Superpotential and Scalar Potential

Eventually all models investigated in chapters 7, 8 are entirely described by the standard

N = 1 supergravity formalism in D = 4. The original D = 10 action (4.1.1) carries way to much information and is not necessary for our purposes. We will now connect the ten-dimensional theory to the four-dimensional effective one by introducing a proper superpotential W.

Consider again the D= 10 type IIB supergravity action (4.1.3). For now we dismiss con- tributions fromSCS+Sloc and come back to this point in section 8.3.1. Assume a constant

axio-dilaton S, such that ∂MS = 0. Furthermore we neglect warping effects because they are subleading at large volume, implying2 _F˜

5 = 0. We also drop the curvature termR in

(4.1.3), which will lead to the familiar Einstein-Hilbert action SEH ∼ R d4x

√

−gE RE as explained in subsection 4.1.3. Then the D= 10 type IIB action reduces to

2π l8 s Z d10x √−G1 2 |G3|2 Re(S). (4.1.11)

Note that we have employed the notation3 _|F

p|2 = _p1_!FM1...MpF

M1...Mp_.

1_{The Poincar´e dual space of the homology groupH}

3is just the cohomology group H3 due to the fact

that our Calabi-Yau three-foldMhas real dimensionD= 6.

2_{The warp factorα=e}2A(y)_{expresses also the size of the 5-form flux [67]:}

˜

F5= (1 +∗10)

dα∧dx0∧dx1∧dx2∧dx3. (4.1.10)

3_∗

10 denotes the 10d Hodge-star operator. It is useful to know the symmetric inner product

Z M Fp∧ ∗10Fp= 1 p! Z M d10x√−g Fµ1...µpF µ1...µp _≡ Z M d10x√−g |Fp|2 . (4.1.12)

In order to perform the dimensional reduction one decomposes the ten-dimensional metric into a D= 4 part G(4) as well as a part G(6) of the D= 6 Calabi-Yau manifold M. As a result we obtain the simple splitting [105]

Z d4x q −G(4) Z M d6x √ G(6) |G3| 2 2Re(S) ! ≡ Z d4x q −G(4)_V flux. (4.1.13)

Remarkably, S. Gukov, C. Vafa, E. Witten [49, 50] proved that the potential Vflux is equal

to the four-dimensional standardN = 1 supergravity potentialVF for choosing a superpo-

tential

W =

Z

M G3∧Ω3. (4.1.14)

with the holomorphic (3,0)-form Ω3 given in (2.3.20). Therefore, this is often called the

Gukov-Vafa-Witten superpotential. Note that the superpotential above is holomorphic. Together with the definition (4.1.6) of the flux G3, the superpotential can be spelled out

explicitly in terms of fluxes and moduli

WGVW= Z M G3∧ Ω3 = Z M (F−iSH)∧XΛαΛ−FΛβΛ= =− fΛXΛ−˜fΛFΛ + iS hΛXΛ−˜hΛFΛ . (4.1.15)

Recall that the scalar potential VF is of the form

VF = M4 Pl 4π e K_KIJ¯_D IW D_J¯W −3 W 2 , (4.1.16)

where the sum runs over all moduli listed in table 2.3. Apparently, the scalar potential only depends on the superpotential W and the K¨ahler potential K or the K¨ahler metric

G_IJ¯=∂I∂_J¯K, respectively. The K¨ahler-covariant derivatives are defined by

DIW =∂IW + (∂IK)W . (4.1.17)

The potential (4.1.16) is known asF-term scalar potential since it contains F-termsFI_{. For} the rest of this thesis we will, however, drop the specification F-term and denote VF ≡V.

F-terms are given by the expression

FI =eK2 KIJ¯D_¯

JW . (4.1.18)

Comments on flux vacua and the string landscape

Moduli are said to be stabilized if they take on values for which the flux-induced scalar potentialV is minimized, i.e. their vevs. This makes sense as the moduli develop a physical mass term at the minima. Minima of the scalar potential are called flux vacua and the

Property Condition Description

no-scale KIJ¯(∂IK)(∂_J¯K) = 4 important for susy breaking and mediation

sum over S, Tα, Ga to the standard model [106]

supersymmetric DIW = 0 ∀moduli general condition of supergravity [107] tachyonic ∃ negative mass Vacua without tachyons often called stable;

eigenstates tachyonic AdS minima are only stable if above Breitenlohner-Freedman bound [108] AdS V|Minimum <0 AdS has negative cosmological constant

proportional to vacuum energy density [109, 110]; Minkowski or dS for V|Minimum =/ > 0

Table 4.1: Properties of flux vacua of the F-term scalar potential (4.1.16).

collection of flux vacua is famously known as string landscape. Characteristic features of flux vacua are summarized in table 4.1. According to this table, supersymmetric flux vacua are always AdS since their scalar potential reduces to V = −eK_|W|2 _{≤ 0. It is}

widely believed that the number of flux vacua is gigantic. As a rough estimate, a flux configuration is determined by the number of flux quanta along the independent 3-cycles, which are 100−300 for a typical Calabi-Yau manifold. Assuming only 10 different flux values, we already obtain the huge number of 10100₋₁₀300 _{flux vacua. Of course, there are}

several constraints like tadpole cancellation that need to be satisfied for a physical vacuum. Nevertheless, more accurate estimates from the literature lead to even larger multiplicities of flux vacua. The huge flux landscape narrows the predictive power of string theory and is thus quite unpopular. Thereto let us point out:

”I would be happy personally if the multiverse [i.e. string landscape]

interpretation is not correct, in part because it potentially limits our ability to understand the laws of physics. But none of us were consulted when the universe was created.” - E. Witten [111] -

In view of later applications to string inflation, let us mention a powerful no-go theorem by J. Conlon [112]. In many models of string inflaton a single axionic modulus is employed as unconstrained inflaton field, whereas all other moduli are stabilized. However, the no- go theorem of Conlon states that there is no tachyon-free supersymmetric flux vacuum consistent with stabilized moduli and unfixed axions. This is an important restriction on constructing inflationary models within string theory. Conlon also points out several loop- holes to the theorem, such as D-term contributions or corrections to the K¨ahler potential. Due to this theorem, the authors of [32] analyzed non-supersymmetric vacua.