Mordell-Weil group and massless matter in F-theory arXiv: 1405.3656 - PowerPoint PPT Presentation

Mordell-Weil group and massless matter in F-theory • arXiv: 1405.3656 with O. Till , C. Mayrhofer , D. Morrison • arXiv: 1406.6071 with L. Lin arXiv: 1307.2902 with J. Borchmann , E. Palti and C. Mayrhofer • arXiv: 1402.5144 with M. Bies , C. Mayrhofer and C. Pehle Timo Weigand Institute for Theoretical Physics, Heidelberg University Strings 2014, Princeton, 06/25/2014 – p.1

The beauty of F-theory F − theory Particle physics ← → Geometry Fruitful connections to particle physics via F-theory GUTs have motivated a lot of recent progress in understanding of 4-dimensional F-theory vacua Two examples of recent advances: 1) The geometry of elliptic fibrations with extra sections • determines the abelian sector of gauge theories • affects the structure of Yukawa selection rules • is responsible for absence of proton decay operators in F-theory GUTs 2) The C 3 -background • determines matter multiplicities • is a crucial input in 4-dimensional F-theory Strings 2014, Princeton, 06/25/2014 – p.2

Outline I.) The Mordell-Weil group in F-theory II.) Non-torsional sections & applications: Standard Models in F-theory (’F-theory non-GUTs’) [Ling,TW’14] III.) Torsional sections [Mayrhofer,Morrison,Till,TW’14] IV.) Gauge data/ C 3 backgrounds and applications [Bies,Mayrhofer,Pehle,TW’14] Strings 2014, Princeton, 06/25/2014 – p.3

Mordell-Weil group 1) Elliptic curve : E = C / Λ τ τ+1 ↔ addition of points b Rational points : a 1 • have Q -rational coordinates ( x, y, z ) in Weierstrass model y 2 = x 3 + fxz 4 + gz 6 , [ x : y : z ] ∈ P 2 2 , 3 , 1 • form an abelian group under addition = Mordell-Weil group E E = Z r ⊕ Z k 1 ⊕ · · · ⊕ Z k n 2) Elliptic fibration : π : Y → B elliptic fiber degenerate ell. fiber Rational section σ : B ∋ b �→ σ ( b ) = [ x ( b ) : y ( b ) : z ( b )] CY four− fold Y base three− • σ ( b ) is a K -rational point in fiber brane S fold B • degenerations in codimension allowed Strings 2014, Princeton, 06/25/2014 – p.4

Mordell-Weil group Mordell-Weil group E ( K ) = group of rational sections • zero-element = zero-section σ 0 : b → [1 : 1 : 0] in y 2 = x 3 + fxz 4 + gz 6 • group law = fiberwise addition Z r ⊕ Z k 1 ⊕ · · · ⊕ Z k n E ( K ) = ���� � �� � free part torsion part Physical significance: • Free part ↔ U (1) gauge symmetries [Morrison,Vafa’96],[Klemm,Mayr,Vafa’98],.. . • Torsion part ↔ Global structure of non-ab. gauge groups ( π 1 ( G ) ) [Aspinwall,Morrison’98], [Aspinwall,Katz,Morrison’00], . . . , [Mayrhofer,Till,Morrison,TW’14] Systematic recent study of U(1)s via rational sections: � extra selection rules, e.g. crucial in F-theory GUTs � window to gauge fluxes and chirality � general interest in any theory with massless U(1)s (landscape studies, . . . ) Antoniadis,Anderson,Bizet,Borchmann,Braun,Braun,Choi,Collinucci,Cvetiˇ c,Etxebarria,Grassi,Grimm, Hayashi, Keitel,Klevers,K¨ untzler,Krippendorf,Oehlmann,Klemm,Leontaris,Lopes,Mayrhofer, Mayorga, Morrison, Park,Palti,Piragua,R¨ uhle,S-Nameki,Song,Valandro,Taylor,TW,.. . Strings 2014, Princeton, 06/25/2014 – p.5

Mordell-Weil group Divisors on elliptic 4-fold ˆ Y 4 → B : • zero-section Z • pullback from base π − 1 ( D b ) • resolution divisors F m , m = 1 , . . . , rk( G ) • rational sections S i Shioda map → NS ( ˆ • homomorphism ϕ : Y 4 ) ⊗ Q E ( K ) [Shioda’89] � �� � � �� � group of sections group of divisors ϕ ( S − Z ) = S − Z − π − 1 ( δ ) + � l i F i , l i ∈ Q • transversality Y 4 [ ϕ ( S − Z )] ∧ [ X ] ∧ [ π − 1 ω 4 ] = 0 X ∈ { Z, F l , π − 1 ( D b ) } � ˆ Strings 2014, Princeton, 06/25/2014 – p.6

Non-torsional sections • [ S ] ≡ [ ϕ ( S − Z )] is non-trivial in H 1 , 1 ( ˆ Y 4 ) • [ S ] serves as the generator of U (1) gauge group C 3 = A ∧ [ S ] A : U (1) gauge potential Engineering extra sections via restriction of complex structure of fibration • Non-generic Weierstrass models, simplest example: [Grimm,TW’10] y 2 − a 1 x y z − a 3 y z 3 = x 3 + a 2 x 2 z 2 + a 4 x z 4 + ✟✟ ✟ a 6 z 6 , [ x : y : z ] ∈ P 2 , 3 , 1 zero-section: [1 : 1 : 0] extra section: [0 : 0 : 1] Generalisations in [Mayrhofer,Palti,TW’12] • Systematic approach via different fiber representations � cf. talk by D. Klevers • rank 1: P 1 , 1 , 2 [4] [Morrison,Park’12] • rank 2: P 2 [3] [Borchmann,Mayrhofer,Palti,TW][Cvetiˇ c,(Grassi),Klevers,Piragua]’13 • rank 3: complete intersections [Cvetiˇ c,Klevers,Piragua,Song’13] • toric hypersurfaces [Braun,Grimm,Keitel’13][Grassi,Perduca’12] Strings 2014, Princeton, 06/25/2014 – p.7

Standard Model applications Explore U(1) selection rules in F-theory Standard Models [Ling,TW’14] SU (3) × SU (2) × U (1) 1 × U (1) 2 ↔ U (1) Y = a U (1) 1 + b U (1) 2 • U (1) 1 × U (1) 2 [Borchmann,Mayrhofer,Palti,TW][Cvetiˇ c,(Grassi),Klevers,Piragua]’13 v w ( c 1 w s 1 + c 2 v s 0 ) + u ( b 0 v 2 s 02 + b 1 v w s 0 s 1 + b 2 w 2 s 12 ) + 0 = +u 2 ( d 0 v s 02 s 1 + d 1 w s 0 s 12 + d 2 u s 02 s 12 ) • SU (2) ↔ W 2 = { w 2 = 0 } SU (3) ↔ W 3 = { w 3 = 0 } g m = g m,k,l w k 2 w l g m ∈ { b i , c j , d k } , 3 × 3 ’toric enhancements’ 3 • fully resolved fibrations over suitable base B Types of matter representations: 5 × ( 3 , 1 ) ↔ ( u c R , e c ( 3 , 2 ) ↔ Q L 3 × ( 1 , 2 ) ↔ ( H u , H d , L ) R ) 6 × ( 1 , 1 ) ↔ ν c R or extra singlet extensions Rich pattern of Yukawa couplings ↔ MSSM + extra dim-4 and dim-5 couplings Strings 2014, Princeton, 06/25/2014 – p.8

Standard Model identifications [Ling,TW’14] Matter spectrum Baryon- and Lepton number violation α : ( 2 I 1 , 3 A 3 ) , ( 2 I 2 , 3 A 2 ) , ( 2 I 3 , 3 A 5 ) ; β : 3 A 4 3 A 3 3 A 2 , possibility no. 5 3 A 4 3 A 5 3 A 5 , 3 A 1 3 A 3 3 A 5 ; δ : 1 (5) 1 (6) 1 (6) , 1 (5) 1 (6) 1 (6) ; 2 , 2 I a = 1 , b = 0; ( H u , H d ) = ( 2 I 1 ) γ : 2 I 1 2 I 2 1 (2) , 2 I 1 2 I 3 1 (1) , 2 I 2 2 I 2 1 (3) , 2 I 2 2 I 3 1 (4) , R ) : ( 3 A 4 , 3 A heavy ( u c R , d c 3 ) 2 I 3 2 I 3 1 (2) ; λ 1 : 2 I 1 ; λ 3 : � ; λ 6 : − ; λ 8 : 1 (2) ; λ 10 : ( 3 A 3 ) ∗ ; R : 3 A R : 3 A 2 , 3 A light u c 1 ; light d c 5 λ 4 : ( 3 A 4 , 1 (2)) , ( 3 A 1 , 1 (1)) ; λ 5 : ( 2 I 2 , 2 I 2) ; λ 9 : ( 3 A 4 , ( 2 I 2) ∗ ) , ( 3 A 1 , ( 2 I 3) ∗ ) ; heavy generations of ( L, ν c R , e c R ) : λ 7 : 3 A 4 ( 3 A 3 ) ∗ 1 (2) , 3 A 4 ( 3 A 2 ) ∗ 1 (3) , ( 2 I 1 , 1 (5) , − ) , ( 2 I 2 , − , 1 (2) ) , ( 2 I 3 , 1 (6) , 1 (1) ) 3 A 4 ( 3 A 5 ) ∗ 1 (4) , 3 A 1 ( 3 A 3 ) ∗ 1 (1) , 3 A 1 ( 3 A 2 ) ∗ 1 (4) , 3 A 1 ( 3 A 5 ) ∗ 1 (2) ; R : 1 (5) , 1 (6) ; light e c R : 1 (3) , 1 (4) light ν c λ 2 : 3 A 4 3 A 4 3 A 3 1 (3) , 3 A 4 3 A 4 3 A 2 1 (2) , 3 A 4 3 A 4 3 A 5 1 (4) , 3 A 4 3 A 1 3 A 3 1 (4) , 3 A 4 3 A 1 3 A 2 1 (1) , 1 µ : 1 (5) 3 A 4 3 A 1 3 A 5 1 (2) , 3 A 1 3 A 1 3 A 3 1 (2) , 3 A 1 3 A 1 3 A 5 1 (1) Red choice: no dim-4 ✘✘✘ ✘ R-parity, no Q Q Q L , u c R u c R d c R e c R Next steps include: • Analysis of chiral spectrum via gauge fluxes (see later) • Analyse distinctive non-pert. features Strings 2014, Princeton, 06/25/2014 – p.9

Torsional sections [Mayrhofer,Morrison,Till,TW’14] • If R is Z k -torsional section: k ( R − Z ) = 0 • Shioda map is homomorphism: 0 = ϕ ( k ( R − Z )) = kϕ ( R − Z ) ∈ NS ( ˆ Y 4 ) ⊗ Q • Since NS ( ˆ a i Y 4 ) ⊗ Q is torsion free: ϕ ( R − Z ) = R − Z − π − 1 ( δ ) + � k F i is trivial i � Ξ k ≡ R − Z − π − 1 ( δ ) = − 1 i a i F i , a i ∈ Z is integer divisor k Relation to group theory: • Lie algebra g ↔ coroot lattice Q ∨ = � F i � Z F i : resol. divisors ↔ codim.-1 • Representation content ↔ weight lattice Λ ↔ codimension-2 fibers • Integer pairing Λ ∨ × Λ → Z gives weights coweight lattice Λ ∨ ⊇ Q ∨ • If Λ ∨ = Q ∨ = � F i � Z → gauge group G 0 : π 1 ( G 0 ) ≈ Λ ∨ Q ∨ = ∅ • Integer Ξ k = − 1 i a i F i → ’k-fractional refinement’ of Λ ∨ � k • dual weight lattice becomes coarser → smaller representation content gauge group G = G 0 / Z k with π 1 ( G ) = Z k Strings 2014, Princeton, 06/25/2014 – p.10

Fibrations with MW torsion Hypersurface fibrations with MW torsion • Z k for k = 2 , 3 , 4 , 5 , 6 , Z 2 ⊕ Z n with n = 2 , 4 and Z 3 ⊕ Z 3 [Morrison,Aspinwall’98] • Toric models: Z 2 , Z 3 , Z ⊕ Z 2 [Braun,Grimm,Keitel’13] • Previous analysis from perspective of SL (2 , Z ) -monodromies: [Berglund,Klemm,Mayr,Theisen’98] • Exemplification of general structure for toric models [Mayrhofer,Morrison,Till,TW’14] General pattern • Codimension-one fibers: Z k -torsion automatically produces non-ab. gauge group factor G 0 / Z k • Codimension-two fibers: matter spectrum greatly restricted due to reduced center of G • Codimension-three fibers: no further selection rules at the level of Yukawa couplings Strings 2014, Princeton, 06/25/2014 – p.11