CC BY
38
UDC 530.1; 539.1
MINIMAL MODELS OF INFLATION IN SUPERGRAVITY AND SUPERSTRINGS
S. V. Ketov a'b'c
a Department of Physics, Tokyo Metropolitan University, Minami-ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan. b KavU institute for the Physics and Mathematics of the Universe (iPMU), The University of Tokyo, Chiba 277-8568, Japan. c institute of Physics and Technology, Tomsk Polytechnic University, 30 Lenin Ave., Tomsk 634050, Russia.
E-mail: ketov@tmu.ac.jp
A novel framework is proposed for embedding the natural inflation into the type IIA superstrings compactified on a Calabi-Yau three-fold. Inflaton is identified with axion of the universal hypermultiplet (UH). The other UH scalars (including dilaton) are stabilized by the CY fluxes whose impact can be described by gauging of the abelian isometry associated with the axion. The stabilizing scalar potential is controlled by the integrable three-dimensional Toda equation. The inflationary scalar potential of the UH axion is dynamically generated at a lower scale in the natural inflation via the non-perturbative quantum field effects such as gaugino condensation. The natural inflation has two scales that allow any values of the CMB observables (ns,r).
Keywords: inflation, supergravity, superstrings.
1 Introduction
The most economical, simple and viable inflationary models are the single-field, (inflaton) theories whose scalar potential is controlled by a one or two parameters. Amongst the most popular models of that type are (i) the Starobinsky inflation [1], the Linde inflation [2], the Higgs inflation [3] and the natural inflation [4].
The closed type II strings give the UV completion of quantized gravity, while the "closed string gravity" consist of the closed string zero modes including metric, dilaton and B-field, all being universally-coupled to other fields. Their effective action (after integration of the string massive modes) gives rise to the (modified) Einstein gravity including the higher-order curvature terms. Those terms in the perturbative string effective action can be computed from either string amplitudes of the massless modes or their equations of motion given by the vanishing RG beta-functions of the Non-Linear Sigma-model describing string propagation in a background of the massless modes. However, the coefficients in front of all Ricci- and scalar- curvature dependent terms in the perturbative gravitational string effective action are ambiguous, because they are defined around the vacuum with the vanishing Ricci tensor. To resolve the ambiguity, one needs a non-perturbative setup for strings. It is usually unavailable, but there are some exceptions where the crucial role is played by extended local supersymmetry. Actually, the N=2 extended local supersymmetry in the critical dimension D=10 is
required for consistency of closed (type II) strings, while their CY compactification gives rise to N=2 local supersymmetry in 4D spacetime. The corresponding low-energy string effective action is given by a matter-coupled N=2 supergravity, while its moduli space M is the direct product MV <g> MH of the moduli space MV of h1,1 N=2 vector multiplets and the moduli space Mh of (1 + h1,2) hypermultiplets, in terms of the CY Hodge numbers h1,^d h1,2 (the UH is represented by 1 in the (1 + h1,2)).
Inflaton can be interpreted as the pseudo-
B
spontaneous breaking of the rigid scale invariance. When f is a scale of spontaneous breaking of the scale invariance, and A is a scale of inflation, a typical pNGb scalar potential takes the form [4]
V (B) = A4
! I B
1 - cos I f
(1)
In string theory, f is of the order of the Mpi, whereas A originates in particle physics dynamically, via gaugino condensation [5]. Our proposal is to identify the axion B
in 4D.
2 UH moduli space
The hypermultiplet moduli space MH of the CY-compactified 4D, type-IIA closed strings is known to be independent upon the CY complex structure but can receive non-trivial quantum corrections. The
perturbative corrections are only possible at the 1-loop string level, being proportional to the CY Euler number [6]. The non-perturbative (instanton) corrections are due to the Euclidean D2-branes wrapped about the CY special (supersymmertic) 3-cycles and due to the solitonic (NS-type) Euclidean 5-branes wrapped about the entire CY space. The 4D instantons due to the wrapped D2-branes are called D-instantons.
In quantum 4D, N=2 closed string theory the non-perturbative UH moduli space is different from the classical UH space, as regards both its topology and its metric, because of the non-perturbative d.o.f. in 4D due to the wrapped branes, and because some UH scalars get the non-vanishing VEVs in quantum theory that break the classical symmetries. In addition, the CY flux quantizaton implies quantized brane charges that can be identified with the Noether charges of the Peccei-Quinn (PQ) symmetries. It is expected that the string duality symmetry, described by the discrete group SL(2, Z), always survives.
The cosmological inflation can be associated with a special region of the quantum UH moduli space. We identify that region by demanding the smallness of the string coupling, where the NS5-brane instantons are suppressed and the axion isometry is preserved.
The quantum gravity corrections are encoded in the quaternionic-Kahler structure of the quantum UH moduli space. When assuming a single isometry survival, the appropriate framework is given by a reformulation of the UH quaternionic-Kahler geometry as the Einstein-Weyl geometry with a negative scalar curvature, defined by [7]
W-bcd = 0
in terms of the two potentials, P and u, and the 1-form ©, in local coordinates (t,p,j, v).
It follows from Eq. (2) that the potential P(p, j, v)
u
P=¿f C- 2pdpu
(5)
whereas the potential u(p, j, v) obeys the 3D nonlinear equation
-(% + dl)u + ay
0
(6)
that is known as the (integrable) SU(<x) or 3D continuous Toda system. Finally, the 1-form © satisfies the linear differential equation [9]
-d A © = (dvP)dj A dp
+ (B^P)dp A dv + dp(Pe-u)dv A dj, (7)
whose integrability condition is just given by Eq. (6). The classical UH metric in the parameterization (4) is obtained by taking
P
3
2 |A|
d A 9 = dv A dj .
const. > 0 , e
P
(8)
3
Rab = 2zAgab , A = const. < 0 , (2)
so that u = 2^. The string coupling is given by the dilaton VEV as gstring = The classical region
of the UH moduli space corresponds to the vanishing
gstring-
The quantum UH moduli space was investigated in Refs. [10-16]. As was found in Refs. [12, 15], a summation of the D-instanton contributions is possible when there is the extended U (1) x U (1) isometry. In this case the UH metric is governed by the Calderbank-Petersen potential F(p,rj) obeying the equation [17]
3
where the W-bcd is the anti-self-dual part of the Weyl p2 (dp + d^) F = - F tensor, and the Rab is the Ricci tensor of the UH moduli space metric gab, with a,b = 1, 2, 3, 4. Given the abelian isometry of the UH metric described by a Killing vector K obeying the equations
(9)
Ka;b + Kb;a = 0 , K2 = gabKaKb > 0,
one can choose some adapted coordinates, in which all the metric components are independent upon one
(t)
states that any such metric with the Killing vector dt can be brought into the form
ds2
dsTod
]p(dt + <)2
+ P [eu(dj2 + dv2) + dp2]|
Its unique SL(2, Z) modular invariant solution is given by the Eisenstein series E3/2. The asymptotical expansion of the Eisenstein series reveals a sum of the classical contribution proportional to p-1/2, the
(3) perturbative string 1-loop contribution proportional to Z(3)p3/2, and the infinite sum of the D-instanton terms indeed [12].
3 CY fluxes and gauging the UH isometry
So far no scalar potential was generated for the UH scalars. As is well known in string theory, the moduli stabilization can be achieved via adding nontrivial fluxes of the NS-NS and RR three-forms in
(4) CY [18], while it amounts to gauging isometries of the
u
UH moduli space in the effective 4D, N=2 supergravity [19]. As the abelian gauge field one can employ either gravi-photon of N=2 supergravity multiplet or a vector field of an N=2 matter (abelian) vector multiplet. As a result, the UH gets a non-trivial scalar potential whose critical points determine the vacua of the theory [19].
The scalar potential arising from the gauging procedure takes the form [20]
V
9
2gabdaWdbW - 6W2
(10)
gab
W
W2 = 1 dK A *dK - 1 dK A dK 3 6
(11)
K=
kadqa of the gauged isometry and the Hodge star (*) in any local coordinates (q) on the UH moduli space.
In the parametrization of Eq. (4) we have the Killing vector Ka = (1,0,0,0) that yields the Killing 1-form
K=
1
v2P
(dt + 0)
(12)
A 2 1 / 3
W2 = T" + 12P (3 + 2|A|P + 2p^p
+ ^ K^P)2 + (dvP)2] .
4 Conclusion
We proposed the inflationary scenario in the 4D quantum gravity given by the type IIA closed strings compactified on a Calabi-Yau three-fold. Inflaton was identified with the axion of the Universal Hypermultiplet.
The other (non-inflaton) scalars of the Universal Hypermultiplet (including dilaton) were stabilized by the CY fluxes whose impact was calculated via the gauging procedure of the UH moduli space axion isometry. The latter survives when the NS5-brane instantons are suppressed, i.e. at a small string coupling
gstring < 1.
After the stabilization by CY fluxes/gauging, the N=2 local supersymmetry in 4D is unbroken, while axion is still massless and has no scalar potential. However, clt cl lower scale the axion can get a scalar potential due to some non-perturbative quantum field theory phenomena such as gaugino condensation. The slow-roll natural inflation can, therefore, take place with the scalar potential (1) whose structure is essentially dictated by the pNGb nature of the axion.
It is worth noticing here that the scalar potential (1) of the natural inflation yields the scalar index ns and the tensor-to-scalar ratio r of the CMB anisotropy as [4]
whose square is given by K2 = gabKaKb = gtt = pipit is straightforward to compute the superpotential squared. We find
1_
8nf 2
/ r \ I/4
, A « 2.2-10 GeV I-) . (14)
' V0.002/ v ;
(13)
The first term in the scalar potential (10) is always positive, whereas the second term is always negative, which is similar to the scalar potential in a generic matter-coupled N=1 supergravity [21]. The Minkowski vacua are determined by the fixed points of the scalar potential, related to the poles of the function Pp2. The existence of meta-stable de Sitter vacua was explicitly-demonstrated in Refs. [14,15].
Therefore, the CMB observables (ns,r) are directly-related to the scales (f, A) of the natural inflation, respectively.
Acknowledgements
The author thanks S. Alexandrov, E. Kiritsis, A. Sagnotti, A. A. Starobinsky and S. Vandoren for discussions. This work was supported by a Grant-in-Aid of the Japanese Society for Promotion of Science (JSPS) under No. 26400252, the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and the Competitiveness Enhancement Program of the Tomsk Polytechnic University in Russia.
n
s
2
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Received 12.10.2014
С. В. Kemoe
МИНИМАЛЬНЫЕ ИНФЛЯЦИОННЫЕ МОДЕЛИ В ТЕОРИЯХ СУПЕРГРАВИТАЦИИ
И СУПЕРСТРУН
Предложено вложение естественной инфляции в ранней Вселенной в теорию IIA суперструн, компактифицированных на многообразиях Калаби-Яу. Инфлатон является аксионом универсального гипермультиплета. Остальные скаляры стабилизированы в результате локализации аксионной симметрии. Метод согласуется с любыми параметрами микроволнового реликтового излучения.
Ключевые слова: инфляция, супергравитация, теория струн.
Кетов С. В., доктор физико-математических наук, профессор. Токийский столичный университет. Minami-ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Япония. Токийский университет. Chiba 277-8568, Япония.
Томский Политехнический Университет.
пр. Ленина 30, 634050 Томск, Россия. E-mail: ketov@tmu.ac.jp
