Recent progress on the

Strings  2014,  Princeton   25th  of  June,  2014   Recent  Progress  on  the      Abelian  Sector  of  F-­‐Theory   Denis  Klevers   arXiv:1303.6970  [hep-­‐th]:  M  Cve8č,  D.K.,  H  Piragua   arXiv:1306.3987  [hep-­‐th]:  M  Cve8č,  A  Grassi,  D.K.,  H  Piragua   arXiv:1307.6425  [hep-­‐th]:  M  Cve8č,  D.K.,  H  Piragua   arXiv:1310.0463  [hep-­‐th]:  M  Cve8č,  D.K.,  H  Piragua,  P  Song   arXiv:1407.nnnn  :  D.K.,  D  Mayorga  Peña,  P  Oehlmann,  H  Piragua,  J  Reuter   arXiv:14nn.nnnn  :  M  Cve8č,  D.K,  H  Piragua,  W  Taylor     F-­‐theory  &  U(1)-­‐symmetries     INTRODUCTION       Why  F-­‐theory?   F-­‐theory                                    =                            Type  IIB   •  ellipQcally  fibered       •  back-­‐reacted     •  regions  with    on       Why  F-­‐theory?   M-­‐theory   Type  II  A        On  T2        Limit  vol(T2)            0   on  S1   F-­‐theory                                    =                            Type  IIB   •  ellipQcally  fibered       •  back-­‐reacted     •  regions  with    on       Why  F-­‐theory?   M-­‐theory   Type  II  A        On  T2        Limit  vol(T2)            0   on  S1   E8xE8  Het     Certain   setups   F-­‐theory                                    =                            Type  IIB   •  ellipQcally  fibered       •  back-­‐reacted     •  regions  with    on       Why  F-­‐theory?   M-­‐theory   Type  II  A        On  T2        Limit  vol(T2)            0   on  S1   E8xE8  Het     Certain   setups   F-­‐theory                                    =                            Type  IIB   •  ellipQcally  fibered       Certain   setups   SO(32)  Het     S-­‐duality   Het/F-­‐theory  duality:  see  Anderson’s  talk   •  back-­‐reacted     •  regions  with    on     Type  I     EffecQve  theories  of  F-­‐theory   Use  F-­‐theory  to  engineer  effecQve  theories:   F-­‐theory   Calabi-­‐Yau  geometry     (+  G4-­‐flux,…)   Geometry                        Physics   N=1  SUGRA  effecQve     theories  in  6D  &  4D   Classifica8on  of  6D  (1,0)  SCFTs     via  F-­‐theory:  see  Vafa’s  talk   Since   are  non-­‐perturbaQve,  they   different  from  those  of   •  used  for  models  of       local  models:    [Donagi,Wijnholt;  Beasley,Heckman,Vafa;  Bouchard,Heckman,Kane,Seo,Shao,Tavanfar,Vafa;     Font,Ibanez;  Randall,Simmons-­‐Duffin;  Hayashi,Kawano,Tsuchiya,Watari,Yamazaki;  Dudas,Pal8;        Cecoi,Cheng,Heckman,Vafa;  Marchesano,Martucci…  many  works]   global  models:    [Blumenhagen,Grimm,Jurke,Weigand;Marsano,Saulina,SchäferNameki;  Cordova;     Grimm,Krause,Weigand…  many  works]       Goals  of  this  talk   Develop  &   of  F-­‐theory:       of  F-­‐theory     ArithmeQc  of  ellipQcally  fibred  CY:                                                                          [Morrison,Vafa]      The  Abelian  sector  of  F-­‐theory  has  been  rather   :       only       Few  early  examples:  [Aldazabal,Font,Ibanez,Uranga;  Klemm  Mayr,Vafa]     Torsion  part:    [Aspinwall,Morrison;  Mayrhofer,Morrison,Till,Weigand]     A  lot  of  recent  progress:  [Grimm,Weigand;Esole,Fullwood,Yau;Morrison,Park;  Cve8č,Grimm,DK;  Braun,Grimm,Keitel;     Lawrie,Schäfer-­‐Nameki;  Borchmann,Mayrhofer,Pal8,Weigand;  Cve8č,DK,Piragua;  Grimm,Kapfer,Keitel;Braun,Grimm,   Keitel;  Cve8č,Grassi,DK,Piragua;  Borchman,Mayrhofer,Pal8,Weigand;  Cve8č,DK,Piragua;  Cve8č,DK,Piragua,Song;     Braun,Collinucci,Valandro;  Morrison,Taylor;  Kuntzler,Schäfer-­‐Nameki]     Unlike  well-­‐studied                                        [Kodaira;  Tate;Morrison,Vafa;  Bershadsky,Intriligator,Kachru,Morrison,Sadov,Vafa;  Candelas,Font,…]                      Recently:  [Esole,Yau;Marsano,Schäfer-­‐Nameki;  Morrison,Taylor;  Cve8č,Grimm,DK,Piragua;                                          Braun,Grimm,Kapfer,Keitel;  Borchman,Krause,Mayrhofer,Pal8,Weigand;  Hayashi,Lawrie,Morrison,                                          Schäfer-­‐Nameki;  Esole,Shao,Yau]     Outline  &  Results   SystemaQc  construcQon  of  Abelian  sectors  in  F-­‐theory    of  general   •   in   Exemplify  explicitly  for   2.  Develop   gauge  group      to  study  such  geometries    spectra    (also  with  non-­‐Abel  groups)  G4-­‐flux   •   of  toolbox:   CICY  in  P    3      (  group),  ellipQc  curves  in  16  2D   •  moduli  space  of  F-­‐theory:         A  very  brief  summary   F-­‐THEORY  COMPACTIFICATIONS     G4-­‐flux  &  4D  maher  chiraliQes   [Cve8č,Grassi,D.K.,Piragua]     3)   for  F-­‐theory  on    X4:   Geometry: (verQcal)  G4-­‐flux  in  HV(2,2)(X4  ,  Z/2              )                    requires  computaQon  of   [Wipen]   of  CY  4-­‐fold Example:   explicitly    for  family  of   G4-­‐flux  is    with   P3 properly  studied  in  M-­‐theory   3D  M-­‐/F-­‐theory  duality:    F-­‐theory  on  S1  =  M-­‐theory •  In     ⇥M AB = Z informaQon  of  verQcal   X4 •  M-­‐/F-­‐duality  for  CS-­‐terms  relates   –  use  to   –  use  to  derive     G4 ^ ! A ^ !B   [Gukov,Vafa,Wipen;   Haack,Louis]   of  F-­‐theory   See  also:  [Hayashi,Grimm;  Cve8č,Grimm,D.K.;   Grimm,Kapfer,Keitel;  Braun,Grimm,Keitel]     extend  earlier  condi8ons:  [Dasgupta,Rajesh,Sethi]   22   ApplicaQon  of  toolbox   APPLICATION  1:  RANK  THREE   CURVES   23   EllipQc  fibraQons  with  three  raQonal   p oints     [Cve8č,D.K.,Piragua,Song]     Similarly   for  ellipQc  fibraQons  with  3  U(1)’s:   Bl3 P3 •  EllipQc  curve  with  rank  3  Mordell-­‐Weil  group:    Calabi-­‐Yau   of  E  over  given  base  B       F-­‐theory  vacua                            points  in             –  miraculous  structure  of  singulariQes:   in  6D  found  for  general  base  B                  6D  Anomalies  cancelled   ✓   24   7 7 Bb B2 ˆ7 S9 ˜ b 4S ˜7 S ˜7˜ [s8 ] · [s18 ] (0,1,1) ˆ+˜Sˆ7 S˜17B2 3[K S + , 122 B ]S b 9([p b ]1 )S91 == [˜ s[˜ ] · S 17S119 ˜ 9 ˆ ˆ ˜˜7 ˆ 1 b b b b b2 ]2bb˜ =7 2[K 2[K ] ˜]S +923([p 3([p ]˜ ))2[K +)S2[K 2[K + 3([p 2[K 3([p )7SS ˜˜77 + ˆ]S)Sˆ+4+ s3 ] · S7 =[s8[˜ ]s· ][s·18 ] xx[s +S4[K 3[K 9([p ]+ 4S]]S 4]S)b2S 7S , ]BBS˜75]]SS 1ˆ bB3([p 773([p (1,1,0) 7S 2+ ]S 7ˆS = ] + ] S + 3([p + 2[K + 3([p B B ˆ ˆ ˜ x = ] ] ) ] 3([p ] 9S 22S 27]79)9S772[K 2)]S,) (1,1,0) 2 (1,1,1) +3([p ] ) S + S 2[K ]S 9([p )S S S S + 6S B B x S ] · [s ] B B B 7 9 3· S (0, 1,1) = [˜ ˜7 [s8 ] 8· [s18 ]18 B x s ] 1 b b b (0, 1,1) ˆ ˆ ˜ ˜ Starting theb counting, the4[K singularities were found in P72 + the bwith 2) 2[K ] )S7 x+(1,1,1) 3([p ]bˆˆ· )SS2[K 2[K 3([p 3]S 1simplest ˆBbb ]S7 1˜ ˜ 2B bb] ˆ ˜ 3([p b b ˆˆ b ˜2 bb 22 as ˆˆ(1,1,1) ˜= ˜]˜ ˜ 2S 2S ˜SBl b 2+ [p2 ] b· S 3S 12]S3[p ==Starting ([p c [p ] · ] S S S · 2S 17S 2˜˜ b2[K b792 ˜, 172B + S S 3[K 9([p ] S + 7S x 4[K ] 3([p ) ] S 3([p ] ) S ] S 3([p ) S ˆ ˜ ˜ ˜ ˆ ] ]) 7bcounting, 1=··S 7b + 2simplest 7+ 29([p 9)S 71B 9as ˆ ˆ ˜ ˆ ˜ ˆ ˜ 99([p 77S 99 77˜ 99]·P ]+ ]1b··])S 299 7]S 2,],2S with the the singularities were found in Bl the B B B + S S 3[K ]S ] )S S S S + 7S ([p ) + [p ] · S + c S + [p ] · S + S 3[p ] S S · S S S + x = 2[K ] 3([p ] ) + 2[K S + 3([p ) + 2[K S + 3([p ] ) b b 2 S S + 2[K ]S + + S + S S 8S , 7 9 ([pb ] 27) + 7˜ + 37 99 229S 27Bb9 27b 79 ˜ B (1,1,0) ˜27 B2+˜2Swere ˜7calculated ˆPtheir ˜S,77 · S(4.46) 7simplest ·S 99 97S B Bsingularities , = ([p [p]b2 ]· 7Smultiplicities · the Sˆ+ c [p · S 3[p · S + 2S nishing of2 two=with coefficients The are from classes Starting the found in Bl as the ˆ ˜ ˜ ˜ ˆ ˜ 2) counting, + 1S [p 2·] directly 2·] S S 7S· S 92 2S b ] + [p c · + ] S + S 3[p ] · S · S + , ˜7 S7their 2 The7ˆmultiplicities 1˜7 +17 2are 911 classes 9b ˜ 12[K bb9 b 2[K ˆˆ+ ˜˜7, (4.43) 2S˜Sˆ7+ ]S29([p 9([p )S 5ˆ2Sˆ3([p + 5ˆˆS (4.46) vanishing of two coefficients directly calculated from 159 22 ]SB3([p 9]bb+ ]b 7+ S59S , 28S 4[K )2279([p ]]S )S 2[K ]]9SS 3([p ˜77 2S= + SS + S 8S (4.46) 77 22˜]]7from 22]]b))S ˆS ˜2[K ˜S (1,1,1) B ]]B3[K B B 7= 1are ] )S )S 973([p 977S xx(1,1,1) 4[K 3([p ] ) 2[K S ) S 2[K 3([p S vanishing of two coefficients The multiplicities directly calculated their classes +7 S ]S ] S + 7S , 7 7 S7B 9 9 B B (4.43) B (4.43 Charges   Proceeding MulQpliciQes   in aqsimilar wayqfor1 the hympermultiplets (1, 0,˜ 1), (0, 1,21) and (1,(4.43) 1, 0) b Loci q Multiplicity ˆ ˜ ˆ Q R S 1S ˆ ] b)S + 5bSˆˆS + 5S˜ S ˜ 8S , b ˜ b 2 S + 2[K ]S + 9([p (4.46) ˆ ˜ + 9([p 99 +2 ]5S 77 S]9S qqB2QS]77Sqq77R3([p Multiplicity xLoci =in of 4[K ] for )B 2[K 3([p )S777S99 (1, 2[K 3([p )S7 (1, 1, 99 1) B ]S + 2[K + 50, Scharge 8S ,2 ](1, (4.46) S 2q (1,1,1) B ]S27] )S B Proceeding a multiplicity similar way the hympermultiplets 1),91,7(0, 1, and 0) we get the orders vanishing Loci Multiplicity Proceeding in a similar hympermultiplets (1, 0, 1), (0, 1) and 1, 0) we the notation x for the of hypermultiplets with Q way R for qthe S (q(q ,q ,q ) s = s = 1 [s ] · [s ] (1,1,-­‐1)   Q R S weused used the notation x for the multiplicity of hypermultiplets with charge 18 18 ,q ,q ) 1 b b b Q R S ˆ ˆ ˜ ˜ sthe ss18 = 11the 1)multiplicity [sˆ8]]B·b· [s ]hypermultiplets whereweweused used the x hypermultiplets with = 18+ xorders =0 ]) +for + ]]S +19([p 2[K ][s Sof 3([p S57S˜ S13[K ]2S,7(4.42) +(4.42) 3([p S7 (q,q ,q we get the orders of 12[K 22 b3([p 2the bcharge 21 2] ) (1,0,1) Q R ˆS117,q ˜Sfor the x2[K multiplicity with B ˜7 B ˜7charge snotation = = [s ]2of we get of vanishing (q ,qvanishing )S S+ 2[K ] ·hympermultiplets )S 5bclasses, SˆSˆ S29] +)3[K 8S (4.46) multiplicities, 18 872]+ ,R q, SqS).).where Calculating the other without expanding the classes, we obtain x9notation = ] 3([p ] ) + 2[K ] S 3([p ] ) S + 3([p ] ) S Q0 R 7similar 9+ 918+ 77 7B (1, 9]obtain (4.42) 2 (1,0,1) B B B s = s = 0 [s [s ] Calculating the other multiplicities, without expanding the we Proceeding in a way for the 0, 1), (0, 1, 1) and (1, 0) 19 19] x( 1,0,1) xthe Charges xwithout x( (0, xand ss9 = ss19 = 00Loci 11 x˜(1,1, 1)without [s99]] ··xhympermultiplets [s (0,1,2) (1,0,2) (0,(1, 1,1) 1, we 1, 2) (0,0,2) (1,1,1) Proceeding in0 a ˜similar for 0, ˜1), 1, 1) (1,x1, 1, 0) , q , q(0,1,2)   Calculating the other multiplicities, expanding classes, 19 S ).Calculating 1[sthe way b 2obtain (q(q the other multiplicities, expanding the classes, we obtain = = [s ] ˆ ˆ 19 19 Q ,QqR ,RqS ) +2 S S + S + 2[K ]S 9([p ] )S S S S S + 6S , QS Q b 7 9 9 we= get the of vanishing x ˆ07Loci ˜ vanishing ˜70x s10 = sget 0Sorders [s [s ] x5S(ˆ701,0,1) Charges x7(1,1, x(0,1,2) x4S x6S x(1,1,1) (1, 0, 1) g= = g+ 0+(1,1, 022x]S 4x20 4, x +2 S˜71 S= )S S S1,1) 1)9 B (1,0,2) (0, ( 91, 1,(0,0,2) 2) (0,0,2) 20 10 Charges Loci x10 x9(0, (1, x1, we the of ss10 ss20= 002[K [s09([p ]]2··]x·[s [s 920 1)B (0,1,2) (1,0,2) (]] 1,0,1) 1,1) (9 + 1, 1, 2) (1,1,1) 7x orders 91 Proceeding in a similar way for the hympermultiplets 0, 1), (0, 1, 1) and (1, 0) = = [s ] (1,0,2)   harges Contained in Loci Multiplicity 10 20 10 20 QS Q RS R ˆ b QS = Q Charges Contained inContained Loci (1, 0, =] 00+]b03([p 02 ]3[K x ) 1Multiplicity 3[K S047 + 0bMultiplicity 3([p ]b )ˆ Sˆ027 + S 1, 1) gorders = g= = 40Sˆx B 02x0+ 00ˆ72 +12[K 192[K 2in Charges (0,1,1) (1,x(0, 0, 1) gthe 0Loci 4+ ]3([p 4]S˜7xB44]S˜70 x00 x11 gin ˆMultiplicity 6= 99B 2[K Charges Contained Loci we get of vanishing = ] + 3([p ) ] ] ) S S + 2[K Charges Loci x x x 7 (0,1,1) ˜ (1,1, 1) (0,1,2) (1,0,2) ( 1,0,1) (0, 1,1) ( 1, 2) x(0,0,2) x(1,1,1 RSB R Q RLoci QR ˜ Charges x x277(0,1,2) x88(1,0,2) x˜(]] 1,0,1) x0(0,B 1,1) x(141,1,are 1,1,0, 1)1) proceed s˜s˜3to = s˜(1, = 00 g RSggsingularities [˜ ss433B]]dP ··04S [s ]0·· [s [s (-­‐1,0,1)   (1,1, 1) 1, 2) 771) 18 0, = s ˜ = [˜ S [s ] hen we to the apparent in the picture The multiplicities Then we proceed the apparent singularities in the picture The multiplicities (0, 1, 1) = g = 0 0 0(0,0,2) 18 0) 0 QS Q (0, 1, = g = 0 0 12 , 11(1,1, b b ( 1, 0, 1) s ˜ = s ˜ = [˜ s ] · S [s ] · [s ] ˜7 27] 0multiplicities ˜= ˆ7 S˜07 dP ˆ ˜47 S0are Then we proceed to the apparent singularities in2Sthe The are 6= QS Q 31)6s 79g 3bS 8[s 18 picture (1, 0, = g 0 0 +3([p ] ) S + + 2[K ]S 9([p )S S S S + 6S ( 1, 0, 1) s ˜ ˜ = [˜ s ] · [s ] · ] b 2 9 9 18 9 B ˜ ˆ ˜ ˆ ˜ QR Q ˆ QR (1, 0, 1) gQ = gand = 0the 032]xB1·(0,1,2) x1(1,0,2) 4b·]xthe 040loci 0x5(11S71,˜Staken 4b6S0x9(0,0,2) 0(1,1,1) 111 +3([p ] ) S + S S + 2[K ]S 9([p )S S S + , 1) s ˆ = s ˆ = [ˆ s S [s ] [s ] Charges Loci x x x 7 9 9 b b b (1,1, 1) ( 1,0,1) (0, 1,1) 1, 2) 7 18 (1, 1, 0) g = g = 0 0 given by the multiplication of the classes subtraction of loci already (4.47) , 1, 1, 1) s ˆ = s ˆ = RS R 1 b ven by the multiplication of the classes and the subtraction of the already taken (1, 1, 0) g = g = 0 ˆ ˆ ˜ ˆ77 + x 8[ˆ 2[K ]2 ]= +) 3([p 2[K ]ˆ S )03[K ]S˜ given by the multiplication the classes of = 2[K +9= 3([p + 2[K )the Sˆ·7718 ]S·˜018 3([p ]44)S˜27] b)S˜7 00 R (0, 1, 1)= g2and 0the 03B+]S 44B1 3([p 11 (0, 1, sˆ6= sˆ= 0= s0]0·3([p ]Sˆ S2ˆ]b]7loci [s ]taken +23([p (1,0,1) B+ Bg 7[s (1,0,1) (0, 31) 79g]gRS 33([p 83[K 1Q b ]subtraction 2) QS 6= 9+ B·2[K (0, 1,(0,-­‐1,1)   1)1) s0, ˆxof s6ˆhypermultiplets = [ˆ s747and ]2+ [s ]already [s ]18 ˆS778[s = 0 0 31, 7B 3+ x = 2[K ] 3([p ] ) + 2[K ] S ) S + ] S + 3([p ]0 )S7(4.47) Finally for the (1, 0, 0), (0, 1, 0) (0, 0, 1) we obtain the degrees [HP: (1, 1) g = g = 0 0 1 1 b b b 2 (1,1,0) 1, 2) s ˜ = s ˜ = or s ˆ = s ˆ = [˜ s ] · S [s ] · [s ] = [ˆ s ] · S ] · [s ] QR Q B B B ˆ ˆ ˜ ˜ (4.47) 9 9 10 20 9 19 , into 1, 2) s ˜ = s ˜ = or s ˆ = s ˆ = [˜ s ] · S ] · [s ] account in the first case For example the singularities at the loci s ˜ = s ˜ = 0, x = 2[K ] + 3([p ] ) + 2[K ] S + 3([p ] ) S + 2[K ] S + 3([p ] ) S to account in the first case For example the singularities at the loci s ˜ = s ˜ = 0, 9 10 20 9 19 7 b [s 7[s loci 7] 227 [s QR into (account at ˜[ˆ s˜79 [s = (1, 1, 0)For gs = 005] S= 11·927[s 304·S˜S 1S 1singularities bthe Bˆ B ( 1,1, (-­‐1,-­‐1,-­‐2)   2) s˜(1,1,0) s˜1, = 0or s6ˆ= ˆ= = 07(1, [˜ ·9([p ·(0, ] ·19 [s 00] 11 3s = RS ˜Q ˆ[ˆ = 91, = 92 20 1,1,in2)the s˜first s˜case = 00) sˆ = 0the [˜ ]s119(0, · ]S [s ]2102·00]][s = sB4we S ]0, How can we center this ˆor ˜ˆ9R92˜ ˆ097]S +2 S + S + 9([p )S Ss + (1, g˜g8example = = 00˜0, 02and 09the = 9for 82[K 10 20 80S 9+7 S 9,6S 7gS 91, 9]˜7·8 94the (0, 1) g7s gS 0s]S 4]S 0Sˆand ] 19[HP: Finally the hypermultiplets 0, 0), (0, 0) 0, 1) obtain degrees ,[HP: +2 S S + + 2[K ] )S S S 6S Finally for the hypermultiplets (1, 0), 0) (0, 0, 1) we obtain degrees B1, 6= 97·table] b 9 ˜ B ˆ ˜ ˆ ˜ ,0, 0, 2) s ˜ = s ˜ = or s ˆ = s ˆ = S S [s ][s ] = · S [s ][s ] ininthe coefficients read = s8s848+ s18 s9ˆ49s )S 0, contain 0,that s(0, ˜77 original =original s˜99 = 0How or sˆ7can sˆ = S+ ] 7S +s Ss ]Sss 9([p S+7+ S + 9read 99Bss 9S 18 16 18 ˆ][s 220 7s= 98 9·20 we 9+ 7s 19 10 atthat in2)the original coefficients ss1ˆ7B= + = (] 9)S s19 ss= )S7= = 0, the read s7QS s9621·63[K + ))b˜ = ss[s s[s )10 = 0, contain QR ,contain ˜( ˆˆ919 18 6]S 16 ˜19 18 19 90 s˜= s= ˜1, = 0or ˆ((S = 0s16 ·999(2(S [s1][s ][s] 20 ] (4.47) How center table] Sˆ7or S 9([p )S S 40]sS˜9= 7S , 92[K center this table] (4.47) = we 7123[K 70 19 ][s 97b]S18 10 14S bS 227]S 299 S (0, 0,0, 2)2) s˜coefficients s ˜ = s ˆ = s ˆ = · S S · S [s 772this 9][s 9ˆ2+ (1, 0) g g 1 ˆ ˜ can 9 19 10 20 (0,0,2)   1 b b x = 2[K ] + 3([p ] ) 3[K ] S + 3([p ] ) S + S + ] S ˆ1,0,1) ˆ7x2+ ˆ0, ˜of Finally for the the hypermultiplets (1, 0,(]S 0), (0, and 1) we obtain the [HP: 0)]1,1) 71,+ degrees (0,1,1) 12 Loci xone xthe x0, x1, x(1,0,1) xb(0,1,1) x(1,1,0) 7x(0,0,2) Bx2After B B 12 (4.44) the loci s = s = with degree the subtraction, the multiplicity the x = 2[K ] + ] ) 3[K + 3([p ) S S 2[K ] S (1,1, 1)3([p (0,1,2) (1,0,2) (0, ((0, 1, 2) (4.44) 1 b b 7 (0,1,1) 18 Finally for hypermultiplets (1, 0), (0, 1, 0) and (0, 0, 1) we obtain the degrees ˆ ˆ ˜ ˜ the loci s = s = with degree one After subtraction, the multiplicity of the B B B e loci s8 = 8s18 =18 with degree one After the multiplicity the x(1,1,1) ] this ]S7 b3([p ]S7 of )S7 [HP: (4.44) 14[K 12[K 12[K b3([p the b3([p ] ) subtraction, ] )S ] (4.47) Q How Q can= (4.44) B B B ˆ ˆ ˜ ˜ we center table] xthe == ]be 3([p ) xtable] 2[K 3([p )(the S1,7 x1,(codimension 2[K ]S7xˆ(0,0,2) 3([px2 ](1,0,1) )S7 x(0,1,1) g(1,1,1) =Loci g12 04[K 1is 02this 11,0,1) 1,x 0(1,0,1) Loci xcan x1) x x x0(1,1,0) x1(1,1,1 11,1) b]x(1,0,2) b2) xany 20]x1,1) x x(0,1,2) x](1S(0, x]S x(1,1,0) B B B(0,0,2) How can we center (1,1, 1) (0,1,2) (0, ˆ(using ˜1,0,1) (1,1, (1,0,2) (0, 1, 2) with charge (0, 1, 1) calculated as st hypermultiplet two multiplicities in table calculated of +3([p S˜7˜+ 21be S 2[K 9([p0,codimension ]1) )S 4S 5degrees S˜x17(0,1,1) S9 +2 6S ast twoThe multiplicities in the table can be calculated Finally for the hypermultiplets (1, 0, 0), 1,Bb0) and we obtain the [HP: bis b(0, (1,1,1)   2] ) 7S + using any 9ˆ of S9˜ hypermultiplet with charge (0, 1, 1) calculated as , ˜ ˆ R R Q Q ypermultiplet with charge (0, 1, 1) is calculated as last two multiplicities in the table can calculated the codimension Q Q +3([p ] ) S + S S + 2[K ]S 9([p ] )S S S S S + 6S , 21 70 7be0 5S 7˜the The last two multiplicities in the table can calculated codimension ˜7 + ˆ17any g=12 00we 1012[K 01 92 ]0)Susing 11of 01 10 1(4.46)11 g9 g= g= 0= 1ˆcenter 1+Bb9([p 17 9, x g99How = =Loci 1]S Sˆ(1,1, S˜b9 x+ 5S 8S 00(0,0,2) 12 g can Sthis 70S9 B ˜712+ ˆ9710+ x xtable] x9(1,0,2) x(0, x(1,1,0) (0,1,2) (+ 1,0,1) 1,1) ( 1, 1, 2) ,x (1,0,1) 1) 5ˆ 1x ˜ sub-loci bx 2 ] )Sx b x ut inineach case we have subtract the right S S 2[K ]S + 9([p S S + S S 8S (4.46) Sto S12 but each case we have subtract the right sub-loci Rgto R ˆ ˜x(0,1,1) 9 9 R R Loci x x x x x x x01(1,1,0 B x = 2[K ] + 3([p ] ) + 2[K ] S + 3([p ] ) S + 2[K ] S + 3([p ] ) S = g = 16 16 1 16 16 1 (1,1, 1) (0,1,2) (1,0,2) ( 1,0,1) (0, 1,1) ( 1, 1, 2) (0,0,2) (1,0,1) (0,1,1) loci,but butinineach each case we have to subtract the right sub-loci g = g = 1 0 1 0 1 7 7 (1,1,0) Q Q 1 b b b B B B g = g = 1 0 1 0 9+ 12+ loci, case we have to subtract the right sub-loci ˜7 ˜S˜7Sx 12 g·12 = g]]Q = 0]1 2+ 3([p ] )b 20+ 2[KB1 1]Sˆˆ 13([p2 ] b)S 0ˆˆ7 + 2[KB 1]S˜˜7 + 3([p ]b )1S = 2[K + Q xx = [˜ s ] · S [s ] · [s (1,1,0) ˜ 12 S B = [˜ s ] · [s ] [s 18 (0, 1,1) S S x = 2[K ] + 3([p ] ) + 2[K ] S + 3([p ] ) S 3[K ] S + 3([p ] ) S 3˜ 18 (0, 1,1) g g = 1 1 Loci x x x x x x x x x x x (1,1,0)   (4.48) 7 7 (1,0,1) = in1(1,1, 16way 1Bthe 160,(0,0,2) 12(1,0,1) (1,0,2) ( 11,0,1)b1 (0, 11,1) 16 ( 1, 16 1, 2) (0,1,1)1 0(1,1,0) 11 R 9R gfound 0from 16(0,1,2) 16 16 10and 12+ [˜ shypermultiplets [sg=89+ ]g·= [s ]=Qg12 ally for the hypermultiplets found the WSF, start with (1, 1, 1) ˆ116 ˆ7 S ˜ ] · S7 g9+ 18 0, 1,1) Proceeding afound similar for hympermultiplets (1, 1), (0, (1, 0) 1(1,1 g12+ = =B + 11) we WSF, 1S 1B7the , 1, 1) nally for= the from the WSF, the charge (1, 1, 1) 1]0ˆ 17S 2˜ b1 ]S 2the b9with b 1) R R Q S S 3[K 9([p )S 4 S S + 12 ˆ ˜ ˜711 1, 7 9 Finally for the hypermultiplets from we start charge (1, 1, B x = 2[K ] + 3([p ] ) + 2[K ] S + 3([p ] ) S 3[K ] S + 3([p ] ) S Proceeding in a similar way for the hympermultiplets (1, 0, 1), (0, 1, 1) and (1, 1, 0) g = g = 1 0 1 0 b Finally for bthe hypermultiplets found from the WSF, we start with the charge (1, 1, 1) g = g = 1 1 1 7 (1,0,1) 1 b b b (4.48) ˆ ˜ ˆ ˜ S S 12 B 12 ˜,7 +16 ˜7 subtraction the multiplicites (4.48) 01 bx b9 16 9B]Sˆ714 b 2˜ + S 3[K ]S ]˜12)S S73([p 4SˆS77 S + 7S b)22After bS 212 = 2[K ]vanishing + )9([p + 2[K ]˜ˆS )ˆ ]9SB 3([p ]b 1)+ S 1are 3([p b+ · 2S ˆ= ˜˜77Bof ˆ793[K ˜ gb9RS(1,0,1) = = 0S 1˜ 16 15 16 7S 71 2 ]S ˆ ˜ ˆ ˜ ˆ ˜ ˜ B we get the orders S Rg Bby B 19 b12[K 216 bS 9+ 12+ = ([p ] + [p ] · S + c · S + [p ] · 3[p ] · S S · S S + 2S , 01 ˆ ˜ ˜ the degree of vanishing of the other loci are given +2 S + S + ]S 9([p ] )S S S + 6S , = ([p ] ) + [p ] · S + c · S + S 3[p ] · S · S S · S 2S scase, case,In the degree of vanishing of the other loci are 2 2 7 1 b b b g g = 16 1 16 16 1 g = g = 0 1 0 9 9 2 7 9 9 = 2[K ] 4[K +multiplicites 3([p ]b )2other + Sare 3([p )by S˜72by ]S7 +]S˜3([p ] )S weAfter ofB=the vanishing 9˜ ˜79 , 1(4.48) B 2[K 72]2[K b the 99+ 12vanishing 12+ In this case, the degree of loci are + 2bb] 3([p 23([p (1,0,1) After subtraction the multiplicites are ˆoforders ˜ ˜other ˆ B B2[K xof ] 2]the 3([p )loci ]Sˆ7given ] )3[K Sˆˆ7S this case, degree vanishing of the given 1˜ 2S ]+ 72S 27] )S (1,1,1) are ˆ ˜ ˜ ˜ B B3[p B = ([p ]b )2the + [pxxget S + [p · S S ] · S · S · + 2S SS7 + S c1 · 1 b b b 2(1,0,1)   ] g·subtraction 7 9 +2 S S + S + 2[K ]S 9([p ] )S S S S S + 6S , b , ˆ97ˆ1 9([p ˆ97ˆ5Sˆ716 21] )bS 94S 3([p ˆ717S3([p ˜77 +2S = g12+ = +2 16 B]S7 16 ,] )bS 0(4.48) 1]12S 1]9S b7 22 2[K 9˜71 4[K ]˜]16 2[K 3([p 2[K ]1˜S7˜ 9+ = Sˆ2[K S 6S ˜x 9]b )x 7˜9 1+ 2Loci ] )S 92 x B B are B(0, 7)b)+ 1, B 1) x(1,1,1) = 4[K ] 3([p 2[K ] S 3([p S + ] S 3([p ] ) S ˆ ˆ ˜ Charges x x x x x b 2 7 7 (1,0,0) (1,1, (0,1,2) (1,0,2) ( 1,0,1) 1,1) ( 1, 2) (0,0,2) After subtraction the multiplicites ˆ ˜ ˜ ˆ ˜ x = 2[K ] + 3([p ] ) 3[K ] S + 3([p ] ) S + S + 2[K ] S BxS B+ 6S 11 B 2b ,x (4.43) 2+ 12 ]]b)S 2ˆ+ (4.43) (0,1,1) ˜ +2 S2B7BQS S Sˆ27Q + 2[K 9([p 55S S(0, S 711,1) Bare Charges x x]S x(0, xb )(99x x(4.48) 1Loci b˜ 72 + 9ˆ 7x 7S 21the S 2[K ]S + 9([p )S Sˆ7ˆ2[K 524 S ,1S˜˜ (4.46)(1,1,1) 1) (1,0,2) 1,0,1) 1, 8S 1, ]792) (0,0,2) (1,1,1) ˜ 7]S 9) 1(1,0,2) bB 2(0,1,2) bS 21(993([p B After subtraction multiplicites 72(1,1, 21,1) 99ˆ+ 7(0,0,2) arges Loci contianed in x x x x x ges Loci contianed in x x x x x x 1 b b b ˆ ˆ x = 4[K ] 3([p ) 2[K ] S 3([p ] S + ] S (1,1, 1) (0,1,2) (1,0,2) ( 1,0,1) ( 1, 1, 2) (1,1, 1) (0,1,2) ( 1,0,1) 2) (0,0,2) ˆ ˜ ˜ 1 b b 7 7 Charges Loci in x]0(0,1,2) x x x x(0,1,1) = 2[K + ])x )B 3[K + 3([p ](0, )+ SBx708S (1,0,0) ˆ071,1) ˜B7 x b ]S Charges Loci contianed x=3([p x x]92(0, x(+ B B71,1) 1,0,1) xcontianed = 4[K ] (1,1, )29]2+ 2[K 3([p S 2[K S21, ]0 (0,0,2) )S(4.46) (1, 0, 1)in gS = g]3([p 0B7](Sˆx]7S 42]S 1) (1,0,2) (+ (]+ 2) Q ˆx ˜x ˆ47 S ˜S B 2[K 1) (0,1,2) (1,0,2) 1,0,1) 1, 2) (0,0,2) = 2[K ]0 3([p ])+ )2S + Sˆ+ 2[K ]S3([p 213([p 72[K 27 0(4.43) 92+ (0,1,1) B1] )S S + 9([p S ,1 ˜1, b5S B B After subtraction multiplicites are (0,1,1)   7B˜ 70S297 (1,(1,0,0) 0, 1)x00 g6QS2 = g6(1,1, 093[K 02+ ˆ ˜ ˆ ˜ 91 4B 1ˆ 941 + b= B b]S 17972the ˜ b 12]S b b4 1˜ 21 b7 S b2 ,˜ ˆ ˆ RS R ˆ ˆ ˜ S S S 2[K + 9([p ] )S S + S S 6S , ˆ ˆ ˜ 1, h = h = 1 9 , 1)1) h = h = 0 +3([p ] ) S + S S + 2[K ]S 9([p ] )S S S S S + 6S x = 2[K ]20˜= + )71127B ]S4B ]2 )S27]0b9+ S ]7S 02 ] 90 = 4[K ] )3[K )0Sb0 2[K 3([p 1, 9]S 2[K 7 94 11, 1, 1)0 g4[K g]7the 09with ]3([p 7b 1+ 13([p 1]S 27 = b)S˜70 0023([p 7˜ (1,0,0) (1, = hxx= = 1B2[K 7++ B B2[K B 20 2˜ 1)1)22the = h1) 100+ 0hypermultiplets 41), B B ˆ3([p ˆ5)S 6R (0,1,1) ˆQ6RS ˜RR7ˆ= 1h 2(0, ˜7+ ˆ 1hympermultiplets b9bsimilar 2)S where(1, we used thehnotation notation x for hypermultiplets with charge where we used x multiplicity of charge = ]S7ˆb9S )7˜102+ 2[K ]S 3([p ]ˆ)S )]S 2[K S 3([p ]+ (0, 1, gS gQR = 41 0)S 0ˆ+ 4]4 06S 22[K 2˜ ˆ17S ˜6S Proceeding in a0]S way for the (1, 0, (0, 1, 1) and (1, 0) S S ]S 9([p ]BBof S S + S 6S , (q ,q ,qS+3([p (q ,q ))92for ]˜ )S˜S˜773([p S + ]S 9([p S S S S 6S , 2ˆ 7b+ 2]9bS 70S+ 7S 191, (1,0,0) ˜ ˜ S Q 7,q 29([p 7 B B 9 7 +3([p ] ) + S 2[K ]S 9([p S S S + , Q B B 1 b b b 9 9 S S 2[K ]S + ] )S S + S , ˆ7 3([p ˜˜7(4.45) 7g6171Q2b2= 1 ]0 2[K ˆ2 1(4.45) 7˜7 3([p 73([p (1, 1, 0)QR g9way =multiplicity 09b2]]bbb))S 0+ 19] (4.45) B 12[K ˜ x0) 4[K ]+ ]2˜2)+ ]0S b)2S 1,00) 7+ 2and (1,0,0) ˆS+ ˆSˆˆ+ ˜2[K B ]expanding B BS(4.45) ˆ ˜1) ˆ0270, ˜we here notation x(qother for of hypermultiplets with charge Proceeding in the (1, 1), (0, 1, (1, x(0,1,0) = 4[K 3([p ]ˆ˜]b012b7722)for 2[K 3([p S ] S we get the orders of vanishing (1, g6= = 0hympermultiplets 091+9([p 0)S 0S x1, =,q 2[K ]ˆˆ)7= 3([p ) + S 3([p ] ) S 2[K ] S + 3([p ] ) qused Calculating the other multiplicities, without the classes, we obtain +3([p S 2222S S + 2[K ]S 9([p ] )S S S S + 6S , (q(q , q, R , ,qqSS).).the Calculating the multiplicities, expanding the classes, obtain 7 ,q1a )similar (1,0,0)   7 7 (1,1,0) S˜˜1˜1the 2[K ]S ] S S + S S 6S , Q R 770 9 B B B Qwe Q R S Bg2]9]S B b b b QR B 9 9 b 1 22+ b b b QR 1 b b B2[K ˆ+ ˜973([p ˜7(4.47) ˜+ ˜˜ ˜ ˜2QR x = + 3([p )Bfactor + S ]= x =of 2[K ]2 ])2[K + ]Sˆ]9([p 3([p )2+ + S + ]QR S S23([p S7]]2 ]+ S + ]factor )S Sˆˆin S+ + S S ,(4.47) x(0,1,0) = 4[K ]is2[K )223([p 2[K ]missing Sˆ2[K 3([p ]+ Sˆ3([p S7ˆ)4of 1is 12+ b7to 78 772[K (1,1,0) inthe the column is related to some missing sˆ2ˆS]ˆ18 gBand = 7S 23([p 7] )S= (1,1,0) 7B 9ˆ 2)and 922[K 11,in column xxthe related to some missing of s]2b3([p g6S6B5]s2[K B B B]77S B 7]S 7]2b]9 7S ˆ2[K ˜6S 9)g (1,1, 1) is 1B 7s B (1,1, 1) we get the orders vanishing The in column x related some missing s s in The in the column x related to some factor of and in g = ˆ ˜ ˜ ˜ x = 4[K ] 3([p ] ) + 2[K ] S ) S S ] S 18 (1,1, 1) 18 (1,1, 1) 7 (0,1,0) q , q ) Calculating the other multiplicities, without expanding the classes, we obtain S S S 2[K ]S + 9([p )S + S S + S 6S , 1 b b b B B B Finally for the hypermultiplets (1, 0, 0), (0, 1, 0) and (0, 0, 1) we obtain the degrees [HP: )˜ +x bS Q R S 7˜ b1]S 2ˆ 9] ) ˜ˆ 92[K 92 +x ˆ0) ˆx ˜7 x 13([p 1+ 7S b 2[K 91,12 ]1,˜)S B ˆB7 S ˜27]7]b )+ ˆ+ ˜3([p Charges Loci x xSˆ(1,0,2) x x(1,1,1) ˆ79]22S˜+ x(1,1,0) =right 2[K 3([p (1,1, 1) (0,1,2) (b9+ 1,0,1) (0, 1,1) (72S 2)[HP: (0,0,2) + S 3[K ]S 9([p ] )S S S 7S , 2ˆ]3([p 2S 7]4 7the 3([p S S 2[K ]S + 9([p ] )S + S S S 6S , Finally for the hypermultiplets (1, 0), (0, 1, and (0, 0, 1) we obtain degrees x = 4[K ] ] ) + 2[K ] S 3([p ) S S 2[K ] S B B 10, b991 24 9 7 7 b 0.Thus Thus in order to obtain the spectrum we have class instead of 7 (0,1,0) B ˆ ˜ ˆ ˜ 1 b b B ˆ ˜ ˆ ˜ in order to obtain the right spectrum we have to add that instead of B B B b b ˆ ˆ ˆ ˜ Charges Contained in Loci Multiplicity How can we center this table] S S 3[K ]S 9([p ] )S S S S S + 7S , R Thus order to obtain the right spectrum we have to add that class instead of Contained in Loci Multiplicity RCharges ==0.0.Thus inin order to obtain the right spectrum we have to add that class instead of ˜ ˆ ˜ ˆ ˜ + S S 3[K ]S 9([p ] )S S S S + 7S , QS Q 7 9 9 (0,1,0)   x = 4[K ] 3([p ] ) + 2[K ] S 3([p ] ) S S 2[K ] S 7 9 9 3([p ] )Sg]67this S + ]x)S +S2ˆ5S S +S42[K S B 22 xB0]S9 11 9ˆ2ˆ (0,1,0) b1B2˜ b2[K 7 S1,9 1˜ B2) B How can we0, center 24[K 7gx 7=B]˜ 7(0, (1, 1)Loci =Stable] 0 +723([p 9x,76S ,0x(1,1,1)1 ˆ9([p ˜76S Charges x x˜0+ x (1,1, (0,1,2) (1,0,2) (4229 1,0,1) x(0,1,0) = 2[KB ]ˆS ]b]bbb))S 2S S 3([p S 2[K ]S 5S 4S]1,9S 1ˆ7b2bS b1 B 74b9S 1)S 2)]S 27)1) + 21B 72 b 3([p 971 49([p 17S 91˜(+ 7S b b(0,0,2) ˆ ˜ ˆ 71,1) B ˜ b ˜ 1 2 ˆ act it ˜ RS R + S S 3[K ]S 9([p ] S S + , ct it ˆ ˆ ˜ 1 b b ˜ ˆ ˜ ˆ ˜ ( 1, 0, 1) s ˜ = s ˜ = [˜ s ] · S [s ] · [s ] x(1,1,1) = 4[K ]6 (1,1,1= 3([p ] ))902S]+ 2[K ]9] 2)1,1) 2[K ]23([p )S˜4([p 70B 7Q 2B2[K 97 770 92[K subtract QS ˆ3]S ˆbb8x ˜97(0,0,2) ˜7b6S 1+3([p ( subtract 1, 0, 1)it.it Contained s˜x s˜ = ]ˆ3([p Sx3([p ·9271,0[s ]2)ˆ]07SS 27 2[K b 20 74[K 7x 27 x(0,0,1) =Loci ](0,1,2) 2[K ]xBS([˜ 4([p )]SS2[s S18 + 2[K ]S ],7)1Sx7 (1,1,0) ]2b1)gb)B9S˜223([p )S + 52[K S S S Charges in (0, 1, 1) g] 3([p = 41,0,1) 0](27Sˆ ˜473([p B 33x= 729([p =Loci 4[K ]24([p S )27]S ]9S 7]s 23([p 7]8 x x273([p x 7)bS 9·7Multiplicity 7x4 9] )S 113 2˜ 718 7S 7]x˜ B2[K B (1,1,1) B (1,0,2) (0, 1, (1,0,1) (0,1,1) = 4[K ] ] ) ] ) ] S )9(1,1,0) S 1=2Bg 10 ˆ 1B b+ (1, 0, 1) g = 0 4 B B ˆ ˜ ˜ 2 (1,1,1) ˆ ˆ ˜ B B Loci x x x x x x x x x x x 3([p ] ) S S S 2[K ]S + 9([p ] )S + S S + S S 6S , x = ] 4([p ] ) 2[K ] S 4([p ] ) S S + 2[K ] S 4([p ] ) S 1) (1,0,2) ( 1,0,1) (0, 1,1) ( 1, 1, 2) (0,0,2) (1,0,1) (0,1,1) b(0,1,2) b 2 7 9 9 QR Q 7+ Q 4[K Q B (1,1, 1 b b b 7 7 (0,0,1) B ˜ ˆ ˜ ˆ ˜ ˆ B ˆ[ˆ ˆ9b788+ ˜176S ˆS RS 1R 1B b]ˆ 26 ]1= )3([p S2g729 2]= 2[K ]S 9([p 5][s S S + 42S0) S3([p , 02 ] )1bS˜7 1(1,1,1) = = 00 0)720S1+ 0b] ))S 1S 04 1b )9S˜ (0, 1) sˆgsimilar = sgˆ = 01ˆg3([p s7s0bs110933+ ]]ˆ0]3([p ··4([p SS [s ]and ]091+ (1, 1, 0) g]]˜ 11(1, 0]7˜·2[K 02B (0,0,1)   ˆS ˜1, = 4[K 4([p ]bS 2[K S )S 2˜S 2[K 4([p Q9= 7b 2[K 2]0 7ˆ 7B ]S 7way 7˜ 18 1,1, sˆfor s ˆ = [ˆ [s · [s ] 1hympermultiplets 12 oceeding in similar way for the hympermultiplets 0, 1), (0, 1, 1) (1, (0, 1,a 1) g = = 0 0 x = 4[K ) ] S ] S ] 1 3g12 7 18 ˆ B B ( 1,(0, 0,in 1) = s ˜ = [˜ · [s · [s ] ceeding aa1) similar the hympermultiplets (1, 1, 0) b 2 7 7(4.46) (1,1,1) ˆ ˜ ˆ gx = 0 1 1 ] )S˜ Proceeding inway a9Qs˜(0,0,1) similar for the hympermultiplets (1, 0, 1), (0, 1, 1) and (1, 1, 0) ˆ ˜ Proceeding in way for the (1, 0, 1), (0, 1, 1) and (1, 1, 0) 3= 7 18 B B B ˆ ˜ ˜ ˆ ˜ S S + 2[K ]S + 9([p ] )S + S S + S 8S , b R R = S S + 2[K ]S + 9([p ] )S + S + S S 8S , (4.46) x 4[K ] 4([p ] ) 2[K ] S 4([p ] ) S + 2[K ] S 4([p 7 9 9 ˆ ˜ ˆ ˜ 7 9 9 S S S + 2[K ]S + 12([p ] )S + S S + S S s 10S 25   B 7 7 (0,0,1) ]b ˆ + bB 90 + b 9ˆ+ gQR =˜0= 1S 1sBB 0˜] (4.47) Q2 9([p +]195= 908S 9] · 9B B2 SB˜72= 2[K S S S (4.46) 4([p R9 0=or 2]1 2119 ˆˆsˆ= ˆ2ˆ771]S ˜297790SˆS77299s+ 1, 1,1, 2) 2) s˜s˜88= =g9Rs˜1, s˜9=90) ··+ S [s ·1)S [s [ˆ s)9s8S8+ ]27]·0Sˆ6·59S [s [s,[s xgS12 4[K )]S + 2[K 4([p 2[K ].S ]b )S˜711 1[sB 19 b0 2S b ˜ g= 1g 0]]S˜·B 17]02 ]4([p 12 8S 8)12]2]]S 990 012([p 10 20 91 ( ( 1, = 0sˆ==or = sˆsˆ72hypermultiplets ·71,4([p [s = [ˆ S [s16 (0,0,1) (1, g0g= =4[K 07+0˜4([p 1[˜ 0]1]6bS 26S12= Ss S 2[K + ]]]S)S + 10S ˆ[ˆ ˆ B 991= 10 20 19 78 7the 9S 27s 90) b 9 x ] ] 2[K ] S ) S + 2[K 4([p ) S b B ˜ ˆ ˜ g = 16 16 16 1 (0, 1, 1) s ˆ = ] · S [s ] · [s ] 2 7 (0,0,1) ˆ ˜ ˆ ˜ Finally for (1, 0, 0), (0, and (0, 0, 1) we obtain the degrees [HP: S 3+ this 7Sapproach, 7S79 + 18 9+ 12+ 2S SB7 + 2[K 2Ssee+]S 2[K ]S 12([p 6Sfirst 1212 S 5S 8S , s 10S (4.46) 1]+)S + 9([p b +6 is variety can be seen as aS determinantal variety, (2.7) Following the 9B ]S)S 7BS ) 1) 6D  maher  spectrum  with  U(1)3   ApplicaQon  of  toolbox   APPLICATION  2:  ELLIPTIC  CURVES  IN   TORIC  VARIETIES   26     EllipQc  fibraQons  with  toric  ellipQc  fibers   polytopes:                               [D.K.,  Mayorga  Peña,  Oehlmann,  Piragua,  Reuter]    associated  to  16  reflexive   F13 F16 F15 F14 F9 F11 F10 F12 F5 F6 F7 F8 F1 F2 F3 F4 For  algorithmic  approach  to  toric     models  &  toric  Mordell-­‐Weil,   see:  [Braun,Grimm,Keitel]   Applying  presented  techniques:     with  given  E  &  arbitrary  base  B   •  determine     •  compute   :  maher  reps,  6D  mulQpliciQes;  4D   27     EllipQc  fibraQons  with  toric  ellipQc  fibers   [D.K.,  Mayorga  Peña,  Oehlmann,  Piragua,  Reuter]     F16 SU(3) Z3 Arrow:   SU(3) x SU(2)2 x U(1) F13 (SU(4) x SU(2)2) rank of total gauge group                                 •  Up  to   SU(2) F11 F10 F8 F6 F4 SU(2)4 x U(1) Z2 Z2 SU(3) x SU(2) x U(1) F14 F15 SU(3) x SU(2) F3 F12 SU(2)2 x U(1)2 SU(2)2 x U(1) F9 SU(2) x U(1)2 SU(2) x U(1) F5 U(1)2 F7 in  fiber     U(1)3 F2 F1 Mordell-Weil rank ,  Mordell-­‐Weil   ,  only      (need  Jacobian  fibraQons)   •  Extremal  transiQons  in  fiber  =   :  worked  out   28   EllipQc  fibraQons  with  toric  ellipQc  fibers   [D.K.,  Mayorga  Peña,  Oehlmann,  Piragua,  Reuter]     F     Arrow:   F F F   in  fiber         F F     Polytopes  dual   F F F F     Sum  rule:   F F    of  gauge  group       of  poly  +  its  dual     F F F       F     Mordell-Weil rank •  Up  to   ,  Mordell-­‐Weil   ,  only      (need  Jacobian  fibraQons)   •  Extremal  transiQons  in  fiber  =   :  worked  out   16 SU(3) Z3 SU(3) x SU(2)2 x U(1) 13 (SU(4) x SU(2)2) rank of total gauge group 10 SU(3) x SU(2) 4 15 SU(2) x U(1) Z2 Z2 SU(3) x SU(2) x U(1) SU(2) 14 2 12 SU(2) x U(1) 11 SU(2) x U(1) SU(2) x U(1) SU(2) x U(1) U(1) 3 U(1) 28   ApplicaQon  of  toolbox   APPLICATION  3:  ENHANCING  U(1)2   29   Higgs-­‐TransiQons  in  F-­‐theory:  U(1)’s            GnA   EllipQc  fibraQons  with  higher  rank  Mordell-­‐Weil  group  crucial  for   understanding  the   compacQficaQons            Can  we  tune  complex  structure  to   to  non-­‐Abel     Rank  1  case  understood:   [Morrison,  Taylor]   Every  6D  F-­‐theory  with      comes   SU(2)  on  Riemann  surface  ⌃        g                adjoint                            V   EV                            U(1)   rk(MW)=0                                                        tuning                        m        oduli                      rk(MW)=1   Geometrically:  transiQon  of     30   Higgs-­‐TransiQons  in  F-­‐theory:  U(1)2            GnA   Enhancement  of  U(1)xU(1):   •  Reduce  MW-­‐rank  to  zero  by   E   [Cve8č,  D.K.,  Piragua,  Taylor]    types  of  possible   uf2 (u, v, w) +  Q,  R  with  origin  P   Y (ai v + bi w) = i=1 31   Higgs-­‐TransiQons  in  F-­‐theory:  U(1)2            GnA   Enhancement  of  U(1)xU(1):   •  Reduce  MW-­‐rank  to  zero  by   E   [Cve8č,  D.K.,  Piragua,  Taylor]    types  of  possible    Q,  R  with  origin  P   uf2 (u, v, w) + (a1 v + b1 w)2 (a3 v + b3 w) = •  rk(MW)=2            1:      P      Q                          0   31   Higgs-­‐TransiQons  in  F-­‐theory:  U(1)2            GnA   Enhancement  of  U(1)xU(1):   •  Reduce  MW-­‐rank  to  zero  by   E   [Cve8č,  D.K.,  Piragua,  Taylor]    types  of  possible   Q,  R  with  origin  P   uf2 (u, v, w) + (a1 v + b1 w)3 = •  rk(MW)=2            1:      P      Q                          0   •  rk(MW)=1            0:      P      R                          0   This  tuned  fibraQon  has build  in:     1.  U(1)xU(1)          SU(3):  set        i      =          1    ,  at    f    2    (0,                b    1  ,    a    1    )      =        0           in  E  at  P=[0,-­‐b1,a1]   2.  U(1)xU(1)          SU(2)xSU(2):  set    f2 (0, b1 , a1 ) =      i      =          0      in  B:          uf        2    (u,            v,        w)            =          0       3.  general  case   :  U(1)2          SU(3)xSU(2)2   31   Summary    of  ellipQc  fibraQons  with  Mordell-­‐Weil  group   •  Developed    to  analyze  these  models   •  6D  maher  spectrum  &  4D  Yukawas:   •  4D  chiraliQes  &  G4-­‐flux:  CY   &   •  Applied  tools  to     with  ell  fibers  as   •  hypersurface  in  dP2:  U(1)2,  also  with  SU(5)   in    Bl        3    (P        3    )    :  U(1)3    in  16  2d  toric  varieQes    can  be    into  GnA,    always  in   way   Outlook   •  ClassificaQon  of  n>3  U(1)’s     [Cve8č,  DK,  Piragua,  Peng  Song]:  work  in  progress   •  HeteroQc  dual  of  F-­‐theory  w/  U(1)’s   [Cve8č,  Grassi,  DK,  Piragua,  Song]:  work  in  progress   32   33   ... Het/F-­‐theory  duality:  see  Anderson’s  talk   •  back-­‐reacted     •  regions  with   on     Type  I     EffecQve  theories  of  F-­‐theory   Use  F-­‐theory  to  engineer  effecQve  theories:... F-­‐THEORY  COMPACTIFICATIONS     F-­‐theory  =  geometry/physics  dicQonary   F-­‐theory  specified  by                  of  ellipQc  fibraQon     co-­‐dim     one  sing  over    S             on. .. IllustraQon:  dP            2    -­‐ellipQc  fibraQons  with  two  raQonal  secQons   TOOLBOX  FOR  STUDYING  F-­‐THEORY   WITH  U(1)’S   17   ConstrucQon  of  C[Cve8č,   Y-­‐ellipQc   fi braQons