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Stabilizing All Geometric Moduli in Heterotic Calabi-Yau Vacua

Lara B. Anderson, James Gray, Andre Lukas, Burt Ovrut

TL;DR

This paper introduces a novel mechanism to stabilize all geometric moduli in smooth heterotic Calabi–Yau vacua without Neveu–Schwarz flux, by combining perturbative effects from carefully chosen gauge bundles with non-perturbative corrections. Moduli are fixed in three stages: first, complex structure via F-terms from a bundle holomorphic only on an isolated locus; second, a subset of Kähler moduli and the dilaton via D-terms; third, remaining flat directions through constrained non-perturbative effects like gaugino condensation and membrane instantons, subject to anomalous U(1) symmetries. In explicit constructions, the first two stages yield a Minkowski vacuum, while the non-perturbative sector lifts the remaining degeneracy to an AdS minimum; however, achieving a complete single-Calabi–Yau realization with all three stages remains an ongoing task. The work highlights the delicate interplay between perturbative bundle data and non-perturbative dynamics, offering a framework to stabilize moduli without flux tuning and guiding future heterotic model-building directions.

Abstract

We propose a scenario to stabilize all geometric moduli - that is, the complex structure, Kahler moduli and the dilaton - in smooth heterotic Calabi-Yau compactifications without Neveu-Schwarz three-form flux. This is accomplished using the gauge bundle required in any heterotic compactification, whose perturbative effects on the moduli are combined with non-perturbative corrections. We argue that, for appropriate gauge bundles, all complex structure and a large number of other moduli can be perturbatively stabilized - in the most restrictive case, leaving only one combination of Kahler moduli and the dilaton as a flat direction. At this stage, the remaining moduli space consists of Minkowski vacua. That is, the perturbative superpotential vanishes in the vacuum without the necessity to fine-tune flux. Finally, we incorporate non-perturbative effects such as gaugino condensation and/or instantons. These are strongly constrained by the anomalous U(1) symmetries which arise from the required bundle constructions. We present a specific example, with a consistent choice of non-perturbative effects, where all remaining flat directions are stabilized in an AdS vacuum.

Stabilizing All Geometric Moduli in Heterotic Calabi-Yau Vacua

TL;DR

This paper introduces a novel mechanism to stabilize all geometric moduli in smooth heterotic Calabi–Yau vacua without Neveu–Schwarz flux, by combining perturbative effects from carefully chosen gauge bundles with non-perturbative corrections. Moduli are fixed in three stages: first, complex structure via F-terms from a bundle holomorphic only on an isolated locus; second, a subset of Kähler moduli and the dilaton via D-terms; third, remaining flat directions through constrained non-perturbative effects like gaugino condensation and membrane instantons, subject to anomalous U(1) symmetries. In explicit constructions, the first two stages yield a Minkowski vacuum, while the non-perturbative sector lifts the remaining degeneracy to an AdS minimum; however, achieving a complete single-Calabi–Yau realization with all three stages remains an ongoing task. The work highlights the delicate interplay between perturbative bundle data and non-perturbative dynamics, offering a framework to stabilize moduli without flux tuning and guiding future heterotic model-building directions.

Abstract

We propose a scenario to stabilize all geometric moduli - that is, the complex structure, Kahler moduli and the dilaton - in smooth heterotic Calabi-Yau compactifications without Neveu-Schwarz three-form flux. This is accomplished using the gauge bundle required in any heterotic compactification, whose perturbative effects on the moduli are combined with non-perturbative corrections. We argue that, for appropriate gauge bundles, all complex structure and a large number of other moduli can be perturbatively stabilized - in the most restrictive case, leaving only one combination of Kahler moduli and the dilaton as a flat direction. At this stage, the remaining moduli space consists of Minkowski vacua. That is, the perturbative superpotential vanishes in the vacuum without the necessity to fine-tune flux. Finally, we incorporate non-perturbative effects such as gaugino condensation and/or instantons. These are strongly constrained by the anomalous U(1) symmetries which arise from the required bundle constructions. We present a specific example, with a consistent choice of non-perturbative effects, where all remaining flat directions are stabilized in an AdS vacuum.

Paper Structure

This paper contains 17 sections, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Perturbative contributions to the potential
  3. F-terms
  4. D-terms
  5. Stages 1 and 2: Minimizing the perturbative potential
  6. Stage 1: Fixing the complex structure
  7. Stage 2: Fixing the Kähler moduli and dilaton
  8. An example
  9. Non-perturbative contributions
  10. Stages 1, 2 and 3: Minimizing the full potential
  11. Separating off Stage 1
  12. Stabilizing the remaining moduli: Stages 2 and 3
  13. A no-go result
  14. Avoiding the no-go result
  15. An example
  16. ...and 2 more sections

Figures (1)

  • Figure 1: Plots of the potential, for the example in Section 5.3 of the text, for various slices through field space. The left hand image presents the potential as a function of $s$ and $t^1$, whereas the right hand image depicts the $s$, $\chi^2$ plane. The plots are color shaded as a function of the height of the potential. Clearly the vacuum is a minimum of the potential in these directions, as confirmed, for all field directions, by a calculation of the eigenvalues of the mass matrix.