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Loop Corrections to Volume Moduli and Inflation in String Theory

Marcus Berg, Michael Haack, Boris Kors

TL;DR

This work analyzes how loop corrections in string theory modify the volume modulus and inflaton potential, focusing on the rho problem that arises when the volume modulus mixes with mobile D3-branes. By performing open-string one-loop calculations in toroidal Type IIB orientifolds with background fields, the authors show that the gauge kinetic function can become holomorphic in the corrected modulus and acquire inflaton-dependent contributions through the non-perturbative sector, enabling moderate fine-tuning of the inflaton mass via flux quanta. In ${ m N}=2$ settings the rho problem is resolved, while in ${ m N}=1$ models the outcome is model-dependent: the ${ m Z}_6'$ construction yields no quadratic inflaton terms at one loop, whereas the ${ m Z}_2 imes{ m Z}_2$ model can generate such terms, offering a route to reducing the inflaton mass. These results provide qualitative guidance for embedding inflation in string theory, illustrating how open-string loop effects can alter moduli stabilization and inflationary dynamics, albeit with caveats related to Kähler potential corrections and warp factors in realistic backgrounds.

Abstract

The recent progress in embedding inflation in string theory has made it clear that the problem of moduli stabilization cannot be ignored in this context. In many models a special role is played by the volume modulus, which is modified in the presence of mobile branes. The challenge is to stabilize this modified volume while keeping the inflaton mass small compared to the Hubble parameter. It is then crucial to know not only how the volume modulus is modified, but also to have control over the dependence of the potential on the inflaton field. We address these questions within a simple setting: toroidal N=1 type IIB orientifolds. We calculate corrections to the superpotential and show how the holomorphic dependence on the properly modified volume modulus arises. The potential then explicitly involves the inflaton, leaving room for lowering the inflaton mass through moderate fine-tuning of flux quantum numbers.

Loop Corrections to Volume Moduli and Inflation in String Theory

TL;DR

This work analyzes how loop corrections in string theory modify the volume modulus and inflaton potential, focusing on the rho problem that arises when the volume modulus mixes with mobile D3-branes. By performing open-string one-loop calculations in toroidal Type IIB orientifolds with background fields, the authors show that the gauge kinetic function can become holomorphic in the corrected modulus and acquire inflaton-dependent contributions through the non-perturbative sector, enabling moderate fine-tuning of the inflaton mass via flux quanta. In settings the rho problem is resolved, while in models the outcome is model-dependent: the construction yields no quadratic inflaton terms at one loop, whereas the model can generate such terms, offering a route to reducing the inflaton mass. These results provide qualitative guidance for embedding inflation in string theory, illustrating how open-string loop effects can alter moduli stabilization and inflationary dynamics, albeit with caveats related to Kähler potential corrections and warp factors in realistic backgrounds.

Abstract

The recent progress in embedding inflation in string theory has made it clear that the problem of moduli stabilization cannot be ignored in this context. In many models a special role is played by the volume modulus, which is modified in the presence of mobile branes. The challenge is to stabilize this modified volume while keeping the inflaton mass small compared to the Hubble parameter. It is then crucial to know not only how the volume modulus is modified, but also to have control over the dependence of the potential on the inflaton field. We address these questions within a simple setting: toroidal N=1 type IIB orientifolds. We calculate corrections to the superpotential and show how the holomorphic dependence on the properly modified volume modulus arises. The potential then explicitly involves the inflaton, leaving room for lowering the inflaton mass through moderate fine-tuning of flux quantum numbers.

Paper Structure

This paper contains 19 sections, 137 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The KKLMMT model for D-brane inflation
  3. One-loop corrections to the volume modulus
  4. The background field method
  5. Tree-level effective action of type I on $\mathbb{T}^2\times$ K3
  6. One-loop threshold corrections in ${\cal N}=2$
  7. Couplings of non-abelian gauge groups
  8. Couplings of abelian gauge groups
  9. Generalization to ${\cal N}=1$
  10. The $\mathbb{Z}_6'$ model
  11. Wilson lines in the $\mathbb{Z}_6'$ orientifold
  12. Results for $\mathbb{Z}_6'$
  13. The $\mathbb{Z}_2 \times \mathbb{Z}_2$ model
  14. Interpretation
  15. ${\cal N}=2$
  16. ...and 4 more sections

Figures (4)

  • Figure 1: The KKLMMT model
  • Figure 2: D-brane gauge kinetic term corrections. a) Non-planar annulus amplitude, b) Planar annulus amplitude, c) Möbius amplitude.
  • Figure 3: $\mathbb{Z}_3$-symmetric Wilson lines in $\mathbb{Z}_6'$
  • Figure 4: The function $\Delta$ of eq. (\ref{['r']}); for positive values the inflaton mass is lowered by the one-loop corrections to the superpotential.