Torsion Classes after Geometric Transition

r 1 1 +B²

Bb⁰_yθ

1

2 e¹∧e³+ coshzie²∧e⁶+ sinhzie²∧e⁴

− r α

1 +B²

AB²b⁰_yθ

1

2 e¹∧e⁴+ coshzie²∧e⁵−sinhzie²∧e³

(4.28) +p

α(1 +B²)Ab⁰_xθ

2

2 e²∧e⁵+ coshzie¹∧e⁴−sinhzie¹∧e⁶ +2ABb⁰_yθ

1

1 +B² sinhzi e³∧e⁵+e⁴∧e⁶

− (1 +αA²B²cos 2hzi)b⁰_yθ

1 +α(1 +B²)b⁰_xθ

2

2√

α(1 +B²) ×

coshzi(e³∧e⁴−e⁵∧e⁶) + sinhzi(e⁴∧e⁵−e³∧e⁶) W₅ = 1

2

√α (1 +A²)b⁰_yθ1+ (1 +B²)b⁰_xθ2

e²+ 1 2

pα(1 +B²)Ab⁰_xθ2e⁴ +1

2 Bb⁰_yθ

1

1 +B²e⁵−1 2

r α

1 +B² AB²b⁰_yθ1 coshzie⁴+ sinhzie⁶

. (4.29)

The vielbeins ei are defined in (4.19). This completely specifies a symplectic structure on the

“non–K¨ahler deformed conifold”.

4.3 Torsion Classes after Geometric Transition

Very similar remarks hold true for the local IIA metric after transition which we termed “non–

K¨ahler resolved conifold”. We can find a symplectic structure, but W₂^± and W₅ are nonzero.

The metric after transition was obtained in (3.50) to be:

ds² = dr²+e^2φ dz−αA(dx−b_xθ2dθ2)−αB(dy−b_yθ1dθ2)2

(4.30) + αA²B²(1 +A²)⁻¹ dθ₁²+ (dx−bxθ2dθ2)²

+ (1 +A²)⁻¹ dθ²₂+ (dy−byθ1dθ1)² .

We take again the ansatz (4.21) for the complex structure but now with real vielbeins

e¹ = dr , e² = e^φ dz−αA(dx−bxθ2dθ2)−αB(dy−byθ1dθ1) e³ = 1

√1 +A²dθ2, e⁵ = 1

√1 +A² (dy−b_yθ1dθ1) (4.31) e⁴ =

r α

1 +A² AB (sinhzi(dx−bxθ2dθ2) + coshzidθ1) e⁶ =

r α

1 +A² AB (−coshzi(dx−bxθ2dθ2)−sinhzidθ1).

This different choice of vielbeins is of course inspired by the resolved conifold [78]. One also findsW₄= 0 automatically. Again,W₁⁺can only vanish ifP(r) is constant and solvingW₁⁻= 0

has the same solution P(r) =−coshzi. There is no choice ofP(r) that would allow for W₅ = 0 orW₂^±= 0. With the choiceP =−coshzi the remaining torsion classes are

W₂⁺ = −1 2

pα(1 +A²)e^φb⁰_yθ1 e²∧e³−coshzie¹∧e⁶−sinhzie¹∧e⁴

−1 2αp

1 +A²Ae^φb⁰_xθ2 e¹∧e⁵−coshzie²∧e⁴+ sinhzie²∧e⁶

(4.32)

−b⁰_yθ

1−αA²B²b⁰_xθ

2

2√

α AB e³∧e⁵+e⁴∧e⁶ W₂⁻ = −1

2

pα(1 +A²)e^φb⁰_yθ1 e¹∧e³+ coshzie²∧e⁶+ sinhzie²∧e⁴ +1

2αp

1 +A²Ae^φb⁰_xθ2 e²∧e⁵+ coshzie¹∧e⁴−sinhzie¹∧e⁶

(4.33)

−b⁰_yθ

1+αA²B²b⁰_xθ

2

2√ α AB

coshzi(e³∧e⁴−e⁵∧e⁶)−sinhzi(e³∧e⁶−e⁴∧e⁵)

W₅ =

pα(1 +A²)

2A e^φb⁰_yθ1 e⁵+1 2αp

1 +A²Ae^φb⁰_xθ2 e⁴+b⁰_yθ

1+αA²B²b⁰_xθ

2

2√

α AB e²,(4.34) where φ is the IIA dilaton which we found to be exactly the same as the IIB dilaton before transition and constant.

We see that the geometric transition maps the torsion classes W₂^± and W5 into themselves.

This can be translated into a statement about G2 torsion classes, using the definition of the three–form (4.16). So, also theG₂ torsion classesX_i are mapped into themselves. But we know that the flop just replaces the usualx¹¹direction with thez–fibration. These two circles are used to lift SU(3) torsion classes to G2 torsion classes and this implies that the G2 torsion classes should not change during the flop.

In conclusion, we have argued that on grounds of supersymmetry we do not expect a half–flat manifold. Our lift includes a constant dilaton, one might therefore expect the torsion classes (4.17) to reduce toW₂⁺6= 0, leading to a half–flat structure. But we also lift other RR fluxes to G–fluxes inM–theory, which means that supersymmetry does not requireG2 holonomy on the 7d manifold. Apart from that, we only have a local metric which does not show supersymmetry (all components and warp factors are approximated by constants). We could, however, find a symplectic structure on this local background which is in agreement with arguments from section 2.4.

Chapter 5

Geometric Transitions in Type I and Heterotic

The same mechanism as that discussed in chapter 3 can be used to go beyond Vafa’s duality chain and construct new transition dual backgrounds in type I and heterotic theory. The F–theory setup takes us naturally to the orientifold corner of type IIB which is basically type I. We only need to perform 2 T–dualities that convert the D7/O7 system into space–time filling D9/O9.

This gives rise to open and closed unoriented strings — type I. From there we can perform another S–duality and obtain heterotic backgrounds, see figure 5.1. These new backgrounds will also be non–K¨ahler, since the B–field enters into the metric when we T–dualize from the IIB orientifold to type I similar to the analysis in chapter 3.

Heterotic SO(32) NS5 on non–K¨ahler

Heterotic SO(32) fluxes on non–K¨ahler geometric

transition?

?

S–duality Type I D5 on non–K¨ahler

? T² IIB D5–branes on orientifold of resolved

geometric transition

IIB fluxes on orientifold of deformed

? T² Type I fluxes on non–K¨ahler geometric

transition?

?

S–duality

Figure 5.1: The heterotic duality chain. Following the arrows we can construct non–K¨ahler backgrounds in type I and heterotic theory that are dual to the type IIB backgrounds before and after transition. This implies that also the new backgrounds are in a sense transition duals.

67

Following these dualities on both sides of the geometric transition will give us backgrounds that are connected to a flop inM–theory (as performed in section 3.3) via a very long duality chain. Therefore, we claim these backgrounds are also transition duals. Supergravity equations of motion and the torsional constraint [30] for heterotic strings pose severe restrictions on the allowed type of fluxes. We provide a toy example that is consistent with the IIB orientifold action, the IIB linearized supergravity equation of motion and the torsional relation in the U–dual¹ heterotic background.

We can also exploit the fact that the local heterotic metric we find after transition has a similar structure as the solution constructed by Maldacena–Nunez [71]. This enables us, for the first time in this thesis, to leave the local limit and propose a global solution in heterotic theory that is consistent with our IIB orientifold setup. For the construction of vector bundles on heterotic and type I backgrounds we refer the reader to [74], where also their behavior under geometric transition has been studied.

5.1 Another F–Theory Setup and IIB Orientifold

Many of the considerations in this section are very similar to the F–theory setup constructed in section 3.2, but it should be immediately clear that we cannot use the same four–fold. What we constructed in section 3.2 was an elliptic fibration over a resolved conifold base with the torus fibers degenerating over (x, θ₁), which means the D7/O7 system extends along (r, y, z, θ₂). To convert this into a space–time filling D9/O9 system, we would have to T–dualize along (x, θ1), butθ₁ does not correspond to an isometry of conifold geometries. We therefore need a different orientation of the F–theory torus.

First we need to define the two directions along which we want to T–dualize. The logical candidates are among the directions (x, y, z), for the same reason we chose them in chapter 3:

they are the isometry direction of the resolved conifold. But z is not an isometry direction of the manifold after transition (being a deformed conifold). So we would like to avoid T–duality alongzand will T–dualize alongxandy. We then need the D7/O7 to extend orthogonal to the T–duality directions, otherwise they do not lead us to type I.

In summary, we start again with a fourfold that is elliptically fibered over the resolved conifold base, but the fiber degenerates over (x, y). This has of course consequences for the IIB orientifold. We now consider

IIB on B

{1,Ω(−1)^FLI_xy}, (5.1)

where B is the base that looks locally like a resolved conifold. This means, the branes are oriented as follows:

D5 : 0 1 2 3 − − − − y θ2

D7/O7 : 0 1 2 3 r z − θ₁ − θ₂. After T–duality along x andy this turns into

D5 : 0 1 2 3 − − x − − θ2

D9/O9 : 0 1 2 3 r z x θ₁ y θ₂, which is consistent with a type I scenario.

1U–duality is the combined action of T– and S–duality.

5.1. ANOTHER F–THEORY SETUP 69 There is a slight problem with this orientifold. The resolved conifold metric is not invariant under I_xy! Therefore, we have to project out certain components of the metric. Recalling the local metric of the resolved conifold base (3.16)

ds² = dr²+ (dz+A dx+B dy)²+ (dx²+dθ²₁) + (dy²+dθ²₂)

we see that thedx dzanddy dzcross terms spoil the invariance underIxy. To eliminate them we have to “untwist” thez–fiber. However, the orientifold action does not require us to eliminate all terms from thez–fibration, we can keep those that are invariant underI_xy, likedx² for example.

We therefore make the generic ansatz

ds²_IIB = dr²+dz²+d1|dz₁|²+d2|dz₂|² (5.2) with the two tori defined as (note that these are different tori than the ones in (2.15))

dz1 = dx+τ1dy , dz2 = dθ1+τ2dθ2. (5.3) This construction in terms of tori is especially convenient since it will preserve supersymmetry (such toroidal orbifold models have already been considered in [118]). We see that when we define (recall that α⁻¹ = 1 +A²+B²)

d1 = (1 +A)², τ1 = 1 1 +A²

AB+i α^−1/2

, d2 = 1, τ2 = i (5.4) we obtain a metric that is precisely (5.2) without the unwanted cross-terms

ds² = dr²+dz²+ (1 +A²)dx²+ (1 +B)²dy²+ 2AB dx dy+dθ₁²+dθ²₂. (5.5) We could generate a larger class of metrics that are related to this orientifold version of the resolved conifold by allowing more generic complex structures on the tori². The only choice we have to require for all of them is

Reτ2 = 0 (5.6)

because the resolved conifold does not have any dθ₁dθ₂ cross term and we want our starting background before transition to be “close” to a resolved conifold. This will enable us to argue for the existence of a contractible 2–cycle. We could, however, also have taken the point of view that “untwisting” thez–fiber should remove all crossterms, also thedx dyterm that comes from thez–fibration. This can be achieved by setting Reτ₁ = 0 and we will also allow for this case, but keep in mind that τ1 can in principle have both real and imaginary part.

Note that the setup we chose is again a model with four O7–planes each with six D7–branes on top and we have a constant complex structure on the F–theory torus, or in other words a constant axion–dilaton in IIB. Not only is it constant, but actually zero, because D7 and O7 charges cancel exactly, so we set as in [73]

χ^IIB = 0, φ^IIB = 0. (5.7)

Adding D5–branes to this background will simply act as a warp factor in IIB [40], i.e. a harmonic function H(r). Since we work in the local limit anyway, we can absorb this into the coordinate differentials as we did in chapter 3.

At the orientifold point we can also make an ansatz for the B–field, which is invariant under Ω(−1)^FLIxy if all its components have precisely one leg along the T–duality directions. We did

2Supersymmetry would then have to be restored by an appropriate choice of fluxes.

not allow for any magnetic NS flux when we constructed the background in 3.2, so let us keep the assumption that there are nodx dz ordy dz components³. Our ansatz will therefore be

B^IIB_{N S} = b_xidx∧dθ_i+b_yjdy∧dθ_j, (5.8) where the coefficients can now depend on (r, z, θ₁, θ₂), since we do not want to T–dualize along z anymore. With the same reasoning we also have to make an ansatz for the RR two–form:

C₂^IIB = c₁dx∧dz+c₂dx∧dθ₁+c₃dx∧dθ₂+c₄dy∧dz+c₅dy∧dθ₁+c₆dy∧dθ₂. (5.9) The coefficients c_i are in general allowed to depend on (r, z, θ₁, θ₂). This implies an RR three–

form fieldstrength

F₃^IIB = Fxz1dx∧dz∧dθ1+Fxz2dx∧dz∧dθ2+Fyz1dy∧dz∧dθ1

+Fyz2dy∧dz∧dθ2+Frxzdr∧dx∧dz+Frx1dr∧dx∧dθ1

+F_rx2dr∧dx∧dθ₂+F_ryzdr∧dy∧dz+F_ry1dr∧dy∧dθ₁ (5.10) +F_ry2dr∧dy∧dθ₂.

As in [73] we ignore the RR four–form for simplicity.

In conclusion, we make the generic ansatz (5.2) for the metric and (5.8) and (5.10) for the fluxes for the IIB background at the orientifold point before transition. We will mostly focus on the theory at the orientifold point, which takes us to type I. But it will be interesting to compare the type I theory that we find after two T–dualities to the IIB theory we would have obtained if we had T–dualized the IIB background away from the orientifold point. They will have many similarities. But note that away from the orientifold point there are more allowed flux components.

Similar remarks hold true for the background after transition, but here we have an F–theory fourfold that is fibered over a base which resembles the deformed conifold. We now want to find an orientifold version of this, too. We will start with the semi–flat limit we obtained from T–duality, because it does not have anydx dθ2 ordy dθ1 crossterms that would not be invariant underIxy. In other words, we start with (3.72), but impose the values for ˜βi as in (3.74), (3.75):

d˜s²_IIB = dr²+ e^−2φ α₀CD

h

dz+α₀αADe^2φdx+α₀αBCe^2φdyi2

+α₀D₁(dx²+ζ dθ₁²) +α₀C₁(dy²+dθ²₂) (5.11) +2α₀α²ABe^2φ(dθ₁dθ₂−dx dy),

with the squashing factor ζ defined in (3.77). At the danger of overloading notation we have introduced two other abbreviations

C₁ = C+α²A²e^2φ, and D₁ = D+α²B²e^2φ (5.12) where C and D were defined in (3.66) and govern the size of the two S² in the non–K¨ahler resolved metric we found in IIA after transition. With this definition the constant α0 from (3.66) becomes

α0 = C1D1−α⁴e^4φA²B². (5.13)

3The analysis of this chapter would not be influenced by allowingdx dz or dy dz components in the B–field.

We would simply acquire also az–dependent twisting of the T–duality fibersxandy.

5.2. NON–K ¨AHLER BACKGROUNDS IN TYPE I THEORY 71

Im Dokument Geometric Transitions on non-Kähler Manifolds (Seite 65-71)