---
abstract: 'A globally supersymmetric model of inflation will not work in a generic supergravity theory because the higher order, nonrenormalisable supergravity corrections destroy the flatness of the inflaton’s potential. In this talk I derive a form for the Kähler potential which eliminates these corrections if $ W = W_\varphi = \psi = 0 $ during inflation (where $W$ is the superpotential, the inflaton $\in \varphi$, and $W_\psi \neq 0$). I then point out that Kähler potentials of the required form often occur in superstrings and that the target space duality symmetries of superstrings often contain $R$-parities which would make $ W = W_\varphi = 0 $ automatic for $ \psi = 0 $.'
author:
- |
    Ewan D. Stewart [^1]\
    Department of Physics\
    Kyoto University\
    Kyoto 606, Japan
title: 'A General Supergravity Formalism for a Naturally Flat Inflaton Potential [^2] '
---

\

Introduction
============

The approximate isotropy of the cosmic microwave background radiation implies that the inflation that inflated the observable universe beyond the Hubble radius must have occurred at an energy scale $ V^{1/4} \leq 4 \times 10^{16}\,$GeV [@Liddle]. Thus models of inflation should be constructed in the context of supergravity [@susy].

However, this immediately leads to a problem. The positive potential energy $V>0$ required for inflation spontaneously breaks supersymmetry.[^3] This would generally be expected to give an effective mass squared $ m_{\rm soft}^2 \sim 8\pi V / m_{\rm Pl}^2 \sim H^2 $ to any would-be inflaton. But the effective mass of the inflaton must be much less than the inflationary Hubble parameter $H$.

In this talk I will present a general formalism that solves this problem.

Notation and basic formulae
---------------------------

I set $m_{\rm Pl}/\sqrt{8\pi} = 1$ throughout. $\phi$ will represent a vector whose components $\phi^\alpha$ are complex scalar fields, and subscript $\phi$ will denote the derivative with respect to $\phi$, so for example $W_\phi$ represents the vector with components $ \partial W / \partial \phi^\alpha $.

The scalar potential in a globally supersymmetric theory is $$V = \left| W_\phi \right|^2 + D{\rm -term} \,,$$ where the superpotential $W(\phi)$ is an analytic function of $\phi$. The first term is called the $F$-term. For simplicity I will ignore the $D$-term in this talk.

The $F$-term part of the scalar potential in a supergravity theory is $$\label{V}
V = e^K \left[ \left( W_\phi + W K_\phi \right)
        K^{-1}_{\bar{\phi} \phi}
        \left( \bar{W}_{\bar{\phi}} + \bar{W} K_{\bar{\phi}} \right)
        - 3 |W|^2 \right] \,,$$ where the Kähler potential $K(\phi,\bar{\phi})$ is a real function of $\phi$ and its hermitian conjugate $\bar{\phi}$.

The Problem {#prob}
===========

At any point in the space of scalar fields $\phi$ we can make a combination of a Kähler transformation and a holomorphic field redefinition such that $\phi=0$ at that point and, in the neighbourhood of that point, the Kähler potential takes the form $$K = \left| \phi \right|^2 + \ldots \,,$$ where … stand for higher order terms. Then the scalar kinetic terms will be canonical at $\phi=0$ and, from Eq. (\[V\]), the scalar potential will have the form $$\begin{aligned}
V & = & e^{ \left| \phi \right|^2 + \ldots }
    \left\{ \left[ W_\phi + W \left( \bar{\phi} + \ldots \right)
    \right] \left( 1 + \ldots \right) \left[ \bar{W}_{\bar{\phi}}
    + \bar{W} \left( \phi + \ldots \right) \right] - 3|W|^2 \right\}
    \,, \\
 & = & V|_{\phi=0} + V|_{\phi=0} \left| \phi \right|^2
    + {\rm other\ terms} \,.\end{aligned}$$ Thus at $\phi=0$ the exponential term gives a contribution $V$ to the effective mass squared of [*all*]{} scalar fields. Therefore, $$\frac{ V''}{V} = 1 + {\rm other\ terms} \,,$$ where the prime denotes the derivative with respect to the canonically normalised inflaton field. But $|V''/V| \ll 1$ is necessary for inflation to work. So a sucessful model of inflation must arrange for a cancellation between the exponential term and the terms inside the curly brackets. This will require fine tuning unless a symmetry is used to enforce it.

A Solution
==========

Divide the vector of scalar fields $\phi$ into two separate vectors, $\varphi$ and $\psi$, with the inflaton contained in $\varphi$: $$\phi = (\varphi,\psi) \,,\;\;\;\; {\rm inflaton} \in \varphi \,.$$ Assume the $R$-parity $$\label{R}
\psi \rightarrow -\psi \,,\;\;\;\; \varphi \rightarrow \varphi
\,,\;\;\;\;W \rightarrow -W \,,\;\;\;\; K \rightarrow K \,,$$ and that during inflation $$\psi = 0$$ (a natural value since the necessary condition $V_\psi = 0$ is then guaranteed by the $R$-parity). Then the $R$-parity ensures that during inflation $$W = W_\varphi = 0 \,.$$ Thus the scalar potential Eq. (\[V\]) simplifies to $$V = e^K W_\psi K^{-1}_{\bar{\psi} \psi} \bar{W}_{\bar{\psi}} \,.$$ Now it becomes possible to choose a form for the Kähler potential that cancels the inflaton dependent corrections to the global supersymmetric potential in a natural way.

Expanding the $R$-parity invariant Kähler potential about $\psi = 0$ gives $$K = A(\varphi,\bar{\varphi}) + \bar{\psi}\, B(\varphi,\bar{\varphi})
    \,\psi + {\cal O} \left( \psi^2 , \bar{\psi}^2 \right) \,,$$ where $A$ is real and $B$ is hermitian. Therefore $$V = e^A W_\psi B^{-1} \bar{W}_{\bar{\psi}} \,,$$ and so to eliminate the inflaton dependent corrections to the global supersymmetric potential we require $$B^{-1} = f(\varphi,\bar{\varphi}) \, C^{-1}(\chi,\bar{\chi}) \,,$$ and $$A = - \ln f(\varphi,\bar{\varphi}) + g(\chi,\bar{\chi}) \,,$$ where $f$ and $g$ are real functions, $C$ is a hermitian matrix, and $\chi$ are non-inflaton $\varphi$ fields. This gives the inflationary potential $$V = e^{ g(\chi,\bar{\chi}) }
    W_\psi\, C^{-1}(\chi,\bar{\chi}) \,\bar{W}_{\bar{\psi}} \,,$$ and the Kähler potential is required to have the general form $$\begin{aligned}
K & = & - \ln f(\varphi,\bar{\varphi})
    + \frac{ \bar{\psi}\, C(\chi,\bar{\chi}) \,\psi }
    { f(\varphi,\bar{\varphi}) } + g(\chi,\bar{\chi})
    + {\cal O} \left( \psi^2 , \bar{\psi}^2 \right) \,, \\
\label{K}
 & = & - \ln \left[ f(\varphi,\bar{\varphi})
    - \bar{\psi}\, C(\chi,\bar{\chi}) \,\psi \right]
+ g(\chi,\bar{\chi}) + {\cal O} \left( \psi^2 , \bar{\psi}^2 \right)
    \,.\end{aligned}$$

Superstring Examples
====================

Orbifold compactifications
--------------------------

The Kähler potential of the untwisted sector of the low-energy effective supergravity theory derived from orbifold compactification of superstrings always contains [@Ferrara] $$K = - \ln \left( S + \bar{S} \right)
- \sum_{i=1}^{3} \ln \left( T_i + \bar{T}_i
- \left| \phi_i \right|^2 \right) \,,$$ where $S$ is the dilaton, $T_i$ are the untwisted moduli associated with the radii of compactification, and $\phi_i$ are the untwisted matter fields associated with $T_i$. Now if we divide the scalar fields into $\varphi$, $\psi$ and $\chi$ fields as follows $$\begin{aligned}
T_1 & \in & \varphi \,, \\
\phi_1 & \in & \psi \,, \\
S \,,\, T_2 \,,\, T_3 \,,\, \phi_2 \;{\rm and}\; \phi_3 & \in &
\chi \subset \varphi \,,\end{aligned}$$ then we get a Kähler potential of the required form \[Eq. (\[K\])\] $$K = - \ln \left( \varphi + \bar{\varphi} - \left| \psi \right|^2
    \right) + g(\chi,\bar{\chi}) \,,$$ and the target space duality symmetries, $$T_i \rightarrow \frac{a_i T_i - i b_i}{i c_i T_i + d_i} \,,\;\;\;\;
\phi_i \rightarrow \frac{\phi_i}{i c_i T_i + d_i} \,,\;\;\;\;
a_i d_i - b_i c_i = 1 \,,$$ contain the desired $R$-parity \[Eq. (\[R\])\] on setting $ b_i = c_i = 0 $, $ a_1 = d_1 = -1 $ and $ a_2 = a_3 = d_2 = d_3 = 1 $.

Fermionic four-dimensional string models
----------------------------------------

The Kähler potential of the untwisted sector of the revamped flipped SU(5) model [@flipped] is [@Lopez] $$\begin{aligned}
\lefteqn{ K = -\ln \left( 1 - \left| \Phi_1 \right|^2
- \left| \Phi_{23} \right|^2 - \left| \Phi_{\overline{23}} \right|^2
- \left| h_1 \right|^2 - \left| h_{\overline{1}} \right|^2
+ \frac{1}{4} \left| \Phi_{1}^{2} + 2 \Phi_{23} \Phi_{\overline{23}}
    + 2 h_1 h_{\overline{1}} \right|^2 \right) } \nonumber \\
 & & \mbox{} - \ln \left( 1 - \left| \Phi_2 \right|^2
- \left| \Phi_{31} \right|^2 - \left| \Phi_{\overline{31}} \right|^2
- \left| h_2 \right|^2 - \left| h_{\overline{2}} \right|^2
+ \frac{1}{4} \left| \Phi_{2}^{2} + 2 \Phi_{31} \Phi_{\overline{31}}
    + 2 h_2 h_{\overline{2}} \right|^2 \right) \nonumber \\
 & & \mbox{} - \ln \left( 1 - \left| \Phi_4 \right|^2
- \left| \Phi_5 \right|^2 - \left| \Phi_3 \right|^2
- \left| \Phi_{12} \right|^2 - \left| \Phi_{\overline{12}} \right|^2
- \left| h_3 \right|^2 - \left| h_{\overline{3}} \right|^2
\right. \nonumber \\
 & & \left. \hspace{3em} \mbox{} + \frac{1}{4} \left| \Phi_{4}^{2}
    + \Phi_{5}^{2} + \Phi_{3}^{2} + 2 \Phi_{12} \Phi_{\overline{12}}
    + 2 h_3 h_{\overline{3}} \right|^2 \right) \,.\end{aligned}$$ Now if we divide the fields as follows $$\begin{aligned}
\Phi_4 \;{\rm and}\; \Phi_5 & \in & \varphi \,, \\
\Phi_3 \,,\, \Phi_{12} \,,\, \Phi_{\overline{12}} \,,\, h_3
    \;{\rm and}\; h_{\overline{3}} & \in & \psi \,, \\
\Phi_1 \,,\, \Phi_2 \,,\, \Phi_{23} \,,\, \Phi_{\overline{23}} \,,\,
    \Phi_{31} \,,\, \Phi_{\overline{31}} \,,\, h_1 \,,\,
    h_{\overline{1}} \,,\, h_2 \;{\rm and}\; h_{\overline{2}} & \in &
    \chi \subset \varphi \,,\end{aligned}$$ then we get a Kähler potential of the required form \[Eq. (\[K\])\] $$K = - \ln \left( 1 - \left| \varphi \right|^2
    + \frac{1}{4} \left| \varphi^{\rm T} \varphi \right|^2
    - \left| \psi \right|^2 \right) + g(\chi,\bar{\chi})
+ {\cal O} \left( \psi^2 , \bar{\psi}^2 \right) \,,$$ and the target space duality symmetries [@Lopez] contain the desired $R$-parity \[Eq. (\[R\])\].

More orbifold compactifications
-------------------------------

A Kähler potential of the form $$K = - \ln \left[
    \left( A_1 + \bar{A}_1 \right) \left( A_2 + \bar{A}_2 \right)
    - \left( A_3 + \bar{A}_4 \right) \left( A_4 + \bar{A}_3 \right)
    \right]$$ often occurs in orbifold compactifications [@Ferrara; @Cvetic; @Cardoso] and is of the required form.

Calabi-Yau compactifications
----------------------------

Kähler potentials of the form $$K = - \ln \left( 1 - |N|^2 - |C|^2 \right)
    + {\cal O} \left( C^2 , \bar{C}^2 \right)$$ occur for subspaces of enhanced symmetry of the moduli space of a simple Calabi-Yau manifold [@Dixon]. They are of the required form \[Eq. (\[K\])\].

Summary
=======

A globally supersymmetric model of inflation will not work in a generic supergravity theory because the higher order, nonrenormalisable supergravity corrections destroy the flatness of the inflaton’s potential. In this talk I have derived a form for the Kähler potential which eliminates these corrections if $ W = W_\varphi = \psi = 0 $ during inflation (where $W$ is the superpotential, the inflaton $\in \varphi$, and $W_\psi \neq 0$). It is encouraging that Kähler potentials of the required form often occur in superstrings and that the target space duality symmetries of superstrings often contain $R$-parities which would make $ W = W_\varphi = 0 $ automatic for $ \psi = 0 $.

Acknowledgements {#acknowledgements .unnumbered}
----------------

I thank D. H. Lyth for his help in simplifying the presentation of this work. I am supported by a JSPS Postdoctoral Fellowship and this work was supported by Monbusho Grant-in-Aid for Encouragement of Young Scientists No. 92062.

[99]{} E. D. Stewart, ‘Inflation, Supergravity and Superstrings’, hep-ph/9405389. A. R. Liddle, Phys. Rev. D49 (1994) 739. For reviews of supersymmetry and supergravity, see H. P. Nilles, Phys. Rep. 110 (1984) 1 and D. Bailin and A. Love, [*Supersymmetric Gauge Field Theory and String Theory*]{}, IOP, Bristol (1994). S. Ferrara, C. Kounnas and M. Porrati, Phys. Lett. B181 (1986) 263. I. Antoniadis, J. Ellis, J. S. Hagelin and D. V. Nanopoulos, Phys. Lett. B231 (1989) 65. J. L. Lopez, D. V. Nanopoulos and K. Yuan, ‘Moduli and Kähler Potential in Fermionic Strings’, hep-th/9405120. M. Cvetič, J. Louis and B. A. Ovrut, Phys. Lett. B206 (1988) 227. G. L. Cardoso, D. Lüst and T. Mohaupt, ‘Moduli Spaces and Target Space Duality Symmetries in (0,2) $\mbox{\boldmath $Z$}_N$ Orbifold Theories with Continuous Wilson Lines’, hep-th/9405002. L. J. Dixon, V. S. Kaplunovsky and J. Louis, Nucl. Phys. B329 (1990) 27.

[^1]: e-mail address: eds@murasaki.scphys.kyoto-u.ac.jp

[^2]: Talk given at the Seventh Marcel Grossmann Meeting on General Relativity, Stanford University, 24-29 July 1994, and based on Ref. 1.

[^3]: After inflation $V$ disappears and so supersymmetry is restored modulo whatever breaks supersymmetry in our vacuum.
