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Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories

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

TL;DR

This work develops a computable framework for stabilizing complex structure moduli in heterotic compactifications using gauge bundles, by mapping the vacuum space where bundle holomorphy constrains moduli into algebraic varieties. It combines Atiyah’s holomorphy criterion with Koszul sequences and Bott–Borel–Weil representations to reduce complex-structure dependence to explicit polynomial constraints, enabling explicit scanning for stabilized loci and revealing a rich multi-branch structure. When stabilized loci yield singular Calabi–Yau geometries, the authors implement splitting conifold-type transitions to resolve singularities and transport the stabilization data to smooth manifolds, while enforcing anomaly cancellation and slope-stability, sometimes at the cost of additional Kähler moduli. An algorithm is provided to identify smooth stabilized loci across splits, and its power is demonstrated on a tetra-quadric example, where a singular branching locus is resolved to a smooth geometry with substantial net moduli reduction. Overall, the paper shows that gauge fields can robustly fix complex structure moduli in a controlled, computable way and that geometric transitions offer a viable route to non-singular, stabilized heterotic vacua with explicit moduli-control mechanisms.

Abstract

We discuss the use of gauge fields to stabilize complex structure moduli in Calabi-Yau three-fold compactifications of heterotic string and M-theory. The requirement that the gauge fields in such models preserve supersymmetry leads to a complicated landscape of vacua in complex structure moduli space. We develop methods to systematically map out this multi-branched vacuum space, in a computable and explicit manner. In analyzing the resulting vacua, it is found that the associated Calabi-Yau three-folds are sometimes stabilized at a value of complex structure resulting in a singular compactification manifold. We describe how it is possible to resolve these singularities, in some cases, while maintaining computational control over the moduli stabilization mechanism. The discussion is illustrated throughout the paper with explicit worked examples.

Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories

TL;DR

This work develops a computable framework for stabilizing complex structure moduli in heterotic compactifications using gauge bundles, by mapping the vacuum space where bundle holomorphy constrains moduli into algebraic varieties. It combines Atiyah’s holomorphy criterion with Koszul sequences and Bott–Borel–Weil representations to reduce complex-structure dependence to explicit polynomial constraints, enabling explicit scanning for stabilized loci and revealing a rich multi-branch structure. When stabilized loci yield singular Calabi–Yau geometries, the authors implement splitting conifold-type transitions to resolve singularities and transport the stabilization data to smooth manifolds, while enforcing anomaly cancellation and slope-stability, sometimes at the cost of additional Kähler moduli. An algorithm is provided to identify smooth stabilized loci across splits, and its power is demonstrated on a tetra-quadric example, where a singular branching locus is resolved to a smooth geometry with substantial net moduli reduction. Overall, the paper shows that gauge fields can robustly fix complex structure moduli in a controlled, computable way and that geometric transitions offer a viable route to non-singular, stabilized heterotic vacua with explicit moduli-control mechanisms.

Abstract

We discuss the use of gauge fields to stabilize complex structure moduli in Calabi-Yau three-fold compactifications of heterotic string and M-theory. The requirement that the gauge fields in such models preserve supersymmetry leads to a complicated landscape of vacua in complex structure moduli space. We develop methods to systematically map out this multi-branched vacuum space, in a computable and explicit manner. In analyzing the resulting vacua, it is found that the associated Calabi-Yau three-folds are sometimes stabilized at a value of complex structure resulting in a singular compactification manifold. We describe how it is possible to resolve these singularities, in some cases, while maintaining computational control over the moduli stabilization mechanism. The discussion is illustrated throughout the paper with explicit worked examples.

Paper Structure

This paper contains 23 sections, 3 theorems, 129 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Vacuum Varieties and Bundle Holomorphy
  3. Complex Structure Dependence and Cohomology
  4. An Example: A Multi-Branch Vacuum Space and its Properties
  5. Splitting Transitions and Resolving Singular Points
  6. Resolving Singularities in the Calabi-Yau Threefold
  7. A Rapid Review of Splitting Transitions
  8. Divisors, Line Bundles, and Cohomology in Splitting Transitions
  9. Line Bundle Cohomology in Splitting Transitions
  10. An Algorithm for Determining Smooth Stabilized Loci in Complex Structure Moduli Space
  11. Obstructions to "Carrying a Bundle Through" a Splitting Transition
  12. An Example: Resolving Singular Branches in a Multi-Branch Vacuum Space
  13. Conclusions and Further Work
  14. Topological Identities in Splitting Transitions
  15. Intersection Numbers
  16. ...and 8 more sections

Key Result

Lemma 1

Suppose that $X$ and ${\hat{X}}$ are two Calabi-Yau three-folds realized as complete intersections in products of projective spaces and related by a "splitting transition" of the type described in (gen_split). Let ${\cal{L}}={\cal O}_{X}(a^{1},\ldots, a^{m})$ be a "favorable" line bundle on $X$--tha

Figures (1)

  • Figure 1: An illustration of the vacuum space

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Lemma