Heterotic Brane World
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1 Heterotic Brane World Hans Peter Nilles Physikalisches Institut Universität Bonn Germany Based on work with S. Förste, P. Vaudrevange and A. Wingerter hep-th/ , to appear in Physical Review D Heterotic Brane World p.1/31
2 Outline Orbifold Compactification Heterotic Brane World p.2/31
3 Outline Orbifold Compactification Early work in 80 s Heterotic Brane World p.2/31
4 Outline Orbifold Compactification Early work in 80 s GUTs without GUT group Heterotic Brane World p.2/31
5 Outline Orbifold Compactification Early work in 80 s GUTs without GUT group New developments Heterotic Brane World p.2/31
6 Outline Orbifold Compactification Early work in 80 s GUTs without GUT group New developments A Toy SO(10) Model with 3 Families Heterotic Brane World p.2/31
7 Outline Orbifold Compactification Early work in 80 s GUTs without GUT group New developments A Toy SO(10) Model with 3 Families Gauge group geography in extra dimensions Heterotic Brane World p.2/31
8 Outline Orbifold Compactification Early work in 80 s GUTs without GUT group New developments A Toy SO(10) Model with 3 Families Gauge group geography in extra dimensions Unification (sin 2 θ W ) Heterotic Brane World p.2/31
9 Outline Orbifold Compactification Early work in 80 s GUTs without GUT group New developments A Toy SO(10) Model with 3 Families Gauge group geography in extra dimensions Unification (sin 2 θ W ) Proton decay Heterotic Brane World p.2/31
10 Outline Orbifold Compactification Early work in 80 s GUTs without GUT group New developments A Toy SO(10) Model with 3 Families Gauge group geography in extra dimensions Unification (sin 2 θ W ) Proton decay Yukawa textures and flavour symmetries Heterotic Brane World p.2/31
11 Outline Orbifold Compactification Early work in 80 s GUTs without GUT group New developments A Toy SO(10) Model with 3 Families Gauge group geography in extra dimensions Unification (sin 2 θ W ) Proton decay Yukawa textures and flavour symmetries Outlook Heterotic Brane World p.2/31
12 Orbifolds of heterotic string Orbifold compactifications combine the success of Calabi-Yau compactification calculability of torus compactification Heterotic Brane World p.3/31
13 Orbifolds of heterotic string Orbifold compactifications combine the success of Calabi-Yau compactification calculability of torus compactification Fields can propagate Bulk (d = 10 untwisted sector) 3-Branes (d = 4 twisted sector fixed points) 5-Branes (d = 6 twisted sector fixed tori) Heterotic Brane World p.3/31
14 Z 3 Example Heterotic Brane World p.4/31
15 Z 3 Example Action of the space group on coordinates X i (θ k X) i + n α e i α, k = 0, 1, 2, i, α = 1,..., 6 Embed twist in gauge degrees of freedom X I (Θ k X) I I = 1,..., 16 Heterotic Brane World p.4/31
16 Classification of Z 3 Orbifold Very few inequivalent models Heterotic Brane World p.5/31
17 Classification of Z 3 Orbifold Very few inequivalent models Case Shift V Gauge Group Gen ` 1 3, 1 3, 2, 05 `08 3 ` 1 3, 1 3, 2 3, 05 ` 1 3, 1 3, 2 3, 05 ` 1 3, 1 3, 06 ` 2 3, 07 ` 1 3, 1 3, 1 3, 1 3, 2 3, 03 ` 2 3, 07 E 6 SU(3) E 8 36 E 6 SU(3) E 6 SU(3) 9 E 7 U(1) SO(14) U(1) 0 SU(9) SO(14) U(1) 9 Heterotic Brane World p.5/31
18 Classification of Z 3 Orbifold Very few inequivalent models Case Shift V Gauge Group Gen ` 1 3, 1 3, 2, 05 `08 3 ` 1 3, 1 3, 2 3, 05 ` 1 3, 1 3, 2 3, 05 ` 1 3, 1 3, 06 ` 2 3, 07 ` 1 3, 1 3, 1 3, 1 3, 2 3, 03 ` 2 3, 07 E 6 SU(3) E 8 36 E 6 SU(3) E 6 SU(3) 9 E 7 U(1) SO(14) U(1) 0 SU(9) SO(14) U(1) 9 as a result of the degeneracy of the matter multiplets at the 27 fixed points Heterotic Brane World p.5/31
19 Classification of Z 3 Orbifold Very few inequivalent models Case Shift V Gauge Group Gen ` 1 3, 1 3, 2, 05 `08 3 ` 1 3, 1 3, 2 3, 05 ` 1 3, 1 3, 2 3, 05 ` 1 3, 1 3, 06 ` 2 3, 07 ` 1 3, 1 3, 1 3, 1 3, 2 3, 03 ` 2 3, 07 E 6 SU(3) E 8 36 E 6 SU(3) E 6 SU(3) 9 E 7 U(1) SO(14) U(1) 0 SU(9) SO(14) U(1) 9 as a result of the degeneracy of the matter multiplets at the 27 fixed points We need to lift this degeneracy... Heterotic Brane World p.5/31
20 Z 3 Orbifold with Wilson lines Torus shifts embedded in gauge group as well X I X I + V I + n α A I α Heterotic Brane World p.6/31
21 Z 3 Orbifold with Wilson lines Torus shifts embedded in gauge group as well X I X I + V I + n α A I α further gauge symmetry breakdown number of generations reduced Heterotic Brane World p.6/31
22 Early work on Z 3 Orbifold Successful model building with three families of quarks and leptons gauge group SU(3) SU(2) U(1) n doublet-triplet splitting mechanism for Yukawa suppression Heterotic Brane World p.7/31
23 Early work on Z 3 Orbifold Successful model building with three families of quarks and leptons gauge group SU(3) SU(2) U(1) n doublet-triplet splitting mechanism for Yukawa suppression Leads to a picture of GUTs without GUT group Incomplete gauge and Higgs multiplets Transparent geometric interpretation Unification of gauge couplings (sin 2 θ W ) Heterotic Brane World p.7/31
24 Things to improve Z 3 orbifold example is simple, but too rigid only fixed points and no fixed tori difficult to get normal grand unified picture no large string threshold corrections continuous Wilson lines too distructive value of sin 2 θ W Heterotic Brane World p.8/31
25 Things to improve Z 3 orbifold example is simple, but too rigid only fixed points and no fixed tori difficult to get normal grand unified picture no large string threshold corrections continuous Wilson lines too distructive value of sin 2 θ W This was the situation around It gave a picture of unification that is somewhat different form the conventional one... Heterotic Brane World p.8/31
26 Work in the 90 s some continuation on orbifold constructions, though not very specific fermionic formulation of heterotic string with very specific (semi) realistic models Type IIB orientifolds D brane constructions intersecting branes Heterotic Brane World p.9/31
27 Work in the 90 s some continuation on orbifold constructions, though not very specific fermionic formulation of heterotic string with very specific (semi) realistic models Type IIB orientifolds D brane constructions intersecting branes This gives a vast variety of models, both with and without supersymmetry in d = 4... small or large compactified dimensions... Heterotic Brane World p.9/31
28 Recent developments Gauge couplings meet at GeV in the framework of the Minimal Supersymmetric Standard Model (MSSM) Heterotic Brane World p.10/31
29 Recent developments Gauge couplings meet at GeV in the framework of the Minimal Supersymmetric Standard Model (MSSM) See-saw mechanism for neutrino sector favours the interpretation of a family of quarks and leptons as a 16 dimensional spinor representation of SO(10) Heterotic Brane World p.10/31
30 Recent developments Gauge couplings meet at GeV in the framework of the Minimal Supersymmetric Standard Model (MSSM) See-saw mechanism for neutrino sector favours the interpretation of a family of quarks and leptons as a 16 dimensional spinor representation of SO(10) Attempts at model building in field theoretic orbifold GUTs in 5 and 6 dimensions Heterotic Brane World p.10/31
31 Recent developments Gauge couplings meet at GeV in the framework of the Minimal Supersymmetric Standard Model (MSSM) See-saw mechanism for neutrino sector favours the interpretation of a family of quarks and leptons as a 16 dimensional spinor representation of SO(10) Attempts at model building in field theoretic orbifold GUTs in 5 and 6 dimensions Can we incorporate this into a string theory description? Heterotic Brane World p.10/31
32 Five golden rules Family as spinor of SO(10) Incomplete multiplets N = 1 superymmetry in d = 4 Repetition of families from geometry Discrete symmetries of stringy origin Heterotic Brane World p.11/31
33 Five golden rules Family as spinor of SO(10) Incomplete multiplets N = 1 superymmetry in d = 4 Repetition of families from geometry Discrete symmetries of stringy origin Such a scheme should incorporate the successful structures of GUTs avoid (some of) the problems Heterotic Brane World p.11/31
34 Five golden rules Family as spinor of SO(10) Incomplete multiplets N = 1 superymmetry in d = 4 Repetition of families from geometry Discrete symmetries of stringy origin Such a scheme should incorporate the successful structures of GUTs avoid (some of) the problems We need more general constructions to incorporate this program into string theory... Heterotic Brane World p.11/31
35 Candidates In ten space-time dimensions... Type I SO(32) Type II orientifolds Heterotic SO(32) Heterotic E 8 E 8 Intersecting Branes U(N) M Heterotic Brane World p.12/31
36 Candidates In ten space-time dimensions... Type I SO(32) Type II orientifolds Heterotic SO(32) Heterotic E 8 E 8 Intersecting Branes U(N) M...or in eleven Horava-Witten heterotic M-theory Type IIA on manifolds with G 2 holonomy Heterotic Brane World p.12/31
37 Z 2 Z 2 Orbifold Example θ 1 θ 2 θ 3 Heterotic Brane World p.13/31
38 Z 2 Z 2 Orbifold Example θ 1 θ 2 θ 3 3 twisted sectors (with 16 fixed tori in each) lead to a geometrical picture of... Heterotic Brane World p.13/31
39 Intersecting Branes Heterotic Brane World p.14/31
40 Z 2 Z 2 classification Case Shifts Gauge Group Gen. 1 ` 1 2, 1, 06 ` `0, 2, 1 2, 05 `08 E 6 U(1) 2 E 8 48 ` 1 2, 1, 06 ` `0, , 1 2, 04, 1 `1, 0 7 E 6 U(1) 2 SO(16) , 0 6 `08 5 4, 1 7 ` 1 4 2, 1 2, 06 SU(8) U(1) E 7 SU(2) , 0 5, 1 `1, `0, 2, 1 2, 05 E 6 U(1) 2 SO(8) , 1 3 2, 1, 0 3 ` 1 2, 1 2, 1, 05 `1, , 1 7 ` 1 4 2, 1 2, 06 SU(8) U(1) SO(12) SU(2) 2 0 Heterotic Brane World p.15/31
41 Z 2 Z 2 with Wilson lines θ 1 A 3 θ 2 A 3 θ 3 Again, Wilson lines can lift the degeneracy... Heterotic Brane World p.16/31
42 Three family SO(10) toy model A 16 θ 1 A A 1 A 3 A 4 A 6 θ 2 A 3 A 5 A 2 A 6 θ 3 16 A 1 A 5 Localization of families at various fixed tori Heterotic Brane World p.17/31
43 Zoom on first torus... e 2 θ 1 e 1 e 2 θ 2 e 2 e 1 θ 3 e 1 Interpretation as 6-dim. model with 3 families on branes Heterotic Brane World p.18/31
44 second torus... e 4 θ 1 e 4 e 3 e 4 θ 2 e 3 e 3 θ families on branes, one in (6d) bulk... Heterotic Brane World p.19/31
45 Three family SO(10) toy model A 16 θ 1 A A 1 A 3 A 4 A 6 θ 2 A 3 A 5 A 2 A 6 θ 3 16 A 1 A 5 Localization of families at various fixed tori Heterotic Brane World p.20/31
46 third torus θ 1 e 6 e 6 θ 2 e 6 e 5 e 5 θ 3 e family on brane, two in (6d) bulk. Heterotic Brane World p.21/31
47 Model building We can easily find models with gauge group SU(3) SU(2) U(1) 3 families of quarks and leptons doublet-triplet splitting N = 1 supersymmetry Heterotic Brane World p.22/31
48 Model building We can easily find models with gauge group SU(3) SU(2) U(1) 3 families of quarks and leptons doublet-triplet splitting N = 1 supersymmetry Key properties of the models depend on geometry family symmetries texture of Yukawa couplings number of families gauge group on branes Heterotic Brane World p.22/31
49 Gauge group geography SO(10) E 6 SO(10) E 6 SO(10) E 6 E 6 E 6 SO(10) SO(10) E 6 Heterotic Brane World p.23/31
50 Gauge group geography SU(5) SO(10) SO(10) SO(10) SU(5) SU(5) SU(5) SU(5) SU(6) SU(2) SU(6) SU(2) SU(6) SU(2) Heterotic Brane World p.24/31
51 Gauge group geography: Pati-Salam SU(8) SU(6) SU(2) E 6 SU(6) SU(2) SU(4) 2 SO(10) SU(6) SU(2) E 6 SU(6) SU(2) SU(8) Heterotic Brane World p.25/31
52 Gauge geography: Standard Model SU(4) 2 SU(3) 2 SU(6) SU(2) SU(3) 2 SU(4) SU(2) 2 SU(4) SU(2) 2 SO(10) SU(5) SU(5) SU(6) SU(2) Heterotic Brane World p.26/31
53 The Memory of SO(10) SO(10) is realized in the higher dimensional theory broken in d = 4 incomplete multiplets Heterotic Brane World p.27/31
54 The Memory of SO(10) SO(10) is realized in the higher dimensional theory broken in d = 4 incomplete multiplets Still there could be remnants of SO(10) symmetry 16 of SO(10) at some branes correct hypercharge normalization R-parity family symmetries that are very useful for realistic model building... Heterotic Brane World p.27/31
55 Proton decay R-parity from SO(10) memory avoids dangerous dimension-4 operators Heterotic Brane World p.28/31
56 Proton decay R-parity from SO(10) memory avoids dangerous dimension-4 operators No proton decay via dimension-5 operators because of doublet-triplet splitting Heterotic Brane World p.28/31
57 Proton decay R-parity from SO(10) memory avoids dangerous dimension-4 operators No proton decay via dimension-5 operators because of doublet-triplet splitting Avoid SO(10) brane for first family: suppressed p-decay via dimension-6 operators Heterotic Brane World p.28/31
58 Proton decay R-parity from SO(10) memory avoids dangerous dimension-4 operators No proton decay via dimension-5 operators because of doublet-triplet splitting Avoid SO(10) brane for first family: suppressed p-decay via dimension-6 operators Thus the proton could be practically stable! Heterotic Brane World p.28/31
59 Unification SO(10) memory provides a reasonable value of sin 2 θ W and a unified definition of hypercharge Heterotic Brane World p.29/31
60 Unification SO(10) memory provides a reasonable value of sin 2 θ W and a unified definition of hypercharge presence of fixed tori allows for large threshold corrections at the high scale to match string and unification scale Heterotic Brane World p.29/31
61 Unification SO(10) memory provides a reasonable value of sin 2 θ W and a unified definition of hypercharge presence of fixed tori allows for large threshold corrections at the high scale to match string and unification scale gauge-yukawa unification from SO(10) memory for third family (on an SO(10) brane) Heterotic Brane World p.29/31
62 Unification SO(10) memory provides a reasonable value of sin 2 θ W and a unified definition of hypercharge presence of fixed tori allows for large threshold corrections at the high scale to match string and unification scale gauge-yukawa unification from SO(10) memory for third family (on an SO(10) brane) no gauge-yukawa unification for first and second family required Heterotic Brane World p.29/31
63 Yukawa textures and family symmetries Yukawa couplings depend on location of Higgs and matter fields Heterotic Brane World p.30/31
64 Yukawa textures and family symmetries Yukawa couplings depend on location of Higgs and matter fields Exponential suppression if fields at distant branes Heterotic Brane World p.30/31
65 Yukawa textures and family symmetries Yukawa couplings depend on location of Higgs and matter fields Exponential suppression if fields at distant branes neutrino masses need some bulk mixing (one might need anti-families) Heterotic Brane World p.30/31
66 Yukawa textures and family symmetries Yukawa couplings depend on location of Higgs and matter fields Exponential suppression if fields at distant branes neutrino masses need some bulk mixing (one might need anti-families) GUT relations could be partially present, depending on the nature of the brane (e.g. SO(10) brane) Heterotic Brane World p.30/31
67 Yukawa textures and family symmetries Yukawa couplings depend on location of Higgs and matter fields Exponential suppression if fields at distant branes neutrino masses need some bulk mixing (one might need anti-families) GUT relations could be partially present, depending on the nature of the brane (e.g. SO(10) brane) family symmetries arise if different fields live on the same brane Heterotic Brane World p.30/31
68 Conclusion E 8 E 8 heterotic compactifications might lead to models that incorporate all the successful ingredients of grand unified theories, while avoiding the problematic ones spinor representations of SO(10) geometric origin of (three) families incomplete multiplets supersymmetric unification R-parity absence of proton decay gauge-yukawa unification (partial GUT relations) discrete family symmetries Heterotic Brane World p.31/31
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