Heterotic type IIA duality with fluxes and moduli stabilization

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1 Heterotic type IIA duality with fluxes and moduli stabilization Andrei Micu Physikalisches Institut der Universität Bonn Based on hep-th/ and hep-th/ in collaboration with Jan Louis, Eran Plati and Gianmassimo Tasinato Hamburg, February 2007

2 Introduction Moduli stabilisation Fluxes Calabi Yau compactifications very particular string background Manifolds with SU(3) structure, T-folds, Generalised Geometry with SU(3) SU(3) structure possible new backgrounds generalise the notion of flux to geometric/non-geometric fluxes. Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 2

3 Generalised backgrounds important for moduli stabilization. Examples: Type IIB + fluxes fix complex structure moduli + dilaton; No superpotential for Kähler moduli Type IIB on mf with SU(3) structure (e.g. half-flat mirror to IIA NS fluxes) generate potential for Kähler moduli. SU(3) structures generate superpotential for Kähler moduli in heterotic compactifications. Look for backgrounds which allow stabilizing moduli in Minkowski ground state. Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 3

4 Plan of the talk Motivate backgrounds from duality arguments Heterotic type IIA duality with fluxes Model for moduli stabilization: type IIA orientifolds on manifolds with SU(3) SU(3) structure Minkowski susy solution Conclusions Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 4

5 Mirror symmetry with NS-NS fluxes Electric NS-NS flux H mirror dual geometric flux dω. W = J(T ) dω depends on Kähler moduli T [Vafa; Gurrieri, Louis, AM, Waldram] dω constant does not depend on complex structure moduli Z. Explicit realization dω i = e i β 0, dα 0 = e i ω i. Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 5

6 Generalization [D Auria, Ferrara, Trigiante, Vaula; Tomasiello; Graña, Louis, Walram] Restore symplectic invariance dω i = p ia β A q A i α A, dα A = p ia ω i, dβ A = q A i ω i, 0 = p [i A q A j]. dω depends on complex structure moduli superpotential: W = J dω p ia T i Z A Crucial coupling for moduli stabilization is it true? Hints from Heterotic type IIA duality Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 6

7 N = 2 supersymmetry in 4d Crucial ingredient: Bianchi identity Heterotic on K 3 T 2 dh = trr R trf F Take F inst solution breaks gauge group to G Coulomb branch: G U(1) n v, n v (Abelian) vectormultiplets M V = SU(1, 1) U(1) SO(2, n v 1) SO(2) SO(n v 1) Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 7

8 Also n h hypermultiplets M H M K3 = SO(4, 20) SO(4) SO(20) Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 8

9 Heterotic/K3 T 2 + flux [Curio, Klemm, Körs, Lüst; Louis, AM] Flux = internal value for F harmonic ω α harmonic two form dual to γ α Flux contribution to Bianchi identity ρ αβ = γ α F I flux = m αi F I flux = m αi ω α, K 3 (trf inst F inst trr R) + m αi m βj ρ αβ η IJ = 0. K3 ω α ω β intersection matrix of K3 signature (3, 19) Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 9

10 Result: N = 2 gauged supergravity b α = B B = b α ω α. γ α Gauge symmetry: δa I = dλ I, & δb = λ I F J η IJ, δb α = m αi λ J η IJ. Potential: V flux = 1 2 h αβm α I m β J (ImN ) 1 IJ + e φ 4v mαi m βj η IJ ρ αβ. in agreement with N = 2 gauged sugra. Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 10

11 Comparison to type IIA/CY 3 with flux RR fluxes axion charged wrt all vectors NS-NS fluxes half of the hyperscalars charged wrt A 0 (graviphoton). [Curio, Klemm, Körs, Lüst] fluxes through P 1 base RR fluxes in IIA. need fluxes that carry one index in vector multiplets (I) and one in hypermultiplets (A). Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 11

12 Type IIA on manifolds with SU(3) structure 3-form gauge potential C 3 with gauge invariance δc 3 = dλ 2, Four-dimensional fields Ĉ 3 = C 3 + A i ω i + ξ A α A ξ A β A with dω i = q A i α A, dβ A = q A i ω i. 4d gauge symm: Λ 2 = λ i ω i, dλ 2 = λ i q A i α A + dλ i ω i δa i = dλ i, δξ A = λ i q A i. Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 12

13 Subtle point: same basis in 4d need to perform an electric-magnetic duality exchange electric with magnetic (geometric) flux non-geometric flux Heterotic typeiia duality with fluxes requires full fledged SU(3) SU(3) structure with non-geometric fluxes. Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 13

14 Susy Minkowski solutions in type IIA Generalize to SU(3) SU(3) structure + orientifold planes. Pure spinors Π ev e J and Π odd e B Ω [Graña, Minasian, Petrini, Tomasiello; Graña, Louuis, Waldram; Benmachiche, Grimm] Need differential operator which lowers the rank of a form D. Consistency condition Superpotential Tadpole condition / Bianchi identities D 2 = 0. W = F ev + DΠ odd, Π ev. DF ev = δ local Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 14

15 Form basis: ev ± and odd ± Expand pure spinors and RR-fields/fluxes in these forms and insert in the above relations to get the 4d superpotential and the constraints on the parameters. Assume large moduli limit cubic N = 2 prepotentials. Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 15

16 Type IIA model W = e 0 + e i T i K ijkm i T j T k m 0 6 K ijkt i T j T k +S(h 0 + h i T i K ijkq i T j T k ) + p a Z a + p ia T i Z a. Constraints m 0 h 0 + m i h i q i e i = Q 0, m 0 p a + m i p ia = Q a, q i p ia = 0. Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 16

17 General remark: no solutions with all moduli stabilised unless non-geometric fluxes are present # Kähler moduli > # of complex structure moduli Simplest case: no complex structure moduli and one Kähler modulus might not capture all the aspects of the general problem. Next to simplest case: one complex structure modulus and two Kähler moduli. Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 17

18 Results Enough parameters in order to solve all equations including W = 0 and the tadpole conditions. Values for geometric moduli bounded by combinations of the orientifold charges can still be large. Can choose fluxes so that < axions >= 0 no ambiguity in tadpole conditions. Vacuum in agreement with 10d susy equation analysis. Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 18

19 Conclusions manifolds with general SU(3)( SU(3)) structure crucial in the network of string dualities certain SU(3) structures dual to gauge field fluxes in heterotic compactifications non-geometric SU(3) SU(3) structure essential for Minkowski vacua in type IIA sensible Minkowski vacua can be obtained Conference on Generalised Geometry and Fluxes, Hamburg 19 Feb 1 Mar 19