Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

String compactifications, SU(3) structuresand smooth compact toric varieties

Magdalena Larfors

University of Oxford

November 6 2013

M. L., D. Lust, D. Tsimpis (1005.2194); M. L. (1309.2953);

J. Gray, M. L., D. Lust (1205.6208)

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

The talk in one slide

SU(3) structure manifolds allow SUSY flux vacua in string theory.Few example manifolds exist - can we construct more?

Physics motivation:models with stable moduli; de Sitter vacua; AdS/CFT.Math motivation:non-complex, non-Kahler geometries.

Construct SU(3) structures on smooth, compact, toric varieties.

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

Outline

1 Motivation

2 SU(3) structure

3 Toric geometry

4 SU(3) construction

5 SU(3) properties

6 Examples

7 Conclusions and outlook

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

Motivation: Heterotic string vacua

M10 = R(1,3) ×M6 with 4D N = 1 vacuum⇔ nowhere vanishing spinor η on M6

Simplest case: η covariantly constant⇔ holonomy group of M6 restricted to SU(3).

⇔ M6 is Calabi–Yau.Candelas, Horowitz, Strominger, Witten:85

X Phenomenology (standard model)7 Moduli stabilization see however Anderson, Gray, Lukas, Ovrut:11...

What are the alternatives?

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

Motivation: SUSY and SU(3) structure

More generally:M10 =M4 ×W M6 with 4D N = 1 effective theory

⇒ nowhere vanishing spinor η on M6

∇Tη = 0: connection with torsion

⇔ structure group of M6 restricted to SU(3).

SU(3) torsion constrained by SUSY, BI, EOM.

Strominger:86, Hitchin:02, Gualtieri:04, Grana et al:05, ...

Remark: SU(3) structure ⊂ SU(3) × SU(3) structure.

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

SU(3) structure

2. FROM G-STRUCTURES TO CALABI-YAU GEOMETRY

U! U"

ea e !a

O(d)

(a) Structure group O(d)

vv v

U! U"

eae !a

O(d ! 1)

(b) A globally defined vector reduces thestructure to O(d ! 1)

Figure 1: A set of non-degenerate tensors describes a G-structure. On the left: in thespecial case of the figure we assume that the structure group is already reduced to O(d)(see example 2.1). On the right: an everywhere non-vanishing vector field v is introduced.Because of the existence of this vector field it is possible to construct a reduced framebundle, where on the overlap between the patches only the rotations that leave the vectorinvariant are allowed as transition functions, i.e. (proper and improper) rotations in aplane orthogonal to the v-axis, making up O(d ! 1). The figure is inspired by a similarone from a talk by Davide Cassani.

A convenient way to describe a G-structure, used a lot by physicists, is via one ormore G-invariant tensors — or spinors as we will see later — that are globally defined onM and non-degenerate. Indeed, since these objects are globally defined it is possible tochoose frames ea in each patch so that they take exactly the same form in all patches. Itfollows that only those transition functions that leave these objects invariant are allowedand the structure group reduces to G or a subgroup thereof, see figure 1.

Note that, typically, such a set of G-invariant tensors is not unique, so that thereare several descriptions of the same G-structure. Furthermore, it is possible that thesetensors are actually invariant under a larger group G!, in which case one can add moretensors to more accurately describe the G-structure. The G-invariant tensors can befound in a systematic way using representation theory. Indeed, one should decompose thedi!erent representations of GL(d,R), in which a tensor on M transforms, into irreduciblerepresentations of G and scan for invariants. These invariants will then correspond tonon-degenerate G-invariant tensors.

If the G-structure is already reduced to SO(d) (see example 2.1) and the manifold isspin, which means one can lift the SO(d) in the transition functions to its double coverSpin(d) in a globally consistent way, we can also consider spinor bundles. We will especiallybe interested in invariant spinors since they are needed to construct the generators ofunbroken supersymmetry.

10

Koerber:10

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

SU(3) structure

M6 orientable with metric: G = SO(6) ⊂ GL(6).

M6 spinnable: SO(6) lifts to Spin(6) ∼= SU(4).

Let η Weyl, positive chirality: η ∈ 4 of SU(4). Choose basis:

η =

000η0

invariant under

(U 03×1

01×3 1

), U ∈ SU(3)

Globally defined η =⇒ G = SU(3).

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

SU(3) structure: Torsion

η ⇔ real two-form J and complex decomposable three-form Ω s.t.

Ω ∧ J = 0, 3i4 Ω ∧ Ω = J ∧ J ∧ J 6= 0

where Jmn = −iη†+γmnη+, Ωmnp = −iη†−γmnpη+

Almost complex structure: Imn ∼ εnk1..k5 ReΩmk1k2ReΩk3k4k5

J,Ω ⇒ metric gmn = ImpJpn Hitchin:00

J,Ω closed ⇔ M6 is Calabi–Yau.

Otherwise non-zero torsion Chiossi, Salamon:02

dJ = − 32 Im(W1Ω) +W4 ∧ J +W3

dΩ =W1J ∧ J +W2 ∧ J +W5 ∧ Ω

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

SU(3) structure compactifications

Remark:many Calabi–Yau → many fluxless compactifications (w. moduli).few examples of SU(3) structure compactifications with flux.

Why so few explicit examples?

SU(3) structure not enough: SUSY, BI and EOM selects Wi

Need explicit construction of J,Ω to compute Wi

Complications:

W1,W2 6= 0 ⇒ Imp not integrable (non-complex).

W1,W4,W3 6= 0: not symplectic (non-Kahler)

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

SU(3) structure compactifications

Why so few explicit examples?

Non-complex manifolds: cannot use tools from algebraic geometry.

Tomasiello:07

CP3 and CP1 over CP2 have two almost complex structures:

Imp ∼ J,Ω: not integrable

Imp ∼ Fubini-Study metric: integrable

ML, Lust, Tsimpis:10

Other manifolds with several almost complex structures?

CP3 and CP1 over CP2 are smooth, compact, toric varieties

→ look at other SCTVs!

Remark: No compact toric Calabi–Yau manifolds exist.

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

Toric geometry

Symplectic quotient description (GLSM)

z i , i = 1, . . . n: holomorphic coordinates of Cn

U(1)s action: z i −→ e iϕaQai z i V a =

∑i Q

ai z

i∂z i

Define moment maps µa :=∑n

i=1 Qai |z i |2 then

M2d = z i ∈ Cn|µa = ξa/U(1)s

is a toric variety (where d = n − s).

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

Toric geometry

3D SCTV Classification by Oda:78

CP3 Q =(1 1 1 1

)CP2 bundles over CP1 Q =

(1 1 a b 00 0 1 1 1

)a, b twist parameters.

CP1 bundles over 2D SCTV Q =

(q n 00 1 1

)q 2D charge matrix, n twist parameters

...

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

Toric geometry

Form Φ on Cn ⇒ well-defined form Φ| = Φ|µa=ξa on M2d if:

Φ is vertical: ιV aΦ = ιV aΦ = 0

Φ is gauge-invariant: LImV aΦ = 0

µ

M2da!

a

a aµ = "

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

SU(3) construction: Local SU(3) ML, Lust, Tsimpis:10

Gaiotto, Tomasiello:09

SU(n) structure on Cn → local SU(3) structure on M2d

On Cn:

JCn = dz i ∧ dz i

ΩCn = dz1∧ ...∧dzn

On M2d :

J = P(JCn) = Dz i ∧ Dz i

Ω = AP(ΠaιV aΩCn)

JCn ∧ ΩCn = 0

ΩCn ∧ Ω∗Cn ∝ J3Cn

dJCn = dΩCn = 0

J ∧ Ω = 0

Ω ∧ Ω∗ = 4i3 J

3

dJ = 0, dΩ = dA ∧ Ω

Ω: complex decomposable but not gauge-invariant (Qa(Ω) 6= 0).

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

SU(3) construction: Local SU(3) ML, Lust, Tsimpis:10

Gaiotto, Tomasiello:09

SU(n) structure on Cn → local SU(3) structure on M2d

On Cn:

JCn = dz i ∧ dz i

ΩCn = dz1∧ ...∧dzn

On M2d :

J = P(JCn) = Dz i ∧ Dz i

Ω = AP(ΠaιV aΩCn)

JCn ∧ ΩCn = 0

ΩCn ∧ Ω∗Cn ∝ J3Cn

dJCn = dΩCn = 0

J ∧ Ω = 0

Ω ∧ Ω∗ = 4i3 J

3

dJ = 0, dΩ = dA ∧ Ω

Ω: complex decomposable but not gauge-invariant (Qa(Ω) 6= 0).

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

SU(3) construction: Local SU(2)

If we can rewrite Ω = iK ∧ ω, where K :

1 is (1,0) (w.r.t. Imp) and vertical P(K ) = K

2 has half the Qa-charge of Ω.

3 can be normalized to K · K∗ = 2

then Ω ∼ K∗ ∧ ω has zero Qa-charge and is well-defined.

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

SU(3) construction: Global SU(3)

J := αJ − i(α+β2)2 K ∧ K∗ and Ω := αβe iγK∗ ∧ ω

then define a global SU(3) structure:

Ω ∧ J = 0, Ω ∧ Ω∗ = 4i3 J

3

Ω is complex decomposable I2 = −1

α, β, γ: nowhere vanishing real functions.

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

Topological constraint ML:13, Dabholkar:13

Well-defined SU(3) structure ⇔ exists K s.t.

1 is (1,0) (w.r.t. Imp) and vertical P(K ) = K

2 has half the Qa-charge of Ω.

3 can be normalized to K · K∗ = 2

Condition 2: Ω must have even Qa-charge.

⇔ first Chern class c1(Ω) even in cohomology.

CP3 3 CP1 over 2D SCTV 3 CP2 over CP1 7c1 = 4D choose twist parameters c1 = (2+a+b)D1 +3D5

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

Topological constraint

SU(3) structure and Stiefel-Whitney classes

For 6-manifolds: SU(3) structure ⇔ orientable and spin.Trivial w1 and w2 (even c1)

Argument due to Bryant, Calabi

Spin(6) ∼= SU(4):spinor bundle S →M6 S ′ = C4 bundle

C4 ∼= R8: nowhere vanishing section over M6

SU(4) reduced to SU(3)

Undo double cover: SO(6) reduced to SU(3).

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

Topological constraint

SU(3) structure and Stiefel-Whitney classes

For 6-manifolds: SU(3) structure ⇔ orientable and spin.Trivial w1 and w2 (even c1)

Argument due to Bryant, Calabi

Spin(6) ∼= SU(4):spinor bundle S →M6 S ′ = C4 bundle

C4 ∼= R8: nowhere vanishing section over M6

SU(4) reduced to SU(3)

Undo double cover: SO(6) reduced to SU(3).

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

SU(3) construction: Summary

SU(3) structures on CP3 and all CP1 bundles over 2D SCTV

J := αJ − i(α+β2)2 K ∧ K∗ and Ω := αβe iγK∗ ∧ ω

The SU(3) structure is not unique:

parametric freedom: α, β, γseveral consistent choices of K possible: K ∼∑αiKi

(3 eigenforms to vertical projection matrix: KiPij = Kj)

Explicit construction of J,Ω: can compute Wi .

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

SU(3) properties: Torsion classesGray, ML, Lust:11, ML:13

Fixed α, β, γ:dK W0

1 ,W03 ,W0

4 dK , dω W02 ,W0

5

W1 = (α + β2)e iγW01

W2 = e iγW02

W3 = (α + β2)W03 + χP

W4 = (α + β2)W04 +

1

4Jyχ

W5 =W05 + d ln(αβ) + Idγ

where χ = d lnα ∧ J + i β2

2 d(lnα− 2 lnβ) ∧ K ∧ K∗

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

SU(3) properties: Metric ML:13

J,Ω ⇒ metric gmn = ImpJpn

Positive definite g restricts parameter range

gmn = α[gmn +

(β2

α − 1)

Re (KmK∗n )]

β2

α

1

1

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

Examples: CP3

Charge Q = (1, 1, 1, 1) Q(Ω) = 4; ξ =∑ |z i |2 6= 0

K = α1(−z2Dz1 + z1Dz2) + α2(−z4Dz3 + z3Dz4)

Vertical (1,0) formQ(K) = 2, p = |K |2 6= 0 ⇔ Q(αi ) = 0, αi 6= 0

Constant αi W01 = 4α1α2

√ξ

3αβp

W02 = (2β2 − α)W0

1

(J + 3iβ2

2K ∧ K∗

)W0

3 = −W04 ∧ (J + iβ2K ∧ K∗)

W04 = 1

2β2 d ln p

W05 = 2d ln p

|K |2 = ξ for αi = ±1: half-flat SU(3) structureTomasiello:07, Koerber et al:08

W2 = 0 for αi = ±1, α = 2β2: nearly-Kahler SU(3) structureNilsson, Pope:84, Sorokin et al:85, Behrndt, Cvetic:04

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

Examples: CP1 over CP1 × CP1

Q =

0 1 0 1 n1 01 0 1 0 0 −n2

0 0 0 0 1 1

ξa =∑

Qai |z i |2 6= 0

Q(Ω) =(2 + n1, 2− n2, 2

)K = α1(−z3Dz1 + z1Dz3) + α2(−z4Dz2 + z2Dz4)

Vertical (1,0) form

Q(K) = 12

(2 + n1, 2− n2, 2

)⇔ Q(α1) = 1

2(2 + n1,−2− n2, 2) ,Q(α2) = 1

2(−2 + n1, 2− n2, 2)

p = |K |2 6= 0⇔ αi 6= 0

α1 = B1z6 , α2 = B2z

5 is a valid choice (Bi 6= 0, real).

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

Examples: CP1 over CP1 × CP1

K = B1z6(−z3Dz1 + z1Dz3) + B2z

5(−z4Dz2 + z2Dz4)

Constant Bi W01 = −i 2B1B2

√detgab

3αβp

W02 =W0

1

(2β2 − α)

(J + 3iβ2

2K ∧ K∗

)+

α2ξ3(−3 |K1|2|K2|2

detgabj + i

|z6|2Dz5 ∧ Dz5

)W0

3 = −W04 ∧ (J + iβ2K ∧ K∗)

W04 = 1

4β2

d ln p +

2(B21 ξ

2−B22 ξ

1)

pRe(z5Dz5)

W0

5 = 2β2W04 + d ln p − 1

2d ln detgab

Exact Lie forms; no half-flat SU(3) structure

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

Conclusions and outlook

Conclusions

3D SCTVs with even c1 allow SU(3) structures.

SU(3) structure constructed using one-form K can compute torsion classes.

Choice of K and parameters change torsion classes.

Metric positivity restricts parameters.

Outlook and WishlistRelation between K and torsion classes efficient computations for SCTVs with many generators.

Classification of SCTV SU(3) structures.

Moduli of SCTV SU(3) structures.

Construct new string vacua on SCTVs.

Strings, SU(3), SCTV

Magdalena Larfors

Motivation

SU(3) structure

Toric geometry

SU(3) construction

Local SU(3)

Local SU(2)

Global SU(3)

Topologicalconstraint

Summary

SU(3) properties

Torsion classes

Metric

Examples

CP3

CP1 over CP1 × CP1

Conclusions andoutlook

Conclusions and outlook

Conclusions

3D SCTVs with even c1 allow SU(3) structures.

SU(3) structure constructed using one-form K can compute torsion classes.

Choice of K and parameters change torsion classes.

Metric positivity restricts parameters.

Outlook and WishlistRelation between K and torsion classes efficient computations for SCTVs with many generators.

Classification of SCTV SU(3) structures.

Moduli of SCTV SU(3) structures.

Construct new string vacua on SCTVs.