THEIR EFFECTIVE PHYSICS Denis Klevers University of Pennsylvania - PowerPoint PPT Presentation

F-THEORY FLUXES & THEIR EFFECTIVE PHYSICS Denis Klevers University of Pennsylvania "New Ideas at the Interface of Cosmology and String Theory” 17 𝑢ℎ of March, 2012 Based on: T.W. Grimm, M. Poretschkin, D.K.: arXiv:1202.0285 [hep-th]; T.W. Grimm, T.-W. Ha, A. Klemm, DK: arXiv:0912.3250 [hep-th], arXiv:0909.2025 [hep-th] .

MOTIVATION F-theory describes a broad class of interesting 4d N=1 Type II string vacua • N=1 SUSY gauge theory with non-abelian (GUT-)gauge group. • Charged chiral matter, Yukawa couplings: Beyond SM model building. • Coupling to gravity in compact geometries. • Duality to heterotic string compactifications. Dynamical objects in Type IIB string theory mapped to F-theory geometry. • F-theory in a Type IIB language: • Inclusion of back-reacted 7-branes (D7, O7,...) with cancelled tadpoles. • Type IIB with non-perturbative coupling regions on non-CY geometry. • F-theory in geometric language: elliptically fibered Calabi-Yau fourfold.

MOTIVATION Requirement of G-fluxes in F-theory • Extra discrete degrees of freedom to specify vacuum. • Required by consistency of compactification: D3-tadpole cancellation Goal of this talk: What is the 4d effective physics of G-fluxes? • Induce superpotential: stabilization of some geometric moduli. • Generation of 4d chirality by appropriate fluxes. • back-reaction of fluxes necessary for full understanding of 4d effective physics: derivation of 7-brane gauge coupling from warping in F-theory.

Introduction F-THEORY WITH FLUXES

FORMULATING F-THEORY F-theory introduced as a geometric SL(2,  ) invariant formulation of Type IIB • Vafa ’96; Review: Denef ‘08 Introduce SL(2,  ) invariant geometric object: two-torus 𝑈 2 with “shape” • parameter τ 𝜐 1 SL(2,  ) acts as a modular transformation on τ leaving 𝑈 2 (conformally) • invariant. τ ↦ 𝑏𝜐 + 𝑐 𝑑𝜐 + 𝑒 𝑔𝑝𝑠 𝑏 𝑐 𝑒 ∈ 𝑇𝑀 2, ℤ 𝑑 Identify axio-dilaton of Type IIB: 𝜐 → 𝑈 2 𝜐 . • −1 τ ≡ 𝐷 0 + 𝑗𝑕 𝑡 “Size” of 𝑈 2 unphysical: disregarded by formally setting vol 𝑈 2 ) → 0 . •

FORMULATING F-THEORY Non-trivial profile of axio-dilaton τ in the presence of 7-branes. • ℝ 2 Monodromy: z 𝜐 ↦ 𝜐 + 1 D7 𝜐 (z) 1 ⇒ 𝜐 𝑨 = 2𝜌ⅈ ln 𝑨 , , Singularity at 𝑨 = 0 . 𝜌 7-branes are global defects of space-time inducing a deficit angle 6 . • Greene,Shapere,Vafa,Yau ’90; Vafa ’96 . 24 7-branes produce a deficit angle 4𝜌 : ℝ 2 compactified to 𝑇 2 . • 𝜐 (z) D7 (p,q) 7 O7 𝑇 2 Tori 𝑈 2 τ) over 𝑇 2 define singular elliptically fibered Calabi-Yau twofold K3. •

4D F-THEORY Construct 4d F-theory vacua by replacing 𝑇 2 → six-dimensional : 𝐶 6d • singular K3 → singular elliptic Calabi-Yau fourfold 𝑌 4 . • F-theory is non-perturbative compactification: strong coupling regions of singular at 𝑨 = 0 . τ and complicated setup of 7-branes on 𝐶 6d . 7-branes are encoded in the singularities of 𝑌 4 : Geometric description of • gauge groups (Tate’s algorithm) including ADE groups. Tate ’75; Bershadsky,Intriligator,Kachru,Morrison,Sadov,Vafa ‘96 Chiral matter and Yukawa couplings appear from multiple intersections of 7- • branes. Katz,Vafa ‘96

FORMULATING F-THEORY • Gauge theory in 8d 𝐶 6d 𝐸 4𝑒 𝐶 6𝑒 in 4𝑒 𝐸 • Matter in 6d 𝐸′ 4𝑒 singular at 𝑨 = 0 . Σ 2𝑒 𝐶 6d in • Yukawas in 4d pt 0𝑒 𝐸 4𝑒 pt 0𝑒 𝐶 6d Σ 2𝑒 in • Recent advances in realistic GUT model building in F-theory based on this Donagi,Wijnholt ‘08; Beasley,Heckman,Vafa ’08; Review: Heckman ‘10 structure. Marsano,Saulina,SchaferNamek ’09; Blumenhagen,Grimm,Jurke,Weigand ’09

PROBE-FLUXES IN F-THEORY • Additional discrete degrees of freedom have be added 𝐻 4 ∈ 𝐼 4 𝑌 4 , ℤ . • Quantized G-flux 𝐻 4 : Witten ‘96 Sethi,Vafa,Witten ‘96 • D3-tadpole. Flux-lines 𝑌 4 𝐻 4 𝐻 4 There are two qualitatively different fluxes on 𝑌 4 • Greene,Morrison,Plesser ‘94 4 𝑌 4 , ℤ Vertical fluxes 𝐼 𝑊 4 𝑌 4 , ℤ • Horizontal fluxes 𝐼 𝐼 superpotential, D-term potential, moduli stablization 4d chirality, warping Goal: Understand the effect of two types of fluxes on the 4d effective action of F-theory

The flux superpotential HORIZONTAL FLUXES

THE FLUXSUPERPOTENTIAL The horizontal flux 𝐻 4 enters the superpotential 𝑋 Gukov,Vafa,Witten ‘99 • 𝐻𝑊𝑋 Becker,Becker ‘96 𝑋 𝐻𝑊𝑋 𝑨 4 ) = 𝐻 4 ∧ Ω 4 𝑨 4 ) 𝑌 4 Ω 4 𝑨 4 is 4,0) -form depending on complex structure moduli on 𝑌 4 . • 𝐻 4 is specified by flux quanta 𝑜 𝐵 along cycles 𝐷 𝐵 • 𝑌 4 𝐻 4 = 𝑜 𝐵 𝐷 𝐵 Π 𝐵 𝑨 4 ) = Ω 4 𝑨 4 ) 𝐻 4 𝐻 4 𝐷 𝐵 𝑋 𝐻𝑊𝑋 is sum of periods Π 𝐵 𝑨 4 ) of 𝑌 4 measuring holomorphic • volumes 𝐻𝑊𝑋 𝑨 4 = 𝑜 𝐵 Π 𝐵 𝑨 4 ) 𝑋 Exact calculation of 𝑋 • 𝐻𝑊𝑋 possible in examples.

THE FLUXSUPERPOTENTIAL A big class of F-theory fourfolds explicitly constructed as toric hypersurfaces • Klemm,Lian,Roan,Yau ’97 5 𝑌 4 = {𝑄 = 0} in a toric variety ℙ Δ Mayr ‘96 𝑋 𝐻𝑊𝑋 𝑨 4 ) exactly calculable: Π 𝑨 4 from Picard-Fuchs differential eqs. • Grimm,Ha,Klemm,DK I ‘09 • Calculation of Type IIB superpotentials in weak coupling limit: flux + brane 𝐽𝐽𝐶 + 𝑋 𝐽𝐽𝐶 𝑋 𝐻𝑊𝑋 𝑨 4 ↦ 𝑋 𝑔𝑚𝑣𝑦 7𝑐𝑠𝑏𝑜𝑓 Grimm,Ha,Klemm,DK I ‘09 Use 𝑋 𝐻𝑊𝑋 for stabilization of complex structure moduli in F-theory and IIB. • Dasgupta,Rajesh,Sethi ‘99; Giddings,Kachru,Polchinski ‘01; Kachru,Kallosh,Linde,Trivedi ’03; Lust,Mayr,Reffert,Stieberger ’05 𝑋 𝐻𝑊𝑋 relevant for fourfold mirror symmetry: Generalizes N=2 prepotential. • Klemm,Pandharipande ‘07 By heterotic/F-theory duality 𝑋 𝐻𝑊𝑋 maps to heterotic superpotentials: • heterotic flux, M5-brane and vector bundle superpotential. Grimm,Ha,Klemm,DK II ‘09

Chirality and flux back-reaction VERTICAL FLUXES

VERTICAL FLUXES & CHIRALITY The vertical fluxes 𝐻 4 define a D-term potential 𝒰 , 𝐾 = Kähler form • 𝒰 t) = 𝐻 4 ∧ 𝐾 𝑢 ∧ 𝐾 𝑢 𝑌 4 Haack,Louis ’01; Grimm ‘10 𝐻 4 determines chirality of 4d matter in rep 𝑆 of gauge group 𝐻 • 𝐽𝐾 𝜖 𝑢 𝐽 𝑢 𝐾 2 𝜓 𝑺 = 𝐵 𝑺 𝒰 ≡ 𝐻 4 , 𝐷 𝑺 = ⋯ … . Matter surface 𝐷 𝑺 Marsano,SchaferNameki ’11; Grimm,Hayashi ‘11 • Derivation of chirality formula in 3d N=2 theory via duality F-theory on 𝑇 1 3d M-th 𝑠𝑓𝑡 4d F-theory on 𝑌 4 theory on 𝑌 4 𝑂 = 1 gauge theory 𝑂 = 2 gauge theory Coulomb branch 𝐻 → 𝑉 1 𝑠𝑙 𝐻) Non-abelian gauge group 𝐻 𝒰 )𝐵 𝐽 ∧ 𝐺 𝐾 2 Chiral matter in rep 𝑆 massive matter no matter, 𝜖 𝑢 𝐽 𝑢 𝐾 In F-theory, a CS-Terms Θ IJ A I ∧ 𝐺 𝐾 generated at 1-loop of massive matter • 𝐽𝐾 Θ 𝐽𝐾 ∼ 𝜓 𝑺 , 2 𝐵 𝑺 … . Θ 𝐽𝐾 ≡ 𝜖 𝑢 𝐽 𝑢 𝐾 𝒰 ⟹ 𝜓 𝑺 = 𝐻 4 1-loop: M-theory: 𝐷 𝑺 Grimm,Hayashi ’11; Grimm, DK in progress

BACKREACTED FLUXES IN F-THEORY In M-theory (=F-theory on 𝑇 1 ) G-flux 𝐻 4 back-reacts on geometry • Δ 𝑌 4 𝑓 3𝐵/2 =∗ 𝑌 4 𝐻 4 ∧ 𝐻 4 ) Becker,Becker ‘96; • Warping: Haack,Louis ‘01 • Change of KK-ansatz by warping: non-closed 3-from 𝛾 Dasgupta,Rajesh,Sethi ‘99; 𝐷 3 = 𝛾 + Harmonic forms Grimm,DK,Poretschkin ‘12 Goal: 𝑦 𝐾 : TN-centers 𝑇 1 𝑇 2 • Solve warp-factor equation and construct 3-form 𝛾 in local model for 𝑌 4 . 𝑦 𝑦 2 𝑦 1 periodic • Understand corrections on 4d effective physics: 7-brane gauge coupling. 𝑨

BACKREACTED FLUXES IN F-THEORY Construct a local model of 𝑌 4 for a stack of k 7-branes as follows • k 7-brane stack k 6-brane stack 𝑇 1 Periodic multi-center M- on divisor 𝑇 on divisor 𝑇 T-duality theory Taub-NUT over S 𝑦 𝐾 : TN-centers 𝑇 2 𝑢 𝑦 𝑦 2 𝑦 1 periodic 𝑇 1 𝑨 Grimm,DK,Poretschkin ‘12 Taub-NUT with k-centers is resolved 𝐵 𝑙 -singularity: 𝑙 resolving 𝑇 2 = • = SU(k) gauge group of k 7-branes.

PERIODIC TAUB-NUT FOR F-THEORY • Explicit construction of metric on periodic Taub-NUT: 𝑒𝑡 2 = 1 𝑊 𝑒𝑢 + 𝑉 2 + 𝑊𝑒𝑠 2 , 𝑠 ∈ ℝ 3 Gibbons-Hawking: 𝑙 𝑙 𝑊 = 1 + 𝑊 , 𝑉 = 𝑉 𝐽 , 𝑒𝑊 𝐽 =∗ 3 𝑒𝑉 𝐽 𝐽 𝐽=1 𝐽=1 𝑊 • 𝐽 explicitly constructed as infinite series 𝑊 𝐽 = log 𝑨 − 𝐿 0 2𝜌 𝑨 𝑜 cos 2𝜌𝑜 𝑦 − 𝑦 𝐽 ) 𝑜>0 Ooguri,Vafa’96 Identification along periodic direction 𝑦 yields elliptic fibration of F-theory 𝑌 4 • 𝑒𝑡 2 = 𝑤 0 𝑒𝑢 + 𝑆𝑓 𝜐𝑒𝑦 2 + 𝐽𝑛 𝜐𝑒𝑦 2 + 𝑒𝑡 ℝ 2 ×𝑇 𝐽𝑛 𝜐 𝜐 𝑨 = 𝜐 0 + 𝑙 Leading axio-dilaton: 2𝜌𝑗 log 𝑨 + ⋯ Recover familiar form of 𝜐 𝑨) for k D7-branes + new corrections. • Grimm,DK,Poretschkin ‘12