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Moduli Stabilisation in Heterotic String Compactifications

Beatriz de Carlos, Sebastien Gurrieri, Andre Lukas, Andrei Micu

TL;DR

This paper investigates moduli stabilization in heterotic string compactifications on SU(3) structure manifolds, focusing first on half-flat mirror manifolds and then on generalised half-flat spaces. It shows that perturbative flux/torsion leads to a Gukov-Vafa-Witten type superpotential that does not fix the dilaton, necessitating nonperturbative gaugino condensation to stabilize the dilaton, with a key requirement that the global superpotential ${|W_0|}$ be small. For generalised half-flat models, the authors derive a GVW-type superpotential that effectively stabilizes Kähler and complex structure moduli and analyze vacua under flux/torsion quantization; they find that small ${|W_0|}$ vacua exist when standard embedding constraints are relaxed, yielding a fraction of vacua compatible with gauge unification. The results indicate that heterotic compactifications can achieve moduli stabilization with weak coupling in a landscape of AdS vacua, analogous in flexibility to type II theories, and highlight the need for further understanding of these manifolds and flux/torsion quantization rules.

Abstract

In this paper we analyze the structure of supersymmetric vacua in compactifications of the heterotic string on certain manifolds with SU(3) structure. We first study the effective theories obtained from compactifications on half-flat manifolds and show that solutions which stabilise the moduli at acceptable values are hard to find. We then derive the effective theories associated with compactification on generalised half-flat manifolds. It is shown that these effective theories are consistent with four-dimensional N=1 supergravity and that the superpotential can be obtained by a Gukov-Vafa-Witten type formula. Within these generalised models, we find consistent supersymmetric (AdS) vacua at weak gauge coupling, provided we allow for general internal gauge bundles. In simple cases we perform a counting of such vacua and find that a fraction of about 1/1000 leads to a gauge coupling consistent with gauge unification.

Moduli Stabilisation in Heterotic String Compactifications

TL;DR

This paper investigates moduli stabilization in heterotic string compactifications on SU(3) structure manifolds, focusing first on half-flat mirror manifolds and then on generalised half-flat spaces. It shows that perturbative flux/torsion leads to a Gukov-Vafa-Witten type superpotential that does not fix the dilaton, necessitating nonperturbative gaugino condensation to stabilize the dilaton, with a key requirement that the global superpotential be small. For generalised half-flat models, the authors derive a GVW-type superpotential that effectively stabilizes Kähler and complex structure moduli and analyze vacua under flux/torsion quantization; they find that small vacua exist when standard embedding constraints are relaxed, yielding a fraction of vacua compatible with gauge unification. The results indicate that heterotic compactifications can achieve moduli stabilization with weak coupling in a landscape of AdS vacua, analogous in flexibility to type II theories, and highlight the need for further understanding of these manifolds and flux/torsion quantization rules.

Abstract

In this paper we analyze the structure of supersymmetric vacua in compactifications of the heterotic string on certain manifolds with SU(3) structure. We first study the effective theories obtained from compactifications on half-flat manifolds and show that solutions which stabilise the moduli at acceptable values are hard to find. We then derive the effective theories associated with compactification on generalised half-flat manifolds. It is shown that these effective theories are consistent with four-dimensional N=1 supergravity and that the superpotential can be obtained by a Gukov-Vafa-Witten type formula. Within these generalised models, we find consistent supersymmetric (AdS) vacua at weak gauge coupling, provided we allow for general internal gauge bundles. In simple cases we perform a counting of such vacua and find that a fraction of about 1/1000 leads to a gauge coupling consistent with gauge unification.

Paper Structure

This paper contains 19 sections, 132 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The heterotic string on half-flat manifolds
  3. The heterotic string on half-flat mirror manifolds
  4. Setup for the extended models
  5. Reduction for the generalised models
  6. General structure of low-energy theories
  7. The models
  8. Gaugino condensation
  9. Quantization of flux and torsion
  10. Vacua of the basic models
  11. The STZ model
  12. The general case
  13. Generalised half-flat models
  14. Relation between locally and globally supersymmetric solutions
  15. The STZ model
  16. ...and 4 more sections

Figures (3)

  • Figure 1: Total number of vacua, $N$, as a function of the range of flux/torsion parameters, $M$ (in logarithmic units).
  • Figure 2: Number of vacua $N$ with $|{\cal W}_0|<w$ as a function of $w$ (in logarithmic units), for flux/torsion parameters in the range $-70,\dots ,70$.
  • Figure 3: Contour plot of the potential, on the $(t,z)$ plane, for the example shown in the text, Eq. \ref{['param']}. The solid (dashed) lines are the $\mathrm{Re} (F_T)= \mathrm{Re} (F_Z)=0$ ($\mathrm{Re} (W_T)= \mathrm{Re} (W_Z)=0$) local (global) SUSY conditions. Note that the lines ${\rm Re} (W_T)=0$ and ${\rm Re} (F_T)=0$ coincide.