1 r , . CY n 3 2 ( 22 h ,z τ, 2 N φ 2 N 2 N  h pc fsc om s3dmninl h litcgenu elliptic The 3-dimensional. is forms such of space the ) p,r 6 ,n, , , 4.1 0 0 0 =4 ch =4  =4 − , , , for , 1 0 χ 1 2 2 1 ch ) 5 N ,( ), CY 72( − ,n, = = = 5 N =2 , 5 0 2 3 =2 4.2 567 − − − , χ p ( 4 and +48 = 1 2 1 ,z τ, 1 ch ch ch ( 0 = efidta h litcgnsi ie by given is genus elliptic the that find we ) ,z τ, − ( 5 N 5 N 5 N χ ch + ) , , , 0 0 0 6 =2 =2 =2 CY ch + ) , , , , χ 3 3 1 2 2 2 .Aanb acigtefis e coefficients few first the matching by Again 1. 0 6 − − − )) − 5 N ch ch ch E 37 ,n, 48( 5 N =2 χ 6 , 5 N 5 N 5 N 0 ( − =2 , , , CY , 0 0 0 τ =2 =2 =2 − 2 3 , , , χ − − − ) ( 5 1 2 φ 1 ,z τ, 2 2 2 3 3 1 ( +24. = − 3 6 + ,z τ, , , ch + 2 ∀ , ) 1 n i  ( ) χ ,z τ,  0 ∈ 5 N )) , . − 0 =2 N , ) E 2 1 . 2 χ ≥ 4 −  1 ( CY itcgnso Y5-folds CY of genus liptic τ . s of ase ch ch ) 5 eti gnie CY (genuine) certain r φ flsadteMathieu the and -folds 5 N 5 N = , − 2 , 0 =2 0 =2 , 2 e hr h Y5- CY the where les , 2 3 − , − K 1 vnhwi arises. it how even ( Y5flswith 5-folds CY r ( 2 1 lyi concrete a is ally ,z τ, 8 h expansion the 48, ,z τ, .T understand To 3. -od hnthe Then 3-fold. , ch + ) ) φ umber 0 , 1 alchoice rall ( ,z τ, 5 N , 0 =2 , .As 3. − ) holds (4.7) (4.9) χ 2 3 ( CY ,z τ, (4.8) 6 s )  Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. h aeta per o mrlmosieis moonshine umbral for appears that case The χ aiod n a banawat fCY lo of of wealth products a taking obtain by may that one see manifolds we above discussion our From osoai rus ic CY Since groups. sporadic to aeElrnumber Euler have .. aaiYumnflso dimension of manifolds Calabi-Yau 4.1.7 neie rmtecnetosdsusdi subsections in discussed connections the from inherited oncint oeto ftesoai groups sporadic the of (one/two) to connection niussraeor surface Enriques a oee osdri n hnteetea aoifrso um of forms Jacobi extremal the when [ and lo if a consider require however would case may general the of treatment a constructed n a osdrtecssweeteC -odi rdc ft of product a is 6-fold CY the where cases the consider may One h xaso fti aoifr ntrsof terms in form Jacobi this of expansion The rdcso w Y-od.Ti aiodwl uoaial s automatically will e manifold above This acquire CY3-folds. will two that of 6-folds CY products of examples 2 explicit to find related be can and on into fall setting obviously by not uncovered does be may which connectio example a particular general exist. One in does if groups sporadic understand and to interesting be would eteape a perfrC -od,ie,Jcb om of forms Jacobi i.e., 4-folds, CY with for 2, appear may examples next 21 CY 25 , hscnetr a rvni [ in proven was conjecture This 6 Z sw aearayse n xml sgvnb h litcgen elliptic the by given is example one seen already have we As 22 = CY pera litcgnr fC aiod rpout there products or manifolds CY of genera elliptic as appear ] χ χ 6 CY 0 − + = = = = ( CY 3 (1)  χ χ χ χ 1728 16ch 6ch CY CY CY CY 8 4 4 · .A led etoe in mentioned already As 0. = ) χ 6 6 6 6 6 N CY θ θ ( h φ 6 N , 2 2 φ 1 4ch =4 , 0 2 ( ( 1 , 3 =4 , 0 3 (2) 3 2 ± , ,z τ, τ, 3 2 , 1 1 ( K 28ch + and 2 ( 6 N ,z τ, 0) 48ch + od,ti ab eahee o n aro Y3flsthat 3-folds CY of pair any for achieved be maybe this holds, , ,z τ, 0 ) =4 Erqe ufc)tmsaC -od ntoecssa cases those In 4-fold. CY a times surface) (Enriques 3 , .AGL 2 2 0 θ θ ) ) + 3 3 ( ( − ,z τ, τ, ± 6 N  6 N , 2 χ 3 3 =4 , − respectively. 4 0) 2 , 23 =4 2 i h mrlmosieconjecture moonshine umbral the via (2) E 2 3 , 0 ) 1 2ch 2 2 ]. 4 56ch + = d θ θ 112ch + ( 4 4 flsfrlarger for -folds τ 6 N ( χ ( ) , τ, ,z τ, 0 φ 1 =4 , 2 1 − 2 .Te h litcgnsis genus elliptic the Then 0. = 0) 2 ) 14ch + 6 N , 2 2 , 1 3 38 =4 6 N ( , , 3 2 ,z τ, 3 =4 d , + 1 fls( -folds ) N . . . 6 N + φ , 4.1.4 1 =4 0 , . . . , hrceshsapae n[ in appeared has characters 4 = 1 2 1 i  > d χ L ( 42ch + CY ,z τ, > d 2 d
. (11) eun Y4flsawy have always 4-folds CY genuine 6 6 aentbe systematically been not have ) .Snefrorexamples our for Since 8. = )wt oeta connection potential with 6) − M , 6 N 2 , 4.1.3 2 =4 E , 12 1 2 fwr.A tr one start a As work. of t 6 M , lpi eu ytaking by genus lliptic 86ch + ( ewe Y6-folds CY between n τ , fteaoecases above the of e atisfy e iesoa CY dimensional wer ) 22 egt0adindex and 0 weight φ 4.1.4 − 3 M , 2 hree of. , 6 N rlmoonshine bral 1 sof us , 23 ( tcertainly It . χ 3 =4 ,z τ, , 0 2 1 .M K 25 + = )) 24 ecan We . ’,three 3’s, K . . . χ 1 ilbe will .The 3. (4.10)  . 0 = 21 ] Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. rjciesae(e Appendix (see space projective i) of: consisting symmetry Alternatively χ fdegree of ne pern nuba onhn.Frti ob h aewe case the be o to form this Jacobi For the to moonshine. rise umbral give in can χ that appearing 8-folds 4 CY index for look we Next exist. wnn ihi elzda emti ymtyo h CY. the of symmetry geometric sym a the as when realized genera m is twined This with the twining calculate developed. also were to space generalized projective weighted in surfaces χ lentvl rigCY trying Alternatively eemnda h ouinof solution the as determined nuba onhn n nsthat finds one moonshine umbral in aihn litcgenus. elliptic vanishing ih eal oaheeb aigpout flwrdimension lower of products taking CY by with achieve to able be might ntepeiu uscin ehv salse rthnst hints first established have we Calabi-Yaus subsections specific previous for the genera In elliptic Twined exa interesting to rise 4.2 give to restrictive too is again This ims hsie ewl aclt h wndelpi gene elliptic twined the the calculate i via will subsection may we this they idea In this i.e, groups. dismiss moonshine, sporadic certain of to kind connected some be in involved be may npriua h ags ls fteeCY these of class largest the particular In .. acltn wndelpi eeafrC yesrae nwei in hypersurfaces CY for genera elliptic twined Calculating 4.2.1 space. projective weighted in n[ In 0 0 CY n bla etrmlilt(iigrs oan to rise (giving multiplet vector abelian one hc olw rmtemr eea rprythat property general more the from follows which 2 = = 80 5 oceeyw osdraC -odi egtdpoetv sp projective weighted in d-fold CY a consider we Concretely hnw nrdc w-iesoa iersgamdlwith model sigma linear two-dimensional a introduce we Then hnoelosfrC 2flsta iers oteJcb om t forms Jacobi the to rise give that 12-folds CY for looks one When o egt0adidx3w a nsubsection in saw we 3 index and 0 weight For × χ h ehd ocluaeelpi eeafrC aiod r manifolds CY for genera elliptic calculate to methods the ] 1 χ rjcieabetspace ambient projective 8 CY = =CY m 3 χ and 2 = CY 5 hc antb h aefrgnieC -od u hc on which but 8-folds CY genuine for case the be cannot which 0 = × P Z d CY i d CY =1 = 3 5 w × K esethat see we CY i 8 3 = n Φ and 3 × χ K ∝ 0 K 3 ( p χ CY × (Φ has 3 CY i D.2 CY 1 d r h ooeeu oriae fteweighted the of coordinates homogeneous the are 5 , . . . , = ) χ × . ) 0 3 χ × χ χ =  0 0 CY CY Φ χ = and 4 = d 3 39 if 2 if 0 1 +2 d 3 oitrsigeape xs nti case. this in exist examples interesting no χ ilhl u since but hold will 0 = flswl l eraie shypersurfaces as realized be all will -folds ,where 0, = ) usit h aeproblem. same the into runs 1 = d d seven is χ sodd is 2 U CY 4.1.6 = 1 ag symmetry), gauge (1) d χ = 3 p htpsil Y6flsdo 6-folds CY possible that . = mples. K afrseicCY specific for ra satases polynomial transverse a is χ a CY hat 3 re osrnte or strengthen to order n 4 × er lmn eare we element metry = toscnesl be easily can ethods lC’.Frttrying First CY’s. al T χ 4 χ 6 N aie shyper- as ealized relpi genera elliptic ir d 2 has srequired. is 0 = ace ( flsfor -folds egt0and 0 weight f (2 = CY χ ol need would CP 5 a appear hat 0 , × )super- 2) but 0 = w d CY +1 d 1 ghted ,...,w -folds. (4.11) > d 3 ) d +2 ∝ 2 e Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. es o hrlmliltwith multiplet chiral a For less. ie iet mlilctv)cnrbto oti refined this to contribution (multiplicative) a to rise gives r oae at located are Xp naeinvco edwl edt n( an to lead will field vector abelian An p ntenx tpw td h endelpi eu endby defined genus elliptic refined the study we step next the In h ueptnilivratunder invariant superpotential The where iii) h usrp in supscript The ii) eedn na xr hmclpotential chemical extra an on depending loehrteter ecnie ed to leads consider we theory the Altogether iglrte o hrlmlilt ihnegative with multiplets chiral for singularities Q rmti ecnoti h tnadelpi eu yintegr by genus elliptic standard the obtain can we this From n[ in (Φ Z soecncnic nsl hrlmliltof multiplet chiral a oneself convince can one As . 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U + τ 1 R Q ( ( 1 charge (1) R Z ) ( τ, cag hs singularities these R -charge wu F ) R ) 2 − . cag 2. -charge tr equation -term Y d i R − 1) 2 =1 +2 ) litcgnso h form the of genus elliptic z w . F 1) R + tn over ating θ i < Q z 1 q ¯ ne the under ( wu L ¯ + τ, θ 0 1 − w ( ( wu ) .Frortheory our For 0. d 8 τ, R 2  and ) R − 2 . z 1) u U + R U sshown As . ∂W/∂X z m mass- ome 1 charge (1) -charge wu 1 gauge (1) + wu ) (4.13) (4.16) (4.17) (4.12) (4.14) (4.15) W ) , = = R Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. h aeo o-bla ymtista emt h Φ the permute that symmetries non-abelian of case The wnn h litcgnsb uha ymtywl edt sh a to lead will symmetry an such of by argument genus elliptic the Twining eu wndb uhasymmetry a such by twined genus yuigteseicpoete fthe of properties specific the using By hsi nytecia multiplet chiral the only is this ec h litcgns( genus elliptic the Hence o ecncnie naeinsmer cigo h hrlmu chiral the on acting symmetry abelian an consider can we Now a ute ipiyti xrsin ial n ban th obtains one CY of Finally genus elliptic expression. the this simplify further can n a edsrbdb rnvrepolynomial transverse a by described be can and h nern sproi under periodic is integrand The udmna oanudrti dnicto r ie by given are identification this under domain fundamental a Z CY d ( ,z τ, = ) θ Z 1 θ CY 1 Z by iη ( d τ, CY ( ( τ α ,z τ, − d ) i ( 3 z ,z τ, o aho h hrlfilsΦ fields chiral the of each for Tr = ) ) g k,ℓ = = m X = ) d Φ : − 4.16 =0 fl hti yesraei egtdpoetv space projective weighted a in hypersurface a is that -fold − u = k,ℓ k,ℓ m m 1 X X i − − = I =0 =0 mu RR k,ℓ k,ℓ m m → X X 1 1 u fteC a ewitnas written be can CY the of ) =( − − z =0 =0 y e  m 1 1 X e − + − k + u g 2 m + 2 y e ℓ π m ( π z ℓτ − ∼ k − n h iglrte r h ouin of solutions the are singularities the and m Y d i i − i α m =1 ℓz +2 2 ℓ + = i + π 1) u Φ Y z d i i Y d i =1 ℓz ) +2 θ =1 − +2 ℓτ θ i /m F 1 + θ i , 1 fnto se .. pedxBo [ of B Appendix e.g., (see, -function L 1 Y d ( i k 41 θ  =1 y , +2 ,e q, θ du θ 1 − J ,e τ, 1 ( 1 ∼ 0 ( θ ,e q, θ ( q 1 = θ ,α τ, 0 2 1 τ ,ℓ k, ℓτ, ,e τ, 1 L π 1 θ ( u 2 ( θ ≤ ( 0 ,α τ, i π 1 τ, ( 1 − ,z τ, 2 ( + i α , ( ( i π τ, 2 8 q ,ℓ k, i α 2 τ, m i π + w w m + ( τ i m i d , . . . , i i i w i − + − ec eoti o h elliptic the for obtain we Hence . k w h ouin hc r within are which solutions The . − + w ( i m m w m i k w q k mu i 1) k m ≤ i i q ( w mu ) k w ( m m + ∈ k i w F k q ) i m ℓ m. i r ) q ( + w y ℓ Z ℓτ + q m ¯ k ) 2 + i w y w m L ℓ m . ¯ i i Y d ℓτ i + i w y 0 ℓ m ℓτ =1 + +2 − − i y a etetdb first by treated be may w m . ) 1 ℓτ olwn oml for formula following e + d 8 i w m z + ) θ −  i ) . 1  1 z + z ( − ) )) τ, θ ) . z 1 z − ( ( )) tpesas ltiplets f ftesecond the of ift ) τ, R 2 z ) R − 2 z 1) + z wu 80 + (4.23) (4.22) (4.20) (4.19) (4.21) (4.18) )one ]) wu ) ) . Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. usle to ourselves osbecneto ewe Y5flsand 5-folds CY between connection possible nste2 the finds edsrbda eeiepltpsi ie ntewbie[ website the of on 727) given 757 is 5 are polytopes there reflexive p precisely as the (more described in set discussed be large techniques A the apply 5-folds. now CY will we this for ing sovosyntcnitn iha nepeaini terms in interpretation an with consistent not obviously is sw aese nsubsection 5-folds in CY seen have of we genera As elliptic Twisted abelia as appear 4.2.2 again symmetries the coordinates diagonalizes new that the In transformation coordinate a performing l files All 4adters fteepnincecet ipycrepn t correspond simply coefficients expansion the bee of has rest beginning the the and at constant 14 multiplicative the is that h oei htoefid Y5flswihtie litcgen elliptic twined which 5-folds for CY experienced finds have we 353 one 19 as that with behaviour a us is by leaves described hope then be This The can space. that ones projective the weighted to in set this restrict to need h aewe h Y5fl sjs rdc of product a just is on 5-fold acts w CY symmetry element the the when of traces case appropriate the the repre by irreducible replaced of are dimensions group to correspond that ficients M asit resof irreps into ways hsmyocri eti iutos o xml o h hyp the for example For situations. space certain projective in occur may this nohreapeta ok nasmlrwyi h hypersurface the is way similar a in space works projective that example other An ecnie the consider we o hsoefid h litcgnstitdb hssymmetry this by twisted genus elliptic the finds one this For Z 24 CY oepantecag ntepeatroemydcmoe355 decompose may one prefactor the in change the explain To . tw, 5 2 A 14 = A ntefiles the in , + M eisbtnwmlile ihafco 69 factor a with multiplied now but series X n · 12 ∞ =1 h Z CP 2 loti xml em interesting. seems example this also CP A 2 K  M (2 n ymtygvnby given symmetry 1 6 ch )ti il(rval)b h ae u lofrgnieC 5 CY genuine for also But case. the be (trivially) will this 3) , 1 6 A 1 , 24 , 2 5 N ) 1 , , , 2  0 3 =2 htwl iers o1 fe h twisting. the after 14 to rise give will that , , 6dRefWH.xxx-xxx.gz , 3 ch 5 2 1 , , 4 ( 9 , ,z τ, 5 N , 4 10 ,n, , 8 =2 pligtesm re ws ( twist 2 order same the Applying . hc a ue number Euler has which 3 2 ch + ) 4.1.5 ( ,z τ, Z 2 ch + ) h litcgnso Y5flssest ml a imply to seems 5-folds CY of genus elliptic the : 5 N ,  0 =2 K , − Φ Φ 2 1 nMtiumosie .. httecoef- the that i.e., moonshine, Mathieu in 3 42 1 2 5 N ( ,n, ,z τ, − → − → =2 M − 3 2 ) 24 .I re oapyormtoswe methods our apply to order In ).  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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. yti ymtygvstececet fte4 eisof series 4B the of coefficients the gives symmetry this by h litcgnr on nti a a lasb pi into of split series be 2A always and can way 1A this the in found genera elliptic The on n tpfrhrw a osdrteodrfu symmetr four order the consider can we further step one Going ftetiiggnr fC -od htaehprufcsin hypersurfaces [ are paper my that In 5-folds CY of spaces. ide genera the to twining leads This the found. of be can examples hypersur specific CY such the More of genus elliptic the twining that finds one Here n litcgnstie yagopelement group a by twined genus elliptic any hsbhvormyarayb xetdfo tnadCTargu CFT standard from expected be already may behaviour This weig the in hypersurface the of genus elliptic CP the give may we ihaptnilynntiilmlile,.,i transform it multiplier,i.e, trivial non potentially a with Z .I 677cssw eeal ofida find to able were we cases 727 16 In transver 3. single a construct to code simple a by used we Then 2. .Frtee177 aiod h litcgnstie ysu by twined genus elliptic the manifolds 167272 these For 4. .Fo h 5 2 xmlso h est [ website the on examples 727 757 5 the From 1. 1 6 CY , tw, 1 , 1 a acltd ncs h aclto oktoln twsa genera. was elliptic it twined long 642 too 13 took at arrived calculation we the method case In calculated. was xetmyb trbtdt h ipiiyo h oeused. code the of simplicity 5-folds the CY to 880 18 attributed to be cases may possible extent of number the reduced This th in polynomial space. projective transverse weighted a ambient by the of descried coordinates be can that 5-folds 5 2 , 1 A , 1 , = 43 + 1 , 3 + + 9 2 wndb h ymtygvni ( in given symmetry the by twined h 37 X X n n 22 · ∞ ∞ =1 =1 h hswsdn ntefloigmanner: following the in done was this ] 6  A A  ch n (2 n ch 5 N A  , 5 N 0 ) ch =2 , ,  0 =2 1 2 , 5 N M ( ch 1 2 ,n, ,z τ, ( =2 ,z τ, 24 5 N 3 2 ,n, =2 ( ch + ) ihitgro afitgrpeatr.A nexample an As prefactors. integer half or integer with ,z τ, 3 2 ch + ) ( ,z τ, Z ch + ) 4 5 N ch + ) : , 5 N 0 =2  , , 0 − =2 , − 5 N 2 1 Φ ,n, Φ ( 2 1 =2 2 43 ,z τ, 5 N ( − 1 ,n, Z ,z τ, − → =2 2 3 → 2 − ( ) ,z τ, ymtyo h rnvrepolynomial. transverse the of symmetry  3 2 ) iΦ  ( g − iΦ ,z τ, 4.24 − ) 1 saJcb form Jacobi a is i  2 , 2 . 2 )  ) i   ch ch . 5 N 81 si ( in as s , 5 N 0 =2 , , 0 eto h 933CY 353 19 the took we ] M =2 3 2 , ( 2 3 ,z τ, 24 ( ,z τ, iercmiainof combination linear a fasseai study systematic a of a ihapeatro 42. of prefactor a with 3.47 ch + ) tdpoetv space projective hted y ch + ) egtdprojective weighted han ch et [ ments aein face φ ),( 0 g otd ythis By borted. hc osome to which , 5 N homogeneous e ,m 2.29 epolynomial. se , 5 N 0 =2 , , Z 0 − =2 .. Γ w.r.t , 11 CP 2 − 2 3 .Twining ). ( 2 3 symmetry , ,z τ, ( 1 6 12 ,z τ, , 1 , 1 since ] ) (4.26) (4.27) , 0  1 ) ( ,  4 | g , 4 | , 4 ) Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. ya lmn fte1 r2 ogunecashstiilphas trivial has class multiplier. congruence non-trivial 2A has or element 1A 2B the of element an by u fte1 4 wndelpi eeaoefid 2 ae tha cases 927 finds one genera elliptic twined f 642 13 the of Out emycos h olwn w ucin sbsseeet f elements g basis as matc elliptic functions respectively twining level two the following of the in choose failure multiplier may We trivial to non rise the give to tre then leads tha that will fact symmetries This the i.e., from differently. follows symmetries, series non-geometric 2B for seri a up 2B found the never to comb f analysis proportional hand our linear something other obtain a the always On give will always multiplier. one trivial will for element genera twining two 2A order an by twining htteeeet for elements they the that that fact the from follows φ That elements. basis few very ossetapossible a suspect to a ge oa2 lmn of element 2B a to agree may M -od a strict a has 5-folds 2 0 a , 24 .I oto h ae ihmultiple with cases the of most In 2. .W eeae ag ubrof number large a generated We 1. .Frtecsswt nyoeobvious one only with cases the For 3. 2 3 ute 1 fteeteoealceceti nee intege even an is coefficient overall the these of 811 Further . f f h etrsaeo osbetie litcgnr o Y5- CY for genera elliptic twined possible of space vector The ae nteerslsoecncnld htnn fteanaly the of none that conclude can one results these on Based n h ucin htapa for appear that functions the and ymtyw rcee sfollows as proceeded we symmetry 2 1 rprinlt utthe just to proportional n acltdtetiigelpi eeafralo them. of all for genera elliptic twining the calculated and f nsa es n ymtyta ed oatie litcgen elliptic twined a to leads that either symmetry with one least g at elliptic twining finds their calculated and symmetries abelian a a 2 ( ( a ,z τ, ,z τ, . 3 = ) 11 = ) − +45  3 M  ch  ch 24  ch M 5 N ch M , 5 N 0 K =2 or 5 N , 24 , 0 ,n, =2 1 2 5 N 24 , =2 r rae yvr e ai lmns[ elements basis few very by created are 3 ,n, ( 1 2 M =2 ,z τ, ymty h eut r ossetwt h dathat idea the with consistent are results The symmetry. 2 3 ( ymty(o h eann 1 h rfco sodand odd is prefactor the 116 remaining the (for symmetry ( ,z τ, 2 3 12 ,z τ, ( ,z τ, ch + ) . ch + ) M f ch + ) 2 ch + ) 12 a .T ute hc fteemnfl elyhv a have really manifold these if check further To ). n h ett elna obntosof combinations linear be to rest the and 5 N , 5 N 0 =2 , 5 N , 0 − =2 ,n, 5 N , =2 − Z ,n, 1 2 K =2 − ( 2 1 2 − ,z τ, 44 3 2 ( a olw rmLma14 n[ in 1.4. 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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. u.As hoeial n ol mgn h iuto wher situation of the copies imagine multiple could to corresponds one theoretically Also out. pc) oee te osblte r lopsil.Frex For certa than (at possible. larger also are groups are that symmetry possibilities discrete other However small space). have 5-folds CY the ymtis(table symmetries o-iemrhcmnfls o l h xmlsw hnpr then th manifol we that examples found conclude the immediately we all not For cases can manifolds. the one non-diffeomorphic of that so some tha numbers For certain Hodge be to manifolds. order different in examples to those for numbers Hodge all follows: as proceed to chose we Hence consuming. examples those all that fact the to related m is which 5-fold CY fcranirdcberpeettosudrasingle a under representations irreducible certain of shaving as hl o tesi ol ipycrepn oteiett.T of identity. combination the linear to a correspond having simply with would those it of others certain for of while part be could studied oasingle a to tl excluded. still ognrt ute xmlsoemyuePL [ PALP use may one examples further generate to 26 = .Fr7 For 1. .Ke hs ae o hc h onaepolynomial Poincare the which for cases those Keep 2. .FrtermiigcssuePL [ PALP use cases remaining the For 4. .Fo h eann ae eptoefrwihtefruaf formula the which for those keep cases remaining the From 3. hsi eesr odto o naporaeplnma oei e Ap see - exit to polynomial appropriate an for condition necessary a is This ytkn hs tp n rie tdzn fnweape.W examples. new of dozens at arrives one steps these taking By h pca ae with cases special The P gives vlae at evaluated i w | i χ M ± ≤ ≤ CY 48 4or 24 24 5 0 hc edt otyrte ag eaieElrnumber. 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(4.31) (4.30) dall ed Z 2 Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. . aigaqatttv nlssoecnosreta for that observe can one analysis quantitative a Making 1.) h wndelpi eeafrteeape bv rsn fr arising above examples the for Z genera elliptic twined the o number the give parenthesis numbers in Hodge numbers different The spaces. jective 1 Table h wnn litcgnr r rprinlt h Atwine 2A the to proportional are genera elliptic twining the pntiigwt the with twining Upon .%ad52 ftecss lofrtececetof coefficient the for Also cases. the of 5.2% and 3.4% . o h ae htatal iesmtigproportional something give actually that cases the For 2.) th coefficients. Furthermore even coefficients. for small ence other than likely more not n bevsta hyhv rfcoswt bouevlela multip the value find we absolute considered with we examples prefactors the for have specifically they that observes one − ec edn odffrn utpiiis sacnrt exam concrete a As pla multiplicities. take not CP different may to cancellation e the leading symmetry hence hence the and kept under calculating invariant be When are will trace that contribute). states longer expectatio only no the genus and space tic mass ell moduli a the in to obtain place contribute state general points which a exist special (at to (at out states cancel symmetry of lot high st a with fermionic expect may and cases bosonic in counts particular genus In elliptic the that fact Z 2 CY 22 χ ue number Euler = 16 6 ymtisoecnmk w observations: two make can one symmetries 5 f − , 17 2 χ χ χ χ 48 a , , 17 = = +24 = +48 = +2 = − h ubro Y5flscntutda yesrae nwei in hypersurfaces as constructed 5-folds CY of number The : , 34 − − 6 f , 58 24 48 2 a , f 62 , 1 50 a , 102 ( ,z τ, f 2 ubro example of number hc a ue number Euler has which a } 22 = ) hl hsmyse upiigi a nesodfo the from understood can it surprising seem may this While . Z +90 2(29) 32 (59) 68 (67) 72 7(24) 27 2 symmetry  ch  ch 5 N , 0 =2 Z 5 N , ,n, 2 1 2 =2 ( ,z τ, 3 2 : (  ,z τ, ch + ) Φ Φ ae with cases 46 2 2 ch + ) − → − → χ 5 N , 0 = =2 5 N , Φ Φ − ,n, =2 − 2 2 5(22) 25 (23) 26 (43) 51 (59) 64 1 2 − , . ( 8adhneelpi genus elliptic hence and 48 ,z τ, 3 2 Z ( 2 ,z τ, ) symmetry f  1 ) A −  litcgnsof genus elliptic d n bevsta eois zero that observes one + moeo h geometric the of one om 2 tswt ieetsign. different with ates licities ote2 the to sn paetprefer- apparent no is e pi eu u ntotal in but genus iptic χ  sta oto those of most that is n . . . eetisre nthe in inserted lement ei h aefashion same the in ce ch grta n.More one. than rger l emyconsider may we ple Ymnflswith manifolds CY f 4 and +48 = nmdl pc)we space) moduli in 5 N , 0 =2 h wnn ellip- twining the , {− eeiecases reflexive 2 3 A ( ,z τ, 42 litcgenus elliptic (4) 4 (4) 4 (4) 4 (6) 6 f ch + ) 2 he pro- ghted a χ , − = M 38 (4.33) 5 N , 24 − 0 =2 f , − 2 48 in a 3 2 , ( ,z τ, (4.32) )  Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. rmalteeosrain ecnecueastrict a exclude can we observations these all From acltdalthe all calculated of multiplicities. o o eea Y5fl thlsta sepg n[ in 6 page (see that holds it 5-fold CY general a for Now nsubsection In 6-folds Calabi-Yau 4.2.3 studied. we cases t litcgnsi just is genus elliptic Its okn tteelpi eu o h uni Y ie sahyp a W as given genus. 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(  ,z τ, )s h have the so 8) . 5 N , 0 ) =2 ubr of numbers ,  − . CP just o 2 3 uninto turn ( ,z τ, 1 4 , (4.35) (4.36) (4.38) (4.37) (4.34) 1 ation , 1 ) f ,  1 2 , a 1 . . Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. hspriua oe is model particular this hr ehv h v ope coordinates complex five the have we where ilstefloigtie litcgenus elliptic twined following the yields .. nyteoealpeatrcags oi hssituatio this in 2 So of series changes. twined prefactor overall the only i.e., ntels eto eaaye ag ls fC -flsfor folds 5- CY of class large a models analysed Gepner we two section and last orbifold the toroidal In A 4.3 iesoa pc.I hscs h rgnof origin the case this from In space. dimensional n h riod r fteform the of are orbifolds the and 2 nti uscinw ilhnesuysc xmls npart s in examples, enlarged models. such have Gepner study CY’s and hence the points will where we space subsection this moduli the In in finite, particular is points In number special way. their any at if known in not representative is is doe it result there and Since 5-folds CY it. of for list evidence any finding without symmetry iceesmer ru fte1-iesoa torus 10-dimensional the of group symmetry discrete oriae r o fti ye emyhwvrlo tcases at t look however exchanges may and We type. moonshine this Conway of not and are Monstrous coordinates in occur that elements , 3 , h oodlobflsw r neetdi r fteform the of are in interested are we orbifolds toroidal The ecos osuya xml where example an study to choose We 4 K , eoti h rdc ftesnua ii fa of limit singular the of product the obtain we 6 3. Z T 10 Z / quintic ( Z g 2 4 ) .AGL 1280 = + 160 = g g g g 3 4 2 1 ( θ ,z τ, : : : : θ 2 2 { { { { ( ( 3 z z z z ,z τ, φ = ) τ, (2). 1 1 1 1 " 0 z , z , z , z , , 0) θ 2 3 ) θ 2 3 φ 2 2 2 2 − 2 θ ( θ z , z , z , z , ( 0 3 ,z τ, 3 5 , τ, ( 1 ( 3 3 3 3 φ τ, . ,z τ, z , z , z , z , 0) 0 ) , 0) 3 3 2 4 4 4 4 θ θ ) z , z , z , z , 3 T θ = 3 θ ( 4 ( 4 10 τ, 5 5 5 5 ( ,z τ, ( − { → } {− → } { → } { → } τ, ,z τ, /G 0) 5 0) ) 48 θ  θ ) 4 = ch 4 # ( ( z z z τ, G ,z τ, 3 N z 1 1 1 T , z z , z , , 0 0) 1 M =2 = , 1 4 − z , . . . , 2 1 ) , /G 2 2 24 ( − z z , , Z + ,z τ, 2 − z 1 , 2 4 a etogtt ietycome directly to thought be can 3 2 θ z − , × θ sgnrtdb h olwn 4 following the by generated is z , ch + ) 2 3 − 5 2 M z T ( , ( ,z τ, T 3 3 on z − τ, 10 24 z , z , 4 6 z , The . /G 0) ewl ee banthe obtain never will we n 4 4 4 ) − K T ymtymgtso up show might symmetry 3 N z , z , z , 3 θ , 10 θ z 0 ihacmlxthree complex a with 3 =2 2 3 , 3 5 5 5 5 For . − clrtria orbifold toroidal icular ( h litcgnsof genus elliptic The . ( ed o nwi this if know not do we } } } } o xs complete a exist not s τ, ,z τ, 1 2 , , , , h xsec of existence the Z T ( where 0) ,z τ, 2 10 ) θ mer groups. ymmetry oodlorbifolds toroidal θ /G 4 G 4 ) (  ( esgso all of signs he τ, 1 ,z τ, , G where 0) = ) = Z G m , 1 G (4.40) (4.39) × m M sa is G = 24 2 Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. litcgnso n ou aihsdet h emoi zer fermionic the to due vanishes torus any of genus elliptic z hscnb aiyse oaiefo h u vraltetwist the all over sum the from arise to seen with easily be can This ou as torus ihasuperpotential a with Φ hc r ihysmercl h oescrepn oorbi to correspond models The symmetrical. highly are which ic t qae oa lmn in element an to squares its Since litcgnsb hseeetoeobtains: one element this by genus elliptic oa to hr ehv introduced have we where rdcso 5rs.1 oisof copies 10 resp. 15 of products and xlie o ocluaeteelpi eu o uhmodels such for [ genus in elliptic moonshine the to calculate to how explained i h nwse etrde o otiuea ehv already have we as contribute not does sector untwisted The . 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( 4 ( 2 j − τ,z + + z , A ( G ) 4 1 riodof orbifold τ,x t 1 4 ) k + k , t t 5 = ) t +1 3 ) ) 4 4 x { → } k (0) h t (0) (0) ) 3 3 ic efidalna obnto of combination linear a find we Since . 2 ( 49 (0) # for − g θ θ = 2 2 ] ] 4 1 ∈ A t X k k τ, τ, i 4 + ) g k z k 1 =2 4 k (0) 1 G ,k 1 1 +1 6= k k k k N , 2 and 1 = [ +1 +2 +2 +1 k ∈ ,k t − X t ileatycag h ino two of sign the change exactly will 2 1 3 k sotie rmacia multiplet chiral a from obtained is 6= uhmnmlmdl ihcentral with models minimal such ( i ∈{ z z 3 G k z 1 4 (

3 2 2 6= 1 4 ti fodr2 wnn the Twining 2. order of is it , ) , z , 3 k t ) , 1 4 k t } 3 ( 1 z , − t ( k k − 1 4 4 1 ( z , 1 4 modes. o .I [ In 2. = + ) 4 1 2 + ) od by folds 5 ) adti a applied was this (and } t k t . 2 k ( ( dscos(sectors sectors ed t icse htthe that discussed − h 1 4 k 1 ) 4 1 htat nthe on acts that ( h t ) 4 1 1 61 Z t ( 1) sntrelated not is ) k − k t 3 , +1 k 15 (0) 2 1 4 83 ( )] n (2) and − ftensor of twas it ] t k 1 4 (4.44) (4.41) (4.42) (4.43) (0) )] f 1 10 a Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. htteeape esuidd o edt noutright an to lead not do studied we examples the that nti eto ehv icse h eut ulse n[ in published results the discussed have we section this In remarks Concluding 4.4 hc r gi iercmiain of combinations linear again are which Y5flsta hwMtiumosie h rva ae wh cases trivial The of product moonshine. any the Mathieu find realiz is show to all able that particular not 5-folds (in are CY we 5-folds space) CY ambient of f projective 642) genera weighted (13 elliptic class twining large the a calculating After characters. ihtehl fti efidtefloigelpi eeafrth for genera elliptic following the find we consider this of help the With ecnie the consider We ihamnssg.Ti eut ntefloigtie ellip twined following the in results This sign. minus a with charge enwwn otieteeelpi eeab ymty h s mul The chiral the symmetry. multiplying a by by acts genera Φ simply elliptic symmetry these the twine where to want now We iial o h oesta rs tseilpit nmodu in points special at arise that models the for Similarly expans interesting an shows 5-folds h CY we of As genus manifolds. CY elliptic dimensional the higher in genus elliptic the → h wndelpi eu fteGpe oe ec becomes hence model Gepner the of genus elliptic twined The Z e Z Gepner 2 tw π c Gepner i α ( h wndelpi eu fasnl iia oe sthen is model minimal single a of genus elliptic twined The . Φ c 5i u ae ie contribution a gives case) our in 15 = ( ,z τ, ( ,z τ, = ) = ) Z K 2 k ihaC -odaeteol xmlsw r bet find. to able are we examples only the are 3-fold CY a with 3 ymtyta ipyat ymliligtecia multipl chiral the multiplying by acts simply that symmetry k 2 + 1 2 + 1 a,b X k Z Z a,b +1 Z X k Z Z =0 (1) tw +1 (2) tw (1) =0 k,α (2) e 15 10 15 10 e π ( ( 6 ( i ,z τ, ( ,z τ, π c ( ,z τ, 6 ,z τ, i ( ,z τ, c a ( + a = ) 77 = ) 35 = ) + b = ) ) = ) b e ) f e 2 π θ θ 6 1 2 50 i − a π − c 6 i ( f c 455 f τ, τ, aτ and 615 ( 1 1 aτ a 2 a k k k k +2 110 + 2 +2 +1 +2 +1 φ 100 + φ +2 f 0 az 0 , z z 2 , az 3 2 ) a 3 2 φ − + oas nti aew conclude we case this in also So . Y φ ) i =1 N ( 0 f 0 f Z , α α , 2 1 2 1 a , k Z
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. , ( , . k,α ,z τ, 37 M i ( rodrtoeeet of elements two order or nfidn onhn in Moonshine finding on ] 24 ,z τ, da yesrae na in hypersurfaces as ed + i genera tic o ntrsof terms in ion isaeweew have we where space li symmetry. aτ v eni particular in seen ave + oesw atto want we models e xmlso genuine of examples iltwt phase a with tiplet + aτ r h Y5-fold CY the ere b )) + mls aeis case implest N b . ) . N (4.50) (4.51) (4.45) (4.49) (4.46) (4.48) (4.47) 2 = et Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. eodCencass hni ed wtotfiebaesour five-brane (without reads it Then classes. Chern second udeascae to associated bundle ls ene oebdnntiilguebnls(instantons bundles gauge non-trivial non-va embed with to manifold a need on we compactifications class for that see we So of oseiydt eodtesgamdlmti on metric model sigma the beyond data specify to orsod oa to to G corresponds connection and a orbifolds show toroidal that like examples algebra, symmetry extended an inh dniyfrtetrefr edsrnt ftehete the of strength field form three dH the for identity Bianchi th of compactification the discuss will we subsection this on In compactifications Heterotic 5.1 Moonshine. Mathieu the to connected to be CY pre matching also the the by side of which invariants II invariants, topological type Witten/Gopakumar-Vafa to the connected On are Moonshine. couplings Mathieu to connected be once oC aiod.I atclrw ilmk s ft of use make will Mat we on which particular compactifications in heterotic In manner relate different to manifolds. a CY present to will connected we invariants section Gromov-Witten this In and Moonshine Mathieu 5 ma and exist still may studies. moonshine further to connections CY4-folds For p not moon does umbral genera of series. elliptic forms twined the twining Jacobi here to also However connections interest cases. find any we in 6-folds moonshine CY Mathieu For in involved are 5-folds CY r,cle h e uesmercidx[ ind index special supersymmetric a new to the related are called ory, couplings hete gravitational the the On theory. and effective prepot the moduli of vector couplings the and gravitational spectrum, heterotic their match comparing will by We tions 3-folds. CY on compactifications on X K = hncmatfigtehtrtcsrn namanifold a on string heterotic the compactifying When 3 and 4 1 × ( tr T F ( 2 R n H-riod.Ordsuso olw [ follows discussion Our CHL-orbifolds. and stefilssrnt fthe of strength fields the is ∧ R ) K − c .S umn p rmti nlssoecno ocuethat conclude cannot one analysis this from up, summing So 3. 2 tr F ( X ( as F = ) ∧ V 1 F c × i ooooy,where cohomology), (in 0 = ) 2 ( V V 2 1 + ) n xrs h odto ntrso h respective the of terms in condition the express and M c 2 24 ( V K 51 2 r gi h rva ae hr n factor one where cases trivial the again are E ) K 3 c , 8 3 85 × × × 1 ( – T E V 89 T 2 1 8 n riod hro,t yeII type to thereof, orbifolds and , 2 = ) .A ewl e hsidxwill index this see will we As ]. ag ed ewietegauge the write We field. gauge n H orbifolds CHL and X c 1 hsi h aesnethe since case the is This . 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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. h eut rsne nsection in presented results the tnadembedding standard oepeieyw edt me ( embed to need we precisely More of ow edt me nttl2 ntnosi the in instantons 24 total in embed to need we so where oncin n h ibr pc tutr eeaie acc generalizes structure space Hilbert the and connection, h two the first/second fte10-dimensional the of aevco ude ihrank with bundles vector have N iua o h opcicto on compactification the for ticular supersymmetric oss fagaiymliltadavco utpe.Te10 The multiplet. vector a and multiplet gravity a of consist 4d effective the in of point hypermultiplets generic a a approaches: At two i) of either follow may One instanto of choices other for to also following the in see will we nti uscinw ilbifl ics o n a bant obtain may one how discuss the briefly of will we subsection this In spectrum The 5.1.1 iesos h pca aeweeoechooses one where case special The dimensions. oncineult h pncneto,ie,and i.e., connection, spin the to equal connection hr h pe ne aestelf n ih eta charg central right and left the labels index upper the where sn h emoi ecito fteeltcsoeotist obtains [ one space latices Hilbert internal these the of for description fermionic the Using po etmvn ooson bosons of moving representation left 4 of up tnsfr1 emoscmn from coming fermions 12 for stands E h hoy o h tnadebdigtefirst the embedding standard the For theory. the 8 E M (0 = n lastly, and o etr otesrcueo h nenlCTaditra Hilb internal and CFT internal the of structure the to turn we Now 8 24 × E G , ilcniu oexist. to continue will 8 E E i )wrdsetsprymtyand supersymmetry world-sheet 4) × stecmuatof commutant the is 8 8 , atr n a etitt 0 to restrict can one factors E n E 8 (1) 8 eeoi tigcmatfidon compactified string heterotic hn2( Then . 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(2) n ih-oig¯ right-moving a and , H steHletsaeo w etmvn ooson bosons moving left two of space Hilbert the is N c internal 24, = 2 ( V pcrmo h eeoi tig[ string heterotic the of spectrum 1 = K r 1 1 + ) ouisaeoecnoti h ubro etrand vector of number the obtain can one space moduli 3 + n = H 85 ( r Z K i 3.3 H ) 2 i r , emosfo h two the from fermions ) ∈ oehrwt h emosfo h fermionic the from fermions 4 the with together 3 in c D 1 (6 92 N T , 2 o h litcgnsof genus elliptic the for 2 K n 2 , N ( 6) K 2 E (1) V ] D o o-tnadebdig ei general in we embeddings non-standard For . eogn oteisatn meddi the in embedded instantons the to belonging 3 hswl ra h ag ru to group gauge the break will This . pctm hoyb iesoa reduction dimensional by theory spacetime 2 = 3 8 2 6 u otesmer ne h xhneof exchange the under symmetry the to Due . n , = ) × H ⊗ ≤ , 52 H (2) T n c Z 2 E (8 2 D (6 ntnosi oesubgroups some in instantons ) uesmerccontribution. supersymmetric 6 = N 8 , 6 efind we ≤ , 0) 0) c 2 uesmer nfu spacetime four in supersymmetry 2 = K 2 ngnrlti ilte edt a to lead then will this general In 12. H ⊗ steHletsaeo h unbroken the of space Hilbert the is ( n E K (1) 3 8 H × )= 3) × E 1 E (8 24 = E T 8 8 , = 0) 8 2 atri rknto broken is factor e,eg,[ e.g., see, , ag ru nti situation. this in group gauge χ SU H ⊗ n , K mednsaconnection a embeddings n 3 E K (2) (2) rigy[ ordingly T (2 4(5.2) 24 = 8 ilhl.Hwvras However hold. will 3 2 scul otegauge the to couple ’s , n estegauge the sets and 0 = G , efloigstructure following he 3) emsls spectrum massless he 94 es. iesoa gravity dimensional 1 , 90 = 95 H , 93 E N , 94 97 D 7 (6 ]. 2 (4 = , scle the called is , r pc of space ert 6) , K 95 98 3 T D G H , smade is ,which ], 2 , 6 1 95 1 )and 4) × n a and × H × (5.3) – 97 D (6 D G H 6 , 0) 2 2 ]. 2 , . Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. yacntn hf ftecia bosons, chiral the of shift constant a by ag ru.Hnei omtswt l h atncret i currents Cartan the all with commutes it Hence group. gauge ae(hr h riodgopi eeae yasnl eleme single a by generated is of group description orbifold the (where case h nrknguegopta en h omtn of commutant the being than group gauge unbroken the h 0dmninlvco utpe si h don repres adjoint the in is ( i.e., multiplet , group vector dimensional 10 multiplets. 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G i γ 1 i x , k + ) γ : h iesoa ag oosaie rmthe from arises bosons gauge dimensional 4 The . ( X 2 g e , X .Dsadn h ue uoopim(which automorphism outer the Discarding ). I k 2 7→ π i E 53 ( s/M R X K 8 Z ( k eopssas decomposes H I ( ouisaewhere space moduli 3 M y i γ + ) 1 ( , eeae ya element an by generated , + g R V iy olei h aia ou fthe of torus maximal the in lie to ) T I k ( G . 2 2 H ) i ) e , R by 1 ( + ) × ( k − H 2 x H i π ) 1 n rvt utpe,3 multiplet, gravity one o adj i γ , s/M x , 2 R ( sthe es g nlguebsn with bosons gauge onal ilarnethemselves arrange will H 2 ( k H in ) ( t,uigtebosonic the using nt), G naino h gauge the of entation i y ∂X ∈ h corresponding The . i ¯ i ) 3 , + tigon string )(5.4) 1) ae ieetrep- different label [0 E E iy ues K n a nyact only can and , 8 8 4 2 a ewrit- be may 3 × )) × π e s , and ) E E ± 2 8 8 0 = g π o our For . K a be may , heterotic i v E H a 3 v , , . . . , 8 T 1 × × (5.6) (5.5) × 4 a E ees H T by = 8 3 2 2 , , Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. h oncint ahe onhn.I sdfie by defined is It moonshine. Mathieu to connection the [ in hr h rc stknoe h aodsco fteinternal the of sector Ramond the over taken the is trace the where ouipeoeta.Acnrlojc hti eddt calc to needed is that object central A gravitation the prepotential. calculating moduli to turn now we section previous 89 Then ie by given by twisted sector fulfil to invariance modular by restricted futher hr ∆( where siltr nthe in oscillators ρ aigdsusdtesetu fthe of spectrum the index discussed supersymmetric Having new The 5.1.2 Here flf n ih etxoperators, cons vertex States right methods. and orbifold left by found of be now can spectrum The vectors d onsof points uhebdig for embeddings such P opeesetu a ewre u [ out worked be can spectrum complete and E ∆( n 4 = nyapasi h aeo siltr ( oscillators of case the in appears only stenwsprymti ne [ index supersymmetric new the is ] i 4 =1 99 = K ,m n, m E .W erdc h xmlsta ilb motn ou nta in us to important be will that examples the reproduce We ]. , r N 3 g n n N i R 2 × ( M R/L V M odd, = M = ) stezr on nryfo h wse etroscillators sector twisted the from energy point zero the is = T ,m n, eeoi opcictossnei onstenme of number the counts it since compactifications heterotic 2 = − 2 g mle that implies 1 = orsodt h ieetpsil ntno embeddings. instanton possible different the to correspond 2 n n N r h siltrnme fthe of number oscillator the are ) and exp ute h utpiiiso ttsstsyn bv mas above satisfying states of multiplicities the Further . 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V c 8 =(22 ) 2 2 egtltie tis It lattice. weight − SO + s fcombinations of ist F soitdto associated CFT , 9) v E P 2 8 egtwith weight (8) K + ) n ti ieby give is it , hr r two are There bles I 16 3 − oml are formula s =1 × mρ BPS-states. 1 P . T 2 I ] , 2 } 3 M even = . . nthe in (5.10) (5.9) (5.7) 6. = 85 – Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. ssoni [ in shown As charges. U(1) various of track kept not have al 3 Table 2 Table N h ntecs ihu n isnlnson lines Wilson any without case the In 116 84 − Z E N w xmlso perturbative of examples Two : new yemliltsetu o ieetebdig with embeddings different for spectrum Hypermultiplet : 6 SU v 3 1 × 3 1 (2 SO ( (2 E q, (9) ag Group, Gauge 3 1 SU E 3 1 , 7 3 1 , 85 (1 SU hf vector shift 1 (2 × × q (14) 1 ¯ (2 6 (2 × SU 2 (3 = ) 2 , ,scin3 oal speaking, morally 3, section ], ag group, Gauge × (3) SO SO × , , , , − hf vector Shift , (9) 1 0 U , 0 0 1 E (8) 4 5 SU 1 1 2 5 7 (1) , , 1 ; (14) (14) × − 5 , 2 ; 2 ; 6 × 1 0 × , 0 , 2 × 3 , 0 × E (2) 6 , , 0 0 ; U i × − 2 0 ; E 1 0 5 7 " 2 ; SU × × 4 7 SU (1) 1 2 ; 8 8 E × 8 P vectormultiplets BPS , ) × , ) , ) 0 U U 0 , 8 0 2 (3) 3 U (2) 0 U 6 7 ) (1) (1) ) 7 ) (1) (1) ) X +2( ( g g Twisted sectors 84 ′ ′ +( 1 1 ( ( ( g g g g 14 84 56 + + ; 1 ′ ′ ′ ′ 64 1 2 0 2 0 , +( Untwisted E g g 2( ; , ; ; ( 1 q 55 +2( ; ′ ′ 27 1 1 +( 1 1 ∆ 8 ; 3 3 sector 1 1 1 ( + ) ( + ) 2( + ) 2( + ) 1 × q ; , ( + ) , 1 ( + ) ∆ ¯ 1 1 1 3 E ( T ; ( , ; ; 28 − ; 1 27 8 1 2 56 1 1 1 1 ) h e uesmercidxfor index supersymmetric new the , ) 1 eeoi riodsetaon spectra orbifold heterotic ) 1 1 P hypermultiplets BPS ; ; ) , 1 +4( , ; ) 27 14 2 ; ; 3( , 2 64 5( 12( 1 1 ; 1 8( ; 1 1 ) ) ) , ; 1 1 27 ) 8 3 , ( + ) 1 Hypermultiplets 1 , ( + ) X 2 , ) ) 2 , , 1 ; 1 +2( ; 1 14 ; 14 ; ; 1 1 9( 1 1 1 4( + ) , 10( + ) 8( + ) , 6( + ) 1 36 4( + ) 1 2 , +18( ; 45( + 9( ; 1 +9( +18( 64 ; 9( 9( 1 Twisted 9( 9( 18( ; 1 q 27 sector ( + ) 1 9 1 ∆ 8 18( + ) 2( + ) 14 56 1 1 1 K ) ; , 1 1 1 , q 1 , , , 1 3 , 2 1 ∆ , ; , ¯ 1 1 ; 2 ; ; 2 as 3 1 , ; 2 14 3 ; ; 1 # 1 1 ; 1 ; ; ; 3 1 ; 1 1 ; ; 1 1 ) 1 ) ) 1 , 14 . ) ) 1 1 ) 1 ) ) ) 1 ) ) ) 9 ) ) , T ; ) ; 1 64 1 4 ; / ) 1 Z ) T ) (5.11) 3 4 We . / ˆ b 9 8 3 2 Z 4 . Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. ie h otiuinfo the from contribution the lines oeta h ag ru a emxmlyHgsdfor Higgsed maximally be can group gauge the that note ttmn [ statement fvco-adhypermultiplets l however and so vector- Doing of embeddings. non-standard to standard from u-Hgsn,teeycagn h ag group gauge the changing thereby (un-)Higgsing, oei ntno ouisae .. hnetewyteinst the way the change i.e., E space, moduli instanton in move nado h oooyo h ag ude .. ( e.g., bundle, gauge the of topology the on and on ( where 5.2.2 on theory the ie h atrsto n( in factorisation the Given ymkn s ftemdlrpoete of properties modular the of use making By lassatisfy always efie o1b aiu arguments various by 1 to fixed be h ibr pc tutr ( structure space Hilbert The K e uesmercidxfcoie noahlmrhcpart holomorphic a into factorizes index supersymmetric new utpes hsi unfie h offiin fthe of coefficient the fixes turn in This multiplets. uesmercidxwl eedo h oooyo h mani the of topology the on depend will index supersymmetric ftecaatr of characters the of 5.13 8 27 n the and 3 × u otegaiainlaoay6d, anomaly gravitational the to Due o h aeof case the For , satal nqeyfie pt utpiaieconstant. multiplicative a to up fixed uniquely actually is ) Z E τ K ,U T, 8 2 hsi qiaett oigaon ntehprutpe m hypermultiplet the in around moving to equivalent is This . 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H K 8 − Z × − Z 3 2 new 2 × n E  , 2 SU V D n 8 ( ( T 24 q, q, (1) 1 2 244, = n h litcgnsof genus elliptic the and 6 otiuin n noaltiesum lattice a into and contributions 2 2 ude ihisatnnmes( numbers instanton with bundles (2) p p q q ¯ ¯ ae h eea om[ form general the takes R 2 L 2 E = ) = ) 6 = = η ( q ( 5.12 n Z p ) q 2 2 1 X ∈ 5.3 E H ) T K p Γ 24 N 1 n , 4 2 R 2 2 3 K n a eueta h omof form the that deduce can one ) ( U , mle that implies ) ( 2 V q + V q ati [ is part 3 2 q ) ) − − | 1 2 eesadfrtenme of number the for stand here m 27 + · p N N L 2 Z . 1 q n ¯ n m 2 H 24 opcictoso h eeoi tigon string heterotic the of compactifications 1 = , 2 1 (2) 1 2 1 p 56 ( 4 swl s( as well as 240 = + R 2 U q, E = q m 85 + Z q 6 ¯ 1 η ( ) / 2 new p ] q , m 6 ( X n ∈ Z ) q na xaso f( of expansion an in 101 Γ 2 E 2 K ) K 2 , 24 , n a aeamc oegeneral more much a make can one + 4 3 2 ihteelpi ouu taking modulus elliptic the with 3 G ( ( q ] q q n 5.41 1 2 1 ) disa xaso nterms in expansion an admits ) 1 (  × p T n L 2 = (1) − (with ) G + p n n , 2 − R 2 n n ooecnesl move easily can one So . (1) ) Z 2 n 2 e (1) (2) nosaeebde in embedded are antons U T 2 − E (1) , 0 = N 2 2 n , .Hwvroemay one However ). 6 T ae h e number net the eaves ngnrltenew the general In . πτ n , Z η ( g | odoe(e.g., one fold 5.13 2 q (2) 2 ( 2 ye-n vector hyper-and 1 = K , eseta the that see We . p )i subsection in 1) = ) (2) q , hscntn can constant This R 2 1 3 E nhne.We unchanged. ) ) ( , , 24 n oWilson no and ) Z o1[ 1 to ) 4 q only. 2 ( oigfrom coming ) K dl pc by space oduli q 3 ) . sgvnin given as 30 infinity). ]. (5.12) (5.13) χ K K 3 3 ) Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. hog h rsneo the of CHL-orbifolds. presence studying the when Through sectors twisted from arise may aeoitdsnevnse u to due vanishes since omitted have pca aus[ values special h atr a eudrto ntefloigwy The way: following the in understood be can factors The where broken ewe h e uesmercidxadto and index supersymmetric new the between eopsto ntrso iesoso reuil repre irreducible of dimensions of numbe terms the also in thereby decomposition and a lines) new Wilson the without that case shows the This (for group. 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E 24 ) q  NS 8 oepeieymkn s of use making precisely More . 5.14 1 4 K θ × X n − η  ∞ 3 =1 ne h discussion. the enter 3 (  ( − E q E θ tnsfrn uhinsertion. such no for stands q η θ n ban connection a obtains one ) ) η 3 R , A ) 8 1 4 ( ( ( (  q q q n q eeoi tigcompact- string heterotic atrcmsfo h un- the from comes factor ) ettosof sentations ) ) etr Hr stands + (Here sectors ) g ) 6  h  ch htesupersymmetric the th 6 fBSsae admits states BPS of r = uesmercindex supersymmetric 6 Z n 3 N Z +1 eeyntjs the just not hereby ,n,l K ell =4 K ell / 3 stegeneraliza- the ss 4 3 ( ( ,l q, ( q, =1 q, 1 hc we which +1) − / − 2 q ( M q 1 q / 1 (5.15) ) 2 / 24 2 (5.16) ) Z ) . N oming (5.14) ning acts Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. vacua. hsato utpreserve must action This o the For on ( riod fhtrtcsrn opcictos[ compactifications string heterotic of orbifolds aaersn the Parametrising and h riodlmtof limit orbifold The e opreserve to der ewl osdreape of examples consider will We reyb n1 an by freely nenlCTdsrbn h eeoi tigon string heterotic the describing CFT internal K section in mentioned wihmycag hnmvn ntemdl pc of space moduli the in moving when change may (which orsod ooeo h olwn iecnuaycasso classes conjugacy nine following the of one to corresponds T H orbifolds CHL eoigtecodntson coordinates the Denoting ob xpitfe,a sgaate yte1 the by guaranteed is as free, point fix be to z 4 29 28 1 sepesbeas expressible is 3 g z , K s then ic uhatmrhssawy aeafie on [ point fixed a have always automorphisms such Since hyaenmdatrCadui oke n yknwowr h rtt first the were who Lykken and Hockney Chaudhuri, after named are They npriua ecnie riod where orbifolds consider we particular In o the For So T ( : 2 n h ooopi -omon 1-form holomorphic the and 3 ) 2 y g ∈ Z h rioddtere antb huh ob bandfro obtained be to thought be cannot theories orbifolded the Z 1 K ′ N 3 + C ( : sotie yteobfl action orbifold the by obtained is 3 aew realize we case 2 a oata ypetcatmrhs o order (of automorphism symplectic a as act to has iy x / Z 1 Z 2 2 x , /N y , 28 2 z N aew olw[ follow we case i 2 ihperiodicity with 3 forder of y , ∼ pc iesprymty ic h cinivle both involves action the Since supersymmetry. time space 2 = + hf noeo h ice of circles the of one on shift T 1 z 2 y , iy 3.3 T i K by + 4 2 4 A A A B A A A B 8A 7B, 7A, 6A, 5A, 4B, 3A, 2A, 1A, ste elzdb riodn with orbifolding by realized then is 3 ) y , / h ypetcatmrhssof automorphisms symplectic the n Z ∼ x 3 N 1 g SU K 4 1 y , e s oceeylt( let Concretely . ( x , riodn yafel ciggopi eesr nor- in necessary is group acting freely a by Orbifolding . 1 e as 3 ( : 4 2 + T 2 oooys tms eantehlmrhc2-form holomorphic the retain must it so holonomy (2) Z ) 2 π z 2 i 2 ∼ 92 ∈ s/ n 1 z , [ y( by T 2 4 [0 70 n osdrapitin point a consider and ] ( e ( x 4 2 y 2 , / , 1 ) e , 1 2 Z x 92 + 7→ π + 3 1 /N h H cinmyb rte as written be may action CHL the ) 1 T x , , ,x π, 102 iy [ ( 38 = 58 2 97 e 2 2 shift. [ ) 2 , ) .W hoetecodntson coordinates the choose We ]. e 104 and ] π 2 3 e , 103 i ∈ y , 2 s 3 πi y z − Z 1 1 [0 ]. 1 ]. e , 2 y , . . . , e , π + N , T K i 2 Z s/ 2 105 − ,y π, 2 csa natmrhs on automorphism an as acts π 1 = 3 .W ee oteeobflsas orbifolds these to refer We 3. 4 2 ( saoe h H riodof orbifold CHL the above, as ) π oehrwt nato nthe on action an with together 3 riod [ orbifolds ti eesr o h cinon action the for necessary is it ] y i s 4 2 3 , z ) + ( + 2 n ) ∈ K . ,y π, 1 iy K n , [0 omasbru of subgroup a form 3 K 4 ) ahautomorphism Each 3). f . )) , 2 3 M ouisae where space, moduli 3 2 ) 38 + s , π ∈ 23 ecodntson coordinates be ) ]. ,y π, N Z 0 = [ 106 td reyacting freely study o 2 on ) m . 4 , . . . , + ] N π K ) 1 = . (5.18) 3 3 T d , 29 4 (5.19) (5.20) (5.17) (5.21) obe to As . M K 6 = K T 3. 23 3 2 Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. efud[ found be hr now where setal uttepoeto onto projection the just essentially utpesaiigfo h wse etr a hne ob the So Only change. of may points untwist unchanged. sectors fixed twisted the is the multiplets from from vector arising arise multiplets of multiplets ( number changed vector been and the not group and has same) group the gauge the is into instantons of ding utpe tl edt eaddt hsspectrum. this to added 1 be the to from need contribution still universal multiplet the from coming multiplets tables i si oteuobfle aeoecnoti h pcrmat spectrum the obtain can one case unorbifolded the K to in As Similarly ii) numb the only changed. multiplet gauge is changes. dimensional hypermultiplets 10 the of from fie number coming vector the 3 multiplet, only dim gravity 10 so 1 the is from orbifolding contribution after universal tiplet the that finds one Then si h nrioddcs (subsection case unorbifolded the in As spectrum The 5.2.1 4 with orbifolding by implemented be now can 3 order h eeeais( degeneracies the above. ways [ two of reduction the dimensional in By constructed i) be again be may theory wse by twisted re h omlso h asso tts( states of masses the of formulas The free. h ag ru ssal ec ewn oHgsteguegroup gauge the Higgs to want we hence small, is group gauge the and ouisaeb F ehd [ methods CFT by space moduli 3 h 1 eaeitrse nfidn h osbetp Ida theorie dual II type possible the finding in interested are We , 1 4 5 om nteorbifolded the on forms ooti h opeesetu h rvt utpe n t and multiplet gravity the spectrum complete the obtain To . and 38 g χ ′ , ( 5 nyde o rdc asessae ic the since states massless produce not does only g 92 npatc nigteda hoyi nyfail hnter the when feasible only is theory dual the finding practice In . g n g ′ g , .Aanw erdc h ae htaeo neett si tab in us to interest of are that cases the reproduce we Again ]. n ( : 5.8 m and h x 1 g 1 D , hne to changes ) ′ 1 x , r stenme ffie onsof points fixed of number the is ) ( 2 = g g 2 m z , n g = ) k , , k 1 ′ r z , o the for 2 M 97 ) 1 K 7→ :Oenest aeit con httenumber the that account into take to needs One ]: = N 1 sgvnb [ by given is 3  N g M m X Z ′ x =0 24 38 − nain ttshsetrd ic h embed- the Since entered. has states invariant 1 + 1 2 1 , + N and X r 5.1.1 =0 59 92 − 2 − 1 3 π .Oetigt oiei httesector the that is notice to thing One ]. χ 5.7 Z 2 x , ( ,tesetu fteCLorbifolded CHL the of spectrum the ), , 3 g tytesm,ol h oml for formula the only same, the stay ) 2 n 65 riod h eutn pcrmcan spectrum resulting the orbifolds z , g , for ] 2 m + N g ′ r 3 1 )∆( 2 = e g d n 2 and lds 1 n + ,m n, , nain under invariant 3 onigtecommon the counting y 2 3 , 5 g e iesoa gravity dimensional 0 (5.24) ) , ′ 2 ninlgaiymul- gravity ensional ro hypermultiplets of er dsco,tegauge the sector, ed 7 z , cini xdpoint fixed is action .. h hf vector shift the i.e., ntecontribution the in . otetere of theories the to s k 2 ubro hyper- of number  sfra possible as far as pca onsin points special hypermultiplets, . e3vector 3 he g m n of ank g (5.23) (5.22) ′ r so , les Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. n n h ultere ftemxmlyHgsdmodels Higgsed maximally the of theories dual the find charges. and U(1) various of track kept not have We orbifolds. CHL orbifolds CHL al 5 Table 4 Table h hre clr cmn rmete etro hypermulti or vector either from (coming scalars charged the Appendix singularities. 30 N ntetp Isd igigcrepn omvn nteC ouisp moduli CY the in moving to correspond Higgsing side II type the On h 116 84 − N E Perturbative : Perturbative : 6 v SU ag ru,shift Group, Gauge F 3 1 × 3 1 (2 SO (2 E E igigi oeb triga on nmdl pc hr al where space moduli in point a at starting by done is Higgsing (9) 3 1 SU 3 1 , 7 × × 3 1 , (1 SU 6 (2 1 (2 SU (14) 1 (2 (3 × ag group, Gauge 2 SO SO × 2 , × (3) hf vector Shift , , , , , − , (9) 1 1 0 U shift (8) 0 0 SU 1 E , 4 5 1 2 (14) (14) 7 5 5 (1) , 1 ; 1 × , 2 ; 2 ; , 6 × 0 , × 0 0 × (2) × 3 , 0 E 6 , , 2 0 ; U × − 5 0 ; E 1 0 2 ; × × SU 7 E E 2 ; SU 4 7 (1) 1 × 8 8 E × 8 , ) , 8 8 , ) U U , ) 0 0 0 8 (2) 0 × × U 2 (3) 3 U 7 (1) (1) 6 7 ) ) ) (1) ) (1) E E 8 8 eeoi riodsetaon spectra orbifold heterotic on spectra orbifold heterotic g g Twisted ( sectors ′ ′ 84 1 1 +( g g g g ( ( ( + + 14 84 56 ′ ′ ′ ′ ; 2 0 2 0 64 +2( 1 +( g g Untwisted 60 ; , ; ; ( ′ ′ +2( ; 3 3 27 1 1 +( 1 1 sector 1 1 ( + ) ( + ) 2( + ) 2( + ) 1 2( , ( + ) , 1 ( + ) 1 ( 1 1 3 ( ; 28 ; ; 27 ) ; 1 56 1 1 1 1 ) , 1 ) +2( 1 1 ; ; ) , 1 2 ; ) 14 27 2 ; ; 3( ) ; 64 ( 4( 6( 1 1 ; 1 1 1 1 ) ) ) , 27 8 1 , ( + ) Hypermultiplets ) 3 , ( + ) 2 , , 2 ) , 1 1 ; +2( ; 1 14 ; ; 14 ; 1 1 1 3( 1 1 2( + ) 4( + ) , 6( + ) , 2( + ) 1 2( + ) 1 36 lt)hv vanishing a have plets) 2 , ; +6( 15( + 3( ; +3( 1 3( 3( 64 30 +6( Twisted ; 3( 3( 1 T T ; 6( 27 1 sector ( + ) silsrtdin illustrated As . 9 1 1 8 1 1 14 56 4 4 2( + ) 6( + ) 1 1 1 1 ; , ) 1 / / c hog conifold through ace , , , , 1 , , 1 3 ; , Z Z 2 2 ; 2 1 1 ; ; 3 2 ; 1 1 , ; 14 ; ; ; 3 4 1 1 ; ; 1 1 ; ; 3 1 ) ; 1 1 1 1 1 ) ) 1 1 ) , 14 ) ) fe re 3 order after 2 order after ) ) 1 ) ) 9 ) ) 1 ) ) , ; ; ) 1 1 64 ; ) 1 ) ) of l ˆ b 9 8 3 2 Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. ru otern fteguegop egv nepii examp explicit an give We Appendix group. the dim gauge in the the done from unb is of multiplets the Higgsing rank vector of the rank of to the number group to the equal reduce being will those This of number the factors, olm ftoevco utpes hrb hs ag gro gauge gr these still gauge Thereby can remaining multiplets. we the vector manner in those this multiplets of Coulomb vector in the group of gauge scalars the the Higgsed have we Once igigde o hneterltv ubro etradhyp and vector of number relative the change not does Higgsing ntables in hr h atto ucin fteshifted the of functions partition the where rvt utpe,ie,3vco utpes h rvt m gravity universa the the multiplets, i.e. from vector hypermultiplets, coming 3 multiplets i.e., multiplet, vector gravity the only with left 5 belo scalars to of VEV branch giving Higgs by the section group in along above gauge moving mentioned the hypermultiplets, Higgs certain may to we Then VEV. n[ in ( Z form general the takes orbifold CHL medn ie ysitvectors shift by given embedding opcicto on compactification ygnrlsn h ehd ecie n[ in described methods the generalising by h e uesmercidxfrteCLobflso the of orbifolds CHL the for index supersymmetric super- new The new the and corrections couplings/threshold Gravitational 5.2.2 n new,N hsi h aefrtetidadsxheape hti after is That example. sixth and third the for case the is this 1 n , 38 2 , ntnosit the into instantons ) 92 = ymti index symmetric .Frtecs where case the For ]. 4 − r l uhta h ag ru a ecmltl ige.Fo Higgsed. completely be can group gauge the that such all are 2 η 20 1 ( q ) Z Z a,b M X E ( K E ( N a,b a,b 8 − =0 8 ′ 3 h 1 ) ) ( r,s ( − × N X q q − = ) =0 = ) N T 1 E 5.1.2 2 v e 8 − α,β α,β hypermultiplets. ihdffrn ntno mednscnb obtained be can embeddings instanton different with X × X K 2 1 M πiab 1 =0 =0 E sraie sa as realized is 3 hscrepnst hnigteebdigo the of embedding the changing to corresponds this γ, 2 8 F e Γ e γ ˜ − 2 ag ru hl oee keeping however while group gauge h xmlso pcrmw aepresented have we spectrum of examples The . − Z i i h e uesmercidxo norder an of index supersymmetric new the , πβa πβa E ( a,b 8 61 P P ) ( 101 I 8 q I 8 =1 =1 E ) × γ 8 .W reyepanterslsgiven results the explain briefly We ]. γ ˜ I I Z atc r ie by given are lattice Y I Y I E =1 ( 8 =1 8 a,b 8 ′ T θ ) θ 4 ( h / h q Z α ) β α β × +2 +2 M +2 +2 liltadteremaining the and ultiplet aγ bγ 2 a b 1 n eea o standard non general and γ ν ˜ γ ˜ I I I I F E i i p r rknto broken are ups u,mvn ln the along moving oup, 8 ( rmultiplets er , ,r ,s b, r, a, , otiuino the of contribution l × nino h gauge the of ension oe ag group. gauge roken ouisae As space. moduli E 8 igigw are we Higgsing e o o the how for les eeoi string heterotic ieaVVto VEV a give ; q n ) Z 1 n , 2 ( N r,s , 2 2 table r h ) (5.26) (5.25) ( nging fixed. − q, U N (1) q ¯ N v ) . , Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. where the lsdepeso o ieetstainhv enobtaine been have situation different for expression Closed stetaeover trace the is emosi h wse aodsco hs emo ubri number fermion whose sector Ramond twisted the in fermions hr h oet r enda n( in as defined are momenta the where where expression ¨ he n ope tutr ouiof moduli structure complex K¨ahler and Γ where n a write may one eetdi h phase the in reflected ed ntecmo xdpit.Te a ewre u o the for out worked be can They points. 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Mathieu of series Thompson fsitvcosadcorresponding and vectors shift of in hr ehv sdtesotadnotation shorthand the used have we where h constant The h ausfor values The ietydtrie h ieec ewe h ubro hyp of number the between difference the determines directly 101 3.3 Z n ttksfu ieetvalues different four takes it and ] o h aeo o-tnadebdig fodr2considere 2 order of embeddings non-standard of case the For β α new,N ≤ g ˜ (0 g (0 nti iuto h e uesmercidxbcmslink becomes index supersymmetric new the situation this in ˜ , , ,r k r, s, 0) 0) ( ( = τ q, 0 = ) N q ¯ 8 f f ≤ ˆ b = ) (0 ( ˆ b r,s α , eed nte1 ieetpsil ntno mednsli embeddings instanton possible different 14 the on depends N , 0) eeatfroreapeaegvni table in given are example our for relevant β , ) ( ( + N − g ( ˜ 1 q q r,s 70 g ˜ (0 = ) = ) and 1  ) r,s N X ,s .For ]. = θ ) − =0 η ( 3 1 ( × τ 4 2 ( N q q = ) η η Z E )  ) 24 24 E ( 2 1 3 N (  4 1 N N r,s , N 2 ( ( ( α 8 6 q q − τ ) samdlrfr fwih ne Γ under 2 weight of form modular a is η ) ) 1) + g ( q ( ˜ = ) 2 = r,s N q, E ( 1

q / ˆ b ) 4 4 ) q ˆ ¯ b 2 E 1 + ( ˆ Z b ∈ 12 , ) q π · , E 3 6 K auscnb on n[ in found be can values ) ell  ( (  , E ( 4 q N 3 E 5 ,s r, ( 12 ,r,s 0 2 3 ) 6 N , q , ( β − ehave we 7 − ) 63 ( q i 9 4 ( ) g ˜ ( τ (5.35) ) q, r,s × , β 1) (0 6= Z ) 2 3 g ˜ ( ) − β , " K r,s ell ( ∂ , τ q 3 τ Z  K 8 9 ) ,r,s ) 2 1 ( , g ˜ ( ln   new,N θ τ r,rk ) )(5.32) 0) [ 3 η 2 ) 2 − ( . := η ( E q 92 + ) q η ( ) ( 4  ) Nτ ( τ ( Z τ ,   a h olwn xaso in expansion following the has q = ) θ η ) 97 K ) 6 ell 4 2 3 ) (  ( Z 3 q . q ] − ,g ) ) K N ell ′ r  ˆ b 3 ,g radvco multiplets vector and er ( 92 ,r,s  4 6 ′ N s q ob u discussion our by So . n h opeelist complete the and 2 .Tevleof value The ]. E ( 1 1) + q, / 4 4 ( dt h MacKay- the to ed n[ in d Z − q ) K )(5.34) 1) ell ! E 0 3 ( N ,r,s N 92  endas defined ) ( ,q q, n finds one ] τ N + 2 1 tdin sted k ) (5.33) (5.36) ˆ b #  also , Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. where aaee ntefloigway following the in parameter hr 1 where eta swl scrangaiainlculnso the of ve couplings the match gravitational to certain want as we well section this as prepotential of tential start moduli the vector at the mentioned and As couplings Gravitational 5.3 and 6 Table htfraltefu o tnadebdig ˜ embeddings standard non four the all for that etr with vectors hog h relation the through hoyaiigfo h eeoi n yeI compactificati II type and heterotic the from arising theory f˜ of a, o re riod n finds one orbifolds 3 order For 5.5.2 ˜ b, f χ f c, ˜ ( r,rk (0 ≤ auso ˜ of Values : iltr u ob h ue hrce fteda Y(e subs (see CY dual the of character Euler the be to out turn will d ). ˜ , 1) ) ,k r, eedo h ntno mednsadaegvni table in given are and embeddings instanton the on depend = = K ≤ as 3 3 3 3 1 3 1 η η 1 1 (1 (2 ,w mte the omitted we 2, 24 24 3 1 3 1 , 3 1 , (1 T − (2  1 (2 c ˜ aE ˜ 4 2 , a, 1 aE ˜ , / , , = − , 1 Shift 0 0 Z ˜ 0 b, 4 4 a ˜ 1 5 7 4 − 6 , 3 E , 2 ; 2 ; E c ˜ 2 ; = 0 0 and 6 2 3 6 6 , , and 8 , 0 ; + 1 0 ; 1 0 + 2 3 1 4 7 7 3 1 2 8 ˜ b 8 N , ) 3 N , ) ˜ 9 46+5 + (456 b E ) 0 3 0 h E N 3 3 2 4 + (48 5 ) 3 E H orbifold. 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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. fteitra ofra edter.Teitra rc ca trace internal The ov theory. taken field is conformal trace internal The the time. of space non-compact transverse the where y, where genus ieaeprubtvl n oplvleat.Cnrtl fo Concretely coup . gravitational exact level The loop one group). perturbatively orbifold are the side of element an or ic ntetp Isd h couplings the side II type the on since nryeetv cintegaiainlcouplings gravitational the action effective energy F fteC nteda yeI ie o h aeweew e l mo all set we where case the For side. ext II to Gopakum type need of ( dual one term from the side in apart on II written CY type be the can the of part to This connection part. a holomorphic make Borcher by to theorem order reduction In lattice the using by evaluated where h ou,oefid htteprl ooopi ati given is part holomorphic purely the that finds one torus, the eeoi tigcmatfidon compactified string heterotic g 32 y ¯ for , oepeieyi stecmlxcnuaeo h niooopi part. antiholomorphic the of conjugate complex the is it precisely More η olciesadfrtedpnec ntevco oui Th moduli. vector the on dependence the for stand collective ( F 1 τ g g X f F ) > g ( 2 ( + ol he i hudhnentb ofsdwt ihrtes the either with confused be not hence should (it sheet world y, r,s Tr F stecmlxcodnt on coordinate complex the is R , ¯ y ) g ¯ hol ( h ,aegvnb h n-opintegral one-loop the by given are 1, = ) τ + ,U T, (i sdfie n( in defined is ) = r h efda ato h rvpoo n h imn tens Riemann the and graviphoton the of part self-dual the are ∂X ¯ 2 + ,tecmlxsrcueadtecmlxfidK¨ahler struct complexified the and structure complex the ), π ) 2 (2 ( 1 2 1 ( − g g c !) − g (0 1) π − 2) 2 ,s 2 1 g ) Z − (0) 2 ( − 1 η d ( N 1) X ζ s τ 5.30 τ 2 =0 S 1 2 (3 − τ ) F 1 24 n = q F  − h ), τ K X Y i m> Z X r,s 2 =1 2 2 L g p η 3 g 0 L r,s 1 − F 0 ( ) Z × Γ τ  e 24 g p , c 2 ( T ) − ( F r,s , T 2 q d ¯ y, 2 2 2 R r,s g Tr L 2 π ˜ 65 2 ) x 0 and i y rs rmtplgclcnrbtoso the of contributions topological from arise ¯

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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. h coefficients The bla ag group gauge abelian lhuh( Although g where eeoi etrmdl rptnili fteform the wi of is case prepotential the moduli considering vector when heterotic concretely More prepotential. 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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. prpit irrClb-a ihhdenumbers hodge with Calabi-Yau mirror appropriate r eae through related are a ihHdenumbers Hodge with Yau enlidcs( indices ternal o yeIBcmatfiain h ubro etradhprm hyper and vector of number the i.e., changed, compactifications IIB type For h clr ftefia utpe rs rmteRR3fr wi 3-form R-R the from arise multiplet ( final dices the of scalars The utpesaiefo ercwt w nenlindices internal two with metric from arise multiplets by M eedwt oecmet ntesrcueo h ouisaeof space moduli the of structure the the to on comments some with end We rvpoo) ial hr are there Finally graviphoton). w nenlidcs hs clr orsodt complexi to corresponding correspond The scalars CY. 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C 1 , µ 1 Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. ( o titytesm.Ruhysekn genus speaking Roughly same. the strictly not interchangea invariants Gopakumar-Vafa and Gromov-Witten h ubro dsic as maps ways’ ‘distinct of number the h uniisw ilwn ocmaet ac eeoi and heterotic match couplings to are compare tions to want prepotential will we the quantities and The couplings Gravitational 5.4.2 subsection. next the in moduli vector the of where hswl edt oetasfrtoefilsadt nitrcino h s the produ of the interaction keep an still to however and terms fields kinetic those The for multiplets. potentials of to lead will This and K¨ahler manifold erco h ope tutr ouisaei xc tboth at exact is space stru moduli complex structure σ to complex correspond the moduli on vector metric the theory IIB type genus ( in tr string metri the at receives and K¨ahler exact space moduli to is correspond space moduli vector moduli the multiplet an vector (perturbative of corrections p metric stringy is dilaton receives the hence compactifications IIB which and IIA type For metry. fBSsae nClb-a opcicto ftetp I th IIA type the [ e.g., of see, invariants, compactification Gromov-Witten Calabi-Yau to related in closely states BPS of g in pootoa opwr of powers to (proportional tions h CY the ssoni [ in shown As CY where utcvrn apn Σ mapping ( multicovering hs reeege as energies free these n oe relvl ewl aemr osyo o ocmueth compute to how on say to more have will We level. tree model - 35 36 1 oaua-aaivrat.Teeivrat on na ex an in count invariants These invariants. 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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. h etrmdl rptnilfratp I compactificatio IIA type prepotential a for the prepotential - moduli invariants vector Gromov-Witten The zero Genus 5.5.1 duality. 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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. rioddvrino h eeoi-yeI ult aeto have duality II heterotic-type with the of version orbifolded e subsection see nerlsmlci ai fproso h irro h ell the of mirror the of expansion of BPS integral periods The of basis symplectic integral eage htisnmrtri ekJcb omadisdenomi its and form Jacobi weak a is numerator its that argued be l-eu eut o h oooia tig[ string topological the for results all-genus aet ufi.Ti hnalw n ocluaetetil in triple the calculate to one allows then This fulfil. to have brdC -od ih2scin lse ntable in (listed 2-sections with 3-folds CY fibered Γ of forms Jacobi degree higher meromorphic be to shown are with fibrations Z coefficients expansion a the irreducible that argued are was fibers it Then all present. where particular in [ considered, in generally More app classes. that manifolds Calabi-Yau elliptic of the fibrations duality, section are II in side detail type more - in heterotic discuss will we As [ In bootstrap Conformal 5.5.2 map. mirror the as well as ae h h oua rnfrain cigo hs argum these on acting transformations Z modular the The base. litcprmtr,weetenmrtr o r ekJacob weak are now numerators the where parameters, elliptic r aoifrso egtzr,weetesmi vr2cce i by 2-cycles denote over here is we which sum constant, the coupling where zero, argument weight elliptic of forms Jacobi are ovlmso ba uvsaeicue,se( see included, are curves fibral of volumes to ftetplgclsrn atto function partition string topological the of exp(2 ilcletvl eoeby fibra denote the collectively to will correspond that K¨ahler parameters complexified epoiemr eal nAppendix in details more provide We top. top. 114 n[ In pcaiigteast ( ansatz the Specializing a egnrlzdt hsstain hr o loparamet also now where situation, this to generalized be can sivrat rs spr ftesmlci oorm grou monodromy symplectic the of part as arise invariant, is N π ieetmto a sdt banalgnsGopakumar-Vaf genus all obtain to used was method different a ] i scin,where -sections, 114 P i twsage htteCsapaigo h yeI iei h C the in side II type the on appearing CYs the that argued was it ] β i t i 5.6 ,where ), N Z scin and -sections hnteagmnsgvnaoefrteepnincoefficient expansion the for above given arguments the Then . top. τ srltdt h opeie oueo h bradtestring the and fiber the of volume complexified the to related is ( ,t τ, λ , N t i i , = ) m steodro h H riod o oemr details more some for orbifold, CHL the of order the is 116 E.6 K 1 = eoeelpi aaeesof parameters elliptic become Z Z N pca ls felpial brdC -od was 3-folds CY fibered elliptically of class special a ] ufcsover surfaces 3 rmteAppendix the from ) 0 top. h , . . . , ( ≤ ,λ τ, moe oesrcueon structure pole a imposes the 4 ) E   1 . , 71 + 1 1 SL ( B Z 115 r hfe oue fcre nthe in curves of volumes shifted are ) (2 top. β λ 5.6 ∈ per selpi argument. elliptic as appears , P , H E.2 ,a es o eti e fcurve of set certain a for least at ], Z 1 X nteuobfle eso fthe of version unorbifolded the in , fteform the of 1 hc ae tpsil oobtain to possible it makes which , , ru sboe oΓ to broken is group ) 1 9 ( .Oefid htfrgnsone genus for that finds One ). 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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. where d epciey oeconcretely More respectively. nodrt aclt h eu n oaua-aainvarian Gopakumar-Vafa one genus the calculate to order In al 7 Table [ one degree base for finds one string heterotic the of 2 order of where o aedge h ofra otta ie h eut[ result the gives bootstrap conformal the 2 degree match. base For perfect a finds one and n nsta tdge eowrt h aeo the of li base we the and characteristic Euler w.r.t. the zero on depend degree only invariants at that finds on string One heterotic the of 3 order of orbifolds CHL to dual ( using (again h rdcin rm( from predictions the aaivrat tgnszr a ecluae sn h stan the using calculated be can zero genus at invariants Vafa aeo the of base hscnb hce gis rdcin rm( from predictions against checked be can This table b 37 d , erfrt 53) 53)i [ in (5.31) (5.30), to refer We etw unt Y3flswt -etos itdi table in listed 3-sections, with 3-folds CY to turn we Next f 10 Z orsodt h ere fthe of degreed the to correspond E φ 1 . d 2 0 eu eoGpkmrVf nainso erezr ihr with zero degree of invariants Gopakumar-Vafa zero Genus : ( B ,λ τ, ed ob fixed be to needs stedvsrascae otetrescin nedtable Indeed section. three the to associated divisor the is 4 3 2 1 0 5 \ K d F = ) 5.35 bainfrtefmle fClb-a manifolds Calabi-Yau of families the for fibration 3 χ 2 η 4 1962 240 + ,( ), (2 4 0 3 0 0 2 0 0 0 1 0 0 0 0 0 0 Z τ 5.45 2 ) ( 5.38 24 ,λ τ, ∆ φ ) Once )). χ χ 2 2 2 − ),( = ) ( 240 + 240 + 2 τ d , 37 1 5.39 114 ) − F (2

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B 30 30 and = χ χ 7 n ˆ b M where , 1 = (5.60) (5.59) (5.58) + 2 (3) 117 ,ν 3 2 K n , of 2 ]. 3 9 4 . Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. ewe the between ftecnomlbosrp sn eu eoGopakumar-Vafa zero finds genus one Using ansatz bootstrap. conformal the of fClb-a manifolds Calabi-Yau of K 8 Table ie fti ult vra over duality this of sides the between duali duality this known in [ well argument involved extension the CY adiabatic the an that using by shown elliptic be an can be ge the to it in theory coupled weakly II is type theory heterotic the that Assuming certai to corresponding manifolds gat CY CY E information dual dual the the use the find may to one finding how sections show - will duality we section II this Type In - Heterotic 5.6 Yta is that CY at bu hsdaiywtotteCLobfl,se .,[ e.g, see, orbifold, CHL the without duality this about facts n yeIAcmatfidon compactified IIA type and n nsta hsmthsterslscluae rmtehe the from calculated results the matches this ( that finds One bantecretaon fuboe pctm supersymmet spacetime unbroken of amount correct the obtain 5.38 8 bainaddge n ihrsett h aeo the of base the to respect with one degree and fibration 3 × Z ),( E d 1 ( F 10 8 5.39 9 8 7 6 5 4 3 2 1 0 ,λ τ, \ oaua-aaivrat fdge eowt epc ot to respect with zero degree of invariants Gopakumar-Vafa : eeoi opcicto on compactification heterotic g = ) ),( 291 K 18353988 170 E 54 88 413280 5.45 brdover fibered 3 8 3 15 10 χ × χ 48 1962 χ χ 2 3 1 χ χ K χ 159217686 + 2 1627330 + χ 5694624 + 55646304 + E 6 ))). η 95454 + 18016 + 4 0 0 0 0 0 240 + ufcsfiee over fibered surfaces 3 8 600000 0 0 0 0 26 + (3 5 4 3 2 1 0 − eeoi tigcmatfidon compactified string heterotic − τ − ) 3 30 24 195 χ ∆ M φ χ 6 − − ( (3) χ 2 P τ 1 P , ,ν 1 ) 1 1 (3 − h eutn emtyo h yeI hoyi a is theory II type the on geometry resulting The . 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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. ˆ h ne of index The ude vrsm ae hc norcs ilbe will case our in lifted which are base, equation some hypersurface over the hyper bundles of a coefficients as a curve the based fiber/elliptic Then one ‘fiber genus a the by constructs first systematically constructed were sections b ulCsfrtecssta a ptnily ecmltl Hig completely be (potentially) can + that (12 cases numbers the for CYs dual ait hti a is that variety -rn htwastersrcino h eu n fibratio one genus the of restriction the wraps that 5-brane a ( found. 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P F 1 [ 1 0 113 + , q F n ..tegnrcClb-a has Calabi-Yau generic the s.t. and 0 - - 115 - - 107 3 - 115 3 2 123 3 1 2 3 2 3 3 3 3 1 q 2 h 2 n oprdwt h predictions the with compared and ] 1 sbs edt h aehtrtcstring heterotic same the to lead base as 8 + ) , 1 q 2 a on ocrepn to correspond to found was 4 = F q 1 h 1 hslast l possible all to leads This . 1 q , 2 1 2 ] . clrtetoCsfound CYs two the icular ufc natrcvariety. toric a in surface bv.Tevleof value The above. h poc’ hti one is That pproach’. F oete fthe of either to n 2 baino ere4 degree of fibration a ob etoso line of sections be to 1 , 1 nswr computed were ants sd .. instanton i.e., gsed, = iga ping 114 ˆ b 9 8 1 P 1 ilsatoric a yields ] × ˆ b K 8 8 9 9 2 br The fiber. 3 on P 1 tta a that ct K ih2- with (5.64) 3 ˆ b × P 1 ˆ b , T 1 ˆ b ’s is 2 2 Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. nignwmosiepeoea nsection In phenomena. moonshine new finding Y3flsta ae3scin hc r itdi table in listed are which 3-sections have that 3-folds CY where hc ilb eeiewhen reflexive be will which htaecmltl igal eecntutdb ehd si methods by constructed were Higgsable completely are that points ∆ such steidxta salse h onhn oncinfor connection el their moonshine analysing the by establishes done that was index This the is groups. dimension sporadic higher and connecting folds phenomena moonshine for looked has insight new of open. lot remain a questions 1978 many phy in and observation McKays mathematicians with for start subject fascinating is Moonshine Outlook and Conclusion 6 subse (see mat match genus that various models of invariants two Gopakumar-Vafa the table of indicated cases and Higgsable number Euler the listed a for order In o genus be all should CYs dual and the 3-sections with that 3-folds know we above discussion yesraei baino egtdpoetv pc wh space projective weighted of 3 fibration exhibits a that in fibration hypersurface one genus a every that noting by start surface r yesrae nafu-iesoa oi min spac ambient toric four-dimensional K a in hypersurfaces are fteabetsae h ulCswr hncntutdb lift by constructed then were CYs dual The space. ambient the of htaegnsoefiee ih3scin n aePcr numb Picard [ have in and classified sections polytopes 3 reflexive with 3-dimensional fibered 4319 one genus are that n (3) bainhst be to has fibration 3 nti hssIhv re ocnrbt oteunderstandin the to contribute to tried have I thesis this In ftepoints the of K n ufcscrepnigt h eeiepltpsta r t are that polytopes reflexive the to corresponding surfaces 3 F {− ∈ n n , 1 K 0 = , baint xs h aeo h bainnest eaHirzeb a be to needs fibration the of base the exist to fibration 3 0 , , 1 1 } , hs a elfe oa4dmninlpltp yadn th adding by polytope 4-dimensional a to lifted be can These . ,s htoecnrsrc h osrcint Y3flsthat 3-folds CY to construction the restrict can one that so 2, P 1 5 h bainhst rs rmacmail oi fibration toric compatible a from arise to has fibration the ytersitvcos o hs ae n nsta the that finds one cases these For vectors. shift their by h 1 , ν ν 1 1 ′ n nadto loehbta exhibit also addition in and 3 = ν ν ν ν ν ∈ ( = 3 2 5 4 1       2∆ ν, − 2 (3) 1) n 0 0 0 1 0 0 0 1 1 hswyoefid 4gnsoefibered one genus 14 finds one way This . ν , 76 − 2 ′ 1 0 0 1 (0 = − 4 , 1 − ae nm ulcto [ publication my on based ,       )(5.66) 1) 130 , n nsta hr xs 3 exist there that finds one ] .Snetebs fthe of base the Since e. scin sbrtoa to birational is -sections 10 r h aeis base the ere K ia o[ to milar ction iit lk.Sneits Since alike. sicists eew aealso have we Here . [ 3 lClb-a mani- Calabi-Yau al engie u still but gained been itcgns which genus, liptic fmosieby moonshine of g htecompletely the ch 10 r2 cnigall Scanning 2. er K 5.5.2 ing .I particular In ]. bain We fibration. 3 ecne hull convex he efiee CY fibered ne 114 K ). surfaces 3 .B our By ]. P 2 (5.65) 37 [ 129 ruch ,I ], ]. e Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. al 10 Table ancss hrb loorudrtnigteo heterotic of the manifol understanding CY our deepened. also dual Thereby the find cases. tain to [ able of were methods we of the cases genera twining some twisted In and variants. twining the connects also between hswsdn ysuyn h H riod fodr2ad3o the of 3 and 2 order of orbifolds [ CHL in the given studying by done was This n loehbtacompatible a exhibit also and fiain nC -od,wihwssont ikMtiumoon Mathieu link of to invariants) shown (Gromov-Witten/Gopakumar-Vafa was invariants which 3-folds, CY on ifications eeao eti oodlobflsa enrmdl a stu established. was be models could Gepner phenomena an moonshine coul orbifolds new moonshine toroidal certain new of no 5- by genera played cases CY role of those special 642) all the (13 in strengthening number However large a analysed. of and genera elliptic twining the form. intersection anti-diagonal with two rank re n ngnrltyt nesadhwC aiod ‘know’ manifolds CY orb how CHL understand to to manifolds Since try CY general dual in the and find order to try might one imminently akhls tsesitrsigas olo o e connect new for look to also interesting seems it holes, back yeI opcictoson compactifications II type           on owr hr r tl ayoe rbesoemgtadd might one problems open many still are there forward Going nsection In K − n 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 n t litcgnsaepeeti aysrn construct string many in present are genus elliptic its and 3 E Polytope 92 h 4gnsoefiee aaiYuthreefolds Calabi-Yau fibered one genus 14 The : − 8 ν n [ and ] × 0 1 0 0 0 1 − E 5 8 0 1 ae nm ulctos[ publications my on based , 114 eeoi tigcmatfiain on compactifications string heterotic − 38 1 ,t logv nepii osrcinfrteemanifolds these for construction explicit an give also to ], n nigterptnilyda Ymnfls hrb one Thereby CY-manifolds. dual potentially their finding and ]           − − − − − − − − ν n (2 1 (1 1 (1 (0 1 1 (0 ( 1 (0 1 (0 1 1 ( 2 (0 2 (0 2 (0 (1 2 (0 2 2 K 3 K − × − 2 bainsc htteplrzto atc sof is lattice polarization the that such fibration 3 , , , , , 1 T , , , , , , , , 0 0 1 1 0 , 0 0 0 2 1 0 0 − , , , , , 2 0 , , , , , , , − − − − − 2 , tgt once otemcott onigof counting microstate the to connected gets it 0) 0) 0) 0) 0) 0) 0) , 1) 1) 1) 0) 1) 1) 0) K 77 nMtiumosie loteelliptic the Also moonshine. Mathieu in 3 31 − − − − − − − 120 132 138 144 150 156 168 χ , 38 ,Itidt epnteconnection the deepen to tried I ], Base K F F F F F F F F F F F F F F 0 0 0 0 1 0 0 1 0 0 1 1 0 1 3 B K × M oGoo-itnin- Gromov-Witten to 3 tp Isrn ult is duality string II -type T osi uhsituations. such in ions h Ymnfls[ manifolds CY the n,ν (3) 2 3 1 idbtas eeno here also but died (2 n yeI compact- II type and addt shift Candidate od a calculated was folds hthv 3-sections have that hn otopological to shine 3 1 , 1 (2 bu moonshine. about sad ae on based and, ds 2 efud thus found, be d , , eeoi string heterotic 0 0 flso higher of ifolds 7 5 os .. for e.g., ions, 2 ; 2 ; es Most ress. , , 0 1 4 7 ) , 0 ncer- in 3 ) 30 ]. Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. oncin omosiet rs nsm fteesituation these of an connections. some physics interesting in of further arise areas many many to in moonshine up to show connections objects modular general In 78 edn opotentially to leading s n ih expect might one d Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. h eeaie aoiteafnto a ewitnas written be can function theta Jacobi generalized functions The theta Jacobi A.2 ∆ efrhrdfieteDdkn t function respectively. eta 6 Dedekind and the 4 define weight further We of forms modular are which nti pedxw ilclettedfiiin fseicmodu define specific we of Throughout definitions work. the this conventions collect in will - used we objects appendix modular this In some of Definition A . iesensre and series Eisenstein A.1 ttasom ne oua rnfrain as transformations modular under transforms It eas en h olwn functions following the define also We esatb enn h iesenseries Eisenstein the defining by start We ec ti oua omo weight of form modular a is it hence on 2 k SL , k (2 2 = , Z , ), 3 , samdlrfr fwih 2 weight of form modular a is 4 E E 6 4 ( ( τ τ θ 1 = ) 240 + 1 = )  a b  ( − ,z τ, 504 = ) ∆ ∆ ∆ η η X X n n η ( 4 8 6 ∞ ( ∞ =1 =1 η -function τ ( ( ( τ X k ( τ τ τ = ) ∈ − )= 1) + = ) = ) = ) 1 1 Z n n τ 1 − q − 3 5 2 1 q = ) 2 1 q q ( q q ihamlile system. multiplier a with 24 n n η η η 1 k n n η η (2 (2 (3 79 + n Y q ( ( ∞ 240 + 1 = 1 = e √ a 2 =1 τ τ τ τ τ E := ) η 12 iπ ) ) ) ) ) − 2 (1 6 8 ( 4 16 8 18 e η k τ E , η iτη π − e ( , , ) (4 − i 2 o Γ for τ ( 8 πiτ 6 k 504 (A.2) ) τ ( h eeaoso oua forms modular of generators the , q + τ ) n y , a 2 16 ) ) q ) q , 1 . b . ( 2160 + = − e k (2 ). 16632 e πiz 2 πiz ) ( k q . + 2 q a 2 2 + ) a bet htare that objects lar + , , ...... (A.6) (A.1) (A.3) (A.5) (A.4) (A.7) Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. n n defines one and 0. Setting functions oeepiil n nstefloigepeso n produc and expression following the finds one explicitly More z θ 0 = i θ θ θ θ ( τ 4 3 1 2 ( ( ( ( := ) , ,z τ, ,z τ, ,z τ, ,z τ, ( y θ θ = ) = ) = ) = ) )i bv entosw bante‘rnae’Jcb thet Jacobi ‘truncated’ the obtain we definitions above in 1) = θ 1 3 i = = = = ( ( ( ,z τ, ,z τ, ,z τ, n − q X X n n − Y Y n n ∞ ∞ + =1 =1 ∈ ∈ X 1 8 = ) = ) i iq Z Z 2 1  0) = n (1 (1 ( ∈ y 1 8 + X y − Z n θ θ  2 1 2 1 − − y q 1) ∈   y + n n i , Z 2 q q n 1 2 1 1 0 0 2 q ( n n y y − n −   2 ) ) 1 = n 2 − 1) ( (   q y 1 2 ,z τ, ,z τ, n 1 + 1  − n 2 2 , . . . , − − 1 2 n Y ) ) ∞  1 2 =1 yq θ , θ , yq y n Y n (1 ∞ =1 80 rm( From . 4 n n q − − n − (1 2 2 4 2 1 2 1 2 ( (     q ,z τ, ,z τ, − n ( + )(1 q + 1 1 = ) = ) n − )(1 A.9 y y θ θ yq − − −   1 1 eimdaeysee immediately we ) n q q 0 1 1 0 yq ( + )(1 n n   − − n ( ( 2 1 1 2 )(1 ,z τ, ,z τ,   representations: t , . y − ) ) − , . 1 y q − n 1 ) q , n ) , θ (A.11) (A.10) (A.12) 1 ( (A.8) (A.9) τ = ) a Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. h orsodn ulbssvcoste fulfil then vectors basis dual corresponding The n h ulmetric dual the and efrhrdfietefloigwa aoiforms Jacobi weak following the define further forms We Jacobi Weak A.3 iean Give atc scalled is lattice A ecletsm ai entosadfcsaotltie fo lattices about facts and definitions basic some lattices collect on We notes Some B e ewl osdrtecsswhere cases the consider will We R for qiaett en nmdlradintegral. and unimodular being Λ to equivalent all atc etr aeee (length) even have vectors lattice stemti nΛ h oueo h ntcl,cnann one containing cell, unit the of volume The by Λ. on metric the is i p,q i , vol v v p , 1 = , , w w + Λ = (Λ) n , . . . , q n ∈ ∈ dmninlvco space vector -dimensional = hc seuvln oΛ to equivalent is which Λ .Teeeet fΛhv expansions have Λ of elements The Λ. φ n p φ φ − 0 ihLrnza ne rdc,i.e., product, inner Lorentzian with 2 0 , | edfiea define we , , 3 2 1 1 det ( ( ( ,z τ, ,z τ, ,z τ, unimodular g g | 2 = ) = ) 4 = ) efrhrdfiethe define further We . ij ∗ = = = = − Λ θ √ y 1 e 1 1 θ θ

η ∗ lattice y 1 y 0+ 10 + ( i 2 2 ( ,z τ, = ( (  · τ 2 + + ,z τ, τ, e = Λ ) θ θ { 6 j if 2 2 0) √ ) w 2 ( ( steivreof inverse the is ) − 2 ,z τ, τ, y vol ob h e fpoints. of set the be to Λ θ θ scalled is Λ . ( ∈ y V 3 3 + y 0) V ( ( e + X ) Λ .I scalled is It 1. = (Λ) i V ,z τ, τ, + =1 i ∗ O n seither is  = ⊂ | O · 0) w O ( 2 < e n ) q ( 81 + Λ j ( ) q i θ θ · e q . e ) 4 4 ∗ = v  ) i ( ( 1 nitga atc scalled is lattice integral An . | , ,z τ, τ, , . . . , n ∈ θ θ δ ullattice dual i ij 3 3 0) R self-dual Z ( ( ∈ ) ,z τ, τ, n , g v Z e ∀ ij 0) ihEcieninrpoutor product inner Euclidean with λ n v · φ ) ) Thereby . w k,m  > ∋ ∈ . 2 pne ytebssvectors basis the by spanned , = v Λ + fweight of fΛ=Λ = Λ if P } = Λ  . integral ∗ i p n lwn [ llowing θ θ =1 4 i 4 e ( ( vol v i I ,z τ, τ, atc on sgiven is point lattice Hence . I ∗ · 0) (Λ hscniinis condition This . w k ) if  I ∗ n ne m index and 95 − ( = ) 2 v ! P ]. · g w p i = ij vol + even p q ∈ +1 = (Λ)) (A.13) (A.14) (A.15) Z v e (B.2) (B.3) (B.1) fall if I i for , · · − w e 1 I j . Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. o h td fCT h vn efda atcsaeo parti charge of central are with lattices CFTs holomorphic self-dual to even, lead the CFTs of study the For vn efda atcsol xs hntedmnini m a is dimension the when exist only lattices self-dual Even, h uoopimgopo h atc ouoteWy-group Weyl- the modulo lattice the system. A-D-E of group automorphism the tnsoti htissots etr ae(length) have vectors shortest its that in out stands h uba ymtygroup’ symmetry ‘umbral the (there dimension [ 24 e.g., to see, details up further in for lattices - such rapidly increases of number the gives oro etr vcoso (length) of (vectors vectors root no elble yteDni iga fterro ytm hyf They table system. in root seen their be of can diagram as Dynkin classification the by labelled be uhltie,cle the called lattices, such rs rmmdfiain ftero ytm nldn ‘glui The including 24. to systems up root sum to the has of factors the num modifications of Coxeter from rank the the arise lattice and given same a the be for to that rules the by algebras n2 iesos hc ilb fpriua motnet u to importance particular of be will which dimensions, 24 In al 11 Table ubro vn efda atcsi ieetdimensions. different in lattices self-dual even, of Number : imirlattices Niemeier ieso ubro lattices of number dimension G 2many 24 2 32 24 16 1 8 L o ahlattice each for 2 ) h eann 3Neee atcscan lattices Niemeier 23 remaining The 2). = 12 82 h otssesaiefo h A,D,E the from arise systems root The . n fteeltie,the lattices, these of One . c 131 qa otedmnino h lattice. the of dimension the to equal L ]) 2 sgvn hc sdfie obe to defined is which given, is ,s npriua tcontains it particular in so 4, = gvcos.I table In vectors’. ng ua neetsnethey since interest cular e o ahfco has factor each for ber lil f8 Table 8. of ultiple fe hi number their after l noa A-D-E an into all fteassociated the of hr r 4of 24 are there s imirlattices Niemeier ec lattice Leech 12 11 , Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. h aodsco ntrt requires unitarity sector Ramond the where eoeahgetwih tt iheigenvalues with state weight highest a denote o h ersnainblnigto belonging representation the for h hrces[ characters The . (Extended) characters C.1 and algebra Superconformal C o h (extended) the For h aodsco through sector Ramond the 39 • oeta u ento of definition our that Note ase BS ersnain xs for exist representations (BPS) Massles F stefrinnme and number fermion the is al 12 Table 132 N – imirro ytmUba ymtyG symmetry Umbral system root Niemeier ch N imirltie n soitduba symmetry. umbral associated and lattices Niemeier : 134 d,h N uecnomlagbawt eta charge central with algebra superconformal 2 = 2 = =2 r ie y(sn h ovnin f[ of conventions the (using by given are ] − 24 A c θ D A D A ,ℓ 11 A A A 1 characters D D ( A A A A ( D D D 15 E E 17 A A A A 10 16 ,z τ, 5 4 7 2 9 2 D ,z τ, D D D 24 2 12 1 24 12 2 24 12 2 8 6 D 4 6 6 4 3 8 8 3 E E 3 4 8 4 6 E 3 6 4 7 iesb iu info h ento sdthere. used definition the from sign minus a by differs ) 4 5 6 E 7 7 8 9 2 Tr = ) 6 | Ω q h i = H ≥ edfiete(graded) the define we h,ℓ 83 e 2 24 (( c π i − τ = eo ewl louse also will we Below . h 1) d 8 ,ℓ h, = 2 F . GL .AGL q 3 GL GL SL L 2 Sym Sym Sym d 8 .Sym Dih Dih M 0 .M w.r.t. ; 2 Z Z Z Z Z Z Z Z − (5) 2 2 2 ℓ 24 2 2 2 2 2 4 4 2 (3) 24 (3) (3) c 12 3 6 4 = 3 3 4 e (2) / 6 2 2 π d 2 L i , zJ 0 d 2 0 ) − and , N 1 74 , hrcesin characters 2 = J d 2 ] 39 0 − Writing . ): c y 2 3 = , . . . , = d e let , 2 − π i (C.1) z ( H In . d 2 | Ω h,ℓ − i Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. oterepresentation the to where nbt ae h hrcesfor characters the cases both In h hrces[ characters The nlgul othe to Analogously C.2 ler ihcnrlcharge central with algebra • • h itnidxo asesrpeetto sgvnby given is representation massless a of index Witten The n for and asv nnBS ersnain xs for exist representations (non-BPS) Massive 1) asesrpeetto xs for exist representation Massless 1) ch ch h N , , ch d, N d,h N and − − 4 = 0 ch =2 =2 d, N , − d d 2 2 d 2 0 For . d, N =4 24 ( ,ℓ and c ℓ ,z τ, 0 =2 ℓ characters ,ℓ> ,ℓ ( r h ievle of eigenvalues the are ,z τ, = ≥ 135 0 ( = ) 0 ch ℓ ( ( d 2 d 2 ,z τ, ,z τ, = ) for 0 6= d, N N ℓ > r ie y(sn ovnin rm[ from conventions (using by given are ] n has one ch 0 =2 − ,ℓ ( = ) H ch ( = ) d,h N aete(rdd hrceso the of characters (graded) the case 2 = ≥ θ 1) h,ℓ =4 ≥ 1 0 d,h N − ( ( d =2 τ, ,z τ, si the in As . 24 hyaegvnby given are they 0 − c − ( i d − − ,ℓ 24 2 c c 1) 1) vn For even. = ( z i) ,ℓ< 3 = η ,z τ, )= 0) = ) ℓ ℓ θ ( + + θ τ 1 0 < ℓ d 1 2 d 2 ( ( η ) Tr = ) d ( q ,z τ, ,z τ, 3 ( ( and , ,z τ, h − τ − ) r ie by given are 0 L i) ) ch = ) η 3 d 8 ( N ) y 0 θ ( i 2 h θ τ 1 H d ( + 1 ( and η ε d 1 ( +1 2 − ) aeuiaiyrequires unitarity case 2 = X h,ℓ 84 = = ( ( ,z τ, > ℓ 3 τ ,z τ, vn nteRmn etraedfie as defined are sector Ramond the in even, ± 1) X (( n ) d,h N 1 d 8 ∈ 3 T ℓ − ) ℓ , − m X =2 + Z ehave we 0 ) 0 − y 3 ∈ 1) y 1) d 2 q c/ ℓ Z ftehgetwih tt belonging state weight highest the of + , ℓ 0 = d d 24 − F − ε 2 2 1 , 1 1 2 q , e X − n for o 0 for n L 4 X , n ∈ > h ℓ> 2 π (1 0 ∈ + 1 2 Z i − ε Z 0 ℓ d , . . . (( q 24 − ( +1 c 2 q = τ, d d 2 ≤ e − d 2 +1) n d 8 y − 4 2 1 − ; π − < ℓ d 2 (1 1 n d 4 i n 2 m 136 ε z zT ℓ (1 . +( 2 n r ie by given are and q ) − + +( 0 − N . = 3 ℓ ℓ − d 2 ) m + )( ]) ℓ yq , − , z ) 2 1 superconformal 4 = d 2 q + 2 2 1 ) n , ( ) n ) 1 2 )(1 n d 2 ( ) h − − ( q 1 − − ( y ≥ y − − d 2 ) y 1 ) +1) ( ( ) , . . . , yq d d 8 yq d ( − − . d m − n 1) 1) n 2 +1 1) n n +2 n − , ) ℓm (C.2) (C.3) (C.7) (C.4) (C.8) (C.5) (C.6) . . ( d 2 . − Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. i having nti uscinw ietegnrldfiiin n ai p basic [ and mostly definitions follow general We the give ifolds. we subsection properties this and In definitions Basic real D.1 their of overview geometry. short toric a and give spaces C also of jective will properties some we summarize particular briefly In will we geometry appendix this toric In and manifolds Calabi-Yau D CY even- the Likewise ch iii ii eswihaeee in even are which ters ch a aihn rtCenclass, Chern first vanishing has X ) disaRcifltmetric. flat Ricci a admits X ) 40 d, N d, N h oooygopo sasbru of subgroup a is X of group holonomy The ) • otyCY Mostly d 0 0 =2 =2 ihtehl fthe of help the With , , flsa en eun CY genuine being as -folds 2 2 3 1 asv ersnaineitfor exist representation Massive ch npriua for particular In where ( ( ,z τ, z τ, d ch d,h N cmlxdmnin n ufiln n fteeuvln pro equivalent the of one fulfilling and dimensions -complex =4 d,n,l N − ch + ) ch + ) =2 24 ch c ,ℓ ( d d, N ,z τ, flsaespoe ohv exactly have to supposed are -folds ( 0 ,z τ, =4 d, N d, N , 0 0 0 ch + ) =2 =2 ( z , , i = ) − − ,z τ, obnto ftemassive the of combination 3 1 2 2 ( ( = ) ,z τ, z τ, ℓ z q d,n, N hsmyb rte as written be may this 0 = h a eepesdi h olwn way following the in expressed be can =2 − N ( = ) ( = ) − θ 95 d 2 1 l +2 ℓ d ( ( hrcescmiain fmassless of combinations characters 4 = 2 ϑ -folds. , ,z τ, τ, − − P,a 137 − − i d 8 2 ( = ) 1) 1) z θ ( 1 ,z τ, ) .ACY A ]. ( d d θ θ τ, +1 +1 2 2 1 1 η = ) c ( ( − 2 φ φ ( ,z τ, 1 > h ,z τ, τ z 0 0 1) = ) , , ) 3 3 2 2 85 3 η X n 2 ) ) ( ( l ( ∈ 2 2 2 ,z τ, z τ, + d 8 τ 1 π Z m X ℓ , d d i ) q − [ 2 3 ∈ fl sacmatK¨ahler manifold compact a is X -fold SU R ) )ch 1 (2 Z  φ =  n P ]. q SU ( N 0 ϑ 4 ch ( d d N , P + 2 1 − 2 3 oooy ewl fe ee osuch to refer often will We holonomy. ) d d 2 2 a =4 d N , ( +1 +1 3 ( hrcescnb rte as written be can characters 2 = ) − ,z τ, 1 2 , d =4 0 3 , . . . , y ) , , ). 2 0 , m 2 0 ℓ ( 40 n P , )ch ( 2 0 ,z τ, ,z τ, y ( ,z τ, + ( d d N d 4 a ) +2) − =4 zto nwihe pro- weighted in ization + . , n r ie by given are and 3 ch + ) ,n, m oete fC man- CY of roperties 2 1 lb-a manifolds. alabi-Yau 2 1 1 + 1 ) ( l − − − d N 2 1 − ϑ yq yq ) =4 perties ( 3 d 2 N , ,z τ, m m 0 +1 , 1 2 . , charac- 2 = ( − ) ,z τ, 2 . ℓ ( ,z τ, ) (C.14) (C.10) (C.11)  (C.9) . + (C.13) (C.12) 2 1 )  , Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. tutda yesrae nwihe rjciesae.W spaces. projective propert weighted and in construction hypersurfaces the as discuss structed will we section this In . aaiYumnflsa yesrae nwihe rjcieam projective weighted in hypersurfaces as manifolds Calabi-Yau D.2 v iv coordinate h og ubr a encl ragdi h ocle og d Hodge called so the in arranged nicely be d can numbers Hodge The through osrcini egtdpoetv space. projective weighted in construction where in hro.W tr ytedfiigageneralized a defining the by projecti start (weighted) We in thereof. hypersurfaces tions as them construct can h og ubr dmnin fteDbal ooooy)ful ) cohomology Dobeault the of relations (dimensions numbers Hodge The a nqenweevnsigholomorphic vanishing nowhere unique a has X ) 2 = disacvratcntn spinor. constant covariant a admits X ) • • • • opc aiod antb osrce ssbaiod of submanifolds as constructed be cannot manifolds Compact , . od ecnie tr r oneeape,w ute hav further we counterexample), a are (tori consider case we the folds be will which K¨ahler manifolds, simply-connected ( h h h h ttkstegnrlform: general the takes it 3 d, in spaces bient λ p,q p, 0 p,q , 0 0 )fr Ω. form 0) ∈ = = = od o n opc,cnetdKalrcmlxmanifol K¨ahler complex connected compact, any for holds 1 = C z h h i h ∗ sn hsato edfiea qiaec relation equivalence an define we action this Using . q,p d d − n h o-eointeger non-zero the and − 01 20 1 q,d p, ycmlxconjugation. complex by 0 − 0 0 0 0 p since λ yPicr duality. Poincare by · 1 1 z H = p z λ ( X ∼ · ( ) z z 0 ≃ ′ z , . . . , , ∃ ⇔ H 1 d − λ p n ∈ w ( 0 0 86 ( = ) X i C ,wihflosb otato ihthe with contraction by follows which ), scle h egto h homogeneous the of weight the called is h ∗ 0 0 0 0 2 λ : , 1 w z 0 ′ z h h = 0 1 1 λ , . . . , 1 1 d , , λ 1 1 fr . Ω -form C · ∗ z h (:= 1 w , 2 n e fC aiod con- manifolds CY of ies olw[ follow e esaeadgeneraliza- and space ve C z n 0 0 { \ ) , 0 o otC mani- CY most for 1 } ∼ cinon action ) C e . ltefollowing the fil 95 on n h oee one however , 1 , aod For iamond. , 0 C 138 = n through o the for ] h .For d. 0 , 1 (D.1) (D.2) (D.3) C 0 = n +1 - Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. o in not oyoil endby defined polynomial, iglr erequire we singular, etr.An vectors. eesr objects. necessary h rtCencasof class Chern first The An -An then: tice endby defined atc N. lattice aldtedge of degree the called h aihn ou faqaihmgnospolynomial quasi-homogeneous a of locus vanishing the ( n h qiaec ls fapiti eoe y[ by denoted is point a of class equivalence the and hc a rvni [ in proven was which yesfcsi oi aite et ewl otyfollow mostly will We discussion. ways. next. different varieties in toric spaces in generalized ambient hypersufaces be toric may in above constructions hypersurfaces The as manifolds Calabi-Yau D.3 nee.TeElrnme of number Euler The integer. fadtoa qiaec eain ftefr ( form the of relations an equivalence coordinates additional additional of introducing by obtained are they iesaewt weights with space tive side oyoil .. httesto l utpe f1 of multiples all of set the that i.e., polynomial, a indeed is e falmlilso 1 of multiples all of set C n +1 r Let oi aite r eeaiain fwihe projectiv weighted of generalisations are varieties Toric dmninlcn scalled is cone -dimensional r dmninlcone -dimensional { \ P n N eesr odto o uhaplnma oeiti htt that is exist to polynomial a such for condition necessary A . 0 } ea be ) / n dmninlcn scalled is cone -dimensional ∼ n hypersurface A . dmninlltieand lattice -dimensional σ p n dp ( 138 z X = /w ( ), z d w X σ { χ ae n[ on based ] = ) [ i w i n λ d ipecekfrti odto st e if see to is condition this for check simple A . i , = [ 1 = ] w in v 1 0 = m p aihsif vanishes ] X 1 ( + N simplicial { P z d X k [ [ nyt odat hold to only 0 = ) R · · · m z =1 ( w n , . . . , ntl eeae by generated finitely , t 0 X = ) sgvnby given is ] : X l + d 139 =1 m · · · [ w λ Y i =0 gcd ]. nwihe rjciesaei endas defined is space projective weighted in ] s n sdfie by defined is , 87 : v fi sgnrtdby generated is it if z s N { basic d 1 | l,k n λ Y 1 R ] − = i }· − ∈ h elvco pc arigtelat- the carrying space vector real the ≥ w m t P i d P t D.2 ∈ − fi sgnrtdb ai fthe of basis a by generated is it if w o 1 for 0 n Z i n w i =1 | i p w , ( w i D.3 z z w − 0 = ) .Te the Then ]. i nodrfor order In . i ≤ p 0 P m .W tr ydfiigthe defining by start We ). ( : 0: [0 (:= n pc.Ruhyspeaking Roughly space. e λ i v [ [ naporaenumber appropriate an d 140 1 w } ≤ · v , . . . , . z 0 / r s w , . . . , = ) , ( } ewn oconsider to want We d lna independent) (linear 141 − · · · λ s ntefollowing the in ] d w egtdprojec- weighted p ∈ n i ( X ],wihis which 0]), : = ] nldsthe includes ) z N ePoincar´e-he ,where ), d R ob non- be to P s , P 0 san is (0) n [ ≥ w (D.4) (D.7) (D.6) (D.5) := ] r d is is Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. udvso ftefan the of subdivision appropriate by removed be can factor where ru smrhcto isomorphic group ultie say sublattice, a sntbsc .. ti pne by spanned is it i.e., basic, not is -A l nescino oe lobln otefn o a we Σ fan a For fan. the to belong also cones of intersection all -A it bounding hyperplane qiaec eain( relation equivalence v rvossnua on srpae yasot aiod hsproced This manifold. smooth a by replaced up’. is point singular previous -A containing not and convex’). 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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. ic ewn h a ocnito ipiilbsccnsw n we cones basic simplicial of polyhedron. consist the to of fan triangulation the want we Since mohb eeiiy For polynomial reflexivity. generic by a smooth for of manifold vanishing CY the smooth by defined hypersurface hc oee r isdb h eei hypersurface. generic the by missed are however which obntra omls lofudb atrv[ Bartyrev by found also formulas, combinatorial scmati n nyi h upr ftefncvr h latt the covers fan the of an simplicial support of of the think if consist may We only fan and the if if compact only subdivi by is and cured if be smooth can is again variety These simplicial. not are which oceeylet concretely Ya usoe atc onso ae fteivle polyhe involved the of faces of points lattice over sums as CY eul.Nwteplnma equation polynomial the Now neously. h ulltieof lattice dual the ulpolyhedron dual U oapi frflxv oyer (∆ polyhedra reflexive of pair a To in scalled is ∆ hncnie h Luet oyoilgvnby given polynomial (Laurent) the consider Then hr ace r le together. glued are patches where where en h Y trigwt lattice a with Starting CY. the define σ N = twssonb atrvta ie aro eeiepolyhed reflexive of pair a given that Bartyrev by shown was It xhnigterl f∆ad∆ and ∆ of role the Exchanging ntenx tpw att pcf h oyoil hs vanis whose polynomials the specify to want we step next the In oeta (∆ that Note . { a ( w z 1 ()= Σ(∆) ∈ z , . . . , reflexive C σ r aaees ne cln si ( in as scaling a Under parameters. are ∆ egnrtdby generated be k ) { ∗ N | z opeertoa a hs oe r h oe vr(D.13) over cones the are cones whose fan rational complete ∗ by i ) n f∆ if , := ∆ ∗ dmninlcone -dimensional ≥ M ∆ ∆. = ∗ ∗ 0 =Hom( := n = } saltieplhdo,ie,i h etcso ∆ of vertices the if i.e., polyhedron, lattice a is { n oe iesoa oe ersn ein foverlap of regions represent cones dimensional Lower . ≤ w { h oi ait a aepitlk singularities, like point have may variety toric the 4 = v f ∈ .For 4. ∆ ∈ M := h ae f∆wt pxa h origin the at apex with ∆ of faces the N v N, 1 , |h R v , . . . , w ∆ X ∗ |h v Z ∈ N i n bv ie iet h irrC.Teeexist There CY. mirror the to rise gives above ∗ ,w v, ∆ .Te edfieteplhdo through ∆ polyhedron the define we Then ). w , ecnaanascaetefndfie by defined fan the associate again can we ) f n a ,Σ Σ, fan a and a ≤ ∆ σ 89 − ≥ i w n − ≥ i swl end efrhrdfiethe define further We defined. well is 0 = srpeetn oriaepths More patches. coordinate representing as h oi ait tefi nue obe to ensured is itself variety toric the 3 hnteascae oriaepthis patch coordinate associated the then , Y i =1 k z 1 i h 1 w,v i , 142 , 1 = i ∀ i +1 ,frteHdenmeso the of numbers Hodge the for ], w k , . . . , (1) ∈ D.8 ∆ = ) i} { n ai oe ny It only. cones basic and f } v . ino h a.Atoric A fan. the of sion ∆ . 1 ice. v , . . . , rnfrshomoge- transforms e aia star maximal a eed r,seas [ also see dra, endb sa is ∆ by defined k a(∆ ra iglcswill locus hing } econsider we , } ∗ , ⊂ ∆ (D.12) (D.10) (D.11) N ∗ the ) 141 R lie ]. Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. Y3flswt -etosfor N-sections with 3-folds CY describe ial l 7,0,7 eeie4dmninlplhdaha polyhedra dimensional 4 reflexive 473,800,776 all Finally ujciemap surjective l 39tredmninlrflxv oyer aebe class been have polyhedra reflexive describ dimensional They three polyhedra. 4319 dimensional All 2 reflexive 16 are There ii) iha prpit eeaiaino h odl-elgro Mordell-Weil the of k generalisation appropriate an with divisors. fibral Morde and the divisors in vertical independent of linear basis are a that with sections of set a and ering bo conformal the summary brief [ a in give will we appendix this In bootstrap Conformal E dimensions. 4 to compactifications string for [ on accessed be can and ytetretpso divisors: i) of types three the by iii) point. and point, a in ru a ntefie hc a eetne otertoa se rational the to extended be may which fiber the p the of in Addition law origin. group the a as section zero the with intersection nbe n ocnnclyietf vr brwt torus, a th with be fiber every to identify section canonically any to choose one can enables One rational). or holomorphic disasection a admits nec br oigaogcoe ah ntebs ftefibra the of base distinguishe the (this in monodromy paths N a closed experience along generically Moving will fiber. each in n baini eea osnthv eto u nya only but section a have not does general in fibration one have h eei fiber generic the yteSid-aeWzrterm[ theorem Shioda-Tate-Wazir the By esatb osdrn nelpial brd3-fold fibered elliptically an considering by start We -sections. etcldivisors vertical etos.I u oainagnsoefiee aiodwith manifold fibered one genus a notation our In sections). ba divisors fibral 114 sections odl-elgroup Mordell-Weil aaiYui adt egnsoefiee vrabase a over fibered one genus be to said is Calabi-Yau A o eu n brdtreod h bv a egeneralised be may above the threefolds fibered one genus For ycneto nelpi bainhsa es n eto ( section one least at has fibration elliptic an convention By N k ,scin -,frtetplgclsrn atto funct partition string topological the for 2-4, sections ], scin nta fscin.Oecnpc ‘zero a pick can One sections. of instead -sections ′ scin with -sections K surfaces. 3 hc a esltinto split be can which , π ainlsections rational X ossigo ainlcre brdoe iio in divisor a over fibered curves rational of consisting : N X D scle litclyfiee.W s h ovninta g a that convention the use We fibered. elliptically called is tms .. tcrepnsto corresponds it i.e., -times, i → = N 81 MW ′ π B N < .Te iet iesoa Ysadhneaeimportant are hence and CY’s dimensional 6 to rise They ]. − 1 ..tegnrcfie over fiber generic the s.t. 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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. oue fcre ntebs.Ti hf slna in linear is shift This base. the in curves of volumes litcfiuain n[ in figurations elliptic ainof tation h uoopi rprisof properties automorphic The th space. in fun uli monodromies under partition behaviour string transformation general topological the the of properties Modular hr n nsta o nelpi baino o eu one genus a for or fibration elliptic an for that finds one where MW H Assume and curves fibral the fibrations one genus to generalized be then may definition This independent sections, where opeie oueo h eei br Further fiber. generic the of volume complexified h olwn a.Satb enn h ne product inner the defining by Start way. following the manner hnfran for Then nqelna obnto ftezero- the of combination linear unique ftebs telte w ilcletvl ecle fibral called function be ratio partition collectively isolated string will from topological two or the latter divisors (the base fibral the of of fibers are that curves h· s.t. , 42 4 ·i ( h notation The M, o nelpial brdC manifold CY fibered elliptically an For reuil uvsi ilaiefo ihrcre nteb the in curves either from arise will M in curves Irreducible . σ ( ( M D E ω Z ) M sotooa otesbpc pne ytoedvsr in divisors those by spanned subspace the to orthogonal is ) f,i .I a nqeyb endi em fisitreto prop intersection its of terms in defined be uniquely can It ). = Z N → top eoe ai ffirldivisors, fibral of basis a denotes τ selpi rgnsoefirto.PrmtieteK¨ahler f the Parametrize fibration. one genus or elliptic is N { ∈ H N Z · [ h ( 148 -section top. 4 -sections; rk , E ( i M, 1 0 ( ( : , G ,wihwsivsiae n .. [ e.g., in, investigated was which ], ,m τ, + 2 H rssfo h Fter-itoay icse nApni f[ of A Appendix in discussed ‘F-theory-dictionary’ the from arises ) Z , D 4 3 ,wihi ooopimfo h odl-el ru to group Mordell-Weill the from homomorphism a is which ), Q ( 146 , t , + ) M E 4 β } λ , D ) exp(2 = , h K¨ahler modulus the , faC -odoedefines one 3-fold CY a of X × i = ) 147 i ′ =1 r r etcldvsr ult h curves the to dual divisors vertical are H m ]. Z 4 Z ( i 0 M top ( · π ,λ τ, σ i ) P a edrvdfo h aefnto interpre- function wave the from derived be can ( → E ) N i i   + ) β H scin etcldvsr n ba divisors fibral and divisors vertical -section, 91 Z i + 1 t 2 i top. 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The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. parameter where where with t relation n sue htteeaen ba iiosa eei poin generic a at divisors fibral no of space are moduli there structure that assumes One h xaso coefficients expansion the h ‘zero The h egtof weight The sotooa oalo hs curves these of all to orthogonal is n h ne ...tetplgclsrn coupling string topological the w.r.t. index the and h ne arxwrt h emti litcparameters elliptic geometric the w.r.t. matrix index The i i , hnasmn nepnina n( in as expansion an assuming Then 1 = D M c φ i 1 h , . . . , β = ( ∗ ( B ( N ,m τ, N π z stefis hr ls ftebs,tenmrtri neleme an is numerator the base, the of class Chern first the is ) -section’ − r eoe h igo oua om o Γ for forms modular of ring the denotes ) { ∈ 1 ij β Z λ , D 1 φ ˜ , = β 1 β ,m λ, i ) ( ( i , B ,m τ, sgvnby given is ∈ N 1 r endt be to defined are ) 1 = M } E · λ , 1 ( n h xoeto ∆ of exponent the and ∗ 0 ( t b , . . . , − = ) i ssitdby shifted is N − = Z M r N )[ − β 1 2 β φ t ˜ η π . 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E ˜ rk( 0 = nt G E is ) (E.11) (E.10) 0 (E.7) (E.6) (E.5) (E.4) (E.8) (E.9) + D Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. ntehprutpes etk h eesr rnhn rule branching necessary the take We hypermultiplets. the in nti pedxw ilepani oedti o h ag gr gauge the how detail some in explain table will we group appendix gauge this In the of Higgsing F nHgsn e o xml [ example for see Higgsing on ehv h olwn ag ru n atrcnet coming content, matter and sector group untwisted gauge following the have we rkngnrtr fteguefils(unn hmmassive) spectrum them matter (turning and group fields gauge gauge following the of generators broken iigavvt one to vev a Giving ieavvt a to vev a give iia obfr a before to Similar h atrcneti aeldb h ersnain under representations the by labelled is content matter The h olwn ag ru n atrsetu fe Higgsing after spectrum matter and group gauge following the ..w aelf u the out left have we i.e. ubr fvco-adhypermultiplets and vector- of numbers emycekthat check may We xliighwt ig h first the Higgs to how explaining o oceeessk ewl td h hr oe ftable of model third the study will we sake concreteness For o hsw oietefloigbacigrules branching following the notice we this For 4 and 5 2( SO a eHgsdb iigvcu xetto aus(e’)to (vev’s) values expectation vacuum giving by Higgsed be can 12 13 SO (12) , SO 3( SO 4( SO n s h rnhn rules branching the use and 1 (13) 12 4( + ) 14 13 N 14 × (13) (14) (14) h , es‘ae’b h rkngnrtr of generators broken the by ‘eaten’ gets , SO ilbreak will ⊃ 1 − 1 4( + ) 4( + ) 1 SO × N U × ⊃ (14) , v 14 1 hre.Cutn ere ffedmw n the find we freedom of degrees Counting charges. (1) SO SO SO 99 1): (12) 235 = ( + ) 1 1 × , (14) (14) , 1): (13) , 152 14 SO 14 U SO 32 (1) ( + ) 13 ( + ) × – − × 1)factor. (14) 1)to (14) 154 , 2 U 7 4i nhne.I h etse we step next the In unchanged. is 64 = 171 14 1 U → , ( + ) 64 (1) 93 N 64 (1) ]. 12 → v , 2 , 2 1 8 and 184 = 1 , SO 32 13 + ( + ) ( + ) ′ 1 , 1) One (13). + 1 , 1 1 ( + ) 1 64 , , 64 , 64 64 → 12( + ) 1 8( + ) N , 32 → 64 h 13 4.W ilsatby start will We 248. = 15( + ) h two the 64 + 1 rm[ from s 1 ilgt‘ae’b the by ‘eaten’ get will , 32 n h euti the is result the and . rmtetitdand twisted the from , 1 u ftemdl of models the of oup SO 1 ) 5 ′ ) . . eoeHiggsing Before . . 1 1)adw find we and (13) , SO 151 1 ) . 1)groups, (14) .Frmore For ]. scalars (F.2) (F.1) (F.4) (F.5) (F.3) Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. 16 tenme nbakt iethe give brackets in number (the ooti h ag ru n spectrum and group gauge the obtain to epoedb breaking by proceed We rnhn rules branching h rnhn rules branching The ecncniu nsmlrmne sn 3( using manner similar in continue can We m and group gauge following the to lead steps similar more Two ntenx tpw ieavvto vev a give we step next the In eoti h olwn ag ru n spectrum and group gauge following the obtain We etn yteboe eeaos h ag ru n spectr and group gauge thus The are generators. Higgsing broken the by ‘eaten’ and 16 SU 4( SU 4( hrb breaking thereby 1 (5) 1 , SO 14 (4) , 14 × 2( + ) (10) × 4( SO 2( + ) SO SU SU SU 4( 1 SO 1 , (10) ⊃ (14) (5) 14 5 (4) (3) , (14) 14 4 , SU 1 2( + ) , ⊃ × ⊃ × 1 2( + ) 3( + ) SU × 2( + ) (5) × SO SU SO SU U SO 4 to (4) U 16 (1) × (14) (4) (14) 5 (3) 3 (1) 1)to (10) 4 , U , , 2 5 1 1 1 , U × 1 : (1) 2 , × , 1 ( + ) 3( + ) 2( + ) × × 5 ¯ , 1 hre niaeta ecngv e to vev a give can we that indicate charge) (1) ( + ) SU U U n s h rnhn rules branching the use and U U 16 1 : (1) 1 : (1) 16 10 (1) 3 ygvn e to vev a giving by (3) (1) SU 94 6 16 3 → , , , → 2 2 1 1 1 5 where (5) , 10 4 5 6 , , ( + ) ( + ) 1 ( + ) 3 10 ¯ 10 → ( + ) , → → 1 → ( n 3( and ) ( − 3 4 3 6 ¯ 1 10 1 4 ( 1 ( ( , 1 , + ) 1 1 − ( 1 64 + ) , , − + ) + ) ( + ) 64 1 2 10 3 27( + ) ( + ) 5 ¯ + ) 5 + ) 18( + ) ( 1 1 , − ( 3 ¯ ( 1 10 3 ( , − 3 ¯ 3 − , 1 + ) 1 6 ( + ) 64 3 , 4 oHiggs to ) 2 ( n n clrwl get will scalar one and 1 64 2 ) ) ) , 1 , 24( + ) , 1 . ) matrti tpof step this after um 1 1 , 21( + ) ( te spectrum atter 4 ( 1 ) − 5 . , ) 5 ) 4 ¯ . , ) 1 n sn the using and , 1 SU 1 , ) 1 . ) 3 com- (3) . (F.10) (F.11) (F.12) (F.9) (F.7) (F.8) (F.6) Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. ltl n n pwith up end and pletely ntesm a emyHgstesecond the Higgs may we way same the In ag ru a eHgsdb n ftecagdsaas hsw Thus scalars. neutral scalars. 64 charged and the group of gauge any by Higgsed Higgsed completely be may group gauge SO 4( 1 , (14) 14 ( + ) × U 1 (1) 95 , 64 2 , 37( + ) SO 1) h two The (14). 1 , 1 ) . U 1 atr nthe in factors (1) n pwt a with up end e (F.13) Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek.

G Character table of M24

Table 13: The character table of M with shorthand notation e = 1 ( 1 + i√n). 24 n 2 − [g] 1a 2a 2b 3a 3b 4a 4b 4c 5a 6a 6b 7a 7b 8a 10a 11a 12a 12b 14a 14b 15a 15b 21a 21b 23a 23b [g2] 1a 1a 1a 3a 3b 2a 2a 2b 5a 3a 3b 7a 7b 4b 5a 11a 6a 6b 7a 7b 15a 15b 21a 21b 23a 23b [g3] 1a 2a 2b 1a 1a 4a 4b 4 5a 2a 2b 7b 7a 8a 10a 11a 4a 4 14b 14a 5a 5a 7b 7a 23a 23b [g5] 1a 2a 2b 3a 3b 4a 4b 4 1a 6a 6b 7b 7a 8a 2b 11a 12a 12b 14b 14a 3a 3a 21b 21a 23b 23a [g7] 1a 2a 2b 3a 3b 4a 4b 4 5a 6a 6b 1a 1a 8a 10a 11a 12a 12b 2a 2a 15b 15a 3b 3b 23b 23a [g11] 1a 2a 2b 3a 3b 4a 4b 4 5a 6a 6b 7a 7b 8a 10a 1a 12a 12b 14a 14b 15b 15a 21a 21b 23b 23a [g23] 1a 2a 2b 3a 3b 4a 4b 4 5a 6a 6b 7a 7b 8a 10a 11a 12a 12b 14a 14b 15a 15b 21a 21b 1a 1a χ1 11111111111111111111111111 χ2 23 7-1 5-1-1 3-131-1 2 2 1 -1 1 -1 -1 0 0 0 0 -1 -1 0 0 χ3 45 -3 5 0 3 -3 1 1 0 0 -1 e7 e¯7 -1 0 1 0 1 e7 e¯7 0 0 e7 e¯7 -1 -1 − − χ4 45 -3 5 0 3 -3 1 1 0 0 -1e ¯7 e7 -1 0 1 0 1 e¯7 e7 0 0e ¯7 e7 -1 -1 − − χ5 231 7 -9 -3 0 -1 -1 3 1 1 0 0 0 -1 1 0 -1 0 0 0 e15 e¯15 0 0 1 1 χ6 231 7 -9 -3 0 -1 -1 3 1 1 0 0 0 -1 1 0 -1 0 0 0e ¯15 e15 0 0 1 1

96 χ7 252 28 12 9 0 4 4 0 2 1 0 0 0 0 2 -1 1 0 0 0 -1 -1 0 0 -1 -1 χ8 253 13 -11 10 1 -3 1 1 3 -2 1 1 1 -1 -1 0 0 1 -1 -1 0 0 1 1 0 0 χ9 483 35 3 6 0 3 3 3 -2 2 0 0 0 -1 -2 -1 0 0 0 0 1 1 0 0 0 0 χ10 770 -14 10 5 -7 2 -2 -2 0 1 1 0 0 0 0 0 -1 1 0 0 0 0 0 0 e23 e¯23 χ11 770 -14 10 5 -7 2 -2 -2 0 1 1 0 0 0 0 0 -1 1 0 0 0 0 0 0e ¯23 e23 χ12 990 -18 -10 0 3 6 2 -2 0 0 -1 e7 e¯7 0 0 0 0 1 e7 e¯7 0 0 e7 e¯7 1 1 χ13 990 -18 -10 0 3 6 2 -2 0 0 -1e ¯7 e7 0 0 0 0 1e ¯7 e7 0 0e ¯7 e7 1 1 χ14 1035 27 35 0 6 3 -1 3 0 0 2 -1 -1 1 0 1 0 0 -1 -1 0 0 -1 -1 0 0 χ15 1035 -21 -5 0 -3 3 3 -1 0 0 1 2e7 2¯e7 -1 0 1 0-1 0 0 0 0 e7 e¯7 0 0 − − χ16 1035 -21 -5 0 -3 3 3 -1 0 0 1 2¯e7 2e7 -1 0 1 0-1 0 0 0 0 e¯7 e7 0 0 − − χ17 1265 49 -15 5 8 -7 1 -3 0 1 0 -2 -2 1 0 0 -1 0 0 0 0 0 1 1 0 0 χ18 1771 -21 11 16 7 3 -5 -1 1 0 -1 0 0 -1 1 0 0 -1 0 0 1 1 0 0 0 0 χ19 2024 8 24 -1 8 8 0 0 -1 -1 0 1 1 0 -1 0 -1 0 1 1 -1 -1 1 1 0 0 χ20 2277 21 -19 0 6 -3 1 -3 -3 0 2 2 2 -1 1 0 0 0 0 0 0 0 -1 -1 0 0 χ21 3312 48 16 0 -6 0 0 0 -3 0 -2 1 1 0 1 1 0 0 -1 -1 0 0 1 1 0 0 χ22 3520 64 0 10 -8 0 0 0 0 -2 0 -1 -1 0 0 0 0 0 1 1 0 0 -1 -1 1 1 χ23 5313 49 9 -15 0 1 -3 -3 3 1 0 0 0 -1 -1 0 1 0 0 0 0 0 0 0 0 0 χ24 5544 -56 24 9 0 -8 0 0 -1 1 0 0 0 0 -1 0 1 0 0 0 -1 -1 0 0 1 1 χ25 5796 -28 36 -9 0 -4 4 0 1 -1 0 0 0 0 1 -1 -1 0 0 0 1 1 0 0 0 0 χ26 10395 -21 -45 0 0 3 -1 3 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. 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Phys. , 12 ora fHg Energy High of Journal 277 21)074 (2013) 1993. , Nv 2007) (Nov, 79 140 [ , 18)1–128 (1981) , 1309.6404 1205.1784 (May, d ]. . . , ora fH of Journal Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. May2011 Oct. 1997 Start of undergraduate studies in Physics, Vienna University of Technology StartUniversity ofundergraduateVienna inPhysics, Technology studies of Mar.2012 Vienna Start ofundergraduate inMathematics, University of studies Work Experience 1 Oct. 1997 Oct. 1998 2000Jan. Mathematics studies in of Continuation Physics, Technology UniversityVienna of Oct. 2010 Oct. 2011 Oct. 2003 Shift study Shift focus 1 Oct. 2003 Oct.2005 drs Rückaufgasse Vienna, Austria 33/3,1190 Austrian SeptemberSince 2016 Vienna, Austria UniversityEducation August 1979 13, Address Nationality PlaceBirth of DateBirth of Personal Information Telephone:66 131 +43699125 [email protected] E-mail: Austria A-1040 Vienna, Wiedner Hauptstrasse 8-10/136, UniversityViennaInstitute ofTechnology Physics, for Theoretical Andreas BANLAKI Nov. 2012 – – Nov. 2011 Oct.2013

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Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. Austria Vienna, Institut, on"Moonshine", Workshop Schrödinger Erwin 10-14 Sep. 2018 Talk Conference at Winterschool CERN on Geneva, CERN, Strings Switzerland and Fields, "Cosmology, School International Helmoltz-DIAS Strings, Summer New 28 Aug.-10 Sep. 2017 ICTP, ICTP Theoryand Spring Related Topics, School onSuperstring 6-10 Feb. 2017 Italy GGI theory Florence, School "AdS3: GGI, and practice", Munich, "Fields Germany Workshop 2017", ASC, Dualities and 16-24 Mar. 2017 27-31 Mar. 2017 09-13 Oct. 2017 BLTP School onPartition Winter Functions JINR, Forms, and Automorphic School on Winter CERN Supergravity, CERN, Strings Gauge and Theory, 2Feb. 28 Jan.- 2018 School, Summer Quantum Gravity, CARGESE Fields and 2018, Strings 12-16 Feb. 2018 on School Winter CERN Supergravity, Gauge Strings and Theory, CERN, Irish Pre-StringsLeuven, College,Leuven, 2019 Belgium 2018 11-23 Jun. 4-8 Feb. 2019 2019 4-9 Jul. PhD Schools

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Die approbierte gedruckte Originalversion dieser Dissertation ist an der TU Wien Bibliothek verfügbar. The approved original version of this doctoral thesis is available in print at TU Wien Bibliothek. arXiv:1711.09698 [hep-th] 10.1007/JHEP02(2018)129 doi: 1802 (2018) 129 JHEP A Banlaki, Skarke, Chowdhury, Schimpf, A. Kidambi, H. M. A. T.Wrase doi 1903(2019) 065 JHEP A. Banlaki, Chowdhury, A. T. Wrase C.Roupec, A. Banlaki, 4. 4. arXiv: 1811.07880 [hep-th] 3. 3. arXiv: 1811.11619 [hep-th] ulctos 1. Publications doi: 02(2020) 082 JHEP A. Banlaki, Chowdhury, Schimpf A. Kidambi, M. A. 2. 2. arXiv: doi: 04(2020) 203 JHEP On Mathi On Calabi-Yau manifolds and sporadic groups sporadic manifoldsCalabi-Yau and Scaling limits of dS vacua limits the dS Scaling swampland and of Heterotic strings on (K3xT^2)/Z_3 and their dual Calabi-Yau threefolds dual theirHeterotic (K3xT^2)/Z_3 on and strings :

10.1007/JHEP02(2020)082 10.1007/JHEP04(2020)203 10.1007/JHEP03(2019)065 10.1007/JHEP03(2019)065

1911.09697 [hep-th] 1911.09697 [hep-th] eu A. Moonshine and Gromov-Witten Moonshine and invariants Chattopadhyaya, A. Kidambi, T. Schimannek, M. Schimpf Kidambi,T. Schimannek, Chattopadhyaya, Schimpf M. A.