Finally, we check the non-perturbative contribution by D- and worldsheet in-stantons in one-modulus case. Their contribution is very complicated, so it is impossible to solve analytically. Therefore we go to the numerical calculations in this section. Our strategy is composed of the following six steps:
CHAPTER X. SEARCH FOR dS VACUA 47
1. Find the functionℰ(𝑡, ℛ)such that
𝜕𝑡𝑉∣
(𝑒𝑡)2= ̃ℎ2ℰ(𝑡,ℛ)
= 0 , (X.30)
to solve (X.2b).
2. Reduce the equation (X.2a) by(𝑒𝑡)2= ̃ℎ2ℰ(𝑡, ℛ)to
𝒬(𝑡, ℛ) = 0 , (X.31)
where the function𝒬(𝑡, ℛ)does not depend on the flux parameters.
3. Calculate the matrix (IX.6) and reduce it to
𝜕𝜕𝑉 = ̃ℎ2⎛⎜
⎝
Φ𝐼𝐽(𝑡, ℛ) 0 0 Ψ𝐼𝐽(𝑡, ℛ)
⎞⎟
⎠ . (X.32)
4. A numerical calculation begins from here. Just as we assume the exis-tence of the Calabi–Yau manifold with the Hodge number(ℎ1,1, ℎ2,1) = (1, 0)and its topological properties above, we choose some values of the number of instantons𝑁i𝑛𝑠𝑡,𝜆2,𝜅, and the Gopakumar–Vafa invariants 𝑛(0)𝑘 (𝑘 > 𝑁i𝑛𝑠𝑡).
5. Find the lower boundsℛcrand 𝑡crforℛ and𝑡, respectively, from the three physical bounds (the three conditions in the table X.1).
6. Consider the implicit curve𝒬(𝑡, ℛ) = 0in the𝑡-ℛplane with the fol-lowing inequalities:
• ℰ(𝑡, ℛ) > 0,
• the matrixΨ𝐼𝐽(𝑡, ℛ)is positive definite,
• the matrixΦ𝐼𝐽(𝑡, ℛ)is positive definite.
If there is an overlap of them, we get local minima.
There is another issue due to a choice of signs of(−1)ℓand(−1)𝑛where the integers𝑛andℓcome from the stabilized axions. We split the last step into two steps. As 6-1st step, we consider the first two conditions in the 6th step. This consideration excludes(−1)ℓ= 1case and specifies the area where meta-stable dS vacua may exist. And we consider the condition onΦ𝐼𝐽in the specified area by the former step as 6-2nd one. As a result,Φ𝐼𝐽can not positive definite in such area. The typical result is drawn in the figure X.4 for 6-1st step and X.5 for 6-2nd step where the parameters are chosen as𝑁i𝑛𝑠𝑡= 4,𝜆2= 0.1,𝜅 = 10,𝑛(0)𝑘 = 100𝑘. Therefore, we conclude that the non-perturbative instanton corrections can not generate a meta-stable dS vacuum in the one-modulus case.
CHAPTER X. SEARCH FOR dS VACUA 48
Figure X.1: The plane𝛾-(𝑟/|𝑐|)is drawn. There are some correspondences be-tween the colored areas and the conditions; the dark grey region with the blue boundary: (X.12), the pink region with the purple boundary: positivity of (X.8) , and the light grey region with the brown boundary: the potential at the ex-tremum (X.9) is positive. The figure below focuses on the intersection point (𝛾⋆ = 14(√17 − 3), 𝑟⋆= |𝑐|2(√17 + 7)). The narrow region attaching this point allows all three conditions.
CHAPTER X. SEARCH FOR dS VACUA 49
0.05 0.10 0.15 0.20 0.25 0.30
-0.5 0.5 1.0 1.5
0.05 0.10 0.15 0.20 0.25 0.30
-0.5 0.5 1.0 1.5
Figure X.2: The vertical axis is the value of 𝜆2
|𝑐| ̃ℎ2(𝑒𝑡)2|𝑟=𝑟+(𝛾)and the horizontal axis is𝛾. The blue curve is given by (X.8) and the red curve is given by the function 𝑓−1(1 + 3𝛾−1)2/3 ∼ 𝑡2 from (X.22). 𝑓 changes their positions. The large𝑓 gives two intersections (in the figure above at𝑓 = 26). Therefor the two extrema of the potential exists. The small𝑓 gives no intersections (in the figure below at𝑓 = 6.5).
CHAPTER X. SEARCH FOR dS VACUA 50
Figure X.3: The horizontal plane is𝛾-(𝑟/|𝑐|). The vertical axis gives the value of the potential rescaled by 3𝜆2|𝑐|𝐶
̃ℎ2 . There is a local maximum at𝛾 ≈ 0.27, 𝑟 ≈ 5.18|𝑐|and a saddle point at𝛾 ≈ 0.14,𝑟 ≈ 2.66|𝑐|.
Figure X.4: The 𝑡-ℛ plane with the following features: the blue region is ℰ(𝑡, ℛ) > 0, the pink region is Ψ𝐼𝐽(𝑡, ℛ)is positive definite, the red curve is𝒬(𝑡, ℛ) = 0, the horizontal green line isℛ = ℛc𝑟, and the vertical green line:𝑡 = 𝑡c𝑟.
CHAPTER X. SEARCH FOR dS VACUA 51
Figure X.5: The same plane as in the figure X.4 with the following features: the blue region describes the positive trace ofΦ𝐼𝐽(𝑡, ℛ), the red region describes the positive determinant ofΦ𝐼𝐽(𝑡, ℛ). The lower figure implies they are not overlapped andΦ𝐼𝐽(𝑡, ℛ)is not positive definite.
XI
Search for Inflation
Here we investigate the possibility of inflation based on our scalar potential in the perturbative approximation in the one-modulus case. In this situation we have
𝑉(ℛ(𝑟), 𝑡) = 𝑒𝒦 4𝑟2 ⎛⎜
⎝
4𝑟(𝑒𝑡)2
ℛ2𝑀2− 2𝑟 − 𝑒−𝒦𝑁𝑒2+ 4 ̃ℎℛ2
𝑒𝒦𝑁𝑡2− 1 −16 ̃ℎ2𝑟 𝑀 ⎞⎟
⎠ , (XI.1) and
𝑒−𝒦 = 8𝒱 − 𝐶, 𝒱 = 𝜅𝑡3
6 , ℛ = √2(𝑟 + 𝑐)
𝜆2 , 𝑁 = 2𝜅𝑡 . (XI.2) Our strategy consists of the following two steps:
1. Search for flux and Calabi–Yau parameters which can make the potential positive,
2. Check the value of slow-roll parameters.
The variablesℛand𝑡have lower-bounds:ℛ > ℛ𝑐and𝑡 > 𝑡𝑐, because the potential diverges at
ℛ𝑐= √ 2𝑐
𝜆2− 4𝜆22 and 𝑡𝑐=3√3𝐶
4𝜅 . (XI.3)
The potential takes the form following near the critical point𝑡 = 𝑡𝑐: 𝑉 = 𝐴(ℛ)
𝑡 − 𝑡𝑐 + 𝑂(𝑡 − 𝑡𝑐) , (XI.4) where the residue is given by
𝐴(ℛ) = ⎡⎢
⎣
(6𝐶)2/3𝑒2𝜆2− 8𝜅2/3 ̃ℎ2(2𝑐 − ℛ2𝜆2+ 4ℛ2𝜆22) 2(6𝐶)2/3𝜅𝜆2(−2𝑐 + ℛ2𝜆2)(2𝑐 − ℛ2𝜆2+ 4ℛ2𝜆22)
⎤⎥
⎦
. (XI.5)
52
CHAPTER XI. SEARCH FOR INFLATION 53
0.0
0.5
1.0
r 0.4
0.6
0.8
1.0 t
-50 0 50 100
0 50
100 150
1
r
0 1 2 3 4
1
t -100 000
-50 000 0 50 000 100 000
Figure XI.1: The potential𝑉(𝑟, 𝑡)of (XI.1) at small values of𝑟and𝑡on the upper side, and at large values of𝑟and𝑡(in terms of the inverse variables1/𝑟and 1/𝑡) in the lower picture. The parameters in (XI.1) are chosen as ̃ℎ = 1, 𝑒 = 1, 𝜆2= 1/2, 𝜅 = 1. We also have𝑐 = −1/(96𝜋)and𝐶 = 𝜁 (3)/(2𝜋3).
we found the 2nd critical valueℛ𝑐(2)where the residue vanishes,𝐴(ℛ𝑐(2)) = 0. The sign of the potential changes at this critical point, as is shown in Figure XI.2. We find
ℛ𝑐(2)= √16𝑐𝜅2/3 ̃ℎ2− (6𝐶)2/3𝑒2𝜆2 2√2√𝜅2/3 ̃ℎ2(1 − 4𝜆2)𝜆2
. (XI.6)
However, for large values of𝑡orℛ, the potential𝑉is not changed drastically.
To check the dependence of the sign of the potential upon the parameters, we consider the following asymptotic expansion near the origin in a plane (1/𝑟, 1/𝑡):
𝑉(𝑟, 𝑡) = − 𝑒2(𝜆2− 1) 2𝜅(4𝜆2− 1)
1 𝑟2
1
𝑡 + 3𝑐𝑒2𝜆2 𝜅(4𝜆2− 1)
1 𝑟3
1
𝑡 −3 ̃ℎ( ̃ℎ − 2) 2𝜅𝜆2
1 𝑟
1
𝑡3 + 𝑂5(1 𝑟,1
𝑡) . (XI.7) When1/4 < 𝜆2 < 1and ̃ℎ < 2, the coefficients of the three leading terms can be positive. Surely, in this case, the potential 𝑉 becomes positive, as is
CHAPTER XI. SEARCH FOR INFLATION 54
0.0 0.2 0.4 0.6 0.8 1.0
t 5000
10 000 15 000 20 000 25 000 V
Figure XI.2: The blue line is a section of the potential𝑉forℛ > ℛ𝑐(2), and the purple line is a section of the potential𝑉forℛ < ℛ𝑐(2), at ̃ℎ = 1, 𝑒 = 1, 𝜆2 = 1/2, 𝜅 = 1. The section is taken at𝑟 = 0.013for the blue line and at𝑟 = 0.0145 for the purple line.
shown in Figure XI.3.
The next step is to check the slow-roll parameters. We found that the (rel-ative) first and second derivatives of the scalar potential areindependentfrom the parameters of𝑉. Indeed,
|𝑉𝑟/𝑉| = 1/𝑟 + 𝑂2(1/𝑟) , 𝑉𝑟𝑟/𝑉 = 2/𝑟2+ 𝑂2(1/𝑟) , (XI.8) and
|𝑉𝑡/𝑉| = 1/𝑡 + 𝑂2(1/𝑡) , 𝑉𝑡𝑡/𝑉 = 2/𝑡2+ 𝑂2(1/𝑡) . (XI.9) The canonical form of the dilaton𝑟and the Kähler modulus𝑡is𝑟 = 𝑒√2𝜑and 𝑡 = 𝑒√1/6𝜒because their kinetic terms are14(𝜕𝑟)2/𝑟2and3(𝜕𝑡)2/𝑡2, respectively, in the large field approximation. The standard slow roll parameters are now calculated as
𝜀 > 13
6 a𝑛𝑑 𝜂 > 2 . (XI.10)
They violate the slow-roll conditions (𝜀 ≪ 1and𝜂 ≪ 1) for cosmological in-flation. Moreover, although the large𝑟and𝑡i.e. a small string coupling and a large Calabi–Yau volume are compatible with the perturbative approximation, it also implies a decompactification to ten dimensions, which is unacceptable for our Universe.
CHAPTER XI. SEARCH FOR INFLATION 55
20 40 60 80 100
1
r 2000
4000 6000 8000 10 000 V
0.5 1.0 1.5 2.0 2.5 3.0
1
t
-10 10 20 V
Figure XI.3: The sliced potential 𝑉 at ̃ℎ = 1, 𝑒 = 1, 𝜆2 = 1/2, 𝜅 = 1. The horizontal axes show values of1/𝑟and1/𝑡, with𝑡 = 0.5(green),𝑡 = 1(olive), 𝑡 = 2(purple) and𝑡 = 10(blue), on the upper side, and with𝑟 = 0.5(green), 𝑟 = 1(olive),𝑟 = 2(purple) and𝑟 = 10(blue), on the lower side, respectively.
XII
Conclusion and Discussion
In this dissertation, we investigated the possibility of meta-stable dS vacua and cosmic inflation in a class of compactified string theories. We considered the gauged𝒩 = 2supergravity in four spacetime dimensions as the low-energy effective action of the type IIA superstrings compactified onrigidCalabi–Yau threefolds. Fluxes were included to gauge an Abelian isometry in the hyper-multiplet moduli space. In this gauged supergravity, we took into account both perturbative and non-perturbative quantum corrections to the scalar po-tential of the vector- and hypermultiplet moduli spaces but ignored back re-action. There were three types of fields involved: axions, dilaton, and Kähler moduli.
We found that the axions are stabilized due to the non-perturbative D-instanton corrections. After this result we considered the remaining fields.
There were four reasons which imply our negative result about existence of the meta-stable dS vacua. First, we considered the physical bounds in the per-turbative approximation. Though some marginal values allow a small region that allows meta-stable dS vacua, this area did not have the large volume and the weak coupling which are needed to ignore the non-perturbative𝛼′- and 𝑔s-corrections. Second and third, we analyzed meta-stability with one Kähler modulus case analytically and with anyKähler moduli case numerically. In the latter analysis, we considered the necessary condition for the existence of meta-stable dS vacua with any Kähler moduli case, given by the Sylvester’s criterium. And we concluded that not only meta-stable dS vacua but also gen-eral vacua do not exist in the case withanynumber of Kähler moduli. Finally, we investigated the compatibility with the cosmic inflationary property of our scalar potential in one-modulus case in the perturbative approximation. Al-though there is a region of flux parameters which allows a positive scalar po-tential and the fields are rolling down in the perturbative region (with the large volume and the small coupling), the values of slow-roll parameters are not ac-ceptable for our Universe and inflation.
As a result, we extended the known “no-go” theorems which forbid dS vacua and inflation in string cosmology in type IIA Case. For example, we took into account both vector- and hypermultiplets. This setting extends the results of [CKVP+85] where only Abelian𝒩 = 2vector multiplets were in-cluded, as well as [GRLS09] where only hypermultiplets were included. In the
56
CHAPTER XII. CONCLUSION AND DISCUSSION 57 context of inflation, the result[HKTT07] is extended beyond the semi-classical approximation for the rigid Calabi–Yau threefolds with the Hodge numbers (ℎ1,1, ℎ2,1) = (1, 0)by including the perturbative and non-perturbative quan-tum corrections.
Our study of perturbative and non-perturbative quantum corrections has more implications. We introduced the gauged supergravity gauging of an Abelianisometry only in the hypermultiplet moduli space. It has been known that the gauging of anon-Abelianisometry gives dS vacua [FTVP02, CDF+14, FST15]. However, both perturbative or non-perturbative quantum corrections already break any non-Abelian symmetry. Therefore, the dS vacua of [FTVP02, CDF+14, FST15] appear only in the classical approximation. In the supergrav-ity, as low-energy effective theory of the superstring theory with quantum cor-rections included, the dS vacua due to the gauged non-Abelian symmetry can not appear.
The Calabi–Yau rigidness was considered because it can make explicit cal-culations possible due to the absence of complex moduli. The rigid Calabi-Yau threefolds have no mirror dual becauseℎ1,1+ 1can not vanish. The lack of knowledge of the Gopakumar–Vafa invariants follows this fact. In the section VII, the non-perturbative potential is obtained by [AB15] inℎ2,1=0̸ case also.
We may further search in the non-rigid case with hope that we can handle even more complicated calculations with the complex moduli. In this case, we should consider stabilization of the complex moduli in addition.
Of course, our result does not imply the failure of the type IIA string cos-mology. Our study is a “theoretical experiment”, as was mentioned in the in-troduction in certain approximations only. Our consideration was devoted to the exact and explicit calculations. To include all D-instanton non-perturbative corrections, we ignored back reaction, excluded non-zero Roman-mass and the magnetic fluxes, but kept the𝒩 = 2supersymmetry. To proceed to the𝒩 = 1 case, which is more adaptable phenomenologically, we hope that development of the mathematical structure like mirror symmetry would make calculation of quantum corrections possible in𝒩 = 1case.
In our study, we also ignored NS5 instantons because a calculation of their contributions to the effective four dimensional𝒩 = 2supergravity is still out of reach. We believe, the NS5 instantons are going to be crucial for further search of dS vacua and inflation in type IIA superstrings compactified on Calabi–Yau manifolds.
Acknowledements
I would like to thank my supervisor Prof. Dr. Sergey V. Ketov for his tolerant and patient guidance for seven years. And I would like to thank also Dr. Sergei Alexandrov in France, for our re-search collaborations. No my dissertation were possible without his previous works.
I would like to thank the referees of my dissertation, Prof. Dr. Os-amu Yasuda in Tokyo Metropolitan University, Prof. Dr Kuniaki Masai in Tokyo Metropolitan University, and Prof Dr. Yoshifumi Hyakutake in Ibaraki University. The discussion with them im-proved my dissertation.
I appreciate a scholarship and financial support that allowed me to attend international confferences in Bangkok and Madrid from Tokyo Metropolitan University.
And I am sincerely greateful to my father and mother for giving me financial and life support. And I would like to thank my part-ner Midori to heal my heart. Last but not least, I would like to express my gratitude to EVERYONE in my life.
58
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