The Toric SO(10) F-theory Landscape

Transcription

1 The Toric SO(10) F-theory Landscape Paul-Konstantin Oehlmann DESY/Hamburg University In preperation arxiv:170x.xxxx in collaboration with: W. Buchmueller, M. Dierigl, F. Ruehle Virginia Tech,Blacksburg July 4th / 13

2 Outline 1 Motivation 2 Construction: Hypersurfaces and SO(10) Tops 3 Matter Charges and Multiplicities 4 Geometric Transitions and Collisions 5 Conclusion and Outlook 2 / 13

3 Motivation The power of F-theory lies within the ability, to geometrize possibly non-perturbative physicsinto the geometry of Torus E fiber over the physical n-dimensional base B n Dimensional Reduction Approach 1 B 1 : d = 8 Gauge Sector ADE singularities ADE Groups Mordell-Weil group Abelian gauge sector [F-theory ] Multi-Sections Discrete Groups 2 B 2 : d = 6 codimension 2 singularities Matter sector 3 B 3 : d = 4 codimension 3 singularities Yukawa couplings Can study IIB non-perturbative string compactifications Study Exceptional gauge/matter content Study more exotic singularities [Grassi,Halverson,Ruehle,Shaneson; Arras,Grassi, Weigand ] 3 / 13

4 Classifications of Gauge theories Stepwise Compactification Approach 1 B 1 : d = 8 Gauge Sector ADE singularities ADE Groups Mordell-Weil group Abelian gauge sector Multi-Sections Discrete Groups 2 B 2, d = 6 codim.2 singularities Matter 3 B 3, d = 4 codim.3 singularities Yukawa s Engineer SU(5) GUTs (+Abelian symmetries to constrain flavor sector) [Dudas,Palti,Borchmann,Weigand,Mayrover,Krause,Cvetic,Klevers,Piragua,Krippendorf,Ruehle,O.,Mayorga,Schaefer-Nameki,Lawrie,Wong] 4 / 13

5 Classifications of Gauge theories Stepwise Compactification Approach 1 B 1 : d = 8 Gauge Sector SU(5)-perturbative ADE singularities ADE Groups Mordell-Weil group Abelian gauge sector Multi-Sections Discrete Groups 2 B 2, d = 6 codim.2 singularities Matter 5, 10-plets perturb 3 B 3, d = 4 codim.3 singularities Yukawa s 5 H, 10 m 10 m non-perturb Engineer SU(5) GUTs (+Abelian symmetries to constrain flavor sector) [Dudas,Palti,Borchmann,Weigand,Mayrover,Krause,Cvetic,Klevers,Piragua,Krippendorf,Ruehle,O.,Mayorga,Schaefer-Nameki,Lawrie,Wong] 4 / 13

6 Classifications of Gauge theories Stepwise Compactification Approach 1 B 1 : d = 8 Gauge Sector SU(5)-perturbative ADE singularities ADE Groups Mordell-Weil group Abelian gauge sector Multi-Sections Discrete Groups 2 B 2, d = 6 codim.2 singularities Matter 5, 10-plets perturb 3 B 3, d = 4 codim.3 singularities Yukawa s 5 H, 10 m 10 m non-perturb Engineer SU(5) GUTs (+Abelian symmetries to constrain flavor sector) [Dudas,Palti,Borchmann,Weigand,Mayrover,Krause,Cvetic,Klevers,Piragua,Krippendorf,Ruehle,O.,Mayorga,Schaefer-Nameki,Lawrie,Wong] Engineer SO(10) GUTs (+Abelian symmetries to constrain flavor sector) Remember:One Standard Model Family fits into 16-plet 4 / 13

7 Classifications of Gauge theories Stepwise Compactification Approach 1 B 1 : d = 8 Gauge Sector SO(10)-perturbative ADE singularities ADE Groups Mordell-Weil group Abelian gauge sector Multi-Sections Discrete Groups 2 B 2, d = 6 codim.2 singularities Matter 10 H, 16 m -plets non-perturb 3 B 3, d = 4 codim.3 singularities Yukawa s 10 H, 16 m 16 m non-perturb Engineer SU(5) GUTs (+Abelian symmetries to constrain flavor sector) [Dudas,Palti,Borchmann,Weigand,Mayrover,Krause,Cvetic,Klevers,Piragua,Krippendorf,Ruehle,O.,Mayorga,Schaefer-Nameki,Lawrie,Wong] Engineer SO(10) GUTs (+Abelian symmetries to constrain flavor sector) Remember:One Standard Model Family fits into 16-plet Start with a very specific 6D SO(10) U(1) SUGRA [Buchmueller, Dierigl, Ruehler, Schweitzer 16] Step to 4D: Study GUT/SUSY-Breaking, Family Generation, Moduli-Stabilization Problem: Neither heterotic nor Morrison-Park U(1) model [Morrison,Park; Taylor et al.] Is it in the Swamp? 4 / 13

8 Toric Classification of SO(10) Theories Strategy to engineer SO(10) theories Start with some Fiber ambient space with a generic gauge group such as a U(1) [Braun,Grimm,Keitel,Klevers ] Enhance the ambient space with a top [Candelas,Font; Bouchard,Skarke] 5 / 13

9 Toric Classification of SO(10) Theories p =s 1 u 3 e s 2u 2 ve s 3uv 2 e s 4v 3 e s 5u 2 we 1 + s 6 uvwe 1 + s 7 v 2 we 1 + s 8 uw 2 + s 9 vw 2 Strategy to engineer SO(10) theories Start with some Fiber ambient space with a generic gauge group such as a U(1) [Braun,Grimm,Keitel,Klevers ] Enhance the ambient space with a top [Candelas,Font; Bouchard,Skarke] 5 / 13

10 Toric Classification of SO(10) Theories p = s 1 e1 2f 0f2 2f 4f 1 u 3 + s 2 e1 2f 0 2f 2 2f 3f 4 f1 2f 5u 2 v + s 3 e1 2f 0 2f 2f 3 f 1 uv 2 + s 4 e1 2f 0 3f 2f3 2f 1 2f 5v 3 +s 5 e 1 f 2 f 4 u 2 w + s 6 e 1 f 0 f 2 f 3 f 4 f 1 f 5 uvw + s 7 e 1 f 0 f 3 v 2 w + s 8 f 2 f 3 f4 2f 1f5 2uw 2 + s 9 f 3 f 4 f 5 vw 2 Strategy to engineer SO(10) theories Enhance the ambient space with a top [Candelas,Font; Bouchard,Skarke] A top is a half-polytope over the ambient of the generic fiber The z > 0 rays give ADE resolution divisors D fi that restrict to a base divisor ω S ADE (B) : ω = 0 D fi = π 1 (ω) 5 / 13

11 Toric Classification of SO(10) Theories Strategy to engineer SO(10) theories Enhance the ambient space with a top [Candelas,Font; Bouchard,Skarke] z=0, the generic fiber: U(1) theory z=1, the outer roots of SO(10) z=2, the roots with Dynkin label 2 Resolution divisors intersect as the affine Dynkin Diagram of SO(10) D fi P 1 j = Ĝ i,j SO(10) (1) 5 / 13

12 Matter and Yukawa couplings Matter loci Go to the (singular) Weierstrass form and find codimension two singularities: f =ω 2 (s 7 s 8 + O(w)) g =ω 3 (s 7 s 8 + O(w)) =ω 7 (s 7 s 8 s 9 (s 1 s 7 s 5 s 9 ) + O(w)) 6 / 13

13 Matter and Yukawa couplings Matter loci Go to the (singular) Weierstrass form and find codimension two singularities: f =ω 2 (s 7 s 8 + O(w)) g =ω 3 (s 7 s 8 + O(w)) =ω 7 (s 7 s 8 s 9 (s 1 s 7 s 5 s 9 ) + O(w)) ω = s 9 : SO(12) point of 10 (3/2) matter ω = s 7 : E 6 point of 16 (1/4) matter 6 / 13

14 Matter and Yukawa couplings Matter loci Go to the (singular) Weierstrass form and find codimension two singularities: f =ω 2 (s 7 s 8 + O(w)) g =ω 3 (s 7 s 8 + O(w)) =ω 7 (s 7 s 8 s 9 (s 1 s 7 s 5 s 9 ) + O(w)) Full global gauge group (SO(10) U(1))/Z 4 [see Ling Lin s talk] Full spectrum: 16 3/4 16 1/4 10 1/2 10 3/ / 13

15 Yukawa Points E 7 Yukawa couplings conventional E 7 type Yukawa s: 16 ( 1/4) 16 (3/4) 10 ( 1/2) non-kodaira at self intersection points: [Esole,Yau, Marsano,Schaefer-Nameki, Collinucci, Etxebarria ] Ê 7 type Yukawa s 16 (3/4) 16 (3/4) 10 ( 3/2) [Etxebarria s talk] SO(14) type Yukawa s 10 ( 3/2) 10 ( 3/2) 1 (3) 7 / 13

16 Yukawa Points E 7 Ê 7 Yukawa couplings conventional E 7 type Yukawa s: 16 ( 1/4) 16 (3/4) 10 ( 1/2) non-kodaira at self intersection points: [Esole,Yau, Marsano,Schaefer-Nameki, Collinucci, Etxebarria ] Ê 7 type Yukawa s 16 (3/4) 16 (3/4) 10 ( 3/2) [Etxebarria s talk] SO(14) type Yukawa s 10 ( 3/2) 10 ( 3/2) 1 (3) 7 / 13

17 Yukawa Points E 7 Ê 7 SO(14) Yukawa couplings conventional E 7 type Yukawa s: 16 ( 1/4) 16 (3/4) 10 ( 1/2) non-kodaira at self intersection points: [Esole,Yau, Marsano,Schaefer-Nameki, Collinucci, Etxebarria ] Ê 7 type Yukawa s 16 (3/4) 16 (3/4) 10 ( 3/2) [Etxebarria s talk] SO(14) type Yukawa s 10 ( 3/2) 10 ( 3/2) 1 (3) 7 / 13

18 Base Independent 6D Sugra We obtain the full 6D SUGRA spectrum, independent of the base: The full spectrum of generic models in polytopes F i i=1..16 have been constructed [Klevers,Mayorga,O,Piragua,Reuter] We are considering a toric SO(10) tuning of those Correct the singlet loci by the SO(10) tuning using resultant techniques 8 / 13

19 Base Independent 6D Sugra We obtain the full 6D SUGRA spectrum, independent of the base: The full spectrum of generic models in polytopes F i i=1..16 have been constructed [Klevers,Mayorga,O,Piragua,Reuter] We are considering a toric SO(10) tuning of those Correct the singlet loci by the SO(10) tuning using resultant techniques Check anomaly cancellation base independent 6D SUGRA spectrum fixed by K 1 b anticannonical class of the base S the divisor class of the SO(10) D A, D B two topological twists of fiber and base 8 / 13

20 Base Independent 6D Sugra We obtain the full 6D SUGRA spectrum, independent of the base: The full spectrum of generic models in polytopes F i i=1..16 have been constructed [Klevers,Mayorga,O,Piragua,Reuter] We are considering a toric SO(10) tuning of those Correct the singlet loci by the SO(10) tuning using resultant techniques Check anomaly cancellation base independent 6D SUGRA spectrum fixed by K 1 b anticannonical class of the base S the divisor class of the SO(10) D A, D B two topological twists of fiber and base Rep Multiplicity 16 3/4 (2K 1 b D A )S 10 3/2 D B S 10 1/2 (3K 1 b D B 2S)S 16 1/4 (D A S)S S(S K 2 b ) Rep Multiplicity 1 3 (K 1 b D A + D B )D B 6(K 1 1 b ) 2 + K 1 b ( 5D A + 4D B 2S) 2 +DA 2 + D A (2D B + S) D B (2D B + 5S) 12(K 1 1 b ) 2 + K 1 b (8D A D B 25S) 1 DB 2 4DA 2 + 6S 2 + D A (D B + 4S) 8 / 13

21 Results of the Classification Summary of the Classification 36 Inequivalent types of models that come with SO(10) and generic additional gauge factors: Additional Z n U(1) U(1) 2 U(1) 3 -Gauge Factors No. Models: Additional Non-Abelian Factors Super Conformal Matter No. Models: Careful choice of base/fibration: Tune away superconformal points and unwanted gauge groups 9 / 13

22 Transitions in theories An organisation principle Often theories are realized by transitions (toric blow-ups/downs): (un)higgs transitions: SO(10) [SU(2) U(1) 2 ] 10 / 13

23 Transitions in theories An organisation principle Often theories are realized by transitions (toric blow-ups/downs): (un)higgs transitions: SO(10) [SU(2) U(1) 2 ] Giving a VEV (1, 2) (1,0) SO(10) [U(1) 2 ] 10 / 13

24 Transitions in theories An organisation principle Often theories are realized by transitions (toric blow-ups/downs): (un)higgs transitions: SO(10) [SU(2) U(1) 2 ] Giving a VEV (1, 2) (1,0) SO(10) [U(1) 2 ] Colling Singularity: A blow-up with one fiber point in a face SO(10) [U(1) 2 ] + Super Conformal Matter [Bershadsky,Johanson 94] Singularities that can only be resolved by a blow-up in the Base 10 / 13

25 Transitions in theories Field Theory perspective: One can exactly track the Hypers, lost in the in transition: [Anderson,Gray,Rhaguram,Taylor] 2n 1 (0,0) + n 16 (1/2, 1/2) + n 10 (1/4,1/2) + n 1 (1,1) = 29 n These hypers are i.e. missing in the irreducible gravitational anomaly Instead we find exactly n super conformal matter points 11 / 13

26 Transitions in theories Change in Kaehler moduli: h 1,1 (Y 3 ) = l( ) 4 Γ=edge l (Γ) } {{ } h 1,1 toric =Cartans+Tensors 2 l(l ) number of points in (interior) of edges Γ, Faces Θ [Batyrev] 12 / 13

27 Transitions in theories Change in Kaehler moduli: h 1,1 (Y 3 ) = l( ) 4 Γ=edge } {{ } h 1,1 toric =Cartans+Tensors 2 l (Γ) + Θ=face l (Θ)l l (Θ o ) } {{ } h 1,1 nt = FrozenTensors l(l ) number of points in (interior) of edges Γ, Faces Θ [Batyrev] h 1,1 nt directly corresponds to superconformal matter/missing tensors 12 / 13

28 Summary and Conclusion We classify SO(10) models in F-theory realized as hypersurfaces of toric tops base independently 36 types of models Computed all Matter Charges/Representations 6D Multiplicities and Anomaly Coefficients Models often related by transitions: Higgs transitions where the SO(10) is an observer Transition with (4,6) points / super conformal matter Toric divisors become a point in a face Superconformal points counted by non-toric Kaehler deformations 13 / 13

29 Summary and Conclusion We classify SO(10) models in F-theory realized as hypersurfaces of toric tops base independently 36 types of models Computed all Matter Charges/Representations 6D Multiplicities and Anomaly Coefficients Models often related by transitions: Higgs transitions where the SO(10) is an observer Transition with (4,6) points / super conformal matter Toric divisors become a point in a face Superconformal points counted by non-toric Kaehler deformations Happy Independence Day and Thank you very much 13 / 13