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

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

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F-THEORY FLUXES & THEIR EFFECTIVE PHYSICS Denis Klevers University of Pennsylvania "New Ideas at the Interface of Cosmology and String Theory 17 of March, 2012 Based on: T.W. Grimm, M. Poretschkin, D.K.: arXiv:1202.0285

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Denis Klevers

University of Pennsylvania "New Ideas at the Interface of Cosmology and String Theory” 17π‘’β„Ž of March, 2012

F-THEORY FLUXES & THEIR EFFECTIVE PHYSICS

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].

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MOTIVATION

  • 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.

F-theory describes a broad class of interesting 4d N=1 Type II string vacua

  • 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.

Dynamical objects in Type IIB string theory mapped to F-theory geometry.

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MOTIVATION

  • Extra discrete degrees of freedom to specify vacuum.
  • Required by consistency of compactification: D3-tadpole cancellation

Requirement of G-fluxes in F-theory

  • 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.

Goal of this talk: What is the 4d effective physics of G-fluxes?

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F-THEORY WITH FLUXES

Introduction

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  • F-theory introduced as a geometric SL(2,) invariant formulation of Type IIB
  • Introduce SL(2,) invariant geometric object: two-torus π‘ˆ2 with β€œshape”

parameter Ο„

  • SL(2,) acts as a modular transformation on Ο„ leaving π‘ˆ2 (conformally)

invariant.

  • Identify axio-dilaton of Type IIB: 𝜐 β†’ π‘ˆ2 𝜐 .
  • β€œSize” of π‘ˆ2 unphysical: disregarded by formally setting vol π‘ˆ2) β†’ 0 .

Ο„ ↦ π‘πœ + 𝑐 π‘‘πœ + 𝑒 𝑔𝑝𝑠 𝑏 𝑐 𝑑 𝑒 ∈ 𝑇𝑀 2, β„€

FORMULATING F-THEORY

1 𝜐

Vafa ’96; Review: Denef β€˜08

Ο„ ≑ 𝐷0 + 𝑗𝑕𝑑

βˆ’1

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  • Non-trivial profile of axio-dilaton Ο„ in the presence of 7-branes.
  • 7-branes are global defects of space-time inducing a deficit angle

𝜌 6.

  • 24 7-branes produce a deficit angle 4𝜌: ℝ2 compactified to 𝑇2.
  • Tori π‘ˆ2 Ο„) over 𝑇2 define singular elliptically fibered Calabi-Yau twofold K3.

FORMULATING F-THEORY

D7 O7

(p,q)7

𝑇2 𝜐(z) 𝜐 ↦ 𝜐 + 1 β‡’ 𝜐 𝑨 =

1 2πœŒβ…ˆ ln 𝑨 ,,

𝜐(z)

z

Monodromy: D7 ℝ2

Singularity at 𝑨 = 0.

Greene,Shapere,Vafa,Yau ’90; Vafa ’96.

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  • Construct 4d F-theory vacua by replacing 𝑇2 β†’ six-dimensional :

singular K3 β†’ singular elliptic Calabi-Yau fourfold π‘Œ4.

  • F-theory is non-perturbative compactification: strong coupling regions of

Ο„ and complicated setup of 7-branes on 𝐢6d .

  • 7-branes are encoded in the singularities of π‘Œ4: Geometric description of

gauge groups (Tate’s algorithm) including ADE groups.

  • Chiral matter and Yukawa couplings appear from multiple intersections of 7-

branes.

4D F-THEORY

singular at 𝑨 = 0.

Tate ’75; Bershadsky,Intriligator,Kachru,Morrison,Sadov,Vafa β€˜96

𝐢6d

Katz,Vafa β€˜96

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  • Gauge theory in 8d

in

  • Matter in 6d

in

  • Yukawas in 4d

in

  • Recent advances in realistic GUT model building in F-theory based on this

structure.

FORMULATING F-THEORY

singular at 𝑨 = 0.

𝐸4𝑒 𝐢6𝑒

𝐢6d

Ξ£2𝑒 pt0𝑒 𝐢6d

𝐢6d

𝐸4𝑒 𝐸′4𝑒 𝐸 4𝑒 Ξ£2𝑒 pt0𝑒

Donagi,Wijnholt β€˜08; Beasley,Heckman,Vafa ’08; Review: Heckman β€˜10 Marsano,Saulina,SchaferNamek ’09; Blumenhagen,Grimm,Jurke,Weigand ’09

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  • Additional discrete degrees of freedom have be added
  • Quantized G-flux 𝐻4:
  • D3-tadpole.
  • There are two qualitatively different fluxes on π‘Œ4
  • Horizontal fluxes 𝐼𝐼

4 π‘Œ4, β„€ Vertical fluxes πΌπ‘Š 4 π‘Œ4, β„€

superpotential, D-term potential, moduli stablization 4d chirality, warping

𝐻4∈ 𝐼4 π‘Œ4, β„€ .

PROBE-FLUXES IN F-THEORY

𝐻4 𝐻4 π‘Œ4 Flux-lines

Witten β€˜96 Sethi,Vafa,Witten β€˜96 Greene,Morrison,Plesser β€˜94

Goal: Understand the effect of two types of fluxes on the 4d effective action of F-theory

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HORIZONTAL FLUXES

The flux superpotential

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  • The horizontal flux 𝐻4 enters the superpotential 𝑋

π»π‘Šπ‘‹

  • Ξ©4 𝑨4 is 4,0)-form depending on complex structure moduli on π‘Œ4.
  • 𝐻4 is specified by flux quanta π‘œπ΅ along cycles 𝐷𝐡
  • 𝑋

π»π‘Šπ‘‹ is sum of periods Π𝐡 𝑨4) of π‘Œ4 measuring holomorphic

volumes

  • Exact calculation of 𝑋

π»π‘Šπ‘‹ possible in examples.

𝑋

π»π‘Šπ‘‹ 𝑨4) = 𝐻4 π‘Œ4

∧ Ξ©4 𝑨4)

THE FLUXSUPERPOTENTIAL

𝐻4 𝐻4 π‘Œ4

𝐻4 = π‘œπ΅ 𝐷

𝐡

Π𝐡 𝑨4) = Ξ©4 𝑨4)

𝐷𝐡

𝑋

π»π‘Šπ‘‹ 𝑨4 = π‘œπ΅Ξ π΅ 𝑨4) Gukov,Vafa,Witten β€˜99 Becker,Becker β€˜96

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  • A big class of F-theory fourfolds explicitly constructed as toric hypersurfaces
  • 𝑋

π»π‘Šπ‘‹ 𝑨4) exactly calculable: Ξ  𝑨4 from Picard-Fuchs differential eqs.

  • Calculation of Type IIB superpotentials in weak coupling limit: flux + brane
  • Use 𝑋

π»π‘Šπ‘‹ for stabilization of complex structure moduli in F-theory and IIB.

  • 𝑋

π»π‘Šπ‘‹ relevant for fourfold mirror symmetry: Generalizes N=2 prepotential.

  • By heterotic/F-theory duality 𝑋

π»π‘Šπ‘‹ maps to heterotic superpotentials:

heterotic flux, M5-brane and vector bundle superpotential.

THE FLUXSUPERPOTENTIAL

𝑋

π»π‘Šπ‘‹ 𝑨4 ↦ 𝑋 π‘”π‘šπ‘£π‘¦ 𝐽𝐽𝐢 + 𝑋 7π‘π‘ π‘π‘œπ‘“ 𝐽𝐽𝐢

π‘Œ4 = {𝑄 = 0} in a toric variety β„™Ξ”

5

Grimm,Ha,Klemm,DK I β€˜09 Klemm,Lian,Roan,Yau ’97 Mayr β€˜96 Grimm,Ha,Klemm,DK I β€˜09 Dasgupta,Rajesh,Sethi β€˜99; Giddings,Kachru,Polchinski β€˜01; Kachru,Kallosh,Linde,Trivedi ’03; Lust,Mayr,Reffert,Stieberger ’05 Klemm,Pandharipande β€˜07 Grimm,Ha,Klemm,DK II β€˜09

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VERTICAL FLUXES

Chirality and flux back-reaction

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  • The vertical fluxes 𝐻4 define a D-term potential 𝒰, 𝐾 = KΓ€hler form
  • 𝐻4 determines chirality of 4d matter in rep 𝑆 of gauge group 𝐻
  • Derivation of chirality formula in 3d N=2 theory via duality
  • In F-theory, a CS-Terms ΘIJAI ∧ 𝐺𝐾 generated at 1-loop of massive matter

VERTICAL FLUXES & CHIRALITY

πœ“ 𝑺 = 𝐡𝑺

π½πΎπœ–π‘’π½π‘’πΎ 2

𝒰 ≑ 𝐻4,

𝐷 𝑺

𝐷 𝑺 = β‹― … . 4d F-theory on π‘Œ4 F-theory on 𝑇1 3d M-th theory on π‘Œ4

𝑠𝑓𝑑

𝑂 = 1 gauge theory 𝑂 = 2 gauge theory Non-abelian gauge group 𝐻 Coulomb branch 𝐻 β†’ 𝑉 1 𝑠𝑙 𝐻) Chiral matter in rep 𝑆 massive matter no matter, πœ–π‘’π½π‘’πΎ

2

𝒰)𝐡𝐽 ∧ 𝐺𝐾

𝐡𝑺

𝐽𝐾Θ𝐽𝐾∼ πœ“ 𝑺 ,

… . Ξ˜π½πΎβ‰‘ πœ–π‘’π½π‘’πΎ

2

𝒰 ⟹ πœ“ 𝑺 = 𝐻4

𝐷 𝑺

Matter surface

Haack,Louis ’01; Grimm β€˜10 Marsano,SchaferNameki ’11; Grimm,Hayashi β€˜11 Grimm,Hayashi ’11; Grimm, DK in progress

1-loop: M-theory:

𝒰 t) = 𝐻4

π‘Œ4

∧ 𝐾 𝑒 ∧ 𝐾 𝑒

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  • In M-theory (=F-theory on 𝑇1) G-flux 𝐻4 back-reacts on geometry
  • Warping:
  • Change of KK-ansatz by warping: non-closed 3-from 𝛾

BACKREACTED FLUXES IN F-THEORY

Ξ”π‘Œ4𝑓3𝐡/2 =βˆ—π‘Œ4 𝐻4 ∧ 𝐻4) 𝐷3 = 𝛾 +

Harmonic forms 𝑇2 𝑨 𝑇1 𝑦

periodic

𝑦1 𝑦2 𝑦𝐾:

TN-centers

Becker,Becker β€˜96; Haack,Louis β€˜01 Dasgupta,Rajesh,Sethi β€˜99; Grimm,DK,Poretschkin β€˜12

  • Solve warp-factor equation and construct 3-form 𝛾 in local

model for π‘Œ4.

  • Understand corrections on 4d effective physics: 7-brane

gauge coupling. Goal:

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  • Construct a local model of π‘Œ4 for a stack of k 7-branes as follows
  • Taub-NUT with k-centers is resolved 𝐡𝑙-singularity: 𝑙 resolving 𝑇2 =

= SU(k) gauge group of k 7-branes.

BACKREACTED FLUXES IN F-THEORY

𝑇1 𝑇2 𝑒 𝑨 𝑦

periodic

𝑦1 𝑦2 𝑦𝐾: TN-centers

Grimm,DK,Poretschkin β€˜12

k 7-brane stack

  • n divisor 𝑇

k 6-brane stack

  • n divisor 𝑇

Periodic multi-center Taub-NUT over S

𝑇1 T-duality M- theory

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  • Explicit construction of metric on periodic Taub-NUT:
  • π‘Š

𝐽 explicitly constructed as infinite series

  • Identification along periodic direction 𝑦 yields elliptic fibration of F-theory π‘Œ4
  • Recover familiar form of 𝜐 𝑨) for k D7-branes + new corrections.

PERIODIC TAUB-NUT FOR F-THEORY

𝑒𝑑2 = 1 π‘Š 𝑒𝑒 + 𝑉 2 + π‘Šπ‘’π‘  2, 𝑠 ∈ ℝ3 π‘Š = 1 + π‘Š

𝐽 𝑙 𝐽=1

, 𝑉 = 𝑉𝐽

𝑙 𝐽=1

, π‘’π‘Š

𝐽 =βˆ—3 𝑒𝑉𝐽

Gibbons-Hawking:

π‘Š

𝐽 = log 𝑨

βˆ’ 𝐿0 2𝜌 𝑨 π‘œ

π‘œ>0

cos 2πœŒπ‘œ 𝑦 βˆ’ 𝑦𝐽 ) 𝑒𝑑2 = 𝑀0 𝐽𝑛 𝜐 𝑒𝑒 + 𝑆𝑓 πœπ‘’π‘¦ 2 + 𝐽𝑛 πœπ‘’π‘¦2 + 𝑒𝑑ℝ2×𝑇 𝜐 𝑨 = 𝜐0 + 𝑙 2πœŒπ‘— log 𝑨 + β‹― Leading axio-dilaton:

Ooguri,Vafa’96 Grimm,DK,Poretschkin β€˜12

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  • Explicit solution of warp-factor eq.
  • Specializial G-flux: 7-brane flux π”Šπ½on S
  • Use explicitly constructed basis of 𝑙 self-dual 2-forms on Taub-NUT
  • Focus on flux with only non-trivial instanton number on S

THE WARP-FACTOR ON TAUB-NUT

Ω𝐽 = 𝑒 π‘Š

𝐽

π‘Š 𝑒𝑒 + 𝑉 βˆ’ 𝑉𝐽 𝐻4 = Ω𝐽 ∧ π”Šπ½, Ξ”π‘Œ4𝑓3𝐡/2 =βˆ—π‘Œ4 𝐻4 ∧ 𝐻4) π”Šπ½ ∧ π”ŠπΎ

𝑇

= π‘œπ½πœ€π½πΎ

Ruback ’86 Grimm,DK,Poretschkin β€˜12

Warp-factor on periodic Taub-NUT analytically determined

𝑓3𝐡/2 = 1 βˆ’ π‘œπ½ π‘€π‘π‘š 𝑇 π‘Š

𝐽 2

π‘Š βˆ’ π‘Š

𝐽

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  • KK-reduction with fluxes and warping: 3d fluctuations 𝐺𝐽 of 𝐻4

11𝑒

  • Altered KK-ansatz due to warping: non-harmonic 3-form 𝛾
  • β€œChern-Simons” form to flux 𝐻4 determines 𝛾
  • 𝛾 depends on moduli 𝑁𝑏 = 𝐷0, 𝑦𝐾) of Taub-NUT
  • Comparison of 3d effective action determined by dimensional reduction of M-

theory on warped π‘Œ4 and F-theory on 𝑇1.

CORRECTIONS TO 4D GAUGE COUPLINGS

π‘’π‘Œ4𝛾 = 𝐻4 = π”Šπ½ ∧ Ω𝐽 𝐷3

11𝑒 = 𝐡𝐽 ∧ Ω𝐽 + 𝛾,

𝐻4

11𝑒 = 𝑒𝐷3 11𝑒 = 𝐺𝐽 ∧ Ω𝐽 + 𝑒𝑁𝑏 ∧ πœ–π‘π‘π›Ύ + 𝐻4

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  • 𝑔

𝐽𝐾 = D7-brane gauge coupling, 3d vector multiplet 𝑦𝐽, 𝐡𝐽

  • Small string coupling 𝑕𝑑 β†’ 0 : full D7-brane gauge coupling + corrections

CORRECTIONS TO 4D GAUGE COUPLINGS

𝑇𝐻4

3𝑒 βŠƒ 𝐻𝐽𝐾𝐺𝐽 βˆ§βˆ— 𝐺𝐾 + 𝑒𝑏𝑐𝐽 𝑁𝑏𝑒𝑁𝑐 ∧ 𝐺𝐽 ℝ3

β‡’ 𝑔

𝐽𝐾= πœ€π½πΎ

𝐾 ∧ 𝐾 + 𝑗𝐷4)

𝑇

βˆ’ π‘—πœπ‘œπ½ + 𝑃 𝑕𝑑)

𝐽𝑛 𝑔

𝐽𝐾 π‘”π‘šπ‘£π‘¦ = Ω𝐽 ∧ πœ–π·0𝛾 ∧ πœ–π‘¦πΎπ›Ύ π‘Œ4

= πœ€π½πΎ 𝐷0𝑕𝑑

βˆ’1π‘œπ½ + 𝑃 𝑕𝑑

𝑆𝑓 𝑔

𝐽𝐾 = 𝑓3𝐡/2Ω𝐽 ∧ Ω𝐾 π‘Œ4

= πœ€π½πΎ π‘€π‘π‘š 𝑇 + 𝑕𝑑

βˆ’1π‘œπ½ + 𝑃 𝑕𝑑)

Warping corrected couplings:

𝑇𝐻4

3𝑒 βŠƒ 𝐻𝐽𝐾𝐺𝐽 βˆ§βˆ— 𝐺𝐾 + 𝑒𝑏𝑐𝐽 𝑁𝑏𝑒𝑁𝑐 ∧ 𝐺𝐽 ℝ3

Warped M-theory:

𝑇𝐺

3𝑒 βŠƒ 𝑆𝑓 𝑔 𝐽𝐾𝐺𝐽 βˆ§βˆ— 𝐺𝐾 + 𝐽𝑛 𝑔 𝐽𝐾 π‘”π‘šπ‘£π‘¦π‘’πœ‚π½ ∧ 𝐺𝐾 ℝ3

F-theory

  • n 𝑇1:

𝐻𝐽𝐾 = 𝑓3𝐡/2Ω𝐽 ∧ Ω𝐾

π‘Œ4

, 𝑒𝑏𝑐𝐽 = Ω𝐽 ∧ πœ–π‘π‘π›Ύ ∧ πœ–π‘π‘π›Ύ

π‘Œ4

Jockers,Louis β€˜04 Grimm,DK,Poretschkin β€˜12

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  • Better understanding of 𝑕𝑑 corrections to gauge coupling.
  • Inclusion of curvature corrections, M2-branes.
  • Relation between instantons and fluxes in F-theory and in 3d.

Outlook

  • F-theory allows non-perturbative constructions of realistic 4d

N=1 gauge theories with chiral matter and Yukawa couplings.

  • G-flux essential in F-theory.
  • Horizontal flux: 4d superpotential 𝑋

π»π‘Šπ‘‹ allows moduli

stabilization, 𝑋

π»π‘Šπ‘‹ calculable by Picard-Fuchs eq. in examples.

  • Vertical flux: generation of 4d chirality and nice connection at
  • ne-loop in 3d between chiral index and CS-terms.
  • Back-reaction of fluxes induces warping & changes KK-ansatz
  • analytic solution of warp-factor eq. on local 7-brane geometry.
  • Calculation of full 4d of gauge coupling function in F-theory.

Summary

Work in progress