Non–K¨ ahler Backgrounds in Heterotic Theory

has the dx dz and dy dz cross–terms. After a little algebra one finds almost exactly the same metric apart from a warp factor fordz², which turns out to be surprisingly simple

dsˆ˜²_IIB = dr²+e^−2φdz²+ζ α0(D+α²B²e^2φ)|dz₂|²

+C(dx−b_xidθ_i)²+D(dy−b_yjdθ_j)². (5.28) After a rescaling

z −→ z⁰ = (α₀CD)^−1/2z (5.29)

this agrees with the type I metric. The only difference is, as before transition, that this IIB background has a non–vanishing BN S field and we do not have to restrict the ans¨atze for the fluxes to be invariant under orientifold operation. But note that we have established a connection between the semi–flat IIB background after transition (5.11) with the type I background after transition (5.26), instead of considering the full IIB background (with restored coshziand sinhzi terms).

We can therefore conclude that the type I backgrounds constructed in (5.17) and (5.26) are transition dual. Let us repeat the argument: Each of the type I metrics is essentially identical to a IIB background which is T–dual to one of the IIB backgrounds discussed in section 3.4. The backgrounds in section 3.4 are transition duals because we found them by following the duality chain. If now one type I background is T–dual to the IIB background before and the other to the IIB background after transition, this implies that they are also transition duals. They are connected via an even longer duality chain than the one we followed in chapter 3.

Both type I backgrounds are non–K¨ahler, because they are T–dual to IIB backgrounds with NS field. It would be interesting to confirm that they are really complex as anticipated in [73, 118], but we cannot show this conclusively if we only know the local metric. Again, we would require knowledge of the global metric of the F–theory fourfold to be able to extend this analysis to global backgrounds.

5.3. NON–K ¨AHLER BACKGROUNDS IN HETEROTIC THEORY 75 backgrounds with torsion. Those have been the subject of intensive study, see e.g. [30, 32, 119, 121] and references in [33], and it will be interesting to see how the new, non–compact models we discuss here, fit into the existing literature.

Before transition, we find the S–dual of the metric (5.17) ds²_het = α^−1/2 dr²+dz²+|dχ₂|²)−2√

α AB(dx−bxθ_idθ_i)(dy−b_yθjdθ_j) +√

α(1 +B²) (dx−bxθidθi)²+√

α(1 +A²) (dy−b_yθjdθj)². (5.31) The torsion three–form and string coupling are found from (5.19)

H^het = F₃^I = F_xz1^IIBdy∧dz∧dθ₂+F_xz2^IIBdy∧dz∧dθ₁−F_yz2^IIBdx∧dz∧dθ₁

−F_yz1^IIBdx∧dz∧dθ₂+F_rxz^IIBdr∧dy∧dz+F_rx1^IIBdr∧dy∧dθ₂ (5.32) +F_rx2^IIBdr∧dy∧dθ1−F_ryz^IIBdr∧dx∧dz−F_ry1^IIBdr∧dx∧dθ2

−F_ry2^IIBdr∧dx∧dθ1

g^het = 1

√α, (5.33)

After transition, taking the S–dual of (5.26), we find d˜s²_het = 1

√ CD

h

dr²+ (α₀CDe^2φ)⁻¹dz²+ζ α₀D₁|dχ₂|²i +

rD

C (dx−b_xidθ_i)²+ rC

D(dy−b_yjdθ_j)² (5.34) g^het = 1

√

CD. (5.35)

This, together with a torsion three–form of the same type as (5.32), specifies the heterotic background we claim to be transition dual to the background obtained as the S–dual of the type I background before transition in (5.31).

Let us repeat the reasoning why we claim these backgrounds to be transition dual. We verified Vafa’s duality chain to the extend that we found a IIB background that has the local structure of a deformed conifold after a series of T–dualities and anM–theory flop. Trusting this duality chain means, both IIB backgrounds are actually transition dual. We then constructed type I backgrounds that are T–dual to an orientifold version of these IIB backgrounds. We also verified that the type I metrics are actually very close to IIB after two T–dualities away from the orientifold limit. Therefore, we can trust that those metrics in type I contain a contractible two– and three–cycle, respectively. Since the heterotic backgrounds possess the same metric as the type I backgrounds (apart from an overall factor given by the coupling) they should also be transition duals.

Both heterotic backgrounds have to fulfill a torsional relation to preserve supersymmetry [30]. With constant dilaton (as we have in the local limit) this torsional relation reads

H^het = ∗dJ (5.36)

with fundamental two–formJ. But the fluxes were already constrained in IIB by the linearized equation of motion (3.54). This implies the following chain of reasoning for the mapping of the fluxes from IIB to heterotic

∗(H_{N S}^IIB) = F₃^IIB −−→^{T xy} F₃^I = H^het = ∗(dJ), (5.37)

where T–duality Txy along x and y imposes relations between the type IIB RR flux F₃^IIB and the type I three–form F₃^I that can be read of from (5.19)

F_rx(z,1,2)^IIB = −F_ry(z,2,1)^I , F_ry(z,1,2)^IIB = F_rx(z,2,1)^I

F_xz(1,2)^IIB = −F_yz(2,1)^I , F_yz(1,2)^IIB = F_xz(2,1)^I . (5.38) Note that the B–field components bxθi and byθj we start with in IIB appear at the end of the chain in heterotic theory in the metric and are contained in dJ. Therefore, this connection is highly non–trivial and might not always be consistent for an arbitrary choice of background fluxes. It means that there has to exist a complex structure on the heterotic metric that is compatible with the T–duality action on the RR forms.

We can demonstrate this for a simple toy example⁵. We will make a quite restrictive ansatz for the fluxes and work strictly in the local limit whereA, B, C, D=constant (and so areα, α0).

Let us choose for the IIB RR two–form

C₂^IIB = c₁(r)dx∧dz+c₄(r)dy∧dz , (5.39) which means there will be only two components in the RR fieldstrength. They are related to the IIB NS–field via the linearized equation of motion (5.21) and we find

F_rxz^IIBdr∧dx∧dz = ∗[(∂_θ2b_yθ1−∂_θ1b_yθ2)dy∧dθ₁∧dθ₂]

= a₁(∂_θ2b_yθ1 −∂_θ1b_yθ2)dr∧dx∧dz

F_ryz^IIBdr∧dy∧dz = ∗[(∂_θ2b_xθ1 −∂_θ1b_xθ2)dx∧dθ1∧dθ2] (5.40)

= a2(∂θ2bxθ1 −∂θ1bxθ2)dr∧dy∧dz ,

where the constants ai contain the numerical factor due to the Hodge star operator ∗ on the six–dimensional IIB metric (5.5). In order to fulfill the supergravity equation of motion we also have to ensure that the IIB NS field strength does not have any other components than those appearing in (5.40). This imposes the requirement that the components bxθi and byθj

are functions of (θ1, θ2) only and not of r or z. Under T–duality these fluxes turn into an RR three–form in type I:

F₃^I = −F_rxz^IIBdr∧dy∧dz+F_ryz^IIBdr∧dx∧dz (5.41) which becomesH^het after S–duality. The question now is: does a complex structure (or rather fundamental two–form) exist for the heterotic background that is compatible with this torsion three–form?

There are of course many complex structures on the real six–manifold that is described by the metric (5.31). One possible choice is to take the real vielbeins

e¹ = α^−1/4dr , e² = α^−1/4dz

e³ = α^−1/4dθ₁, e⁴ = α^−1/4|τ₂|dθ₂ (5.42) e⁵ = α^−1/4

r1 +A²

2 ((dy−b_yidθ_i) +γ₂(dx−b_xidθ_i)) e⁶ = α^−1/4

r1 +A²

2 ((dy−b_yidθ_i) +γ₃(dx−b_xidθ_i))

5This differs from the example considered in [73].

5.3. NON–K ¨AHLER BACKGROUNDS IN HETEROTIC THEORY 77 with the coefficientsγi being determined by the metric to

γ₂ = −AB±α^−1/2

1 +A² , γ₃ = −AB∓α^−1/2

1 +A² . (5.43)

With the canonical choice of complex structure as in (4.5) the fundamental two–form becomes J = e¹∧e²+e³∧e⁴+e⁵∧e⁶ = {J₁}_bij=0+{J₂} (5.44) where we have explicitely separatedJinto a B–field independent partJ₁and a part that contains the IIB B–field components bxθi and byθj, given byJ2. Since we work in the local limit

dJ1 = 0 (5.45)

trivially. One might expect such a splitting to be always possible, since in the absence of any flux a K¨ahler background maps to another K¨ahler background under T–duality and only switching on NS flux creates torsion. This is of course correct, but a splitting of the fundamental two–form is only possible if we know the “right” complex structure on the K¨ahler manifold, in other words if we know the K¨ahler form. Not any choice of real vielbeinse_i will lead to a closed J₁. These issues have been discussed in chapter 4.

For the local limit this splitting is trivially always possible. But keep in mind that the choice (5.42) with the complex structure imposed byJ is by no means unique. We view this choice as an illustrative example. For the non–closed part we find

J₂ = b_yidx∧dθ_i−b_xjdy∧dθ_j−(b_xθ1b_yθ2 −b_xθ2b_yθ1)dθ₁∧dθ₂ (5.46) up to an overall minus sign related to the sign ambiguity in γ_i in (5.43). The torsional relation H^het=∗(dJ₂) then implies

H_rxz^hetdr∧dx∧dz = ∗[−(∂_θ2b_xθ1−∂_θ1b_xθ2)dy∧dθ1∧dθ2]

= −a₁(∂θ2bxθ1 −∂θ1bxθ2)dr∧dx∧dz

H_ryz^hetdr∧dy∧dz = ∗[(∂_θ2byθ1 −∂θ1byθ2)dx∧dθ1∧dθ2] (5.47)

= a₂(∂_θ2b_yθ1 −∂_θ1b_yθ2)dr∧dy∧dz

with all other components vanishing because bij does not depend on r or z. We have again included numerical factorsai to incorporate the Hodge star operator, they are exactly the same as in (5.40). We would like to match this torsion form with the type I three–form (5.41), which requires

−F_rxz^IIB = H_ryz^het, F_ryz^IIB = H_rxz^het. (5.48) ComparingF^IIB given by the supergravity equation of motion in (5.40) andH^hetgiven through the torsional relation in (5.47), we see that the first and second identity require

−a₁ = a₂ and a₂ = −a₁, (5.49)

respectively. The constantsa₁anda₂ are determined by the Hodge star operator, which is given on a six–manifold by

∗(dx^µ1 ∧dx^µ2 ∧dx^µ3) = 1 3!

p|g|^µ1µ2µ3_ν1ν2ν3dx^ν1 ∧dx^ν2 ∧dx^ν3. (5.50) Since a1 is determined by ^yθ1θ2_rxz, a2 is determined by ^xθ1θ2_ryz and ^yθ1θ2rxz = −^xθ1θ2ryz, this seems perfectly consistent. We conclude that the choice of flux and complex structure in

our toy example is consistent with the duality chain (5.37) when a1 = −a₂. The precise value of a₁ could be found from the metric (5.31), but we will not do so here.

In summary, we found new non–compact, non–K¨ahler manifolds with local metric (5.31) and (5.34), that are related via S–duality to the type I backgrounds constructed in the last section.

We argued the type I backgrounds to be transition duals, therefore also the heterotic non–K¨ahler backgrounds should show geometric transition. We demonstrated for a specific choice of fluxes that this background fulfills the torsional relation with torsion three–form (5.32), which was in turn related to the RR three form flux in the IIB orientifold. This IIB flux was also shown to fulfill the linearized supergravity equation of motion.

We now turn to the question if we can find a global background that reduces in the local limit to the ones we constructed here.

Im Dokument Geometric Transitions on non-Kähler Manifolds (Seite 74-78)