Calabi-Yau Fourfolds with non-trivial Three-Form Cohomology

Transcription

1 Calabi-Yau Fourfolds with non-trivial Three-Form Cohomology Sebastian Greiner arxiv: , (T. Grimm, SG) Max-Planck-Institut für Physik and ITP Utrecht String Pheno 2017 Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

2 Motivation Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

3 Motivation F-theory effective action on elliptically fibered Calabi-Yau fourfold Y 4 : N = 1 gauged supergravity in (3 + 1) dimensions S (4) 1 = M 3,1 2 R(4) 1 K F I J DMI D M J 1 2 Re(f ) ΛΣF Λ F Σ 1 2 Im(f ) ΛΣF Λ F Σ Massless spectrum includes complex scalars from non-trivial harmonic three-forms of Y 4 What can we learn about their dynamics? Can we construct explicit examples? Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

4 Outline Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

5 Outline Three-Forms on Calabi-Yau Fourfolds Calabi-Yau Fourfolds as Toric Hypersurfaces Metric for Three-Form Moduli of a Toric Hypersurface Outlook Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

6 Geometry and Topology of CY 4 Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

7 Geometry and Topology of CY 4 Characteristic feature: SU(4)-holonomy Covariantly constant Weyl-spinor: η = 0, γ 9 η = η N = 2 3d M-theory vacua! (no flux) N = 1 4d F-theory vacua! (no flux) Hodge diamond: h 1,1 0 0 h 2,1 h 2,1 0 1 h 3,1 h 2,2 h 3,1 1 0 h 2,1 h 2,1 0 0 h 1, Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

8 Expansion of the Three-Form Consider scalar three-form moduli arising from C 3 H 2,1 (Y 4 ) H 1,2 (Y 4 ) Kinetic term: dc 3 dc 3 Hodge-star depends on moduli! precisely: ψ = ij ψ, ψ H 2,1 (Y 4 ), J H 1,1 (Y 4 ) Choose three-form basis Ψ l depending on complex structure moduli! Ψ l (z, z) = 1 2 Re(f )lm (α m i f mk β k ) H 1,2 (Y 4 ) Three-form deformations: C 3 = N l Ψ l + c.c. h 2,1 l=1 h 2,1 complex scalars N l bosonic part of chiral multiplets Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

9 Three-Form Ansatz Suggested by [Grimm 10]: (l, k, m = 1,..., h 2,1 ) Ψ l = 1 2 Re(f )lm (α m i f mk β k ) H 1,2 (Y 4 ) Properties: α l, β k basis of H 3 (Y 4, R) topological f lm (z) holomorphic (three-form periods) Assume: β l β m = 0 Result: Ψ l z K = Ψl z K Ψm Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

10 Three-Form Ansatz Suggested by [Grimm 10]: (l, k, m = 1,..., h 2,1 ) Ψ l = 1 2 Re(f )lm (α m i f mk β k ) H 1,2 (Y 4 ) Properties: α l, β k basis of H 3 (Y 4, R) topological f lm (z) holomorphic (three-form periods) Assume: β l β m = 0 Advantage: Result: Ψ l z K = Ψl z K Ψm M Al k = ω A α l β k topological intersection numbers Y 4 Ψ l Ψ k = 1 2 Re(f )lm M Am k v A Goal of our work: Calculate f and M! K M new Re(N) l Re(f ) lm M Am k v A Re(N) k Im(N) l axionic! Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

11 Toric Calabi-Yau Fourfold Hypersurfaces Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

12 Toric Calabi-Yau Fourfold Hypersurfaces Now Calabi-Yau fourfold Y 4 smooth toric hypersurface Toric divisors D i, codimension one submanifolds of Y 4 invariant under torus action Non-trivial three- and two-forms from divisors D i via Gysin-morphism of their inclusion ι i : D i Y 4 [Danilov;Mavlyutov] Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

13 Gysin-Morphism ι i : H n (D i, C) PD H 6 n (D i, C) (ι i ) 1 H 6 n (Y 4, C) PD H n+2 (Y 4, C) Respect Hodge decomposition: ι i : H 0,0 (D i ) H 1,1 (Y 4 ), ι i : H 1,0 (D i ) H 2,1 (Y 4 ). For smooth toric Calabi-Yau hypersurfaces Y 4 H 0,0 (D i ) H 1,1 (Y 4 ), i H 1,0 (D i ) H 2,1 (Y 4 ), i Construct H 1,0 (D i ) for toric divisors D i of Y 4! H 0,1 (D i ) H 1,2 (Y 4 ). Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14 i

14 Divisors with non-trivial One-Forms D: π E H 1,0 (D) H 1,0 (R) For H 1,0 (D) 0: Toric divisor D fibration α R Fiber: E toric surface, (h i,j (E) = 0 for i j) Base: R Riemann surface Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

15 Three-forms of Calabi-Yau Fourfold Hypersurface Construct three-forms: E l E m with A = (l, a) ψ A = ι l (γ a ) H 2,1 (Y 4 ) π l π m ι l : D l Y 4, γ a H 1,0 (R) γ a γ b R Complex structure dependence of ψ A determined by H 1,0 (R) Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

16 Three-forms of Calabi-Yau Fourfold Hypersurface Construct three-forms: E l E m with A = (l, a) ψ A = ι l (γ a ) H 2,1 (Y 4 ) π l π m ι l : D l Y 4, γ a H 1,0 (R) γ a γ b R Complex structure dependence of ψ A determined by H 1,0 (R) Normalized basis: γ a = α a +i f ab (z)β b H 1,0 (R), α a, β b H 1 (R, Z) Metric on H 1,0 (R): i R γ a γ b = 2 Re(f ) ab > 0, f ab = f ba Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

17 Three-Form Metric for Fourfold Hypersurface Recall metric on H 2,1 (Y 4 ) Q AB = ψ A ψ B = i J ψ A ψ B, J H 1,1 (Y 4 ) Y 4 Y 4 Two- and three-forms from toric divisors via Gysin-map Triple intersection of toric divisors! E p Only divisors that fiber over same R contribute! E l Em D l D m D p = M lmp R γ a R with M lmp = #E l E m E p (Generalized sphere tree) Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

18 Three-Form Metric for Fourfold Hypersurface Metric on H 2,1 (Y 4 ) Q AB = 2v l M lmp Re(f ) ab E p A = (m, a), B = (p, b) Reorder indices to find previous result E l Em Q AB = 2v Σ M ΣA C Re(f ) CB γ a R v Σ M ΣA C linear dependence on Kähler moduli v Σ Re(f ) CB complicated dependence on complex structure moduli z K Toric data exchanged by mirror symmetry! Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

19 Outlook Next step: Discuss F-theory physics on elliptically fibered CY 4 and their weak coupling limit on CY 3 Construct examples with three-forms such that weakly coupled IIB has Wilson line moduli on a D7 wrapping a four-cycle of CY 3 (axions) Odd moduli, scalars from B 2, C 2 (axions) Metric on H 2,1 (Y 4 ) determines decay-constants! Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

20 Outlook Next step: Discuss F-theory physics on elliptically fibered CY 4 and their weak coupling limit on CY 3 Construct examples with three-forms such that weakly coupled IIB has Wilson line moduli on a D7 wrapping a four-cycle of CY 3 (axions) Odd moduli, scalars from B 2, C 2 (axions) Metric on H 2,1 (Y 4 ) determines decay-constants! Future directions: Construct bases with three-forms (bulk U(1)) Generalize to complete intersections Include G 4 -flux and calculate superpotential... Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14

21 Thank you for your attention! Questions? Sebastian Greiner (MPP Munich) Three-forms of CY 4 Blacksburg, Juli / 14