F-Theory duals of heterotic K3 orbifolds
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- Emilia Stefaniak
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1 F-Theory duals of heterotic K3 orbifolds Fabian Ruehle Deutsches Elektronensynchrotron DESY Hamburg String Pheno /08/2014 Based on Ludeling, Ruehle: [ ]
2 T 4 /Z 2 Orbifold Details θ : (z 1, z 2 ) (e 2πi/2 z 1, e 2πi/2 z 2 ) = ( z 1, z 2 ) Singularities: 4 4 = 16 Z 2 Gauge group: E 7 SU(2) (E 8 ) Spectrum: [(56, 2) + 4(1, 1)] U + [8(56, 1) + 32(1, 2)] T Note: (56, 1), (1, 2) pseudo-real half-hypers Complex structures τ1,2, radii b 1,2 of tori unfixed [cf. talk by Vaudrevange] (spectrums from [Honecker,Trapletti]) Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 1
3 T 4 /Z 3 Orbifold Details θ : (z 1, z 2 ) (e 2πi/3 z 1, e 2πi/3 z 2 ) Singularities: 3 3 = 9 Z 3 Gauge group: E 7 U(1) (E 8 ) Spectrum: [(56) 1 + (1) 2 + 2(1) 0 ] U + [9(56) + 63(1)] T Note: States differ by U(1) charges, all full hypers CS fixed by rotation to τ 1,2 = e 2πi/3, radii b 1,2 unfixed Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 2
4 T 4 /Z 4 Orbifold Details θ : (z 1, z 2 ) (e 2πi/4 z 1, e 2πi/4 z 2 ) = (iz 1, iz 2 ) Singularities: 2 Z 4, 1 Z 2 per T 2 Gauge group: E 7 U(1) (E 8 ) Spectrum: [(56) 1 + 2(1) 0 ] U + [9(56) + 64(1)] T Note: States differ by U(1) charges, 5 (56) 0 are 10 half-hypers CS fixed by rotation to τ 1,2 = i, radii b 1,2 unfixed Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 3
5 T 4 /Z 6 Orbifold Details θ : (z 1, z 2 ) (e 2πi/6 z 1, e 2πi/6 z 2 ) Singularities: 1 Z 6, 1 Z 3, 1 Z 2 per T 2 Gauge group: E 7 U(1) (E 8 ) Spectrum: [(56) 1 + 2(1) 0 ] U + [9(56) + 64(1)] T Note: States differ by U(1) charges, 3 (56) 0 are 6 half-hypers CS fixed by rotation to τ 1,2 = e 2πi/3, radii b 1,2 unfixed Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 4
6 Introduction to F-theory Constraints for heterotic duality F-theory: Introduce extra torus whose CS encodes varying Type II axio-dilaton CY 3-fold [Vafa] Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 5
7 Introduction to F-theory Constraints for heterotic duality F-theory: Introduce extra torus whose CS encodes varying Type II axio-dilaton CY 3-fold [Vafa] F-theories with (perturbative) heterotic dual have 1 tensor multiplet dilaton Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 5
8 Introduction to F-theory Constraints for heterotic duality F-theory: Introduce extra torus whose CS encodes varying Type II axio-dilaton CY 3-fold [Vafa] F-theories with (perturbative) heterotic dual have 1 tensor multiplet dilaton For duality need special fibration structure [Morrison,Vafa] elliptic fibration over base B 2 K3 fibration over P 1 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 5
9 Introduction to F-theory Constraints for heterotic duality F-theory: Introduce extra torus whose CS encodes varying Type II axio-dilaton CY 3-fold [Vafa] F-theories with (perturbative) heterotic dual have 1 tensor multiplet dilaton For duality need special fibration structure [Morrison,Vafa] elliptic fibration over base B 2 K3 fibration over P 1 Take base Hirzebruch surface F N (P 1 fibration over P 1 ) Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 5
10 Introduction to F-theory Fiber K3 CY threefold Het. K3 Base B = N Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 5
11 Introduction to F-theory Constraints for heterotic duality F-theory: Introduce extra torus whose CS encodes varying Type II axio-dilaton CY 3-fold [Vafa] F-theories with heterotic dual have 1 tensor multiplet (dilaton) For duality need special fibration structure [Morrison,Vafa] elliptic fibration over base B 2 K3 fibration over P 1 Take base Hirzebruch surface F N (P 1 fibration over P 1 ) Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 5
12 Introduction to F-theory Constraints for heterotic duality F-theory: Introduce extra torus whose CS encodes varying Type II axio-dilaton CY 3-fold [Vafa] F-theories with heterotic dual have 1 tensor multiplet (dilaton) For duality need special fibration structure [Morrison,Vafa] elliptic fibration over base B 2 K3 fibration over P 1 Take base Hirzebruch surface F N (P 1 fibration over P 1 ) Gauge instantons embedded as (12 + N, 12 N) into E 8 E 8 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 5
13 Introduction to F-theory Constraints for heterotic duality F-theory: Introduce extra torus whose CS encodes varying Type II axio-dilaton CY 3-fold [Vafa] F-theories with heterotic dual have 1 tensor multiplet (dilaton) For duality need special fibration structure [Morrison,Vafa] elliptic fibration over base B 2 K3 fibration over P 1 Take base Hirzebruch surface F N (P 1 fibration over P 1 ) Gauge instantons embedded as (12 + N, 12 N) into E 8 E 8 Second E 8 unbroken in standard embedding N = 12 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 5
14 Introduction to F-theory Weierstrass description of elliptic fibration Equation: y 2 = x 3 + fxz 4 + gz 6 (f, g sections of base F 12 ) Discriminant: = 4f g 2 [cf. talks Palti, Cvetic, Schafer-Nameki] j-function: j(τ) = f 3 / j(i) = 1, j(e 2πi/3 ) = 0 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 6
15 Introduction to F-theory Weierstrass description of elliptic fibration Equation: y 2 = x 3 + fxz 4 + gz 6 (f, g sections of base F 12 ) Discriminant: = 4f g 2 [cf. talks Palti, Cvetic, Schafer-Nameki] j-function: j(τ) = f 3 / j(i) = 1, j(e 2πi/3 ) = 0 Scaling s t u v x y z f g λ µ ν Expand f, g in u, v, s, t f = c 56 v 8 + c 44 uv 7 + c 32 u 2 v 6 + c 20 u 3 v 5 + c 8 u 4 v 4 g = d 84 v 12 + d 72 uv 11 + d 60 u 2 v 10 + d 48 u 3 v 9 + d 36 u 4 v 8 + d 24 u 5 v 7 + d 12 u 6 v 6 + d 0 u 7 v 5 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 6
16 T 4 /Z 2 Orbifold case Ansatz Heterotic side: E 7 SU(2) E 8 Spectrum: (56, 2) + 8(56, 1) + 32(1, 2) + 4(1, 1) Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 7
17 T 4 /Z 2 Orbifold case Ansatz Heterotic side: E 7 SU(2) E 8 Spectrum: (56, 2) + 8(56, 1) + 32(1, 2) + 4(1, 1) F-theory side Restrict f, g and relate c i, d j s.t. III, I 2, II appear: f = u 3 v 4 (...), g = u 5 v 5 (...), = u 9 v 10 (u + p 12 v) 2 red Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 7
18 T 4 /Z 2 Orbifold case Ansatz Heterotic side: E 7 SU(2) E 8 Spectrum: (56, 2) + 8(56, 1) + 32(1, 2) + 4(1, 1) F-theory side Restrict f, g and relate c i, d j s.t. III, I 2, II appear: f = u 3 v 4 (...), g = u 5 v 5 (...), = u 9 v 10 (u + p 12 v) 2 red Problems Fractional instantons: 24/16 = 3/2 per Z 2 fixed point Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 7
19 T 4 /Z 2 Orbifold case Ansatz Heterotic side: E 7 SU(2) E 8 Spectrum: (56, 2) + 8(56, 1) + 32(1, 2) + 4(1, 1) F-theory side Restrict f, g and relate c i, d j s.t. III, I 2, II appear: f = u 3 v 4 (...), g = u 5 v 5 (...), = u 9 v 10 (u + p 12 v) 2 red Problems Fractional instantons: 24/16 = 3/2 per Z 2 fixed point GG not broken by instantons Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 7
20 T 4 /Z 2 Orbifold case Ansatz Heterotic side: E 7 SU(2) E 8 Spectrum: (56, 2) + 8(56, 1) + 32(1, 2) + 4(1, 1) F-theory side Restrict f, g and relate c i, d j s.t. III, I 2, II appear: f = u 3 v 4 (...), g = u 5 v 5 (...), = u 9 v 10 (u + p 12 v) 2 red Problems Fractional instantons: 24/16 = 3/2 per Z 2 fixed point GG not broken by instantons Both (56, 1), (1, 2) at fixed points too singular at codim 2 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 7
21 T 4 /Z 2 Orbifold case Ansatz Heterotic side: E 7 SU(2) E 8 Spectrum: (56, 2) + 8(56, 1) + 32(1, 2) + 4(1, 1) F-theory side Restrict f, g and relate c i, d j s.t. III, I 2, II appear: f = u 3 v 4 (...), g = u 5 v 5 (...), = u 9 v 10 (u + p 12 v) 2 red Problems Fractional instantons: 24/16 = 3/2 per Z 2 fixed point GG not broken by instantons Both (56, 1), (1, 2) at fixed points too singular at codim 2 {u = 0} {u + p 12 v = 0} expect 12 (56, 2), not 1 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 7
22 T 4 /Z 2 Orbifold case Ansatz Heterotic side: E 7 SU(2) E 8 Spectrum: (56, 2) + 8(56, 1) + 32(1, 2) + 4(1, 1) F-theory side Restrict f, g and relate c i, d j s.t. III, I 2, II appear: f = u 3 v 4 (...), g = u 5 v 5 (...), = u 9 v 10 (u + p 12 v) 2 red Problems Fractional instantons: 24/16 = 3/2 per Z 2 fixed point GG not broken by instantons Both (56, 1), (1, 2) at fixed points too singular at codim 2 {u = 0} {u + p 12 v = 0} expect 12 (56, 2), not 1 { red = 0} {u + p 12 v = 0} expect multiples of 12 for hypers (1, 2), not 32 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 7
23 Duals for T 4 /Z 2 Orbifold Alternative way of looking at the orbifold geometry Smooth fiber torus in the bulk of the base (away from singularities) Four fiber singularities over each base singularity [Braun,Ebert,Hebecker,Valandro; Buchmüller,Louis,Schmidt,Valandro] Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 8
24 Duals for T 4 /Z 2 Orbifold Idea Find Weierstrass description of heterotic model w/ base P 1 (s, t) and fiber T 2 (x, y, z) 1 Pick one section (i.e. one pillow corner) Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 9
25 Duals for T 4 /Z 2 Orbifold Idea Find Weierstrass description of heterotic model w/ base P 1 (s, t) and fiber T 2 (x, y, z) 1 Pick one section (i.e. one pillow corner) 2 Blow up fiber singularity it hits (replace Z 2 singularity w/ P 1 ) 3 Blow down other finite fiber component (original fiber pillow) Same method also used in [Braun,Ebert,Hebecker,Valandro] Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 9
26 Duals for T 4 /Z 2 Orbifold Results for heterotic Weierstrass Get Weierstrass model with four D 4 singularities Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 10
27 Duals for T 4 /Z 2 Orbifold Results for heterotic Weierstrass Get Weierstrass model with four D 4 singularities Heterotic Weierstrass: y 2 = x 3 + f H 8 xz4 + g H 12 z6 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 10
28 Duals for T 4 /Z 2 Orbifold Results for heterotic Weierstrass Get Weierstrass model with four D 4 singularities Heterotic Weierstrass: y 2 = x 3 + f8 Hxz4 + g12 H z6 Need vanishing (f8 H, g 12 H, H 24 ) = (2, 3, 4 points f8 H = αp2 4, g 12 H = βp3 4, H 24 = (α3 + β 2 )p4 6 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 10
29 Duals for T 4 /Z 2 Orbifold Results for heterotic Weierstrass Get Weierstrass model with four D 4 singularities Heterotic Weierstrass: y 2 = x 3 + f8 Hxz4 + g12 H z6 Need vanishing (f8 H, g 12 H, H 24 ) = (2, 3, 4 points f8 H = αp2 4, g 12 H = βp3 4, H 24 = (α3 + β 2 )p4 6 j(τ) = (f8 H)3 / H 24 = α3 /(α 3 + β 2 ) CS τ not fixed Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 10
30 Duals for T 4 /Z 2 Orbifold Results for heterotic Weierstrass Get Weierstrass model with four D 4 singularities Heterotic Weierstrass: y 2 = x 3 + f8 Hxz4 + g12 H z6 Need vanishing (f8 H, g 12 H, H 24 ) = (2, 3, 4 points f8 H = αp2 4, g 12 H = βp3 4, H 24 = (α3 + β 2 )p4 6 j(τ) = (f8 H)3 / H 24 = α3 /(α 3 + β 2 ) CS τ not fixed Results for F-theory Weierstrass c 8 = αp 2 4, d 12 = βp 3 4 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 10
31 Duals for T 4 /Z 2 Orbifold Results for heterotic Weierstrass Get Weierstrass model with four D 4 singularities Heterotic Weierstrass: y 2 = x 3 + f8 Hxz4 + g12 H z6 Need vanishing (f8 H, g 12 H, H 24 ) = (2, 3, 4 points f8 H = αp2 4, g 12 H = βp3 4, H 24 = (α3 + β 2 )p4 6 j(τ) = (f8 H)3 / H 24 = α3 /(α 3 + β 2 ) CS τ not fixed Results for F-theory Weierstrass c 8 = αp 2 4, d 12 = βp 3 4 d 24 Instanton position d 24 = γp 6 4 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 10
32 Duals for T 4 /Z 2 Orbifold Results for heterotic Weierstrass Get Weierstrass model with four D 4 singularities Heterotic Weierstrass: y 2 = x 3 + f8 Hxz4 + g12 H z6 Need vanishing (f8 H, g 12 H, H 24 ) = (2, 3, 4 points f8 H = αp2 4, g 12 H = βp3 4, H 24 = (α3 + β 2 )p4 6 j(τ) = (f8 H)3 / H 24 = α3 /(α 3 + β 2 ) CS τ not fixed Results for F-theory Weierstrass c 8 = αp 2 4, d 12 = βp 3 4 d 24 Instanton position d 24 = γp 6 4 Extra I 2 locus c 20 = κp 5 4 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 10
33 Duals for T 4 /Z 2 Orbifold Results for heterotic Weierstrass Get Weierstrass model with four D 4 singularities Heterotic Weierstrass: y 2 = x 3 + f8 Hxz4 + g12 H z6 Need vanishing (f8 H, g 12 H, H 24 ) = (2, 3, 4 points f8 H = αp2 4, g 12 H = βp3 4, H 24 = (α3 + β 2 )p4 6 j(τ) = (f8 H)3 / H 24 = α3 /(α 3 + β 2 ) CS τ not fixed Results for F-theory Weierstrass c 8 = αp 2 4, d 12 = βp 3 4 d 24 Instanton position d 24 = γp 6 4 Extra I 2 locus c 20 = κp4 5 Numerical coefficients α, β, γ, κ related amongst each other to lead to factorization Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 10
34 Duals for T 4 /Z 2 Orbifold Spectrum E 8 at v = 0, E 7 at u = 0, I 2 at (u + 6p 3 4 v) = 0, I 1 at (...) = 0 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 11
35 Duals for T 4 /Z 2 Orbifold Spectrum E 8 at v = 0, E 7 at u = 0, I 2 at (u + 6p 3 4 v) = 0, I 1 at (...) = 0 Nothing intersects v =0, everything else u =p 4 =0 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 11
36 Duals for T 4 /Z 2 Orbifold Spectrum E 8 at v = 0, E 7 at u = 0, I 2 at (u + 6p4 3v) = 0, I 1 at (...) = 0 Nothing intersects v =0, everything else u =p 4 =0 Quantization in multiples of 12 broken to multiples of 4 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 11
37 Duals for T 4 /Z 2 Orbifold Spectrum E 8 at v = 0, E 7 at u = 0, I 2 at (u + 6p 3 4 v) = 0, I 1 at (...) = 0 Nothing intersects v =0, everything else u =p 4 =0 Quantization in multiples of 12 broken to multiples of 4 Get (56) s by deforming I 2 away and matching w/ SE 4 half-hypers at u = p 4 = 0 (56, 2) 16 half-hypers at u + 6p4 3 v = (...) = 0 (56, 1) Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 11
38 Duals for T 4 /Z 2 Orbifold Spectrum E 8 at v = 0, E 7 at u = 0, I 2 at (u + 6p 3 4 v) = 0, I 1 at (...) = 0 Nothing intersects v =0, everything else u =p 4 =0 Quantization in multiples of 12 broken to multiples of 4 Get (56) s by deforming I 2 away and matching w/ SE 4 half-hypers at u = p 4 = 0 (56, 2) 16 half-hypers at u + 6p4 3 v = (...) = 0 (56, 1) Get (1) s from parameters in the Weierstrass equation In total 9: 5 from p4 and α, β, γ, κ 5 are related four singlets Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 11
39 Duals for T 4 /Z 2 Orbifold Spectrum E 8 at v = 0, E 7 at u = 0, I 2 at (u + 6p 3 4 v) = 0, I 1 at (...) = 0 Nothing intersects v =0, everything else u =p 4 =0 Quantization in multiples of 12 broken to multiples of 4 Get (56) s by deforming I 2 away and matching w/ SE 4 half-hypers at u = p 4 = 0 (56, 2) 16 half-hypers at u + 6p4 3 v = (...) = 0 (56, 1) Get (1) s from parameters in the Weierstrass equation In total 9: 5 from p4 and α, β, γ, κ 5 are related four singlets Get (2) s from parameters that destroy I 2 locus smooth SE 69 overall in c20, c 8, d 24, d 12, d 0 Subtract 1 scaling, 4 singlets preserving I2 64 half-hyper (2) Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 11
40 Duals for T 4 /Z 3 Orbifold Results for heterotic Weierstrass Get Weierstrass model with three E 6 singularities Heterotic Weierstrass: y 2 = x 3 + f8 Hxz4 + g12 H z6 Need vanishing (f8 H, g 12 H, H 24 ) = ( 3, 4, 8) f8 H 0, g 12 H = βp4 3, H 24 = (β2 )p3 8 j(τ) = (f8 H)3 / H 24 = 0 CS fixed to τ = e2πi/3 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 12
41 Duals for T 4 /Z 4 Orbifold Results for heterotic Weierstrass Get Weierstrass model with two E 7, one D 4 singularities Heterotic Weierstrass: y 2 = x 3 + f8 Hxz4 + g12 H z6 At E 7 need vanishing (f8 H, g 12 H, H 24 ) = (3, 5, 9) At D 4 need vanishing (f8 H, g 12 H, H 24 ) = (2, 3, 6) Combined vanishing (8, 13, 24) f8 H = αp3 1 q3 1 r 1 2, g 12 H 0 j(τ)=(f8 H)3 / H 24 =(f 8 H)3 /(f8 H)3 =1 CS fixed to τ = i Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 13
42 Duals for T 6 /Z 6 Orbifold Results for heterotic Weierstrass Get Weierstrass model with one E 8, E 6, D 4 singularities Heterotic Weierstrass: y 2 = x 3 + f8 Hxz4 + g12 H z6 At E 8 (f8 H, g 12 H, H 24 ) = ( 4, 5, 10) At E 6 (f H 8, g H 12, H 24 ) = (3, 4, 8) At D 4 (f8 H, g 12 H, H 24 ) = ( 2, 3, 6) Combined vanishing ( 9, 12, 24) f8 H 0, g 12 H = βp5 1 q4 1 r 1 3 j(τ) = (f8 H)3 / H 24 = 0 CS fixed to τ = e2πi/3 Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 14
43 General gauge sector with different instanton embedding Construction Embedding instantons as (12 + N, 12 N) F N Transition from F N to F N±1 blowup/blowdown in base Base Blowup introduces extra tensor multiplet whose scalar component encodes NS5 brane position in S 1 /Z 2 in the Hořava Witten theory [Hořava,Witten] NS5 brane leaves one E 8 brane, travels through bulk, recombines with other E 8 brane [Seiberg,Witten;Morrison,Vafa] Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 15
44 Conclusion
45 Conclusion Approaches to duality Problem with usual approach in singular case: Fractional instantons, unbroken GG, FPs Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 16
46 Conclusion Approaches to duality Problem with usual approach in singular case: Fractional instantons, unbroken GG, FPs Finding the duals Blow up corner, blow down pillow Weierstrass polynomial coefficients from het. side Fixes complex structure of tori as needed by orbifold Z N orbifolds have central node w/ multiplicity N Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 16
47 Conclusion Approaches to duality Problem with usual approach in singular case: Fractional instantons, unbroken GG, FPs Finding the duals Blow up corner, blow down pillow Weierstrass polynomial coefficients from het. side Fixes complex structure of tori as needed by orbifold Z N orbifolds have central node w/ multiplicity N Presented arguments that ALL 6D orbifold models are connected by F-Theory by choosing polynomials and base F N Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 16
48 Conclusion Approaches to duality Problem with usual approach in singular case: Fractional instantons, unbroken GG, FPs Finding the duals Blow up corner, blow down pillow Weierstrass polynomial coefficients from het. side Fixes complex structure of tori as needed by orbifold Z N orbifolds have central node w/ multiplicity N Presented arguments that ALL 6D orbifold models are connected by F-Theory by choosing polynomials and base F N Outlook Apply to 4D models Use description w/o section Compare to M-Theory w/ frozen singularities Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 16
49 Thank you for your attention! Fabian Ruehle (DESY) F-Theory duals of heterotic K3 orbifolds String Pheno (07/08/2014) 17
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