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Torsion, Duality and the Wilderness of Orientifolds Torsion, Duality and the Wilderness of Orientifolds Iñaki García-Etxebarria 26th September 2014 in collaboration with B. Heidenreich and T. Wrase arXiv:1210.7799, 1307.1701 and work to appear Seiberg (IR) duality: Various different UV theories flow to the same IR fixed point. For example, it may happen that one description becomes strongly coupled in the IR (confining description) while a dual description becomes free in the IR (confined description). In many (but not all) cases we can explicitly identify various dual perturbative descriptions of the same theory. In some cases we can only identify dual descriptions of the IR fixed points. Intro Field theory Microscopics Torsion The Wilderness Conclusions What is duality? Duality comes in two flavors: Exact duality: The same quantum theory has various different descriptions, which become tractable in different regions of parameter space: AdS/CFT T-duality and mirror symmetry IIB/ = 4 (Montonen-Olive) S-duality N Seiberg (IR) duality: Various different UV theories flow to the same IR fixed point. For example, it may happen that one description becomes strongly coupled in the IR (confining description) while a dual description becomes free in the IR (confined description). In many (but not all) cases we can explicitly identify various dual perturbative descriptions of the same theory. In some cases we can only identify dual descriptions of the IR fixed points. Intro Field theory Microscopics Torsion The Wilderness Conclusions What is duality? Duality comes in two flavors: Exact duality: The same quantum theory has various different descriptions, which become tractable in different regions of parameter space: AdS/CFT T-duality and mirror symmetry IIB/ = 4 (Montonen-Olive) S-duality N For example, it may happen that one description becomes strongly coupled in the IR (confining description) while a dual description becomes free in the IR (confined description). In many (but not all) cases we can explicitly identify various dual perturbative descriptions of the same theory. In some cases we can only identify dual descriptions of the IR fixed points. Intro Field theory Microscopics Torsion The Wilderness Conclusions What is duality? Duality comes in two flavors: Exact duality: The same quantum theory has various different descriptions, which become tractable in different regions of parameter space: IIB/ = 4 (Montonen-Olive) S-duality N Seiberg (IR) duality: Various different UV theories flow to the same IR fixed point. In many (but not all) cases we can explicitly identify various dual perturbative descriptions of the same theory. In some cases we can only identify dual descriptions of the IR fixed points. Intro Field theory Microscopics Torsion The Wilderness Conclusions What is duality? Duality comes in two flavors: Exact duality: The same quantum theory has various different descriptions, which become tractable in different regions of parameter space: IIB/ = 4 (Montonen-Olive) S-duality N Seiberg (IR) duality: Various different UV theories flow to the same IR fixed point. For example, it may happen that one description becomes strongly coupled in the IR (confining description) while a dual description becomes free in the IR (confined description). Intro Field theory Microscopics Torsion The Wilderness Conclusions What is duality? Duality comes in two flavors: Exact duality: The same quantum theory has various different descriptions, which become tractable in different regions of parameter space: IIB/ = 4 (Montonen-Olive) S-duality N Seiberg (IR) duality: Various different UV theories flow to the same IR fixed point. For example, it may happen that one description becomes strongly coupled in the IR (confining description) while a dual description becomes free in the IR (confined description). We find evidence for the existence of new Montonen-Olive dualities in a large class of = 1 theories. N Intro Field theory Microscopics Torsion The Wilderness Conclusions What is duality? Duality comes in two flavors: Exact duality: The same quantum theory has various different descriptions, which become tractable in different regions of parameter space: IIB/ = 4 (Montonen-Olive) S-duality N Seiberg (IR) duality: Various different UV theories flow to the same IR fixed point. For example, it may happen that one description becomes strongly coupled in the IR (confining description) while a dual description becomes free in the IR (confined description). In many (but not all) cases we can explicitly identify various dual perturbative descriptions of the same theory. In some cases we can only identify dual descriptions of the IR fixed points. Intro Field theory Microscopics Torsion The Wilderness Conclusions Montonen-Olive = 4 duality N Given a 4d = 4 field theory with gauge group G and gauge N 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.) Intro Field theory Microscopics Torsion The Wilderness Conclusions Field theories from solitons If we want to construct four dimensional field theories from string theory, we want solitons with a four dimensional core. These can be constructed in type IIB string theory via D3 branes. 2 We also have that g4d = gs. Intro Field theory Microscopics Torsion The Wilderness Conclusions Field theories from solitons If we want to construct four dimensional field theories from string theory, we want solitons with a four dimensional core. These can be constructed in type IIB string theory via D3 branes. 2 We also have that g4d = gs. Intro Field theory Microscopics Torsion The Wilderness Conclusions Field theories from solitons Key idea Since the resulting theory is determined by the geometry, one can determine robust results without knowing much of the dynamical details of the duality acting on the core of the soliton. One just needs to know how the duality acts at infinity (which is just the simple action on IIB). We then reconstruct the dual theory as that living in the soliton with the right (dual) charge as infinity. Intro Field theory Microscopics Torsion The Wilderness Conclusions The duality as seen from string theory For example, = 4 U(N) theory is the low energy description of N 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. Intro Field theory Microscopics Torsion The Wilderness 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. Intro Field theory Microscopics Torsion The Wilderness 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, = 4 U(N) theory is the low energy description of N 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. Intro Field theory Microscopics Torsion The Wilderness 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, = 4 U(N) theory is the low energy description of N 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). Intro Field theory Microscopics Torsion The Wilderness 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, = 4 U(N) theory is the low energy description of N 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. Intro Field theory Microscopics Torsion The Wilderness 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.
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