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Heterotic Moduli Stabilisation

Michele Cicoli, Senarath de Alwis, Alexander Westphal

TL;DR

The paper analyzes moduli stabilization for weakly coupled heterotic string theory on Calabi–Yau manifolds up to $\alpha'$ corrections, proposing a two-step approach that first fixes complex-structure moduli and the dilaton with fluxes and gaugino condensation, then stabilizes Kahler moduli through threshold corrections, worldsheet instantons, and higher-derivative effects to obtain Minkowski or de Sitter vacua. It shows how SUSY can be spontaneously broken along Kahler directions, computes the resulting moduli mass spectrum and soft terms, and demonstrates that anisotropic compactifications can align the GUT scale with unification expectations. The work highlights challenges in tuning the cosmological constant within this framework and discusses special cases—such as few complex-structure moduli or holomorphic gauge bundles on sub-loci—that could yield low-energy SUSY, as well as the potential utility of non-CY or geometric-flux scenarios for CC control. Overall, it provides a structured, two‑scale stabilization program that yields LVS‑like large-volume minima in heterotic settings and analyzes the resulting phenomenology and model-building implications.

Abstract

We perform a systematic analysis of moduli stabilisation for weakly coupled heterotic string theory compactified on manifolds which are Calabi-Yau up to alpha' effects. We review how to fix all geometric and bundle moduli in a supersymmetric way by fractional fluxes, the requirement of a holomorphic gauge bundle, D-terms, higher order perturbative contributions to W, non-perturbative and threshold effects. We then show that alpha' corrections to K lead to new stable Minkowski (or dS) vacua where the complex structure moduli Z and the dilaton are fixed supersymmetrically, while the fixing of the Kahler moduli at a lower scale leads to spontaneous SUSY breaking. The minimum lies at moderately large volumes of all geometric moduli, at a perturbative string coupling and at the right value of the GUT coupling. We also give a dynamical derivation of anisotropic compactifications which allow for gauge coupling unification around 10^16 GeV. The gravitino mass can be anywhere between the GUT and TeV scale depending on the fixing of the Z-moduli. In general, these are fixed by turning on background fluxes, leading to a gravitino mass around the GUT scale since the heterotic 3-form flux does not contain enough freedom to tune W to small values. Moreover accommodating the observed value of the cosmological constant (CC) is a challenge. Low-energy SUSY could instead be obtained in particular situations where the gauge bundle is holomorphic only at a point-like sub-locus of Z-moduli space, or where the number of Z-moduli is small (like orbifold models), since in these cases one may fix all moduli without turning on any quantised flux. However tuning the CC is even more of a challenge in these cases. Another option is to focus on non-complex manifolds since these allow for new geometric fluxes which can be used to tune W and the CC, even if their moduli space is presently only poorly understood.

Heterotic Moduli Stabilisation

TL;DR

The paper analyzes moduli stabilization for weakly coupled heterotic string theory on Calabi–Yau manifolds up to corrections, proposing a two-step approach that first fixes complex-structure moduli and the dilaton with fluxes and gaugino condensation, then stabilizes Kahler moduli through threshold corrections, worldsheet instantons, and higher-derivative effects to obtain Minkowski or de Sitter vacua. It shows how SUSY can be spontaneously broken along Kahler directions, computes the resulting moduli mass spectrum and soft terms, and demonstrates that anisotropic compactifications can align the GUT scale with unification expectations. The work highlights challenges in tuning the cosmological constant within this framework and discusses special cases—such as few complex-structure moduli or holomorphic gauge bundles on sub-loci—that could yield low-energy SUSY, as well as the potential utility of non-CY or geometric-flux scenarios for CC control. Overall, it provides a structured, two‑scale stabilization program that yields LVS‑like large-volume minima in heterotic settings and analyzes the resulting phenomenology and model-building implications.

Abstract

We perform a systematic analysis of moduli stabilisation for weakly coupled heterotic string theory compactified on manifolds which are Calabi-Yau up to alpha' effects. We review how to fix all geometric and bundle moduli in a supersymmetric way by fractional fluxes, the requirement of a holomorphic gauge bundle, D-terms, higher order perturbative contributions to W, non-perturbative and threshold effects. We then show that alpha' corrections to K lead to new stable Minkowski (or dS) vacua where the complex structure moduli Z and the dilaton are fixed supersymmetrically, while the fixing of the Kahler moduli at a lower scale leads to spontaneous SUSY breaking. The minimum lies at moderately large volumes of all geometric moduli, at a perturbative string coupling and at the right value of the GUT coupling. We also give a dynamical derivation of anisotropic compactifications which allow for gauge coupling unification around 10^16 GeV. The gravitino mass can be anywhere between the GUT and TeV scale depending on the fixing of the Z-moduli. In general, these are fixed by turning on background fluxes, leading to a gravitino mass around the GUT scale since the heterotic 3-form flux does not contain enough freedom to tune W to small values. Moreover accommodating the observed value of the cosmological constant (CC) is a challenge. Low-energy SUSY could instead be obtained in particular situations where the gauge bundle is holomorphic only at a point-like sub-locus of Z-moduli space, or where the number of Z-moduli is small (like orbifold models), since in these cases one may fix all moduli without turning on any quantised flux. However tuning the CC is even more of a challenge in these cases. Another option is to focus on non-complex manifolds since these allow for new geometric fluxes which can be used to tune W and the CC, even if their moduli space is presently only poorly understood.

Paper Structure

This paper contains 28 sections, 151 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Heterotic framework
  3. Tree-level expressions
  4. Corrections beyond leading order
  5. Higher derivative effects
  6. Loop effects
  7. Non-perturbative effects
  8. Moduli-dependent Fayet-Iliopoulos terms
  9. Supersymmetric vacua
  10. Tree-level scalar potential
  11. Chern-Simons action and gauge bundle moduli
  12. Corrections beyond tree-level
  13. Step 1: $Z$ and $S$ stabilisation by gaugino condensation
  14. Step 2: $T$ stabilisation by worldsheet instantons and threshold effects
  15. Tuning the Calabi-Yau condition
  16. ...and 13 more sections

Figures (2)

  • Figure 1: Sketch of the leading order scalar potential $V\sim|\partial_C W|^2+\ldots = |Z|^2 |C|^2+\ldots$ as a function of the complex structure moduli (summarily denoted by $Z$) and the gauge bundle moduli (summarily denoted by $C$) as arising at the second order in $W$ schematically as $W=W_{{\rm CS},(2)}C^2\sim (Z+{\cal O}(Z^2))\, C^2$.
  • Figure 2: $V$ versus ${\cal V}$ assuming the parameters listed in the text which give rise to a near-Minkowski vacuum with $\langle {\cal V} \rangle = 20$ and a cosmological constant of small magnitude compared to height of the barrier set by $m_{3/2}^2M_P^2$.