Dualities & Supersymmetry Breaking I~nakiGarc´ıa-Etxebarria 10th March, 2014 A bit of a pickle for string theory: much of the work so far has been on (mostly) supersymmetric string theories (particularly on the formal side, but pheno is also often build on mostly susy backgrounds). Perhaps a good opportunity to revisit some old, and not so old, ideas on non-supersymmetric string backgrounds. Introduction S-duality Tachyons USp(32) Conclusions Motivation Despite good theoretical reasons to assume the existence of supersymmetry, we do not have any direct experimental evidence for it. Perhaps a good opportunity to revisit some old, and not so old, ideas on non-supersymmetric string backgrounds. Introduction S-duality Tachyons USp(32) Conclusions Motivation Despite good theoretical reasons to assume the existence of supersymmetry, we do not have any direct experimental evidence for it. A bit of a pickle for string theory: much of the work so far has been on (mostly) supersymmetric string theories (particularly on the formal side, but pheno is also often build on mostly susy backgrounds). Introduction S-duality Tachyons USp(32) Conclusions Motivation Despite good theoretical reasons to assume the existence of supersymmetry, we do not have any direct experimental evidence for it. A bit of a pickle for string theory: much of the work so far has been on (mostly) supersymmetric string theories (particularly on the formal side, but pheno is also often build on mostly susy backgrounds). Perhaps a good opportunity to revisit some old, and not so old, ideas on non-supersymmetric string backgrounds. All aspects of a unified theory, so things look good... but plethora of vacua, so finding our vacuum is difficult. No similar classification for non-supersymmetric theories. How many higher dimensional string theories? Dualities between them? How many vacua? Introduction S-duality Tachyons USp(32) Conclusions Classification In 10d we have 5 susy string theories (6? [Sethi: 1304.1551]). but plethora of vacua, so finding our vacuum is difficult. No similar classification for non-supersymmetric theories. How many higher dimensional string theories? Dualities between them? How many vacua? Introduction S-duality Tachyons USp(32) Conclusions Classification In 10d we have 5 susy string theories (6? [Sethi: 1304.1551]). All aspects of a unified theory, so things look good... No similar classification for non-supersymmetric theories. How many higher dimensional string theories? Dualities between them? How many vacua? Introduction S-duality Tachyons USp(32) Conclusions Classification In 10d we have 5 susy string theories (6? [Sethi: 1304.1551]). All aspects of a unified theory, so things look good... but plethora of vacua, so finding our vacuum is difficult. Introduction S-duality Tachyons USp(32) Conclusions Classification In 10d we have 5 susy string theories (6? [Sethi: 1304.1551]). All aspects of a unified theory, so things look good... but plethora of vacua, so finding our vacuum is difficult. No similar classification for non-supersymmetric theories. How many higher dimensional string theories? Dualities between them? How many vacua? This classification typically is not physical, T-duality can mix 1 $ 2 and the 2 $ 3 distinction can be an artifact of weak coupling. Introduction S-duality Tachyons USp(32) Conclusions Non-supersymmetric string theories One can imagine different classes of non-susy string backgrounds 1. Defect susy breaking: a susy bulk, with (mutually) non-susy defects. 2. Bulk susy breaking: non-susy bulk, but built from susy ingredients which are non-susy when taken together. 3. Intrinsically non-supersymmetric: no susy in sight in the bulk. All of these come in stable and unstable flavors. Introduction S-duality Tachyons USp(32) Conclusions Non-supersymmetric string theories One can imagine different classes of non-susy string backgrounds 1. Defect susy breaking: a susy bulk, with (mutually) non-susy defects. 2. Bulk susy breaking: non-susy bulk, but built from susy ingredients which are non-susy when taken together. 3. Intrinsically non-supersymmetric: no susy in sight in the bulk. All of these come in stable and unstable flavors. This classification typically is not physical, T-duality can mix 1 $ 2 and the 2 $ 3 distinction can be an artifact of weak coupling. Introduction S-duality Tachyons USp(32) Conclusions Outline 1 Introduction 2 Strong/weak duality for realistic 4d theories 3 Tachyon condensation and stable vacua in non-susy theories 4 Into the wilderness 5 Conclusions Introduction S-duality Tachyons USp(32) Conclusions Outline 1 Introduction 2 Strong/weak duality for realistic 4d theories Montonen-Olive duality A new class of 4d dualities with N = 1 N = 0 dualities 3 Tachyon condensation and stable vacua in non-susy theories 4 Into the wilderness 5 Conclusions Introduction S-duality Tachyons USp(32) Conclusions Montonen-Olive N = 4 duality Given a 4d N = 4 field theory with gauge group G and gauge coupling τ = θ + i=g2, there is a completely equivalent description with gauge group G_ and coupling −1/τ (for θ = 0 this is g $ 1=g). Examples: G G_ U(1) U(1) U(N) U(N) SU(N) SU(N)=ZN SO(2N + 1) Sp(2N) Very non-perturbative duality, exchanges gauge bosons with monopoles! (So, the usual field theory tools are not particularly illuminating here.) For example, N = 4 U(N) theory is the low energy description of N D3s on flat space. Using the duality dictionary, one gets _ 2 U(N) = U(N).(gYM = gs) More interestingly, SO(2N + 1) is the low energy theory for 2N D3s on top of a Og3−. Applying the duality dictionary, this is 2N D3s on top of a O3+, which at low energies gives SO(2N + 1)_ = Sp(2N). Beautiful field theory insights follow trivially from the duality dictionary. For example, the gauge boson ! monopole map follows easily from the F 1 ! D1 duality dictionary entry. Introduction S-duality Tachyons USp(32) Conclusions Montonen-Olive duality from string theory Just \engineer" the field theory one wants in string theory, and apply the IIB S-duality dictionary to the construction. More interestingly, SO(2N + 1) is the low energy theory for 2N D3s on top of a Og3−. Applying the duality dictionary, this is 2N D3s on top of a O3+, which at low energies gives SO(2N + 1)_ = Sp(2N). Beautiful field theory insights follow trivially from the duality dictionary. For example, the gauge boson ! monopole map follows easily from the F 1 ! D1 duality dictionary entry. Introduction S-duality Tachyons USp(32) Conclusions Montonen-Olive duality from string theory Just \engineer" the field theory one wants in string theory, and apply the IIB S-duality dictionary to the construction. For example, N = 4 U(N) theory is the low energy description of N D3s on flat space. Using the duality dictionary, one gets _ 2 U(N) = U(N).(gYM = gs) Beautiful field theory insights follow trivially from the duality dictionary. For example, the gauge boson ! monopole map follows easily from the F 1 ! D1 duality dictionary entry. Introduction S-duality Tachyons USp(32) Conclusions Montonen-Olive duality from string theory Just \engineer" the field theory one wants in string theory, and apply the IIB S-duality dictionary to the construction. For example, N = 4 U(N) theory is the low energy description of N D3s on flat space. Using the duality dictionary, one gets _ 2 U(N) = U(N).(gYM = gs) More interestingly, SO(2N + 1) is the low energy theory for 2N D3s on top of a Og3−. Applying the duality dictionary, this is 2N D3s on top of a O3+, which at low energies gives SO(2N + 1)_ = Sp(2N). Introduction S-duality Tachyons USp(32) Conclusions Montonen-Olive duality from string theory Just \engineer" the field theory one wants in string theory, and apply the IIB S-duality dictionary to the construction. For example, N = 4 U(N) theory is the low energy description of N D3s on flat space. Using the duality dictionary, one gets _ 2 U(N) = U(N).(gYM = gs) More interestingly, SO(2N + 1) is the low energy theory for 2N D3s on top of a Og3−. Applying the duality dictionary, this is 2N D3s on top of a O3+, which at low energies gives SO(2N + 1)_ = Sp(2N). Beautiful field theory insights follow trivially from the duality dictionary. For example, the gauge boson ! monopole map follows easily from the F 1 ! D1 duality dictionary entry. Introduction S-duality Tachyons USp(32) Conclusions Montonen-Olive duality from string theory There are four versions of the O3 plane in string theory, distinguished by discrete RR and NSNS 2-form fluxes B2;C2 in the transverse space: [Witten:hep-th/9805112] 3 5 H (S =Z2; Ze) = Z2 : (0; 0) : O3− + ND3s −! SO(2N) (0; 1) : Og3− + ND3s −! SO(2N + 1) (1; 0) : O3+ + ND3s −! USp(2N) (1; 1) : Og3+ + ND3s −! USp(2N) IIB SL(2; Z) exchanges the configurations. Under S-duality Og3− ! O3+ : SO(2N + 1) ! USp(2N) Introduction S-duality Tachyons USp(32) Conclusions Discrete torsion argument for N = 4 The charge of the system is classified by the cohomology on the 5 5 S =Z2 = RP that surrounds the configuration. For fields even under the orientifold action, we have: • 5 H (RP ; Z) = fZ; 0; Z2; 0; Z2; Zg ; while for fields odd under the orientifold action: • 5 H (RP ; Ze) = f0; Z2; 0; Z2; 0; Z2g : This is (co)homology with local coefficients. Working on the S5 covering space k ⊗ C ' γk ⊗ γC. For coefficients in Z we have γk = k while for coefficients in Ze we have γk = −k. Ordinary (co)homology theory otherwise: H• = ker @= im @.