Discussion

and

Ce^I₀ = (δ^I_J−Oe^IAOˇ^A_J)C^J₀, Oˇ^A_IOe^IB =δ^AB, (5.5.75) this strategy eventually brings us to the effective four-dimensional action

S_IIB = Z

M^1,3

1

2R⁽⁴⁾?1⁽⁴⁾−dφ∧?dφ− e^−4φ

4 dB∧?dB−g_ijdtⁱ∧?d¯t^j−g_abdU^a∧?dU^b +1

2ReM_ABF^A₂ ∧F^B₂ +1

2ImM_ABF^A₂ ∧?F^B₂ +1

2∆e_IJdeC^I₀∧?deC^J₀ +1

2(∆⁻¹)^AB

d(O_AICe^I₂) +O_AIeC^I₀dB

∧?

d(O_BJCe^J₂) +O_BJeC^J₀dB +

d(OAICe^I₂) +OAIeC^I₀dB

∧

e^2φ(∆⁻¹)^AB(Oe^T)BJ

NJKdeC^K₀

+1

2dB∧eC^I₀S_IJdeC^J₀

−

O_AIeC^I₂−G_AfluxB

∧

dC^A₁ +¹₂O^A_JeC^J₂− ¹₂G^A_fluxB

+V_scalar?1⁽⁴⁾

(5.5.76) with

V_scalar = V_NSNS+V_RR

= +e^2φ

2 V^I(O^T)_I^AMABO^B_JV^J+e^2φ

2 W^A(Oe^T)_A^INIJOe^J_BW^B

−e^2φ

4KW^AS_ACO^C_I

V^IV^J+V^IV^J

(O^T)_J^DS_DBW^B +e^4φ

2

G^A_flux+Ce^I₀(O^T)_I^A MAB

G^B_flux+O^B_JeC^J₀

(5.5.77)

Comparing this action to its IIA counterpart (5.5.49), one can again construct a set of mirror mappings by extending (5.3.81) to

tⁱ ↔ U^a, g_ij ↔ g_ab, MAB ↔ NIJ, h^1,1 ↔ h^1,2,

V^I ↔ W^A, S_IJ ↔ S_AB

C^I_n ↔ C^A_n, G^I_flux ↔ G^A_flux, O^A_I ↔ Oe^I_A,

(5.5.78)

once more confirming preservation of IIA ↔ IIB Mirror Symmetry in the simultaneous presence of geometric and nongeometric fluxes.

5.6 Discussion 111

Scalar Potential

In section 5.2 we derived the scalar potential of four-dimensional N = 2 gauged super-gravity from dimensional reductions of the purely internal type IIA and IIB double field theory action on a Calabi-Yau three-foldCY₃. Building upon the elaborations of [78, 79], we extended the discussed setting by formally including generalized dilaton fluxes and relaxing the primitivity constraints. This modification revealed a more general form of the reformulated double field theory action, which shows a strong structural resemblance of supergravity compactifications on SU(3)×SU(3) structure manifolds [199].

In section 5.3 it was then exemplified through K3× T² how the framework can be generalized beyond the Calabi-Yau setting. This was done by utilizing the features of generalized Calabi-Yau and K3 structures [147, 203] to enable a special-geometric description of the K3 ×T² moduli space. The dimensional reduction led to a scalar potential term resembling that of N = 4 gauged supergravity formulated in the N = 2 framework of [202]. An important stechnical step here was to exploit the properties of K3 and T² to formally construct K3×T² analogues to the structure forms of CY₃,

e^bCY3+iJCY3 ←→e^bK3+iJK3∧e^bT2+iJT2, e^bCY3 ∧Ω_CY3 ←→ e^bK3∧Ω_K3

∧ e^bT2 ∧Ω_T²

, (5.6.1)

whereJ denotes the K¨ahler form of the respective manifold and Ω its holomorphic volume form. While the constructed scalar potential involves characteristic structures of N = 4 gauged supergravity, relating the result to its standard formulation explicitly turned out to be a nontrivial task and might be an interesting direction for future research. It is to be expected that the discussion for arbitrary manifolds allowing for a generalized Calabi-Yau structure in the sense of [147, 203] follows a similar pattern.

An essential novelty of the approach discussed in these sections is its capability of describing generalized dilaton fluxes and non-vanishing trace-terms of the NS-NS fluxes.

While their inclusion into Calabi-Yau compactifications is only a formal generalization, their contribution becomes nontrivial in the K3×T² setting. In light of the previous works [214, 215], it is to be expected that such fluxes serve as a ten-dimensional origin of non-unimodular gaugings in the N = 4 gauged supergravity framework. This was also briefly discussed in section 4.2.3 of [149] in a double field theory context. Integrating the dilaton flux operators into the twisted differential of double field theory did not require including a rescaling charge operator as done in [215], which is in accordance with the result of [199] for SU(3)×SU(3) structure manifolds.

Relation to four-dimensional N = 2 Gauged Supergravity

In section 5.5 we reconstructed the full bosonic part of the four-dimensional N = 2 gauged supergravity action by including the kinetic terms into the Calabi-Yau setting.

Our results replicate the action constructed in [202] and illustrate how the simultaneous presence of all NS-NS and R-R fluxes not only gives rise to gaugings in the effective four-dimensional theory, but also requires dualizing a subset of the axions in order to account for all fluxes. Turning off half of the fluxes correctly led to the standard formulation of

N = 2 gauged supergravity, which could be further reduced to its ungauged version when setting the remaining fluxes to zero.

Our analysis of the R-R sector strongly resembles that of [199] for SU(3)×SU(3) manifolds. An essential difference of the approach considered in this chapter is that the field strengths are automatically determined by the double field theory action. This leads to a slightly altered formulation of the action in which the ten-dimensional origin of the four-dimensional fields becomes evident. A subset of these fields thereby appears only in particular combinations with the fluxes, which eventually leads to the same physical degrees of freedom as obtained in the SU(3)×SU(3) framework.

Taking up our discussion at the end of section 3.3, our result shows that double field theory provides a natural ten-dimensional origin for previously isolated gauged super-gravities. In the considered setting, it can thus serve as the missing link to complete the

“web of (gauged) supergravities” from figure 3.2 to a new form as illustrated in figure 5.1.

Type II SUGRA Type II DFT

with fluxes

= 2 gauged

SUGRA = 2 SUGRA

dimensional

reduction dimensional

reduction strong constraint,

vanishing fluxes

gauging

N N

Figure 5.1: A “web of supergravities”. Double field theory serves as the missing link between ten-dimensional supergravities and gauged four-dimensional supergravities.

Mirror Symmetry

A final interesting result of our analysis is the recovery of Mirror Symmetry. Both the CY3 and the K3×T² setting featured a set of IIA ↔ IIB mirror mappings of their effective actions that involved a characteristic exchange of roles between the K¨ahler-class and complex-structure moduli. As was to be expected in light of the conjectured equiv-alence of T-duality and Mirror Symmetry [61], this was also accompanied by mappings between the geometric and non-geometric fluxes. In all cases, the double field theory framework provided a nicely-interpretable notation of the mirror mappings as simple in-terchangings between ten-dimensional poly-forms encoding the different types of moduli and the theory-specific R-R fields.

5.6 Discussion 113

Open Questions and Future Directions

An interesting task to pursue in future research would be to to derive the remaining four-dimensional gauged supergravities from double field theory. A natural next step is thereby to analyze how the framework can be applied to the full action compactified on K3×T². Since dimensional reductions on Calabi-Yau three-folds lead to a partially dualized formulation of gaugedN = 2 supergravity, an important question in this context is whether a similar construction has to be applied to its N = 4 analogue. Similarly to the Calabi-Yau setting, it would make sense to also address these questions with a view to compactifications onSU(2) structure manifolds [222–224] and to elaborate the analogies between both frameworks. Other possible directions include extensions of the orientifold setting discussed in [78] or dimensional reductions of heterotic double field theory.

Chapter 6

Type IIB Flux Vacua and Tadpole Cancellation

We have seen in the previous chapters that (flux) compactifications of type II theories give rise to (gauged)N = 2 supergravities in four dimensions. As discussed in the beginning of this thesis, the most commonly used methods of model building in string theory require the amount of supersymmetry to be reduced to N = 1. One way this problem can be addressed is by introducingorientifold projections and D-branes. In conjunction with constraints arising from the presence of fluxes, such compactifications come with a variety of further restrictions which greatly affect the space of allowed background configurations.

This chapter will focus on the role of such consistency constraints in type IIB theory compactified on the orientifold T⁶/Z² ×Z². Up to minor changes, the contents of this chapter are mostly quoted in verbatim from the author’s work [80].

Im Dokument Dualities, extended geometries and the string landscape (Seite 122-127)